Question: For the curve given by \( y^2 = x^3 \), the tangent at a point \( P \) intersects the curve again at a point \( Q \). If the lines \( OP \) and \( OQ \) make angles \( \alpha \) and \( \beta \) with the x-axis, respectively, find the value of \( \frac{\tan \alpha}{\tan \beta} \).

Answer: -2


Question: Compute the line integral \( \oint_C \left( xy + \ln(x^2 + 1) \right) \,dx + \left( 4x + e^{y^2} + 3\arctan(y) \right) \,dy \) where \( C \) is the boundary of the region \( D \) in \( \mathbb{R}^2 \) consisting of points \((x,y)\) satisfying \(x^2 + y^2 \leq 1\) and \(y \geq 0\), oriented counterclockwise. Use Green's Theorem to evaluate the integral.

Answer: 2\pi


Question: A train with infinitely many seats stops at countably many villages, one for each positive integer, in increasing order, and then finally arrives at the city. At the first village, two women board the train. At each subsequent village, one woman leaves the train, and two other women board the train. How many women arrive at the city?

Answer: \infty


Question: Evaluate whether the following statement is true: $$\int_{-1}^{1} \frac{1}{x} \mathrm{d}x = \lim_{\epsilon\to 0^{+}} \left( \int_{-1}^{-\epsilon} \frac{1}{x} \mathrm{d}x + \int_{-\epsilon}^{\epsilon} \frac{1}{x} \mathrm{d}x + \int_{\epsilon}^{1} \frac{1}{x} \mathrm{d}x \right) = 0.$$

Answer: False


Question: Consider a function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as $f(x) = 4x$ for rational $x$ and $f(x) = x + 6$ for irrational $x$. The set $E$ consists of all points in $\mathbb{R}$ where $f$ is continuous. Determine the limit points of the set $E$.

Answer: \emptyset


Question: On the Riemann surface $X = \{ (z, w) \in \mathbb{C}^2 : z^3 + w^3 = 2\}$, choose a local coordinate $t$ in a neighborhood of the point $(1, 1)$. Determine the order of zero of the function $f(t) = z(t)w(t) - 1$ at the point $(1, 1) \in X$. Provide your answer as an integer.

Answer: 2


Question: Evaluate the line integral \( \oint_C y^3 \, ds \), where \( C \) is the curve formed by the intersection of the sphere \( x^2+y^2+z^2=1 \) and the plane \( x+y+z=0 \). Use a suitable parametrization to find the value of the integral.

Answer: 0


Question:
Given the sequence defined by the partial sum of cosines:
\[ s_m = \sum\limits_{k = 0}^m \cos k = \frac{\sin\left(m + \frac{1}{2}\right)}{2\sin\frac{1}{2}} + \frac{1}{2} \]

Define the sequence \( a_n \) as:
\[ a_n = \sum\limits_{m = 0}^n \left[\frac{\sin\left(m + \frac{1}{2}\right)}{2\sin\frac{1}{2}} + \frac{1}{2}\right] \]

Using the identity:
\[ \sum\limits_{m = 0}^n \sin\left(m + \frac{1}{2}\right) = \frac{1 - \cos(n + 1)}{2\sin\frac{1}{2}} \]

We find:
\[ a_n = \frac{1 - \cos(n + 1)}{2\left(\sin\frac{1}{2}\right)^2} + \frac{(n + 1)}{2} \]

Consequently:
\[ a_n = \frac{\cos(n + 1) + (n + 1)\cos 1 - n - 2}{2(\cos 1 - 1)} \]

Calculate the Cesàro sum, which is the limit:
\[ \lim_{n \to \infty} \frac{a_n}{n} \]

Show that this limit evaluates to \( \frac{1}{2} \).

Answer: \frac{1}{2}


Question: Given a Poisson process \(\{N(t), t \geq 0\}\) with rate \(\lambda\), let \(T_1\) denote the time of the first event. Determine the conditional probability \(P[T_1 < s \mid N(t) = 1]\) for \(0 < s \leq t\).

Answer: \dfrac{s}{t}


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{\{x\}^{2n}-1}{\{x\}^{2n}+1} \) for \( 0 < \{x\} < 1 \), where \( \{x\} \) represents the fractional part of \( x \).

Answer: -1


Question:
Let $X, Y$ be locally convex spaces over $\mathbb{R}$ and $f: X \to Y$. Suppose $f$ is Fréchet differentiable at a point $x_{0} \in X$. Given an interval $0 \in I \subset \mathbb{R}$, we call a smooth curve $\gamma: I \to X$ an integral curve for $x_{0}$ if $\gamma(0) = x_{0}$. Determine if the following statement is true:

$$\frac{d(f\circ \gamma)(t)}{dt}\bigg{|}_{t=0} = 0$$

for every integral curve $\gamma$ for $x_{0}$ implies that $x_{0}$ is a critical point for $f$, in the sense that its derivative at $x_{0}$ is zero.

Answer: True


Question: What is the probability that a monkey, with only two keys representing zero and one, will eventually type the complete works of Shakespeare in binary?

Answer: 1


Question: Does there exist a model of ZF set theory (without atoms) that contains an amorphous set?

Answer: Yes


Question: Determine whether the series \(\sum_{k=1}^{\infty} \binom{\frac{1}{2}}{k} (-1)^k\) converges, and if it does, find its limit.

Answer: -1


Question: Calculate the integral \( \int_{|z|=3} \frac{e^{\frac{1}{1-z}}}{z} \, dz \) using residues. Consider the residue at \( z = \infty \) and explain its role in evaluating the integral.

Answer: 2\pi i


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{e^{2\sin x}\cosh x-1}{\sqrt[3]{x(\cosh x-\cos x)}}$$ without using L'Hospital's rule.

Answer: 2


Question: Let $k$ be a field and $A$ a finitely generated $k$-algebra. Suppose there is an isomorphism of $k$-algebras \( A \cong A/I \) for some ideal \( I \). Can we conclude that \( I=0 \)?

Answer: I = 0


Question: Let \(A = XX^T\) be an \(n \times n\) matrix where \(X \in \mathbb{R}^n\). Given that \(\|X\|^2\) is an eigenvalue of \(A\), determine if it is the greatest eigenvalue.

Answer: Yes


Question: Determine whether the domain \( D = \{f \in L^2(\mathbb{R}) : \frac{d^2f}{dx^2} \in L^2(\mathbb{R})\} \) is dense in \( L^2(\mathbb{R}) \). Justify your answer.

Answer: Yes


Question:
Evaluate the limit of the expression as \( n \to \infty \):
\[ \lim _{n \rightarrow \infty}\left(\frac{2}{2^{2}-1}\right)^{\frac{1}{2^{n-1}}} \left (\frac{2^{2}}{2^{3}-1}\right)^{\frac{1}{2^{n-2}}} \cdots\left(\frac{2^{n-1}}{2^{n}-1}\right)^{\frac{1}{2}} \] using Stolz's theorem.

Answer: \dfrac{1}{2}


Question:
Let $E \subseteq \mathbb{R}^n$ be an open subset. Let $f:E \to \mathbb{R}$ be a differentiable function, and suppose that $\nabla f$ is uniformly continuous. Determine whether $f$ is uniformly differentiable. Specifically, does there exist, for any $\epsilon >0$, a $\delta > 0$ such that for all $a,x \in \mathbb{R}^n$,

$$\frac{|f(x) - f(a) - \nabla f (a)\cdot (x-a)|}{|x-a|} <\epsilon$$

whenever $|x-a|<\delta$?

Answer: Yes


Question: Assume $f:G\to K$ is a group homomorphism, and $S\subset G$ is a subset of elements such that the restriction $f|_S$ is the trivial homomorphism. Let $N\leq G$ denote the smallest normal subgroup that contains $S$. Prove or disprove that $f$ is trivial on the entire $N$. \( \text{True or False?} \)

Answer: A


Question: Determine if a proof can be written in the M-system with fewer than 30 lines for the statement $\lnot$(P -> $\lnot$ Q) $\vdash$$_M$ P.

Answer: Yes


Question: Evaluate the improper integral: $$\int_{-\infty}^0 \left( \frac{5x}{x^2+4} - \frac{20}{4x-1} \right) \, dx$$

Answer: 15 \ln 2


Question: Determine whether the linear functional $J: L^1[0,1] \rightarrow \mathbb{R}$, defined by \[ J(f) = \int_0^1 (x^2 + 1)f(x)\,dx \] is bounded on the set $\{f \in L^1[0,1] : \|f\|_1 = 1, f(x) \geq 0\}$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $f:[0,1]\to [0,1]$ is a nondecreasing function with $f(0)=0$ and $f(1)=1$. Define $g:[0,1]\to [0,1]$ by $g(y)=\min\{x\in[0,1]\mid f(x)\ge y\}$. If $g$ is strictly increasing, does it imply that $f$ is continuous? Answer 'yes' or 'no'.

Answer: yes


Question: Suppose a function $f$ satisfies $f(f(x)) = x^4$ for all real numbers $x$ and $f(1) = 1$. Find the value of $f(0)$.

Answer: 0


Question: Find the last digit of a 6-digit number with distinct digits that remains the same when multiplied by 2, 3, 4, 5, and 6, but with the digits rearranged.

Answer: 7


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

Answer: 1


Question: Determine whether the ratio of $q$-smooth numbers can ever approach zero.

Answer: Yes


Question: Determine whether there exists an almost continuous function that cannot be approximated uniformly by any continuous function.

Answer: Yes


Question: Let $X$ be the vector space of $\mathbb{C}$-valued functions on a finite group $G$. Consider the two actions on $X$ defined by $(g.x)(h) = x(g^{-1}h)$ and $(g \cdot x)(h) = x(hg)$. Are these two representations equivalent, meaning they are similar up to some automorphism of $X$?

Answer: Yes


Question: Assume $a_n > 0$ and $\lim\limits_{n \to \infty}\dfrac{a_1+a_2+\cdots+a_n}{n}=a<+\infty$. Find the limit \( \lim\limits_{n \to \infty}\dfrac{a_1^p+a_2^p+\cdots+a_n^p}{n^p} \) where $p > 1$. \( \boxed{} \)

Answer: 0


Question: Evaluate the limit: $$ \lim_{n\to\infty} \frac{4^nn!}{(3n)^n} $$

Answer: 0


Question: Determine a formula for the value of Legendre polynomials at $x=0$ for odd values of $n$. Provide your answer as a general expression.

Answer: 0


Question: Find the general solution to the differential equation: \[ t y y'' - t y'^2 + 2y y' = 0 \]

Answer: y = C_1 e^{C_2/t}


Question: Determine the smallest value of $a$ such that the function \( f_a(x) = \frac{(2x)!}{(x!)^a} \) forms a bell curve shape.

Answer: 3


Question: Find the expression for the non-constant rational polynomial $f(x)$ such that $f(f(x)) = 3f(x)^4 - 1$.

Answer: 3x^4 - 1


Question: Determine whether there exists a sequence of discrete random variables \((X_{1}, X_{2}, \dots)\) that converges almost surely to a continuous random variable with a uniform distribution on the interval \([0,1]\).

Answer: Yes


Question: Evaluate the double sum \( \sum_{x=-\infty}^{\infty} \sum_{y=-\infty}^{\infty} e^{-\frac{x^2+y^2}{2}} \).

Answer: 2\pi


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{\ln(1 - \sin x) + x} {x^2}$$ without using l'Hôpital's rule.

Answer: -\dfrac{1}{2}


Question: Suppose $K$ and $L$ are fields contained in a larger field, and let $KL$ be the smallest subfield containing both. If $KL$ is a finite extension of both $K$ and $L$, must $K$ be finite over $K \cap L$?

Answer: Yes


Question:
Determine whether the minimal polynomials of a matrix $A$ and its transpose $A^T$ in $
abla_{n\times n}(\mathbb{F})$ are the same.

Answer: Yes


Question: Find the minimum distance between the circle \(x^2+y^2=9\) and the ellipse \(2x^2+10y^2+6xy=1\).

Answer: 2


Question: Determine the limit of the function \( f(x) = \frac{\sin x + x\lfloor x \rfloor}{x\cos x - x} \) as \( x \) approaches \( a_n = \sqrt{n+1} - \sqrt{n} \) and \( n \to \infty \).

Answer: -\infty


Question: How many distinct composition series does the group \( Z_{30} \) have?

Answer: 6


Question: Determine if there exist numbers in the set $A = \{1, 5, 9, 13, 17, 21, 25, 29, 33, \ldots\}$, which consists of all natural numbers of the form $4n + 1$, that are squares in the natural numbers but are not $A$-squares. An $A$-square is a number in $A$ that can be expressed as the square of another number in $A$. Provide a justification for your answer.

Answer: Yes


Question: Determine the expected size of the largest connected component of the graph $H(s_1, s_2)$ as a function of $n$, where $s_1$ and $s_2$ are two binary strings of length $n$. The graph $H(s_1, s_2)$ is constructed based on the principles of Hitomezashi Sashiko, a form of decorative mending.

Answer: \Theta(n)


Question: Determine whether any vector bundle $V$ of rank $n$ on a projective variety $X$ has a filtration by line bundles. Answer true or false.

Answer: False


Question: Evaluate the limit: \[ \lim_{x\to\ \frac{\pi}{2}^-}\left(\frac{1}{x-\frac{\pi}{2}} + \tan x\right) \] as \( x \) approaches \( \frac{\pi}{2} \) from the left.

Answer: 0


Question: Calculate the value of the line integral of the function \( g(x, y) = 4x^3 + 10y^4 \) along the straight line segment from the origin, \( (0, 0) \), to the point \( (1, 2) \) in the \( xy \)-plane. Express your answer in simplest form.

Answer: 33\sqrt{5}


Question: Evaluate the integral \( I = \oint_C \frac{\cos(\pi z)}{(z-2)^3} \, dz \), where \( C \) is the circle \( |z|=1 \), using the derivative theorem.

Answer: 0


Question: In a $5$-regular graph $G$ with $m$ edges and a girth of at least $6$, determine the number of paths of length $5$.

Answer: 256m


Question: Determine whether the generalized Laguerre function $L^{\alpha}_n(x)$ can be defined for $\alpha = -1$ with all $n \geq 0$ and $x > 0$.

Answer: Yes


Question: Using Rouché's theorem, determine the region where all zeros of the polynomial $f(z) + g(z)$, where $f(z) = z^5$ and $g(z) = -3z^2 - 1$, lie. Specifically, find the smallest value of $|z|$ such that $|f(z)| > |g(z)|$ for $|z| > 2^{2/3}$.

Answer: 2^{2/3}


Question: Calculate the mass of the region in the first quadrant where the tile lies between $1 < xy < 3$ and $1 < x^2 - y^2 < 3$, given the density function $\sigma(x, y) = (x^2 + y^2)(x^4 + y^4)$.

Answer: 26


Question: Evaluate the line integral \( \int_{\gamma}|z-1||dz| \), where \( \gamma:[0,2\pi]\to \mathbb{C} \) is defined by \( \gamma(t)=e^{it} \).

Answer: 8


Question: Is it possible for an $n \times n$ matrix to have only one eigenvalue that corresponds to $n$ linearly independent eigenvectors? Answer yes or no.

Answer: Yes


Question: Calculate the expected value of the random variable $X_n$ in the sequence defined as follows: $X_1$ follows a uniform distribution on the interval $(0,1)$, $X_2$ is uniform on $(X_1, X_1 + 1)$, and for $n \geq 3$, $X_n$ is uniform on $(X_{n-1}, X_{n-1} + 1)$. Use the fact that the expectation of a uniform random variable on $(a, b)$ is $\frac{b+a}{2}$. Provide your answer in terms of $n$.

Answer: \dfrac{n}{2}


Question:
Evaluate the limit of the following expression as $N$ approaches infinity:
\[ \frac{N(N-1)(N-2)\ldots 1}{(N+ \frac{1}{M})(N-1 + \frac{1}{M})(N-2 + \frac{1}{M})\ldots(1 + \frac{1}{M})} \]
where $M$ is a constant.

Answer: 0


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

Answer: \sin 1


Question:
Evaluate the integral \( \int_{\mathbb{R}^n} f(Ax) e^{i \left\langle y, x \right\rangle} \, dx \) where \( f: \mathbb{R}^n \rightarrow \mathbb{R}^n \) is a continuous function with a finite integral, \( A \) is an invertible \( n \times n \) matrix, and \( \left\langle \cdot, \cdot \right\rangle \) is the standard inner product in \( \mathbb{R}^n \). Which of the following expressions represents the integral correctly?

A. \( \int_{\mathbb{R}^n} f(x) e^{i \left\langle (A^{-1})^T y, x \right\rangle} \frac{dx}{|\det A|} \)

B. \( \int_{\mathbb{R}^n} f(x) e^{i \left\langle A^T y, x \right\rangle} \frac{dx}{|\det A|} \)

C. \( \int_{\mathbb{R}^n} f(x) e^{i \left\langle (A^T)^{-1} y, x \right\rangle} \, dx \)

D. \( \int_{\mathbb{R}^n} f(x) e^{i \left\langle A^{-1} y, x \right\rangle} \frac{dx}{|\det A|} \)

Answer: A


Question: Determine the number of distinct polynomial functions from \( \mathbb{Z}_2^n \) to \( \mathbb{Z}_2 \).

Answer: 2^{2^n}


Question: Let \( R \) be an integral domain and \( S \) a multiplicative set in \( R \) such that the localization \( S^{-1}R \) is a local ring. Determine whether there exists a prime ideal \( \mathfrak{p} \subset R \) such that \( S^{-1}R \cong R_{\mathfrak{p}} \) as \( R \)-algebras.

Answer: Yes


Question: Evaluate the limit: $$\lim_{t\to0^+}\frac{3\sqrt{t}\ln t}{\sqrt{1+t\ln t}-\sqrt{1-t\ln t}}$$

Answer: \infty


Question: Evaluate the triple integral \( \iiint \frac{1}{\sqrt{x^2+y^2+(z-2)^2}} \, dV \) over the solid region \( E \) bounded by the sphere \( x^2+y^2+z^2 = 1 \).

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


Question: Let $V$ be a vector space of dimension 6 over a field $F$. Determine whether there exists a linear operator $T \in A(V)$ such that $\text{Ker}(T) = \text{Range}(T)$. Provide a justification for your answer.

Answer: Yes


Question:
Given two games with the same number $n$ of players, let $\mathcal{A}:=\{Y_1,Y_2\}$ be the set of equilibria of game 1, and $\mathcal{B}:=\{X_1,X_2\}$ be the set of equilibria of game 2. Assume the selection rule of game 1 is independent of the selection rule of game 2. Consider the probabilities:

$$ Pr(\text{players pick $Y_1$ in game 1}| \text{the set of equilibria of game 1 is $\mathcal{A}$}) $$

and

$$ Pr(\text{players pick $X_1$ in game 2}| \text{the set of equilibria of game 2 is $\mathcal{B}$}) $$

Does the independence assumption imply that

$$ Pr(\text{players pick $Y_1$ in game 1}| \text{the set of equilibria of game 1 is $\mathcal{A}$})\times Pr(\text{players pick $X_1$ in game 2}| \text{the set of equilibria of game 2 is $\mathcal{B}$})= Pr(\text{players pick $Y_1$ in game 1}, \text{players pick $X_1$ in game 2}| \text{the set of equilibria of game 1 is $\mathcal{A}$}, \text{the set of equilibria of game 2 is $\mathcal{B}$}) $$?

Answer: Yes


Question: For any function $f$ satisfying $|f(x)| \le Cx^3$ for some constant $C$, determine if it is necessarily true that $f'(0) = 0$. Justify your answer.

Answer: Yes


Question: Find the smallest possible value of the expression \(|z+2+i|+|z-2-i|+|3-z|+|2-z|\) for a complex number \(z\). Choose from the following options: \(2, 3, 5, 6\).

Answer: 6


Question: Determine the discrete logarithm, \( \operatorname{dlog}_x (x^2 + 1) \), in the ring \( \Bbb Z_5[x]/\langle x^3 + x + 1 \rangle \).

Answer: 30


Question:
Given the function $f(x,y)$ defined as follows:

$$ f(x,y) = \begin{cases} \frac{(2x-y)}{x+y}, & \text{if } (x,y)\ne (0,0) \\ 0, & \text{if } (x,y)=(0,0) \end{cases} $$

Determine if $f_{xy}(0,0) = f_{yx}(0,0)$ at the point $(0,0)$. Calculate the partial derivatives $f_x$ and $f_y$ for $(x,y)\ne (0,0)$, and analyze the limits to evaluate $f_x(0,0)$ and $f_y(0,0)$. Provide your answer as either "Yes" or "No".

Answer: No


Question: Find the poles and calculate the residues of the function \( \frac{e^{5z}}{(z+i\pi)^3} \).

Answer: -\dfrac{25}{2}


Question: What is the probability of selecting $r \leq n$ vectors $a_i \in \mathbb{R}^n$ such that they are linearly independent?

Answer: 1


Question: Determine whether a computer with unlimited resources can decide if a number is prime in a finite amount of time.

Answer: Yes


Question: Evaluate the Cauchy integral \(\oint_{|z-\pi|=\pi} \frac{\sin(\pi z)}{z^2-\pi^2}\,dz\) where the contour is the circle centered at \(\pi\) with radius \(\pi\).

Answer: i \sin(\pi^2)


Question: Find the remainder when \(3^{3^{3^{3^{3^{3}}}}}\ldots \) (with 2021 threes) is divided by 100, using Euler's theorem.

Answer: 87


Question: Determine the order of the following differentiably redundant functions: (a) $e^{2x} + e^{x} \sin(\frac{\sqrt{5}x}{2})$, (b) $e^{x} + \sin(2x)$, (c) $e^{-\frac{x}{3}}\cos(\frac{\sqrt{7}}{3} x)$. Which function grows the fastest as $x \to \infty$?

Answer: a


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

Answer: \dfrac{2}{3}


Question: Find the limit of the sequence \( b_n = e^{-n}\cdot \prod_{k=1}^n(1+\frac{1}{k})^k \) as \( n \to \infty \) using the inequality \( \log a_n \leqslant -\frac{1}{2} H_n +\frac{1}{3}\frac{\pi^2}{6} \).

Answer: 0


Question: Evaluate the integral \( \int_0^1 10x^{\sin x + \cos x} \, dx \) and find the greatest integer less than or equal to its value.

Answer: 4


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\sin(\tan(x))-\sin(\sin(x))}{\tan(\tan(x))-\tan(\sin(x))} \] without using L'Hôpital's rule or Maclaurin series.

Answer: 1


Question: Find the maximum value of \( f(x,y,z) = x^2y^2z^2 + x^2y^2 \) subject to the constraint \( x^2 + y^2 + z^2 = 2 \).

Answer: 1


Question: Determine whether a measurable function $f$ on a finite measure space that is not in $L^p$ for all $p > 1$ must also satisfy $f \notin L^1$. Provide a justification for your answer.

Answer: No


Question: Evaluate the complex integral \( \oint_C \frac{(z^3 -2z)}{(z^2 + 2z +2)} dz \) where \(C\) is the circle centered at the origin with radius 2.

Answer: 0


Question:
In a group of 20 children, there is a group of \( n \) grandfathers, each of whom is a grandfather to at least one child. The following conditions apply:
1. Every pair of children shares at least one grandfather from the group.
2. Each grandfather has at least two grandchildren from the group.

Determine the maximum possible value of \( n \), the number of grandfathers in the group, that satisfies these conditions.

Answer: 190


Question: Evaluate the limit: \[ \lim_{x \rightarrow 0^+} \frac{\int_{0}^{x^2} (1 + \sin(t))^{1/t} \, dt}{x \sin(x)} \]

Answer: e


Question: Evaluate the limit: $$\lim_{R\to\infty} \left(\int_{-R}^{R} \cos(x^2) \, dx\right)^2$$ using a double integral.

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit \(\lim_{x\to0}\frac{\cos(1-\sin(x))}{x^4e^x}\) without using L'Hopital's Rule or Series.

Answer: \infty


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

Answer: \dfrac{\pi}{2}


Question: Find a complex function that outputs only real values, similar to the function \( f(z) = |z| \).

Answer: f(z) = |z|^2


Question: In the context of Taylor expansion, determine whether the Hessian matrix at a local minimum is positive semidefinite. Answer true or false.

Answer: True


Question: Let the homogeneous system of linear equations $px+y+z=0$, $x+qy+z=0$, $x+y+rz=0$ where $p,q,r$ are not equal to $1$, have a non-zero solution. Find the value of $\frac{1}{1-p} + \frac{1}{1-q} + \frac{1}{1-r}$.

Answer: 1


Question: Provide an example of a positive operator $S$ in the algebra of all bounded linear operators on a complex Hilbert space $E$, such that $S$ is neither injective nor has a closed range.

Answer: S


Question: Evaluate the triple integral \( \int\int\int_{D} y \, dV \), where \( D \) is the region below the plane \( z = x + 1 \), above the \( xy \) plane, and between the cylinders \( x^2 + y^2 = 1 \) and \( x^2 + y^2 = 9 \).

Answer: 0


Question: Suppose $h(x)$ is a function that satisfies the differential equation $h(x) - h''(x) = 0$. Define $k(x) = h^2(x) - h'^2(x)$. Given that $k(5) = 12$, find the value of $k(10)$. \( \text{Provide your answer in a box: } \boxed{} \)

Answer: 12


Question: Let $X$ be a topological space and $C$ be a connected subset of $X$. Suppose $\{X_\alpha\}$ is a collection of open sets that are disjoint from one another, and their union is $X$. Does there exist an index $\alpha$ such that $C \subset X_\alpha$?

Answer: Yes


Question: Find the edge-connectivity and vertex-connectivity of the Petersen graph.

Answer: 3


Question: Determine the number of ways to arrange 6 pairs of plants in a row such that no two plants from the same pair are adjacent.

Answer: 168422400


Question: Determine the smallest positive integer \( n \) such that there exist \( n \times n \) matrices \( A \) and \( B \) over \( \mathbb{R} \) satisfying \( (AB)^k = 0 \) and \( (BA)^k \neq 0 \) for a given positive integer \( k \).

Answer: 2k


Question: Calculate the residue at $z=0$ for the complex function $\frac{1}{z^2\sin z}$, given that $z=0$ is a pole of order 3.

Answer: \dfrac{1}{6}


Question: Evaluate the limit: $$\lim_{x\to \infty} \int_1^x x\,e^{t^2-x^2} \,dt$$

Answer: \dfrac{1}{2}


Question: Calculate \( L = \lim\limits_{n \to \infty} \frac{1}{n}\cdot\log\left(3^\frac{n}{1} + 3^\frac{n}{2} + \dots + 3^\frac{n}{n}\right) \).

Answer: \log 3


Question: Evaluate the limit: $$\lim\limits_{x \to 0^{\large +}}\mrm{Li}_{x}\pars{1}$$

Answer: -\dfrac{1}{2}


Question: Consider the set $X = \mathbb{N} \times \mathbb{Q}$ with the subspace topology induced by the usual topology on $\mathbb{R}^2$. Let $P = \{(n, \frac{1}{n}) : n \in \mathbb{N}\}$. Determine whether the set $P$ is open, closed, both, or neither in this topology.

Answer: closed


Question: In a convex decagon, no three diagonals intersect at a single point inside the decagon. Determine the number of line segments formed by the intersections of the diagonals.

Answer: 455


Question: Let $x_1, x_2, \ldots, x_n$ be real numbers in the interval $[-1, 1]$ such that $\sum_{i=1}^{n} x_i^3 = 0$. Determine the minimum value of $B = \sum_{i=1}^n x_i^2$.

Answer: 0


Question: Let \( b_n=\frac{1}{n}\sqrt[n]{1^n+3^n+\ldots+(2n+1)^n} \). Determine the limit of \( b_n \) as \( n \to \infty \).

Answer: 2


Question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous and strictly convex function. Can $f$ have more than two roots, i.e., points $x$ such that $f(x) = 0$? Provide a justification for your answer.

Answer: No


Question:
Given the function $f:\mathbb{R}\mapsto\mathbb{R}$ with the properties:
1. $f(x) = f(1-x)$ for all $x \in \mathbb{R}$.
2. $\lim\limits_{x\to 2}{\frac{f(x)+4}{x-2}}=1$.

Find $\lim\limits_{x\to -1}{f(x)}$. \(\boxed{\text{Your answer here}}\)

Answer: -4


Question: Identify the name of the smallest ordinal number that is not ordinally similar to any subset of any chain in an infinite partially ordered set $S$, within the contexts of ZF and ZFC set theories.

Answer: \omega_1


Question: Given that $f(x) \in L^2 [-\pi, \pi]$ and $\int_{-\pi} ^ {\pi }f(x)e^{-inx}dx = 0$ for all integers $n$, find the value of $\int_{-\pi} ^ {\pi }|f(x) |^2 dx$. \(\boxed{0}\)

Answer: 0


Question: Determine whether an integral domain can have a non-zero element $a$ such that $a^n = 0$ for some positive integer $n$. Provide a justification for your answer.

Answer: No


Question: Determine the values of $m$ and $n$ for which a $\mathbb{Z}/m\mathbb{Z}$-module structure can be defined on $\mathbb{Z}/n\mathbb{Z}$.

Answer: n \mid m


Question: Let $\mu$ be a strictly positive measure, and let $f:X\rightarrow[0,\infty]$ be an integrable function such that $\int_Xfd\mu=1$. Calculate the limit: \[ \lim_{n\rightarrow \infty}\int_Xn\log\biggl(1+\frac{f(x)}{n}\biggl)d\mu. \]

Answer: 1


Question: Evaluate the integral \( \int_{-\pi}^\pi \left( \sum_{n=1}^\infty \frac{\cos(nx)}{2^n} \right)^2 \, dx \).

Answer: \dfrac{\pi}{3}


Question: Evaluate the limit using L'Hopital's rule: $$\lim_{x\to0}\left(\left(\frac1x\right)^{x^x}-\frac1x\right)$$

Answer: -\infty


Question: Find the minimum value of the function \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the constraint \( yz + zx + xy = 3a^2 \).

Answer: 3a^2


Question: Can a compact topological manifold be covered by a single chart?

Answer: No


Question:
Let $a$, $b$, and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = abc$. Find the minimum value of the expression:

$$a^5(bc-1) + b^5(ca-1) + c^5(ab-1).$$

Answer: 5832


Question: Evaluate the limit: \[\lim_{x\to+\infty}x\left[\left(\dfrac{a_{1}^{\frac{1}{x^2}}+a_{2}^{\frac{1}{x^2}}+\cdots+a^{\frac{1}{x^2}}_{n}}{n}\right)^{x^2}-\sqrt[n]{a_{1}a_{2}\cdots a_{n}}\right]\]

Answer: 0


Question: Let $f$ be a differentiable function such that $f(0)=0$. Determine the necessary and sufficient condition for the function $g(x)=|f(x)|$ to be differentiable at $x=0$.

Answer: f'(0) = 0


Question: Determine the value of $q$ such that the distance between the point $P(u, v)$ and the circle defined by $x^2 + 4x + y^2 = 0$ is equal to the distance between $P(u, v)$ and the point $M(2, 0)$. The point $P$ satisfies the equation $u^2 - \frac{v^2}{q} = 1$. The circle has a center at $(-2, 0)$ and a radius of $2$. Find $q$.

Answer: 3


Question: Let \(X_1, X_2, \ldots\) be independent random variables with \(\mathbb{E}(X_i) = \mu\) for all \(i\). Let \(N\) be a random variable that is independent of all the \(X_i\) and satisfies \(\mathbb{E}(N) = k\). Find \(\mathbb{E}\left(\prod_{i=1}^{N}{X_i}\right)\) in terms of \(\mu\) and \(k\).

Answer: \mu^k


Question: In triangle $ABC$, where $AD$ is the median, it is given that $AD=\frac{abc}{b^2-c^2}$ with $b>c$ and $\angle C=23^{\circ}$. Can $\angle B$ be uniquely determined with this information? Provide a justification for your answer.

Answer: Yes


Question: Find the value of \( k \) if \( z = \left| \begin{array}{ccc} 3 + 2i & 1 & i \\ 2 & 3 - 2i & 1 + i \\ 1 - i & -i & 3 \end{array} \right| \) and \( \left| z + \overline{z} \right| = k \left| z \right| \), where \( \overline{z} \) is the conjugate of \( z \).

Answer: 2


Question:
For which of the following functions does the series \( \sum_{x \in S(f)} \frac{1}{x} \) converge, where \( S(f) = \{ x : x > 0, f(x) = x \} \)?

- \( f(x) = \tan x \)
- \( f(x) = \tan^2 x \)
- \( f(x) = \tan \sqrt{x} \)
- \( f(x) = \sqrt{\tan x} \)
- \( f(x) = \tan 2x \)

Answer: C


Question: Suppose $f(x)$ is a differentiable function on the entire real line $\mathbb{R}$ and satisfies $\lim_{x\rightarrow \infty} f(x) = \lim_{x\rightarrow -\infty} f(x) = 17$. Can Rolle's Theorem be applied to demonstrate the existence of a point $\xi \in (-\infty, \infty)$ such that $f'(\xi) = 0$?

Answer: Yes


Question: Determine the range of values for \( \alpha \) such that the function \( c(u_1,u_2) = 1 + \alpha(1- 2u_1)(1- 2u_2) \), where \( u_1, u_2 \in (0,1) \), is a valid copula density.

Answer: [-1, 1]


Question: Does there exist a set \( S \subset \mathbb{R}^2 \) such that its set of limit points in \( \mathbb{R}^2 \) contains the line \([0,1] = \{(x,0): 0 \le x \le 1\}\) and no three points in \( S \) are collinear?

Answer: Yes


Question: Determine the maximum number of vertices of an \((n + 3)\)-dimensional hypercube that can be visited by a polygonal chain of size \(n\).

Answer: n + 1


Question: Find the value of \( \liminf_{n\to\infty}\left(-1-\frac{2}{n}\right)^n \).

Answer: -e^2


Question: Is it possible to cover all the irrational numbers in the interval $[0,1]$ with a finite sequence of open intervals such that the total sum of the lengths of these intervals is less than 1?

Answer: No


Question: Determine whether an entire function $f: \mathbb{C} \rightarrow \mathbb{C}$, for which the preimage of every complex number $a$ forms a bounded set, must necessarily be a polynomial. Provide a justification for your answer.

Answer: Yes


Question: Determine the smallest integer $n$ such that $n! > 10^{17}$ without using a calculator or the gamma function.

Answer: 19


Question: Evaluate the double integral \( \iint_D x\,dxdy \), where \( D \) is the disk centered at \((2,3)\) with radius \(2\), using polar coordinates.

Answer: 8\pi


Question: Evaluate the limit: $$\lim_{x \to \infty}{x^a a^x}$$ where $0<a<1$. Provide a justification for your answer.

Answer: 0


Question: Given a convergent sequence $\{x_n\}$ with limit $x$, if there is no constant subsequence in $\{x_n\}$, does the limit $x$ become a limit point of the range set of $\{x_n\}$? Answer yes or no and justify your answer.

Answer: Yes


Question: Determine whether the following statement is true for all real numbers: If the product of two numbers is greater than a third number, then there exist two other numbers whose product is equal to the third number and are both less than the original two numbers.

Answer: False


Question: Determine the upper bound for the length of the shortest word that can distinguish two non-equivalent NFAs, one with $n$ states and the other with $k$ states.

Answer: 2^{n + k} - 1


Question: Determine the cardinality of the set of $\mathbb{Q}$-morphisms from the algebraic closure of $\mathbb{Q}$ to $\mathbb{C}$. Provide your answer as a cardinal number.

Answer: 2^{\aleph_0}


Question: Let $A$ be a nonzero $n \times n$ matrix such that $A^k=0$ for some positive integer $k$. Determine the maximum possible number of non-zero eigenvalues of $A$. Express your answer as an integer.

Answer: 0


Question: Let $T: \ell_2 \to \ell_2$ be defined by $T((x_1, x_2, \ldots, x_n, \ldots)) = (x_2 - x_1, x_3 - x_2, \ldots, x_{n+1} - x_n, \ldots)$. Find the norm of $T$ using the inequality $|a-b|^2 \leq 2(a^2 + b^2)$. \( \boxed{\text{Answer}} \)

Answer: 2


Question: Given the sequence \( \{a_k\} \) of integers where \( a_1 = 1 \) and the recurrence relation \( a_{m+n} = a_m + a_n + mn \) holds for all positive integers \( m \) and \( n \), determine the value of \( a_{12} \).

Answer: 78


Question: Determine whether the set \( T = \{(a,b) \mid g(a) = b\} \) can be dense in \([0,1] \times [0,1]\) if the function \( g: [0,1] \to [0,1] \) is discontinuous.

Answer: Yes


Question: Evaluate the limit without using L'Hopital's Rule: $$\lim\limits_{x\rightarrow 0}\frac{\sinh^{-1}(\sinh(x))-\sinh^{-1}(\sin(x))}{\sinh(x)-\sin(x)}$$

Answer: 1


Question: Determine whether the space $L_p(X, \mu, V)$ is separable given that $X$ is a Polish space, $V$ is a separable normed space, and $\mu$ is a $\sigma$-finite measure.

Answer: Yes


Question: Define the function \( f(x,y,z) = \frac{x\sin x + y\sin y + z\sin z}{x^2 + y^2 + z^2} \) at \((0,0,0)\) such that it is continuous at that point.

Answer: 1


Question: Determine whether the expression \(a^2 + b^2 + 2ac\) can be a perfect square for integers \(a, b, c\) given that \(c \neq \pm b\).

Answer: Yes


Question: Determine whether the limit \( \lim_{{n \to \infty}} \sum_{{k=1}}^{\infty} \frac{1}{(2k-1)^{2n}} \) converges to 1.

Answer: 1


Question: Find the limit of the probability density function of the lognormal distribution with parameters \( \mu \) and \( \sigma^2 \) as the random variable approaches infinity.

Answer: 0


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

Answer: e


Question: Determine the number of positive integers less than or equal to $10^9$ that are neither perfect squares, perfect cubes, nor perfect seventh powers.

Answer: 999967395


Question: Find the number of Sylow $11$-subgroups in a group of order $5^2 \cdot 7 \cdot 11^2$. Express your answer as a single integer.

Answer: 1


Question: Evaluate the infinite series \( \sum \limits_{k=1}^{\infty} \frac{\zeta(2k)-1}{k} \), where \( \zeta(s) \) is the Riemann zeta function defined as \( \sum \limits_{n=1}^{\infty} n^{-s} \).

Answer: \ln 2


Question: Calculate the area enclosed by the curve formed by the intersection of the plane $z=2$ with the ellipsoid $\frac{x^2}{25} + \frac{y^2}{1} + \frac{z^2}{5} = 1$ in the $xOy$ plane.

Answer: \pi


Question:
Evaluate the integral sum given that \( a^5 - b^5 = 1 \):
\[ \int_1^a \sqrt[5]{x^5-1}\,dx + \int_0^b \sqrt[5]{x^5+1}\,dx \]

Answer: ab


Question: In a category with directed colimits and finite limits, determine if the directed colimit of a subobject can be expressed as a colimit of subobjects of the original objects in the direct system. Answer with 'True' or 'False'.

Answer: False


Question: Evaluate the limit: $$\lim_{x\to{0}}\left(\frac{3}{x^4}(\arccos{x}-\arcsin{x})\right)^{1/x^2}$$

Answer: \infty


Question: Evaluate the limit \( \lim_{x \to 0} (\sin x)^{\frac{1}{x}} + \left(\frac{1}{x}\right)^{\sin x} \).

Answer: 1


Question: Determine whether the following statement is true: \[ \mathbb{E}[X_{t_{1}}X_{t_{2}}]=\int_{\mathbb{R}}\int_{\mathbb{R}}x\,\mathrm{d}F_{t_{1},t_{2}}(x),\quad x=(x_{1},x_{2})\in\mathbb{R}^{2}. \]

Answer: False


Question: Evaluate the limit using L'Hopital's Rule: $$\lim_{x \to 0} \frac{\ln(1+\ln(1+x))}{\ln(1+\ln(1+\frac{1}{x}))}$$

Answer: 0


Question: Given that the period of a pendulum is related to its length by the equation \( T = 2\pi\sqrt{\frac{L}{g}} \), and the rate of change of the pendulum's length with respect to temperature is directly proportional to the length with a proportionality constant \( k \), find the rate of change of the period \( T \) with respect to temperature.

Answer: \dfrac{kT}{2}


Question: Determine the maximum order of a permutation in the symmetric group $S_{10}$. Additionally, find the smallest positive integer $n$ such that a bijection $f: \{1, 2, 3, \ldots, 10\} \rightarrow \{1, 2, 3, \ldots, 10\}$ satisfies $f^n = f$ but $f^k \neq f$ for all $1 \leq k < n$. Provide your answer as a single integer.

Answer: 30


Question: Does there exist a totally ordered field in which the rational numbers \( \mathbb{Q} \) are not dense?

Answer: Yes


Question: Does the convergence of the series \(\sum \frac {a_n}{n}\), where \(a_n > 0\) for all positive integers \(n\), imply that \(\lim_{m\to \infty}\sum_{n=1}^m \frac{a_n}{m+n} = 0\)?

Answer: 0


Question: Find all real roots of the polynomial equation \( P(x) = (2x-5)^{2017}+(2x-5)^{2015}+(x-4)^{2017}+(x-4)^{2015}+3x-9 = 0 \).

Answer: 3


Question: Calculate the probability that the sum of the squares of two randomly chosen natural numbers $x$ and $y$, each uniformly distributed in the range $(0, N)$, is divisible by 35 as $N$ approaches infinity.

Answer: \dfrac{9}{1225}


Question: Evaluate the limit: $$\lim_{x\to \infty} \frac{\ln (x^4+1) \ln (3e^x+1)}{\ln (x^5+1) \ln (x^4+6x)}$$

Answer: \infty


Question: Compute the expected value \( \text{E}[X] \) of a random variable \( X \sim \mathcal{N}(\mu, \sigma^2) \) using the density function \( f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)^2\right) \). Use the substitution \( y = \frac{x-\mu}{\sigma} \) to simplify the calculation.

Answer: \mu


Question: What is the maximum number of points that can be placed inside a ball of radius 1 in \(\mathbb{R}^d\) such that the distance between any two points is at least \(\sqrt{2}\)?

Answer: 2d


Question: Let $f(x)$ be a twice differentiable function. If there exists a point $c$ such that $f''(c) = 0$, does it imply that there are no two values of $a,b \in \mathbb{R}$ for which $(b-a)f'(c) = f(b)-f(a)$? Answer with 'Yes' or 'No' and justify your answer.

Answer: No


Question: Determine the exact condition for the parameters $a$, $b$, and $n$ in the equation $(a+b)^n = \left(\frac{1}{a-b}\right)^n$, where $a$ and $b$ are positive real numbers and $n$ is a positive integer, using the binomial theorem for negative exponents.

Answer: a^2 - b^2 = 1


Question: In how many ways can a $5 \times 5$ matrix be formed with elements either $0$ or $1$ such that the sum of elements in each row and column is exactly $4$?

Answer: 120


Question: Calculate the work done by the vector field \( \mathbf{F}(r,\theta) = -4 \sin \theta\ \mathbf{i} + 4 \sin \theta\ \mathbf{j} \) when a particle moves from the point \((1,0)\) to the origin along the spiral path described by the polar equation \( r = e^{-\theta} \).

Answer: \dfrac{8}{5}


Question: Evaluate the limit \( \lim\limits_{x\to 0^-} \frac{\ln(x)}{\cot(x)} \) using the standard limit \( \lim\limits_{x\to 0^+} x\ln(x) = 0 \).

Answer: 0


Question: Let \( G \) be a cubic graph with \( q \) edges and girth \( \ge 4 \). Determine the number of paths of length 3 in \( G \).

Answer: 4q


Question: Consider the spaces $X$ and $Y$ with the Sorgenfrey topology. Let the function $f: X \to Y$ be defined as follows: $f(x) = 1$ if $x < 0$ and $f(x) = 2$ if $x \geq 0$. Determine whether the function $f$ is continuous.

Answer: Yes


Question: Given a matrix $A$ such that $A^2 = A$ and its characteristic polynomial is $(x-1)^j x^{n-j}$, where $0 \le j \le n$, determine the rank and trace of the matrix $A$. Express your answer as a single number.

Answer: j


Question: Evaluate the limit \[ L = \lim_{n \to \infty} \left( 15 \sqrt[n]{c_n} + 10 \sqrt[n]{d_n} \right) \] where \[ c_n = \int_0^1 (1-x^4)^n \, dx \] and \[ d_n = \int_0^1 (1-x^5)^n \, dx. \]

Answer: 25


Question: Evaluate the limit: \[ \lim_{n\to\infty} \frac{n! + n^n}{1 + \ln(n)} \] Given that \(n! = o(n^n)\), determine if the limit diverges to \(+\infty\).

Answer: +\infty


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

Answer: 0


Question: Does a continuous function \(\omega: H \rightarrow H\), where \(H\) is a compact topological space, always have a non-empty fixed subset in \(H\)?

Answer: Yes


Question: Determine the isomorphism type of the quotient group $U_{26}/\langle 5 \rangle$, where $U_{26}$ is the group of units under multiplication modulo 26, the order of $U_{26}$ is 12, and the order of the subgroup generated by 5 is 4.

Answer: \mathbb{Z}_3


Question: Suppose $X$ and $Y$ are two independent real random variables such that the distribution of $X+Y$ is the same as the distribution of $X$. Does this imply that the distribution of $Y$ is degenerate at $0$?

Answer: Yes


Question: Calculate the integral \( \oint_{\gamma} \sqrt{1+x^3} \, dx + 2x \, dy \) for the curve \( \mathcal{C} \) defined by the vector function \( \vec{r}(t) = (1-t^2)\vec{i} + (t-t^3)\vec{j} \) with \( t \in [0, 1] \), using Green's theorem. The curve \( \gamma \) is smooth, closed, simple, and oriented counterclockwise. Express the integral in terms of a double integral over the region \( D \) enclosed by \( \gamma \).

Answer: 2 \iint_D dA


Question: Determine the maximum value of the function \( h(x) \) defined as follows: \[ h(x) = \begin{cases} \min\{g(t):1 \le t \le x\}, & \text{if } 1 \le x < 4 \\ \max\{g(t): 6 \le t \le x\}, & \text{if } 6 \le x \le 9 \end{cases} \] where \( g(x) = \begin{cases} x^2 - 4x, & \text{if } 1 \le x < 3 \\ 2, & \text{if } 3 \le x < 6 \\ x^2 + 2x, & \text{if } 6 < x \le 9 \end{cases} \]

Answer: 99


Question: Determine the number of independent components of the Riemann curvature tensor \( R_{iklm} \) in a 4-dimensional spacetime.

Answer: 20


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left\{ \frac{n!}{\prod_{k=1}^{n} H_k} \right\}^{\frac{1}{n}} \] where \( H_k = \sum_{i=1}^{k} \frac{1}{i} \).

Answer: \infty


Question: Solve the equation \(-x^3 + x + 2 = \sqrt{3x^2 + 4x + 5}\).

Answer: -1


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

Answer: 0


Question: Given that \(\alpha\), \(\beta\), and \(\gamma\) are the roots of the cubic equation \(x^3 + x - 1 = 0\), find the value of \(\alpha^8 + \beta^8 + \gamma^8\).

Answer: -6


Question: Given that $x,y \in\mathbb{R}$ satisfy the equation $x^2 + y^2 - 6x + 8y + 24 = 0$, find the greatest value of the expression $$\frac{16\cos^2(\sqrt{x^2+y^2})}{5} - \frac{24\sin(\sqrt{x^2+y^2})}{5}.$$

Answer: 5


Question: Let $a$ and $b$ be complex numbers such that $|a| = |b| > 1$. If the sequence $(a^n - b^n)$ is bounded, prove that $a = b$.

Answer: a = b


Question:
Calculate the limit of the derivative of the function \(f:\Bbb R \to \Bbb R\) defined as:

\[f(x) = \begin{cases}
x^2 \left\lfloor \dfrac{1}{x} \right\rfloor & \text{if } x \neq 0 \\
0 & \text{if } x = 0
\end{cases}\]

Find the value of:

\[\lim_{x \to 0} f'(x)\]

Answer: 2


Question: Evaluate the limit: $$\lim_{k \to \infty} \frac{(2k)!}{2^{2k} (k!)^2}$$.

Answer: 0


Question: Determine the cardinality of the set \([0,\Omega) \setminus (\bigcup \{ U_\lambda : \lambda < \Omega\text{ is limit} \})\) given that \(U_\lambda = [0, \Omega)\) for all limit ordinals \(\lambda\), where \(\Omega\) is the first uncountable ordinal with the order topology.

Answer: 0


Question: Compute the Galois group \( \operatorname{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}(\sqrt{2})) \).

Answer: \mathbb{Z}/2\mathbb{Z}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{(n+1)!e^{n+1}}{(n+1)^{n+1}}. \]

Answer: \infty


Question: Determine if the limit \( \lim_{x\to0} x \int_x^1 \frac{\sin t}{t^2} dt \) exists using an appropriate theorem.

Answer: 0


Question: Evaluate the contour integral \( \oint_C \frac{\cos z}{z^3} \space dz \) where the contour does not enclose any poles.

Answer: 0


Question: Determine whether there exists a real number $r$ such that $\pi^r$ is a rational number.

Answer: Yes


Question: Determine the number of edges in the line graph L(G) of a graph G, given that G is regular, planar, and its line graph is planar but not isomorphic to G.

Answer: 12


Question: Find the limit of the product \( P_{2m-1} \) as \( m \to \infty \), where \( P_m = \prod_{n=1}^{m}\left[1+\frac{(-1)^{n+1}}{2n-1}\right] \).

Answer: \sqrt{2}


Question: What is the maximum number of points that can be specified in a projective transformation while preserving collinearity and incidence?

Answer: 4


Question: Find the sum of the infinite series \(\sum_{k=1}^\infty \frac{(5k+7)}{k^3+4k^2+3k}\).

Answer: \dfrac{31}{9}


Question: Determine the number of homomorphisms from the group $\mathbb{Z}_n$ to the group $\mathbb{Q}$. \( \boxed{1} \)

Answer: 1


Question: Determine whether the following statement is true or false: For any two complex $n \times n$ matrices $A$ and $B$, if $AB$ is a nilpotent matrix, then $BA$ is also nilpotent.

Answer: True


Question: Solve the equation \(x^{\log(x)-1 } = \frac{1}{\sqrt[4]{10}}\).

Answer: \sqrt{10}


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

Answer: 1


Question: Find the sum of the coefficients of the terms containing $x^4t^2$ in the expansion of $(x^2 + y + 2t + 3k)^{12}$ using the multinomial expansion formula.

Answer: 778567680


Question: Let $S$ be the ellipsoid defined by \( \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2=1, \) with the outward normal vector \( \vec{n}. \) Compute the surface integral \( \int\!\!\!\int_S \vec{F}\cdot \vec{n}\,dA \) for the vector field \( \vec{F}=(-a^2y, b^2x, z^2). \)

Answer: 0


Question: Evaluate the integral \( \int_{C} \cos\left(\frac{z}{2}\right)\,dz \), where \( C \) is the contour defined by \( \gamma(t) = t + i\sqrt{\pi^2 - t^2} \) for \( -\pi \leq t \leq \pi \).

Answer: 4


Question: Arrange 80 people in a network such that each person listens to exactly 12 other people, and each person is listened to by exactly 12 others. Assume that listening to and being listened to by the same person is allowed. Determine if such an arrangement is possible.

Answer: Yes


Question: Given an nxp matrix A and a pxm matrix B, if the kernel of A is equal to the image of B, determine the properties of the product AB.

Answer: AB = 0


Question: Determine whether the following statement is true or false: Every infinite set has a countably infinite subset.

Answer: A


Question: Determine whether the converse of the following statement is true: If $X$ and $Y$ are path connected, then $X \times Y$ is path connected.

Answer: True


Question: Compute the expected value \( \mathbb{E}[\tilde{X}_t] \), where \( \tilde{X}_t = (1-t)\int_0^t \frac{1}{1-s} \, dW_s \) for \( 0 \leq t < 1 \) and \( \tilde{X}_t = 0 \) for \( t = 1 \).

Answer: 0


Question: Given the equations $a+b=23$, $ax+by=79$, $ax^2+by^2=217$, and $ax^3+by^3=691$, determine the value of $ax^4+by^4$. Provide your answer as a single number.

Answer: 1993


Question: Find the smallest positive integer $n$ such that $n^8 + (n + 1)^8$ is a composite number.

Answer: 2


Question: Evaluate the limit \( \lim_{x \to +\infty } \int_{x^2}^{x^3} t \sinh^{-1}(t^4) \, dt \) and find a finite non-zero limit or the main part for the corresponding power function.

Answer: 6 x^6 \ln x


Question: Let $M$ be a maximal ideal in $\mathbb{Z}[x]$. Determine whether $M$ intersects with $\mathbb{Z}$ non-trivially.

Answer: Yes


Question: Find the limit of the series as \( n \) approaches infinity: \[ \lim_{n \to \infty} \left(\frac{1}{n+1} + \frac{1}{n+2} + \frac{1}{n+3} + \dots + \frac{1}{2n}\right). \]

Answer: \ln 2


Question: In a finite group $G$ of order $n$, if $a^n = e$ for all elements $a \in G$, does this imply that the order of each element $a$ is equal to $n$?

Answer: No


Question: Given that \(|z-i| \le 2\) and \(z_1 = 5 + 3i\), find the maximum value of \(|iz + z_1|\).

Answer: 7


Question: Let $A$ be a linear operator on a normed space $X$. If for all $x, y \in X$, the condition $\langle Ax, y \rangle = \langle x, Ay \rangle$ holds, does it follow that $A$ is continuous? Provide a justification for your answer.

Answer: No


Question: Determine whether the following statement is true or false for matrices $A, B \in M_n(\mathbb{R})$ with $n \geq 2$: If $(A \cdot B)^\ast = B^\ast \cdot A^\ast$ holds for invertible matrices $A$ and $B$, then it holds for all matrices $A$ and $B$. Provide your answer as either "True" or "False."

Answer: True


Question: Determine the range of the expression \( \frac{|x+y|}{|x|+|y|} + \frac{|y+z|}{|y|+|z|} + \frac{|z+x|}{|z|+|x|} \) for real numbers \( x, y, \) and \( z \).

Answer: [1, 3]


Question: Consider a Banach space $X$ and the closed unit ball $B$ of $X^*$, equipped with the weak-* topology. Alaoglu's theorem states that $B$ is compact. If $X$ is separable, then $B$ is metrizable and sequentially compact. Is the converse true? If $B$ is sequentially compact, does it imply that $X$ is separable?

Answer: Yes


Question: Evaluate the flux of the vector field \( F = (x^2, xy) \) through the unit circle \( C \), oriented counterclockwise, by computing the line integral \( \oint_C F \cdot n \, ds \).

Answer: 0


Question: Determine whether a bounded sequence \( \{a_n\} \) that has no two subsequences converging to different limits must necessarily converge. Provide a justification for your answer.

Answer: Yes


Question: Let \( f \) be a function satisfying the conditions \( \int_{1}^{4}f(x)\,dx=0 \) and \( -2\leq f(x)\leq 2 \). Find the minimum value of \( \int_{1}^{4}\frac{f(x)}{x^2}\,dx. \)

Answer: -\dfrac{9}{10}


Question: Find the Ramsey number for the graphs $(m+1)K_4$ and $K_4$, given that $R(mK_4,K_4) + 4 \leq R((m+1)K_4,K_4)$ for $m \geq 2$. Provide your answer as a single number.

Answer: 4m + 18


Question: Evaluate the limit \(\lim_{(x,y) \to (0,0)} \frac{x \sin(y)}{x^2 + |y|}\), if it exists.

Answer: 0


Question: Determine whether the following statement is true or false: For holomorphic Jacobi forms, if the Taylor development begins with $\alpha \cdot \Delta(\tau)\cdot z^{8}$ or $\beta\cdot \Delta(\tau) \cdot E_{4}(\tau)\cdot z^{12}$, the weight cannot be less than 12.

Answer: True


Question: If Rouche's theorem conditions are satisfied for two complex functions, where $|f(z)| < |g(z)|$ holds along a contour and both functions are analytic within and on the contour, does this imply that $g(z)$ and $f(z) + g(z)$ have an equal number of zeros, including those on the contour?

Answer: Yes


Question: Find the formula for $f(n)$ given the recurrence relation $\ln(1+f(n)) = \ln(1+f(n-1)) + \ln(1+f(n-2))$ with initial conditions $f(0) = 0$ and $f(1) = 1$.

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


Question: Given two distinct vectors $u$ and $v$ in $\mathbb{R}^n$ such that $u \neq \lambda v$ for any $\lambda \in \mathbb{R}$, determine whether there exists a symmetric positive definite matrix $A \in \mathbb{R}^{n\times n}$ such that $u^TAv=0$. The vectors $u$ and $v$ are then said to be $A$-orthogonal or $A$-conjugate vectors.

Answer: Yes


Question: Let \(X_1, X_2, \ldots, X_n\) be a sequence of i.i.d. random variables with the common characteristic function \(f(t) = e^{-t^2 - \sqrt{|t|}}\). Determine the value of \(\alpha\) such that \(\eta_n = n^{\frac{1}{\alpha}}\) and find the limiting distribution of \(\frac{X_1 + X_2 + \cdots + X_n}{\eta_n}\) as \(n \to \infty\).

Answer: \frac{1}{2}


Question: Determine whether the following statement is true or false: As \( n \) increases through the integers, the number of primes in the open interval \((n, 2n)\) can stay the same, increase by one, or decrease by one, but never change by two or more.

Answer: True


Question: Evaluate the integral \( \int_3^4 \sqrt{(x - 3)(4 - x)} \, dx \) using the property that replacing \( x \) with \( 7 - x \) results in the same integral. Choose the correct answer from the following options: (a) \( \frac{\pi}{4} \), (b) \( \frac{\pi}{8} \), (c) \( \frac{\pi}{2} \), (d) none.

Answer: B


Question: Let $K$ be a finite extension of $\mathbf{Q}_p$. Determine whether the following statement is true: The ramification degree $e_K$ of $K$ over $\mathbf{Q}_p$ has $p^n$ as a divisor if and only if $K(\zeta_p)$, where $\zeta_p$ is a $p$th root of unity, contains $\zeta_{p^{n+1}}$. Provide a justification for your answer.

Answer: True


Question: Suppose $h_{k}(x) > 0$ and $\sum_{k=1}^{l} h_{k}(x) = 1$, where $h_{k}(x)$ are continuous functions. If $\sum_{k=1}^{l} h_{k}(x) g_{k}(x) = 0$ for all $x \neq 0$, where $g_{k}(x)$ are polynomials, determine if it is true that $\sum_{k=1}^{l} h_{k}(x) f_{k}(x) > 0$ whenever $f_{k}(x) > 0$ for all $x \neq 0$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exists a topological space $X$ such that every subset of $X$ is either open or closed, but the only subsets that are both open and closed (clopen) are $X$ itself and the empty set $\varnothing$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{h \to 0} \frac{1}{h} \int_{-\infty}^{\infty} g\left(\frac{x}{h}\right) f(x) \, dx \] where $g:\mathbb{R} \to \mathbb{R}$ is a continuous function such that $g(x) \geq 0$ for all $x \in \mathbb{R}$, $g(x) = 0$ if $|x| \geq 1$, and \( \int_{-\infty}^{\infty} g(t) \, dt = 1 \). $f:\mathbb{R} \to \mathbb{R}$ is a continuous function.

Answer: f(0)


Question: Determine the cardinality of the set of all bijective functions from \(\mathbb{N}\) onto \(\mathbb{N}\).

Answer: 2^{\aleph_0}


Question: In a measure space $(\Omega, F, P)$, if $X$ is a random variable and $N$ is a sub sigma algebra of $F$, with respect to which sigma algebra is the almost everywhere (a.e.) equivalence class of the conditional expectation $E(X|N)$ defined: $F$ or $N$?

Answer: N


Question: Determine the genus of the graph $G$ with vertex set $V(G)=\{u_1,\cdots,u_8,v_1,\cdots,v_5\}$ and edge set $E(G)=\{u_1u_3,u_1u_4,u_1u_5,u_1u_6,u_1u_7,u_1u_8,u_1v_2,u_1v_3,u_2u_3,u_2u_4,u_2u_5,u_2u_6,u_2u_7,u_2u_8,u_2v_2,u_2v_3,u_3u_5,u_3u_6,u_3u_7,u_3u_8,u_3v_4,u_3v_5,u_4u_5,u_4u_6,u_4u_7,u_4u_8,u_4v_4,u_4v_5,u_5v_1,u_6v_1,u_7v_1,u_8v_1.\}$.

Answer: 1


Question: Evaluate the integral \( \int_{0}^{\infty}\frac{\cos(x) - \sin(x)}{x^{1/2}}\,\mathrm{d}x \) using complex analysis.

Answer: 0


Question: Find the minimal Hausdorff compactification $K$ of the interval $[0, \infty)$ such that the function $f(x) = \frac{x}{1+x}$ can be continuously extended to $K$.

Answer: [0, 1]


Question: For the sequence $u_n$, where $u_1 = 1$, $u_2 = 2$, and $u_{n+1} = 3u_n - u_{n-1}$, define $v_n$ as the sum $\sum\limits_{k=1}^{n} \cot^{-1}(u_k)$. Given that $u_n = \sqrt{5} \cdot \left({{\left(\frac{1+\sqrt{5}}{2}\right)}^{2n-1}}+{{\left(\frac{\sqrt{5}-1}{2}\right)}^{2n-1}}\right)$, find the limit $\lim\limits_{n\to\infty} v_n$. \(\text{Express your answer as a single number.}\)

Answer: \dfrac{\pi}{2}


Question:
Consider the set $X = \{(x,y) \in \mathbb{R}^2 : 2x^2 + 3y^2 = 1\}$. Endow $\mathbb{R}^2$ with the discrete topology, and $X$ with the subspace topology. Determine which of the following statements is true:

A. $X$ is a compact subspace of $\mathbb{R}^2$ in this topology.
B. $X$ is a connected subspace of $\mathbb{R}^2$ in this topology.
C. $X$ is an open subspace of $\mathbb{R}^2$ in this topology.
D. None of the above.

Answer: C


Question: Determine the radius of convergence for the series solution of the Legendre differential equation given by \((1-x^{2})\frac{\mathrm d^{2}{y}}{\mathrm d{x}^{2}} - 2x \frac{\mathrm d{y}}{\mathrm d{x}} + l(l+1)y = 0\). The series solution has the form \(y(x) = c_{1}y_{1}(x) + c_{2}y_{2}(x)\) with \( y_{1} = x + \sum_{n = 1}^{\infty} \left [ \frac{(-1)^{n}}{(2n+1)!} \prod_{k=1}^{2n} \left ( l+(-1)^{k}k \right ) \cdot x^{2n+1} \right ]\) and \( y_{2} = 1 + \sum_{n = 1}^{\infty} \left[ \frac{(-1)^{n}}{(2n)!} \prod_{k=1}^{2n} \left ( l+(-1)^{k}(k-1) \right ) \cdot x^{2n} \right]\). Given that applying the Ratio Test to the general term of \(y_1\) gives \(L = x^2\), find the radius of convergence.

Answer: 1


Question: Evaluate the limit \( \lim_{n\to\infty} \frac {1}{n^k}\binom{n}{k} \), where \( k \) is a positive integer.

Answer: \dfrac{1}{k!}


Question: Let $m$ and $n$ be positive integers. Suppose we have $m$ homogeneous polynomial equations with complex coefficients, involving $n$ variables, where each variable appears in every polynomial. If $m < n$, do these $m$ equations always have at least one non-trivial common root in the field of complex numbers?

Answer: Yes


Question: In a game, three players, $A$, $B$, and $C$, compete to earn the highest score. The score of each player is represented by a Poisson distributed random variable: $A$ with parameter $\lambda_A = 2$, $B$ with $\lambda_B = 3$, and $C$ with $\lambda_C = 4$. The scores are independent. Given that the total combined score of the players is $A + B + C = 6$, with the scores being $1, 2$, and $3$, respectively, but not knowing which player got which score, calculate the conditional probability that player $C$ had the highest score, i.e., $C = 3$. \( \boxed{?} \)

Answer: \dfrac{40}{81}


Question: Determine whether the following statement is true or false: Let $(\Omega, \mathcal{F})$ be a measurable space and $f: \Omega \rightarrow \mathbb{R}$ be a measurable function. If $S_n = \{\omega \in \Omega: f(\omega) > a_n\}$ is measurable for all $n \in \mathbb{N}$, where $\{a_n\}$ is a sequence of real numbers, then the set $S = \cup_{n\geq 1} S_n$ is also measurable.

Answer: A


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\cos(x^2+\frac{1}{x})-\cos(\frac{1}{x})}{x}$$ without using L'Hopital's rule.

Answer: 0


Question: Determine the maximum value of the integral \(\int_c^d \frac{\cos x}{x^2} \, dx\) where \(0 < c < d\).

Answer: \infty


Question: Let \( A \) be an \( n \times n \) real strictly upper triangular matrix. If the entry \( a_{nn} \) is changed to a non-zero real number \( z \), will the Jordan block sizes of the resulting matrix remain the same for all non-zero \( z \)? Answer with 'Yes' or 'No'.

Answer: No


Question: Determine the number of integer values of $b$ such that the cubic equation $x^3 + b^2x^2 + 2x + 3 = 0$ has integer solutions.

Answer: 1


Question: Find the 55th number of the form \(3^a + 3^b + 3^c\) where \(a, b, c\) are distinct positive integers. Determine the sum \(a + b + c\) for this number.

Answer: 20


Question: Determine the bias of the estimator \(\bar{\theta} = 2\bar{X} - 1\), where \(\bar{X}\) is the sample mean of \(n\) independent random variables drawn from a uniform distribution \(U[0,1+\theta]\).

Answer: 0


Question: Determine the number of 9-digit positive integers that do not contain any substring of three consecutive repeating digits.

Answer: 843160671


Question: Evaluate the limit: $$\lim_{n\to\infty} \sum_{i=1}^n\sum_{j=1}^n\left[\frac{1}{n^2}\ln\left(1+\frac{1}{n^2}\sin\left(\frac{i}{n}\right)\cos\left(\frac{j}{n}\right)\right)\right]$$

Answer: 0


Question: Calculate the limit of the infinite product: $$\lim_{n \to \infty}\prod\limits_{k=1}^{n} \left(1+\frac{1}{2k-1} \right)$$

Answer: \infty


Question: Calculate the limit \(\lim_{x \to \infty} \left(1-\frac{1}{x}\right)^{e^x}\).

Answer: 0


Question: Determine whether a unital commutative C* algebra can be constructed with exactly five non-trivial projections.

Answer: No


Question: Identify a non-algebraically closed field of characteristic 2 in which the equation \( z^2 + f = 0 \) has no solution.

Answer: \mathbb{F}_2(t)


Question: Find the remainder when \(7^{50}\) is divided by 2023 using the "divide and conquer" method.

Answer: 1477


Question: Calculate the norm of the operator \( A:L^{1}[1,2] \rightarrow L^{1}[0,1] \), defined by \( Af(x) = \int_{1}^{2} \frac{f(t)}{x+t} \, dt \).

Answer: \ln 2


Question: Determine the number of homomorphisms from the symmetric group $S_7$ to the alternating group $A_8$. Provide your answer as a single integer.

Answer: 316


Question: Determine the order of the pole of the function \(\frac{\mathrm{Log}(z)}{(z-1)^4}\) at \(z=1\), where \(\mathrm{Log}(z)\) is the principal branch of the logarithm.

Answer: 3


Question: Given a ring $S$ with identity and a specified number of elements, determine the number of ring homomorphisms from $\mathbb{Q}[x]$ to $S$. Provide your answer as a single integer.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to\infty} (e^x+x)^{4/x}.$$

Answer: e^4


Question: Find the value of $x$ that satisfies the equation $2^{x^2-3}=x^{-1/3}$ using a numerical approximation method.

Answer: 1.660


Question: Determine whether there exists a non-abelian finite group in which more than half of the elements have order 2.

Answer: Yes


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

Answer: f(x) = x


Question: Given that \( f(x) \) is differentiable at \( x=0 \) with \( f(0)=0 \), \( f'(0)=1 \), \( f''(0)=2 \), and \( f^{(n)}(0)\le \frac{n!}{2^{n-3}} \) for \( n\ge 3 \), find the smallest value of \( m \) such that \( f(0.5)\le m \).

Answer: \dfrac{11}{12}


Question: Approximate the definite integral \( \int_{0}^{0.3} \frac{x^2}{1+x^4} \, dx \) to six decimal places using a power series expansion.

Answer: 0.008969


Question: Determine the time complexity of the recurrence relation \( T(n)=3^k\cdot T\left(\dfrac n{2^k}\right)+c\cdot n\cdot\sum_{i=0}^{k-1}3^i \) where \( c \) is a constant, \( T(1)=m \) for some \( m > 0 \), and \( k = \log_2 n \). Express your answer in Big O notation.

Answer: O(n^{\log_2 6})


Question: Evaluate the integral \( \int_\gamma \frac{1}{z-i}dz \) where \( \gamma \) is a circle with radius 2 centered at the origin, using Cauchy's Integral Formula or alternative methods.

Answer: 2\pi i


Question: Determine the order of the pole of the product of two holomorphic functions \( h(z) \) and \( j(z) \), where \( h(z) \) has a zero of order \( m > 0 \) and \( j(z) \) has a zero of order \( n > 0 \).

Answer: m + n


Question: Determine the range of values of $m$ for which there exists a $p > 0$ such that \( f(m,p) = \frac{2}{m^2} - \sum_{k=m}^{m+p} \frac{1}{(2k-1)^2} < 0 \).

Answer: m \geq 7


Question: Determine the prime powers $q$ for which the map \( \varphi: \Bbb{F}_q^2 \to \Bbb{F}_q^2 \) defined by \( (x,y) \mapsto (x^q,y^q) \) is an \( \Bbb{F}_q \)-linear transformation.

Answer: q


Question:
Given the function \(f(x)=(x^2+x+1)(ax+b)\), find \(f'(3)\) given that:
\[\lim\limits_{x\to1}\frac{f(x)-f(1)}{x-1}=3\]
\[\lim\limits_{x\to2}\frac{x^3-8}{f(x)-f(2)}=1\]

Answer: 27


Question: Calculate the integral \(\int_{-\infty}^0 x^3 \exp( - x^2/4) \, dx\) using integration by parts.

Answer: -8


Question:
Given the recursion:

$$J(1) = 1$$
$$J(2n) = 2J(n) - 1,$$ for $n \ge 1$
$$J(2n + 1) = 2J(n) + 1,$$ for $n \ge 1$

If $J(n) = 2(n - 2^m) + 1$, where $n = 2^m + l$, determine the value of $J(2^{m+1} + l)$. Provide your answer as a single integer.

Answer: 2l + 1


Question: Find a basis for the annihilator of the subspace \( \textsf{W} = \operatorname{span}(\{\phi_0,\phi_1\}) \subseteq (\mathbb{C}_{\le 2}[x])^* \), where \( \phi_a(f) = f'(a) \) for all \( f \in (\mathbb{C}_{\le 2}[x])^* \).

Answer: \{1\}


Question: Find the hundredth derivative of the function \( f(x) = \frac{x^2}{1+x^4} \) at \( x = 0 \).

Answer: 0


Question: Determine the value of $k$ for which the quadratic form associated with the bilinear form, given by the matrix \( A = \begin{pmatrix} 0 & 0 & k \\ 0 & k & 0 \\ k & 0 & 0 \end{pmatrix} \), has a signature of \((1, 2)\).

Answer: k < 0


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty}\left(\sqrt{n+a}-\sqrt{n}\right) x^n \), where \( a > 0 \).

Answer: 1


Question: Suppose $f(x, y)$ is defined on some neighbourhood of $(0, 0)$. If $u(x, y) = \sin f(x, y)$ is continuous at $(0, 0)$, does it imply that $f(x, y)$ is continuous at $(0, 0)$? Provide a justification for your answer.

Answer: No


Question: Determine if there exists a sequence defined by $A_0=0$, $A_1=1$, and $A_{n+2}=pA_{n+1}-qA_{n}$, where $p$ and $q$ are fixed integers, that contains an infinite number of perfect cubes.

Answer: Yes


Question: Let $g(x) = \sum b_nx^n$, with $b_n \leq 0$, have a radius of convergence of $1$. If $\lim_{x \to 1^-} g(x) = \tau$, does it follow that $\sum b_n$ converges and $\tau = \sum b_n$?

Answer: Yes


Question: Evaluate the integral \( \int_0^1 10x^{\sin x + \cos x} \, dx \) and find the greatest integer less than or equal to the result.

Answer: 4


Question: Find the arc length of the curve defined by the parametric equations \(x(t) = t\sin(2t)\), \(y(t) = t\cos(2t)\), and \(z(t) = \frac{4}{3}t\sqrt{t}\) from \(t=0\) to \(t=2\pi\).

Answer: 4\pi^2 + 2\pi


Question: Evaluate the limit \( \lim_{r\rightarrow 0} \frac{1}{r} \int_{C_{r}} \frac{\sin(z)}{z}dz \), where \( C_r \) is the contour \( re^{i\theta} \) for \( 0 \leq \theta \leq \pi \).

Answer: -2


Question: Determine the number of real solutions to the equation: $$\sin(\sin(\sin(\sin(\sin(x)))))=\frac{x}{3}$$

Answer: 3


Question: Evaluate the contour integral \( \int_{C}{\frac{f(z)}{z^{3}}dz} \), where \( f(z) = \sum_{k=0}^{\infty} k^4 \left(\frac{z}{4}\right)^k \) and \( C \) is the circle centered at the origin with radius \( \pi \) oriented counterclockwise.

Answer: 2\pi i


Question: Given a function $f$ that is continuously differentiable at $0$, and a monotonically decreasing sequence $\{x_n\}$ approaching $0$ such that $f(x_n)f(x_n+1)<0$, does this condition ensure the continuity of $f$ in a neighborhood of $0$? Answer yes or no.

Answer: Yes


Question: Determine the limit of the expression \( \lim_{{n \to \infty}} \left( \frac{u_n}{v_n} \right)^{2040} \), where \( (u_n)_n \) and \( (v_n)_n \) are positive numerical sequences satisfying \( (1+\sqrt{2})^n = u_n + v_n\sqrt{2} \), with \( n \in \mathbb{N} \).

Answer: 2^{1020}


Question: Evaluate the line integral \( \int_{C}{z\,\mathrm dx+x\,\mathrm dy-y\,\mathrm dz} \) where \( C \) is the path defined by \( \gamma(t)=(t^2, 1-t, t) \) for \( t \in [-1,1] \).

Answer: -\dfrac{4}{3}


Question: Determine a lower bound for the cardinality of the set \( A \), where \( A = \{ R \subset \mathbb{N} \times \mathbb{N} : R \text{ is transitive} \} \).

Answer: 2^{\aleph_0}


Question: In the space $C[a,b]$, consider the norm $\left \| f \right \|$ defined as $$\left \| f \right \|=\max\left \{ \sup\left | f \right |,\sup\left | f{}' \right | \right \}.$$ Determine whether this norm satisfies the triangle inequality, i.e., is it true that $\left \| f + g \right \| \leq \left \| f \right \| + \left \| g \right \|$ for all functions $f$ and $g$ in $C[a,b]$?

Answer: Yes


Question: Determine whether the series \( \sum_{n=1}^{\infty} \frac{\cos(nx)}{n^b} \) converges uniformly on the interval \([0, 2\pi]\) for \(b > 1\).

Answer: Yes


Question: Evaluate the volume integral \( \iiint_K (x^2 + y^2) \, dx \, dy \, dz \) using a different coordinate system, where the region \( K \) is defined by the inequalities \( 0 \leq x^2 + y^2 \leq z^2 \) and \( 0 \leq z \leq 1 \).

Answer: \dfrac{\pi}{10}


Question: Find the minimum value of the expression \(|z| + |z - 1| + |z - 2|\) for a complex number \(z\).

Answer: 2


Question: Find the limit of the expression \( A = \lim_{n \to \infty} \frac{1}{n}\left((m+1)(m+2) \cdots (m+n)\right)^{\frac{1}{n}} \).

Answer: \dfrac{1}{e}


Question: Find the last two digits of the factorial of 75 when expressed in base 3.

Answer: 00


Question: Solve the partial differential equation \(v_x^2 + v_y^2 = v^2\) with the initial condition \(v(0,y) = 1\) using the method of characteristics.

Answer: e^x


Question: Determine whether there exists a polynomial $p(x, y) \in \mathbb{R}[x, y]$ such that $p(\sin 1, \cos 1) = 0$. Provide a justification for your answer.

Answer: Yes


Question: Determine all natural numbers \( n \geq 1 \) such that \( n^2 = 1 + (n-1)! \).

Answer: 5


Question: Evaluate the integral of the function \( f(x) \) over the interval from 0 to 2, where \( f(x) = 0 \) if \( x = \frac{n}{n+1} \) for some natural number \( n \), and \( f(x) = 1 \) otherwise.

Answer: 2


Question: Given a continuously differentiable function $f: \mathbb{R} \to \mathbb{R}$, suppose the limit $\lim_{x\to \infty}(f(x) + f'(x)) = L$ exists, with $0 < L < \infty$. If $\lim_{x\to \infty} f'(x)$ also exists, must it be true that $\lim_{x\to \infty} f'(x) = 0$? Answer yes or no and justify your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \to \pi/2} \frac{\sqrt[3]{ \cos x} - \sqrt[4]{ \cos x}}{\sin^2x}$$ using asymptotic simplifications or Taylor series, but not L'Hopital's rule.

Answer: 0


Question: Calculate the volume of the region bounded by the xy-plane, the paraboloid \( z = x^2 + y^2 \), and the cylinder \( x^2 + y^2 = 2 \).

Answer: 2\pi


Question: Do the surreal numbers have the property that every possible total order can be embedded into them?

Answer: Yes


Question: Solve the following system of equations: \[x^2+y^2=98\] \[\sqrt{25+(x+2)^2}+\sqrt{4+(y-5)^2}=7\sqrt{2}\]

Answer: (-7, 7)


Question: Consider a modified Cantor set construction where, instead of removing the middle-third open interval, you remove the middle-$\alpha$ open interval with length $0<\alpha<1$ at each step. Will the resulting set have measure zero?

Answer: Yes


Question: Find the minimum value of \(|z| + |z-1| + |z-2|\) where \(z\in\mathbb{C}\).

Answer: 2


Question: Is it possible to find a non-zero linear transformation \( S \) from a vector space \( V \) to itself such that \( TS = 0 \), given that there is a non-zero vector \( v \) in \( V \) for which \( T(v) = 0 \), where \( T \) is also a linear transformation from \( V \) to \( V \)?

Answer: Yes


Question: Evaluate the limit: $$A = \lim_{n \to \infty} \left(\binom{n}{0} \binom{n}{1} \binom{n}{2}\cdots \binom{n}{n}\right)^{1/((n)(n+1))}$$

Answer: \sqrt{e}


Question: Determine if the function \( f(x) = x^3 \cos\left(\frac{1}{x}\right) \) with \( f(0) = 0 \) is differentiable at \( x = 0 \). If it is, find \( f'(0) \).

Answer: 0


Question: Find the derivative of the inverse function \( f^{-1}(x) \) at \( x = -1 \) for the function \( f(x) = x^3 - 3x^2 - 1 \) where \( x \geq 2 \).

Answer: \dfrac{1}{9}


Question: Calculate the determinant of an $n \times n$ matrix $A$ where the diagonal elements alternate between -2 and -1, and all off-diagonal elements are -1. Specifically, find the determinant of matrix $A$ when $n=53$. \( \boxed{?} \)

Answer: 0


Question:
Solve for \( x \) in the equation:
\[ 3^{\frac{x-1}{2}} - 2^{\frac{x+1}{3}} = 2^{\frac{x-2}{3}} + 3^{\frac{x-3}{2}} \]

Answer: 5


Question: Calculate the determinant of an $m \times m$ matrix $B = (b_{ij})$, where each element $b_{ij}$ is defined as $b_{ij} = \min(i, j)$. Provide your answer as a single numerical value.

Answer: 1


Question: Given $S_0 = \{A, B\}$, where $A$ and $B$ are subsets of a set $M$ with a non-empty intersection, define $S_{i+1}$ recursively as the set of all subsets of $M$ that can be obtained by taking the union, intersection, or complement of any two elements in $S_i$. Determine the maximum possible cardinality of $\bigcup S_i$ as $i$ tends to infinity.

Answer: 16


Question: In a normed space, consider a vector $x$ and two subspaces with trivial intersection whose direct sum spans the entire space. Is the norm of the projection of $x$ onto one of the subspaces always less than or equal to the norm of $x$? Answer with 'Yes' or 'No'.

Answer: No


Question: Find the closure of the set \( M = \{(a_1, a_2, \ldots) : \text{all but finitely many } a_i = 0\} \) in the space \( l^{\infty} \).

Answer: c_0


Question: Consider a Brownian motion $B_t$ starting at $0$. Compute the probability $P(T_1 < T_{-1} < T_2)$, where $T_a = \inf\{t \geq 0 \mid B_t = a\}$.

Answer: \dfrac{1}{6}


Question: Evaluate the limit: $$\lim_{x \to \infty}x\int_0^{1/x}e^t \cos(t) \space \text{d}t.$$

Answer: 1


Question: Evaluate \(\int\int (x-y-z) \, dS\), where \(S\) is the boundary of the unit ball \(B\). Explain how symmetry helps in solving this problem.

Answer: 0


Question: Evaluate the integral: $$\int_{\pi/6}^{\pi/3} \frac {\sqrt[3]{\sin x}} {\sqrt[3]{\sin x} + \sqrt[3]{\cos x}}dx.$$

Answer: \dfrac{\pi}{12}


Question: What is the probability of never losing an infinite game where you start with a single four-sided die numbered 1 to 4? In each round, rolling a 1 loses the die, a 2 keeps the die, and a 3 or 4 gains an additional die. You lose the game when you have no dice left.

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$ \lim_{n \to \infty} \sum_{k=0}^{3n} \binom{k+n-1}{n-1} \left(\frac{1}{4} \right)^{n} \left(\frac{3}{4}\right)^{k} $$

Answer: \dfrac{1}{2}


Question: Evaluate the limit of the sum \(\lim_{n\rightarrow \infty}\sum^{(n+1)^2}_{k=n^2}\frac{1}{\sqrt{k}}\) using the definition of a Riemann integral.

Answer: 2


Question: Suppose $\mathbb{K}_1$ is a subfield of $\mathbb{K}_2$, and $A$ and $B$ are square matrices of the same size with entries in $\mathbb{K}_1$. If there exists an invertible matrix $P_2$ with entries in $\mathbb{K}_2$ such that $P_2^{-1}AP_2 = B$, does there exist an invertible matrix $P_1$ with entries in $\mathbb{K}_1$ such that $P_1^{-1}AP_1 = B$? Provide a justification for your answer.

Answer: Yes


Question: Find the coefficient of \(x^{40}\) in the expansion of \(\sum_{n=1}^{\infty}(t(n)+1) x^{2n^2}\), where \(t(n)\) is the highest power of 2 which divides \(n\).

Answer: 0


Question: Determine a bound on the parameter \(a\) in terms of \(b\) for the convergence of the integral \[ \int \frac{1}{(x^2+y^2)^a}\frac{1}{(x^2+y^2+z^2+w^2)^b} \, dx\, dy\, dz\, dw \] over an open set containing the origin in \(\mathbb{R}^4\), where \(a > 0\) and \(b > 0\) are real parameters. Consider how polar coordinates can be effectively utilized in this analysis.

Answer: a < \min(1, 2 - b)


Question: Calculate the work done by the vector field \( F(x,y,z) = 2xy\mathbf{i} + x^2\mathbf{j} + z\mathbf{k} \) on a particle moving along the curve \( C \) parameterized by \( \mathbf{r}(t) = (t, t^2, t^3) \) from \( t = 0 \) to \( t = 1 \).

Answer: \dfrac{3}{2}


Question: Calculate the limit: $$\lim_{x\to\infty}\frac{x^{\ln x}}{(\ln x)^x}$$ without using L'Hopital's rule.

Answer: 0


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty}\frac{1}{n}\sum _{k=1}^{\left\lfloor\frac{n}{2}\right\rfloor} \cos\left(\frac{k\pi}{n}\right)$$ where $\left\lfloor\frac{n}{2}\right\rfloor$ represents the largest integer less than or equal to $\frac{n}{2}$.

Answer: \dfrac{1}{\pi}


Question: Determine whether the sequence defined by \( a_n = \frac{1}{\ln(\ln(2^{2^n}))} \) converges.

Answer: 0


Question: Find the limit of \( \frac{x^2 - \log(1+x^2)}{x^2 \sin^2 x} \) as \( x \to 0 \).

Answer: \dfrac{1}{2}


Question: Let $(a_1, a_2, \ldots, a_{2014})$ be a permutation of $(1, 2, 3, \ldots, 2014)$. Define the sequence $S_k$ as $S_k = a_1 + a_2 + \cdots + a_k$ for $k = 1, 2, 3, \ldots, 2014$. Determine the maximum value of $|A|$, where $A$ is the set of indices $i$ such that $S_i \equiv 1 \pmod{2}$.

Answer: 1511


Question: Determine the cardinality of the set of all partitions of the set \( \mathbb{N} \) of integers.

Answer: 2^{\aleph_0}


Question: Given a smooth manifold $M$ of dimension $n$ and a point $p \in M$, consider a smooth chart $(U, \phi)$ around $p$ with $\phi: U \to \widehat{U} \subseteq \mathbb{R}^n$ being a diffeomorphism. If the component functions are denoted by $x^i: U \to \mathbb{R}$, determine the coordinate representation $\widehat{x^j}$ of $x^j$ in terms of $\phi$. How does this relate to the coordinates $(a^1, \dots, a^n)$ in $\widehat{U}$?

Answer: \widehat{x^j} = a^j


Question: Consider a metric space $(M,d)$ and a function $f: M \rightarrow \mathbb{R}$. If $f$ is $K$-Lipschitz, determine whether the map $I_f: M \rightarrow \mathbb{R}$ defined by $I_f(\mu) := \int_M f \, d\mu$ is also $K$-Lipschitz.

Answer: Yes


Question: Given the conditions $x_1 + x_2 + x_3 = 0$, $y_1 + y_2 + y_3 = 0$, and $x_1y_1 + x_2y_2 + x_3y_3 = 0$, determine the value of the expression \[ \frac{x_1^2}{x_1^2 + x_2^2 + x_3^2} + \frac{y_1^2}{y_1^2 + y_2^2 + y_3^2}. \]

Answer: \dfrac{2}{3}


Question: Suppose that $x_1, x_2, x_3, x_4$ are the real roots of a polynomial with integer coefficients of degree 4, and $x_1 + x_2$ is rational while $x_1 x_2$ is irrational. Is it necessary that $x_1 + x_2 = x_3 + x_4$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Suppose we have the set $[0,1]$ equipped with the Lebesgue measure. A measurable function $f$ is said to be non-decreasing on a full-measure set if there exists a set $E$ with $\mu(E)=1$ such that for all $y,z\in E$, if $z\leq y$, then $f(z)\leq f(y)$. If $f$ is not non-decreasing on a full-measure set, does there exist an $x\in[0,1]$, a subset $A\subseteq [0,x]$, and a subset $B\subseteq[x,1]$, both $A$ and $B$ with positive Lebesgue measure, such that $f(z)>f(y)$ for every $z\in A$ and $y \in B$?

Answer: Yes


Question: Determine whether the integral \( \int_0^2 \frac{5\,dx}{\sqrt{2x-x^2}} \) converges or diverges.

Answer: 5\pi


Question: Determine whether the following statement is true or false: For a non-decreasing function $f$, the total variation of $f$ on an interval $[a, b]$ is equal to the integral of its derivative on the same interval.

Answer: B


Question: Evaluate the integral \( \oint_{|z|=b}\frac{e^{5z}}{z^{3}-8}dz \) where \( b < 2 \).

Answer: 0


Question: If $f$ and $g$ are monic polynomials in $\mathbb{Z}[x]$, does their (monic) greatest common divisor in $\mathbb{Q}[x]$ necessarily have coefficients in $\mathbb{Z}$? Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate the double integral \( \iint_D xy \, dA \) where \( D \) is the region bounded by the line \( y = x - 1 \) and the parabola \( y^2 = 2x + 6 \).

Answer: 36


Question: Solve the partial differential equation \( \frac{\partial g(x, y)}{\partial y} = B g(x, y) \) with the boundary condition \( g(x, x) = g(y, y) = 1 \), where \( B \) is a constant.

Answer: e^{B(y - x)}


Question:
Calculate the derivative \(\frac{dT}{dt}\), where \(T\) is the total kinetic energy of a fluid flow with velocity \(\underline{u}\), pressure \(p\), and constant density \(\rho\) within a closed region with a solid boundary. The fluid motion satisfies the Euler Equations: \(u_t + (u \cdot \nabla)u = \frac{-1}{\rho} \nabla p\) and \(\nabla \cdot u = 0\). Given the integral representation for \(T\):

\[T = \frac{1}{2}\rho\iiint_Vu^2dV\]

and the expression:

\[\frac{dT}{dt} = \rho\iiint_V u \cdot u_t dV\]

complete the derivation using vector identities and the divergence theorem.

Answer: 0


Question: Determine the value of $t$ such that the sum of the probabilities $P(T > -t)$, $P(T > 0)$, $P(T > t)$, and $P(T > 2t)$ for a random variable $T$ from a Student-$t$ distribution with 7 degrees of freedom equals 1.75.

Answer: 0.356


Question: Find the value of \( n \) that makes the equation \( \binom{n}{n/2} \approx 2^n\sqrt{\frac{2}{0.02 \pi}} \) true.

Answer: 50


Question: Provide an example of a commutative Noetherian ring with no nonzero nilpotent elements such that its localization at a maximal ideal is not a field.

Answer: \mathbb{Z}


Question: Is knowing all proper implications of an unknown statement $p$ sufficient to uniquely determine $p$? Answer yes or no.

Answer: Yes


Question: Is the dual space of the Banach space $L^1(\mu)$ for a semifinite measure $\mu$ equal to $L^\infty(\mu)$?

Answer: No


Question: Find the remainder when \(5^{5555}\) is divided by \(10,000\) using modular arithmetic and the method of exponentiation by squaring.

Answer: 8125


Question: Evaluate the improper integral \( \int_a^b \frac{dt}{\sqrt{(a-t)(b-t)}} \) for \(a,b\in\mathbb{R}\) and \(a<b\).

Answer: \pi


Question: Calculate the integral \( \int_{1/e^2}^{e^2} \frac{\arctan \left(\frac{1}{t}\right)}{t} \, dt. \)

Answer: \pi


Question: Calculate the volume of the solid formed by rotating the curve \( y = e^{-x^2} \) about the \( y \)-axis from \( x = 0 \) to \( x = \infty \) using the disk method.

Answer: \pi


Question: Find the multiplicative inverse of \(2^{2000}\) modulo \(2017\) using Euler's criterion.

Answer: 992


Question: Let $(X,F,\mu)$ be a finite measure space. Let $f_n$ be a sequence of measurable functions from $X$ to $\mathbb{R}$ such that $f_n\geq 0$ almost everywhere with respect to $\mu$. If $\lim_{n\to \infty}\int_X f_n \,d\mu=0$, does it follow that $f_n$ converges to $0$ almost everywhere with respect to $\mu$? Provide a justification for your answer.

Answer: No


Question: Evaluate the limit \( \lim_{n\to\infty}\int_{1}^{1+\delta}\frac{n}{x^2(1+x^n)}dx \) using Lebesgue's Dominated Convergence Theorem or the Sandwich Theorem, where \( \delta > 0 \) is a fixed number.

Answer: \ln 2


Question: Evaluate the limit: $$\lim_{n \to \infty}\left(\sqrt[3]{\frac{\sin n}{n}+n^3-n^2}-\sqrt[3]{n^3+n}\right)$$

Answer: -\dfrac{1}{3}


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2}\log(\cos(\theta))\cos(\theta)\,\mathrm{d}\theta \).

Answer: 2 \ln 2 - 2


Question: Determine the radius of convergence for the series \( F(x)= \sum_{n=0}^\infty \frac{n!x^n}{n^n} \).

Answer: e


Question: Consider two permutations $a$ and $b$ in the symmetric group $S_n$, where $a$ has the cycle type $(k_1,k_2,...,k_t)$ and $b$ has the cycle type $(l_1,l_2,...,l_t)$, with the cycle type of $b$ being different from that of $a$. Can the equality $|C_{S_n}(a)| = |C_{S_n}(b)|$ occur?

Answer: Yes


Question: Find the derivative of the function \( g(x) = \int_{\cos{x}}^{\sin{x}} \sqrt{1-t^2} \,\mathrm{d}t \) for \( x \in \left[0,\dfrac{\pi}{2}\right] \).

Answer: 1


Question: Solve the equation \((\log_2 3)^x + (\log_3 5)^x = 2 (\log_3 4)^x\).

Answer: 0


Question: In a model $M[G]$ obtained from a countable transitive model $M$ with $M \models \text{CH}$ by adding $\aleph_2$ many Cohen reals with forcing, we have $M[G] \models 2^{\omega} = \aleph_2$. Determine the value of $cov(null)^{M[G]}$. \(cov(null)\) is the minimum cardinality of a set of nullsets of $\mathbb{R}$ such that their union equals $\mathbb{R}$. What is this value in the model $M[G]$?

Answer: \aleph_2


Question: Consider infinite cardinals \(\kappa\) and \(\lambda\). A random function \(\phi: \kappa \times \lambda \rightarrow \{0,1\}\) is constructed by flipping a fair coin for each element of the domain. Determine the probability that the image of \(\phi\) includes a constant function.

Answer: 0


Question: Is it possible to define a Borel \(\sigma\)-algebra on a topological space that is not second countable?

Answer: Yes


Question: Evaluate the integral \( \int_{\pi/8}^{3\pi/8} \frac{dx}{1 + (\tan x)^{\sqrt{2}}} \).

Answer: \dfrac{\pi}{8}


Question: Determine the value of \( k \) for which both series \( \sum\limits_{n=1}^\infty\left(\frac{3}{k}\right)^n \) and \( \sum\limits_{n=1}^\infty\frac{(3-k)^n}{\sqrt{n+3}} \) converge. Choose from the following options: None, 2, 3, 4, 5.

Answer: 4


Question: Find the determinant of a matrix similar to $A_n$, where the entries of the form $(i, i+3)$ are equal to $5$.

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ \mathrm{d}x \) assuming \( a > 0 \).

Answer: \pi a


Question: Determine the smallest integer \( n \) such that \( |P_{n}(x) - f(x)| < 10^{-4} \) for all \( x \in [0,1] \). Use \( P_{n}(0.2) \) to approximate \( f(0.2) \) and compare it to the actual value of \( f(0.2) \).

Answer: 7


Question: Evaluate the triple integral \(\iiint_{B} (2x+2y+2z)\,dxdydz\) over the region \(B=\{(x,y,z):x^2 + y^2 + z^2 \leq R^2\}\) using a coordinate transformation.

Answer: 0


Question: Determine the number of continuous functions \( f: [0,1] \to [0,\infty) \) that satisfy the inequality \( \int_0^x f(t) \, dt \geq f(x) \) for all \( x \in [0,1] \).

Answer: 1


Question: Evaluate the line integral \( \int_C \textbf{f} \cdot d\textbf{r} \) using Green's Theorem, where \( \textbf{f}(x,y)=(ye^{xy}+y^2\sqrt{x})\textbf{i}+(xe^{xy}+\frac{4}{3}yx^{\frac{3}{2}})\textbf{j} \) and \( C \) is the ellipse given by \( (x-1)^2+\frac{y^2}{9}=1 \), oriented counterclockwise.

Answer: 0


Question: Determine whether it is always possible to find a rational number $c$ such that the polynomial $P(x) + c$, where $P(x)$ is in $\mathbb{Q}[x]$, becomes irreducible in $\mathbb{Q}[x]$. Provide a justification for your answer.

Answer: Yes


Question: In a triangle with integer sides, one angle is fixed at 120 degrees. If the longest side is 28 and the product of the other two sides is 240, find the sum of all the sides of the triangle.

Answer: 60


Question: Let $(x_n)$ be a sequence in $\mathbb{R}$. Suppose that every infinite subsequence of $(x_n)$ which omits infinitely many terms of $(x_n)$ converges. Does this imply that $(x_n)$ converges? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Evaluate the limit: $$\lim_{x \to 0^+} \left( e^x - 1 \right)^{\frac{\tan^2 x}{\sqrt[3]{x^2}}}$$ without using l'Hôpital's rule or Taylor series expansions. Use only basic limit properties and known limits, such as $\lim_{x \to 0}\frac{\sin x}{x} = 1$.

Answer: 1


Question: How many polynomials with coefficients of $-1$ or $1$ have only real roots and a degree greater than 3?

Answer: 0


Question:
Consider the set $X = \left\{(x,y) \in \mathbb{R}^2 : |x| \leq 1, |y| \leq 1\right\}$ and the function $f: X \rightarrow \mathbb{R}$ defined by
$$f(x,y) = \frac{x\cos(x) + y\sin(y)}{x^2 + y^2 + \alpha}$$
where $\alpha > 0$. Determine the range of $f(x,y)$ and identify it as one of the following:

(a) not a compact set
(b) a bounded open set
(c) a connected open set
(d) a connected closed set

What is the nature of the range of $f(x,y)$?

Answer: D


Question: In triangle $\triangle ABC$, given that $\cos A \cos B \cos C = \frac{1}{3}$, determine if the value of $\tan A \tan B + \tan B \tan C + \tan C \tan A$ is constant.

Answer: 4


Question: Let $M$ be an $m$-dimensional manifold embedded in $\mathbb{R}^k$, with $m>n$. If every smooth function $f: M \rightarrow \mathbb{R}^n$ has regular values that form an open subset of $\mathbb{R}^n$, must $M$ be a compact manifold? Provide a justification for your answer.

Answer: Yes


Question: Consider the topology \(\tau = \{U \subseteq \mathbb{Z}: \mathbb{Z} \setminus U \text{ is finite or } 0 \notin U\}\) on the set of integers \(\mathbb{Z}\). Determine which of the following properties \((\mathbb{Z}, \tau)\) has: (A) compact but NOT connected; (B) connected but NOT compact; (C) both compact and connected; or (D) neither compact nor connected.

Answer: A


Question: Evaluate the line integral \( \int_C G(x,y,z) \, ds \) along the curve defined by the parametric equations \( x= \frac{1}{3} t^3, \quad y=t^2, \quad z=2t \), where \( t \) is in the closed interval \([0,1]\), for the function \( G(x, y, z) = 4xyz \). Calculate the integral and express the answer rounded to two decimal places.

Answer: 1.06


Question: Calculate the limit: $$ \lim_{n\rightarrow \infty} \dfrac{P_{n-1}\cdot P_{n+1}}{P_{n}^{2}} $$ where $P_{n}$ is the product of the numbers in the $n$th row of Pascal's Triangle.

Answer: e


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

Answer: -1


Question: Determine whether every diagonalizable matrix with distinct eigenvalues can be expressed as the product of two commuting matrices. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \(\lim_{(x,y) \to \infty} \frac{x+2y}{x^2 - 2xy + 2y^2}\) if it exists.

Answer: 0


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

Answer: O(2^k)


Question: Determine whether the following statement is true or false: For any graph $G$ with chromatic number $n$, the complete graph $K_n$ on $n$ vertices always has a smaller genus than $G$. Provide a justification for your answer.

Answer: False


Question: Evaluate the contour integral \( \oint_C \frac{e^z}{4z^2+9\pi^2}\ dz \) where \(C\) is the circle \(|z|=10\). Use the residue theorem to compute the integral.

Answer: -\dfrac{i}{3}


Question: Evaluate the limit \( \lim_{n\to\infty} nx(1-x^2)^n \) for a given \( 0 < x < 1 \).

Answer: 0


Question: Find the limit of the function \( g(x) = \left[\left(\frac{1}{2e}\right)\left(1-\frac{1}{x}\right)^x\right]^x \) as \( x \to \infty \).

Answer: 0


Question: In a labyrinth game, the largest room measures 10x10x8 and the smallest room measures 4x4x4. Assuming there is no empty volume between rooms and adjacency is defined by overlapping faces of at least 1 square, determine the maximum number of adjacent rooms possible.

Answer: 24


Question: Find the largest integer \( m \) such that \( a^{12} \equiv 1 \pmod{m} \) for all integers \( a \) that are relatively prime to \( m \).

Answer: 65520


Question:
Find the minimum distance of the ternary linear code with the following parity check matrix:

$$H=\begin{pmatrix} 1 & 0 & 2 & 1 & 2 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 2 \\ 2 & 2 & 1 & 1 & 0 \end{pmatrix}.$$

Answer: 4


Question: Evaluate the integral: $$\int_{0}^{2} \mathrm{~d} z \int_{0}^{\left(2 z-z^{2}\right)^{\frac{1}{2}}} \mathrm{~d} y \int_{0}^{\left(2 z-z^{2}-y^{2}\right)^{\frac{1}{2}}}\left(x^{2}+y^{2}+z^{2}\right)^{-\frac{1}{2}} \mathrm{~d} x$$

Answer: \dfrac{\pi}{3}


Question: Evaluate the integral \(\int_e{\frac{1}{z^2+4}}dz\) on the ellipse \(4x^2+(y-2)^2=4\).

Answer: \dfrac{\pi}{2}


Question: Which of the following numbers could be the exact number of elements of order 15 in a group: 1200, 1202, or 1204?

Answer: 1200


Question: Evaluate the limit using L'Hopital's Rule: $$\lim\limits_{x\to 0}{\left( \frac{(1+x)^x}{x^xe}\right)}^x$$

Answer: 1


Question: Determine the dimension of the local ring at a closed point of an integral scheme of finite type over an algebraically closed field.

Answer: \dim X


Question: Determine the limit of the ratio of the number of odd composite positive integers less than or equal to \( n \) that are divisible by 3 to the total number of odd composite positive integers less than or equal to \( n \) as \( n \) approaches infinity.

Answer: \dfrac{1}{3}


Question:
Calculate the conditional variance of the random variable $X$ given that $X > 2$ and $Y > 2$ for the joint probability density function:
\[ f_{XY}(x,y)=\begin{cases} 2e^{-(x+2y)}, & x>0,y>0\\ 0, & \text{otherwise}\\ \end{cases}\]

Answer: 1


Question: Determine whether the following statement is true or false: If the probability density function of a random variable $Z$ is the convolution of the density functions of the random variables $Y_1, Y_2, ..., Y_n$, then $Z$ can be expressed as the sum of $Y_1, Y_2, ..., Y_n$, where $Y_i$ are independent and identically distributed.

Answer: B


Question: Four dogs start at the corners of a square $ABCD$ with side length $30$ meters, running anti-clockwise. Dog $A$ chases dog $B$, dog $B$ chases dog $C$, dog $C$ chases dog $D$, and dog $D$ chases dog $A$. Each dog runs at a constant speed of $7$ meters per second. The pursuit stops when a dog reaches the center of the square. Determine the duration of the pursuit.

Answer: \dfrac{30}{7}


Question: Given the bases $B_1 = \{1+x,1-x,1+x^2\}$ and $B_2 = \{1+2x,x,1+x+x^2\}$ for the vector space $P_2(\Bbb R)$, and the linear transformation $T$ represented by the matrix $M(T,B_1,B_2) = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 1 \\ 1 & 1 & -1 \\ \end{pmatrix}$, determine $T(1-x+3x^2)$.

Answer: -7 - 7x - 5x^2


Question: For a continuous function $f:[a,b] \to \mathbb{R}$ and a point $x_0 \in [a,b]$, determine if the limit \( \lim_{h \to 0} \frac{\int_{x_0}^{x_0+h} f(x) \, dx}{h} \) exists and equals \( f(x_0) \).

Answer: f(x_0)


Question: Determine the smallest number of terms $N$ required in the series $\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}$ such that the remainder $R_N = \sum_{n=N+1}^{\infty} \frac{1}{n(\ln n)^2}$ is less than 0.1.

Answer: 22027


Question: Let \( G \) be a finite group, and \( P \) a finitely generated projective \( \mathbb{Z}[G] \)-module. By a theorem of Swan, \( P \otimes \mathbb{Q} \cong \mathbb{Q}[G]^n \) for some \( n \). Does the same hold for \( \mathbb{R} \)?

Answer: Yes


Question: What is the minimum number of students that must be lined up in such a way that there is either a sequence of 10 students in increasing order of height or a sequence of 12 students in decreasing order of height, given that no two students have the same height?

Answer: 100


Question: Evaluate the line integral \( \int_{\alpha} f\, ds \), where \( f(x, y) = x^2y^2 \) and the path \( \alpha(t) = (\cos t, \sin t) \) is defined for \( 0 \leq t \leq 4\pi \). The path \( \alpha(t) \) traces a circle twice. Determine the value of the integral.

Answer: \dfrac{\pi}{2}


Question: Given a function $f:\mathbb{R}\to \mathbb{R}$ that is infinitely differentiable on $\mathbb{R}$, suppose there exists $\delta>0$ such that for all $0<x<\delta$, $f(a-x)<f(a)<f(a+x)$. Does there exist an interval $I\subset\mathbb{R}$ with $a\in I$ such that $f(x)$ is strictly increasing for all $x, y\in I$ with $x<y$? Answer with 'Yes' or 'No'.

Answer: Yes


Question: Given three non-negative numbers \(a, b, c\) such that \(a + b + c = 3\) and \(a^2 + b^2 + c^2 = 5\), find the maximum possible value of \(a^4b + b^4c + c^4a\).

Answer: 16


Question: In the figure, $AD = AB$, and $	riangle DAB$, $	riangle DCB$, and $	riangle AEC$ are all right triangles with $	heta = 90^\circ$. Given that $AE = 5$ and quadrilateral $ABCD$ is cyclic, determine the area of quadrilateral $ABCD$. Use the properties of cyclic quadrilaterals and the Pythagorean or Ptolemy's Theorem to find the necessary side lengths.

Answer: 50


Question: Determine whether the following statement is true or false: For all $r > 0$, the expected value of the integral $\int_{X-r}^{X+r} f(x) \, dx$ (where $X$ is a random variable with probability density function $f$) is greater than or equal to the expected value of the integral $\int_{X-r}^{X+r} g(x) \, dx$ for any other probability density function $g$. Provide a justification for your answer.

Answer: False


Question: Given two matrices $M_1 \in \mathbb{R}^{m \times n_1}$ and $M_2 \in \mathbb{R}^{m \times n_2}$, where $m > n_1 > n_2$ and each has linearly independent columns, and positive vectors $\alpha \in \mathbb{R}^{n_1}$ and $\beta \in \mathbb{R}^{n_2}$ with all components strictly greater than zero such that $M_1 \alpha = M_2 \beta$, determine if there always exists a matrix $N \in \mathbb{R}^{n_1 \times n_2}$ satisfying $M_1 N = M_2$. Provide a justification for your answer.

Answer: Yes


Question: Given a right-angled triangle ABC in Cartesian coordinates with A on the y-axis at (0, √33), B on the x-axis at (4, 0), and C determined by the Pythagorean theorem. An arbitrary point O lies to the right of C, with known lengths connecting O to B and A as √32 and 5, respectively. Is it possible to calculate the length q, connecting O to C, without using the Sine or Cosine Law, only using the given lengths? Answer with 'Yes' or 'No'.

Answer: Yes


Question: Determine whether the sequence \( \sin(\pi \sqrt{n^2 + 2n}) \) converges.

Answer: 0


Question: Find the supremum of the function \( \frac{F(x)(1-F(x))}{n} \) for \( x \in \mathbb{R} \), where \( F(x) \) is a cumulative distribution function and \( n \) is a positive integer.

Answer: \dfrac{1}{4n}


Question:
Evaluate the limit:
\[L=\lim _{x \to 0}\frac{2+\tan \left(e^{x}-\cos x\right)-e^{x}-\cosh x}{x\left(\sqrt{1+2 x}-\sqrt[3]{1+3 x}\right)}\]

Using the generalized binomial expansion, we have:
\[\sqrt{1+2 x}-\sqrt[3]{1+3 x}=\left(1+\frac{1}{2}2x+\frac{\frac{1}{2}\left(\frac{1}{2}-1\right)}{2 !}(2 x)^{2}+\cdots\right)-\left(1+\frac{1}{3}3 x+\frac{\frac{1}{3}\left(\frac{1}{3}-1\right)}{2}(3 x)^{2}+\cdots\right)\]
\[\implies \sqrt{1+2x}-\sqrt[3]{1+3x}=\frac{x^2}{2}+O(x^3)\]
Thus:
\[L=\lim _{x \rightarrow 0} \frac{2+\tan \left(e^{x}-\cos x\right)-e^{x}-\cosh x}{x^{3}\left(\frac{\frac{x^{2}}{2}+O\left(x^{3}\right)}{x^{2}}\right)}\]
\[L=2 \lim _{x \rightarrow 0} \frac{2+\tan \left(e^{x}-\cos x\right)-e^{x}-\left(\frac{e^{x}+e^{-x}}{2}\right)}{x^{3}}\]

Given:
\[e^{x}=1+x+\frac{1}{2} x^{2}+\frac{1}{6} x^{3}+\frac{1}{24} x^{4}+\ldots\]
\[\cos x=1-\frac{1}{2} x^{2}+\frac{1}{24} x^{4}-\frac{1}{720} x^{6}+\ldots\]
\[e^x-\cos x=x+x^2+\frac{x^3}{6}+O(x^4)\]
\[\tan x=x+\frac{1}{3} x^{3}+\frac{2}{15} x^{5}+...\]
\[\tan(e^x-\cos x)=x+x^2+\frac{x^3}{6}+\frac{1}{3}x^3+O(x^4)\]
\[e^x+\cosh x=2+x+x^2+\frac{x^3}{6}+O(x^4)\]

Using these expansions, find an alternate way to evaluate \(L\).

Answer: \dfrac{2}{3}


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

Answer: \pi \ln 2


Question: Evaluate the limit \( \lim_{{n \to \infty}} (n+1)I_n \), where \( I_n = \int_0^1 \frac{x^n}{x+1} \, dx \) for \( n > 0 \).

Answer: \dfrac{1}{2}


Question: Calculate \(1234^{5678} \pmod{1235}\) using modular arithmetic by splitting the base into smaller factors.

Answer: 1


Question: Evaluate the integral \( \int \frac{e^x \sin x}{\sinh x + \cosh x} \, dx \).

Answer: -\cos x + C


Question: If $f_{n}$ is a sequence of real-valued strictly increasing functions that converges pointwise to $f$, must $f$ be strictly increasing?

Answer: No


Question: If matrices $A$ and $B$ are similar to matrices $C$ and $D$, respectively, is $A + C$ similar to $B + D$? Answer yes or no.

Answer: No


Question: Evaluate the complex sum \(\sum_{k=1}^{10}\left(\sin\left(\frac{2k\pi}{11}\right)+i\cos\left(\frac{2k\pi}{11}\right)\right)\) using only trigonometry.

Answer: -i


Question: Find the dimension of the ring \( \mathbb{Q}[a,b,c]/(ab-c, ac-b, bc-a) \).

Answer: 0


Question: Find the expected value of the intersection $|K \cap R|$ between two randomly selected subsets $K$ and $R$ of a finite set $N$ with size $n$, where $K$ and $R$ have fixed sizes $k$ and $r$, respectively. Express this expected value as a function $P(k, r, n)$.

Answer: \dfrac{kr}{n}


Question: Determine the minimum perimeter of a triangle with integer sides \(a > b > c\) such that \(3^a, 3^b, 3^c\) all leave the same remainder when divided by 10000.

Answer: 3003


Question: Evaluate the sum \( \displaystyle\sum_{n=1}^{\infty}\frac{1}{(2n-1) \cdot 4^{n-1}} \).

Answer: \ln 3


Question: Find the general form of all polynomials \( q(x) \) that satisfy the functional equation \( q(x-1) = q(x) - 3x(x-1) - 1 \) for all real \( x \).

Answer: x^3 + C


Question: Compute the conditional expectation \( E(W_6 | W_2 , W_4) \) where \( W \) is a standard Brownian motion.

Answer: W_4


Question: Let $u_{n+1} = \frac{1}{\tanh^2(u_n)} - \frac{1}{u_n^2}$ for all $n \in \mathbf{N}$, with the initial term $u_0 = a > 0$. Determine the limit of the sequence $(u_n)$ as $n$ approaches infinity.

Answer: \dfrac{2}{3}


Question: Determine whether the following statement is true or false: Let $f$ be a differentiable function on $\mathbb{R}$. If $f \circ f \circ f$ is strictly increasing and $f \circ f$ is strictly decreasing, then $f$ is strictly decreasing.

Answer: True


Question:
Determine the total number of ordered pairs \((x, y)\) of non-negative integers that satisfy the equation:
\[ x!y! = x! + y! + 2 \]

Answer: 0


Question: Determine the values of $p$ and $q$ for which the function $z^{\frac{p}{q}}$ is a holomorphic injection on the annulus $\{z : r_1 < |z| < r_2\}$.

Answer: p = \pm q


Question: Determine the number of solutions to the equation \( \int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0 \) on the interval \([0, 3]\).

Answer: 1


Question: Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$ and $P$ be a Sylow $p$-subgroup of $G$. If the exponent of $P$ is at most $p^m$, where $p^{m-1}<n\le p^m$, does it follow that all non-trivial elements of $P$ have order at most $p^m$? Answer 'yes' or 'no' and justify your answer.

Answer: yes