Question: Determine the point on the hyperboloid \( z = \frac{x^2}{4} - \frac{y^2}{25} \) that is farthest from the point \((3,0,0)\).

Answer: (2, 0, 1)


Question:
Determine the total number of solutions to the following system of equations:

1. \( f(x) = 2x(\log_2 x + 1) = 16 \)
2. \( g(u) = (u+1)(u-2) = \ln(u+1) \)

Answer: 3


Question: In an experiment with a 20% success rate, what is the probability of obtaining 50% or more successes in a sample of 100 trillion trials?

Answer: 0


Question: Is it always true that \( \langle x, |A|x \rangle \) is real-valued for all \( x \) when \( A \) is a compact operator on a complex Hilbert space?

Answer: Yes


Question: Prove using the definition of limit that \(\lim_{(x,y)\to(0,0)}\frac{(1-\cos(xy))\sin y}{x^2+y^2} = 0\).

Answer: 0


Question: Given the functions \( f(x) = e^{x+1} - 1 \) and \( g(x) = 100|f(x)| - \sum_{k=1}^n|f(x^k)| \), where \( n \in \mathbb{N} \), determine the sum of all possible integer values of \( n \) such that \( g(x) \) is differentiable over the entire set of real numbers.

Answer: 39


Question:
Consider the function $u \in C^2(B(0,1)) \cap C^0(B[0,1])$ that satisfies the boundary value problem:

$$\left\{ \begin{array}{rcl} -\Delta u + u^2 & = &g(|x|), \ \ \mbox{in} \ \ B(0,1);\\ u(x) & = & 1, \ \ \mbox{on} \ \ \partial B(0,1) \end{array}\right.$$

where $g(|x|) \geq 0$ is a $C^1$ function. Does the maximum value of $u$ in $B[0,1]$ depend on the function $g$?

Answer: Yes


Question: Find the sum of all possible values of $a+b+c$ given that $a$, $b$, and $c$ are prime numbers satisfying the equation $a + b^2 = 4c^2$.

Answer: 31


Question: Let $G$ be a finite group and $p$ a prime number. Determine whether the number of elements in $G$ with order coprime to $p$ is necessarily coprime to $p$. Provide a justification for your answer.

Answer: Yes


Question:
Given a triangle $\triangle ABC$ with sides $a, b, c$, satisfying the equation:
\[ a^2 + b + |\sqrt{c-1} - 2| = 10a + 2\sqrt{b-4} - 22 \]
Determine the type of triangle $\triangle ABC$. Choose from the following options:
A) Isosceles triangle with unequal legs and base
B) Equilateral triangle
C) Right triangle
D) Isosceles right triangle

Note: $b \geq 4$ and $c \geq 1$.

Answer: B


Question: Consider a connected undirected simple non-planar graph $G$ with $15$ vertices. If removing any edge from $G$ results in a planar graph, determine the number of edges in $G$. \( \boxed{} \)

Answer: 40


Question: Determine whether the following statement is true or false: Let $f$ be a non-negative, measurable, integrable function on $[0,1].$ Then $\lim_{a \rightarrow 0}\int_0^a f(x) dx = 0$. If the statement is false, provide a counterexample.

Answer: True


Question: Determine if the function \( F(x,y)=e^{x+y} \) is differentiable at the point \((0,0)\) using the definition of differentiability.

Answer: Yes


Question: In the context of vector spaces, if $V$ is a $k$-vector space, whether finite or infinite dimensional, can any subspace $U \subseteq V$ be expressed as a direct summand of $V$? Answer with 'Yes' or 'No'.

Answer: Yes


Question: Suppose $f : (0,\infty) \to \mathbb{R}$ is locally $H^1$ and \(\int_0^\infty (|f'(t)|^2 + |f(t)|^2) e^{-t} dt\) is finite. Determine if \(\lim_{t \to \infty} e^{-t} |f(t)|^2 = 0\).

Answer: 0


Question: Given that each potato has an equally-likely probability of dying, living, or living and producing a clone, what is the probability that an infinite number of potatoes will survive indefinitely?

Answer: 0


Question: Determine whether a function that is not differentiable at the origin can still belong to the $L^p(\mathbb{R}^n)$ space for some $p \geq 1$.

Answer: Yes


Question: Determine the points $a \in \mathbb{C} \setminus \{-1\}$ at which the function $g(z) = \frac{\overline{z}^2}{1+z}$ is complex-differentiable.

Answer: 0


Question: Let $f: \mathbb{H} \to \mathbb{C}$ be a non-zero weight-$0$ cusp form for $\text{SL}_2(\mathbb{Z})$, defined as $f(z) = \sum\limits_{i=1}^\infty a_i\exp(2\pi iz)$. Determine if it is true that, due to $f$ being a cusp form, $\lim\limits_{y\to\infty}f(x+iy)=0$ uniformly in $x$ for $|x|\leq\frac{1}{2}$. Provide a justification for your answer.

Answer: True


Question: Determine the number of real solutions for the equation $x^{x^{2023}} = 2023$.

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to 0^-}\frac{\arccos(x)-\frac{\pi}{2}}{x \ln x}$$

Answer: 0


Question: Determine the rank of the matrix $B$ where $b_{ij} = \sin (i + j)$. \(i\) and \(j\) are the row and column indices, respectively.

Answer: 2


Question: Determine whether every open cover of a closed and bounded interval \([a, b]\) has a finite subcover.

Answer: Yes


Question: Find the largest positive constant \( c \) such that \[ c \leq \frac{\sup_{t\in [0,1]} |x+ty| }{|x|+|y|} \] for all real numbers \( x \) and \( y \).

Answer: \dfrac{1}{3}


Question: Determine the largest topology on a set $X$ that contains a given family of subsets $\mathcal{S}$.

Answer: \mathcal{P}(X)


Question: Let $f_n \to f$ uniformly locally on $\mathbb{R}$. All these functions are from $\mathbb{R}$ to $\mathbb{R}$, are bijective, increasing, continuous, and smooth. Furthermore, $f_n$ and $f$ pass through the origin. Does it follow that $f_n^{-1} \to f^{-1}$ uniformly locally?

Answer: Yes


Question: In category theory, must a directed graph have arrows connecting each point to itself to be considered a category?

Answer: Yes


Question: Compute the volume of the solid bounded by the sphere $x^2+y^2+z^2=4$ and the plane $z=-1$. Express the limits of the $y$ integral in terms of $z$ and the $x$ integral in terms of $y$ and $z$.

Answer: 9\pi


Question: Calculate the flux of the vector field \( F = (0, 0, z) \) over the spherical surface defined by \( x^2 + y^2 + z^2 = 9 \), where the normal vector is positively oriented. Use spherical coordinates to parameterize the sphere and determine the correct expression for the surface integral.

Answer: 36\pi


Question: Express the sum \( X = \sum_{i = 1}^{2k}\frac{(-1)^{i+1}}{i} \) in a more elegant form, where \( k \) is an even integer.

Answer: H_{2k} - H_k


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{ib \sin{\theta}} \, d\theta \) where \( b \) is a constant, using analytic methods.

Answer: 2\pi J_0(b)


Question: If $A$, $B$, and $C$ are the roots of the cubic equation $ax^3+bx^2+cx+d=0$ such that $3c=5a+b+d$, find the value of \[ \frac{(A+B+C)^2(1+A+B+C)-(A^3+B^3+C^3)+ABC(A+B+C+3)}{(A+B+C)}. \]

Answer: 5


Question: Find the order of the semigroup generated by two elements $x$ and $y$ with the relations $x^2 = y^2 = 0$, $xyx = x$, and $yxy = y$.

Answer: 5


Question: Determine whether the following statement is true or false: If \(\{s_n\}_{n=1}^\infty\) is a sequence in \(\mathbb{R}\) such that the difference \(|s_{n+1} - s_n|\) converges to zero, then the sequence \(\{s_n\}\) is Cauchy. Justify your answer.

Answer: False


Question: Determine the cardinality of the set \( S = \{ h: D \rightarrow D \mid h \text{ is analytic in } D \text{ and } h(z)^2 = \overline{h(z)} \text{ for all } z \in D \} \), where \( D = \{ z \in \mathbb{C} : |z| < 1 \} \).

Answer: 1


Question: Determine the order of an element \( \beta \) in a field of characteristic 2, given that its minimal polynomial is \( g(x) = x^4 + x^3 + 1 \).

Answer: 15


Question:
Given a dataset $(y_1, ..., y_n)$ with each $y_i \in \mathbb{R}^p$ and a $p \times q$ matrix $V$ having $q$ orthogonal unit vectors as columns, solve the following optimization problem:
$$\min_{V} \sum^N_{i=1} || y_i - V V^T y_i ||^2$$

Answer: V


Question: Given an odd prime $p$ and integers $b$ and $n$, find the number of solutions to the congruence $x^2 \equiv b \pmod{p^n}$, where $b$ is a quadratic residue modulo $p$. Provide your answer as a single integer.

Answer: 2


Question: Find the general form of the function $g(x): \mathbb{R} \to \mathbb{R}$ that satisfies the inequality $|g(x+2y) - g(x-2y) - 4y| \le 4y^2$ for all $x, y \in \mathbb{R}$.

Answer: g(x) = x + C


Question: Evaluate the integral: $$\int_{\frac{\pi}{12}}^{\frac{\pi}{4}}\dfrac{8\cos2x}{(\tan x+\cot x)^3}\,dx$$

Answer: \dfrac{15}{128}


Question: In a metric space $M$, consider a connected subset $X \subset M$ where for all $y, z \in X$, the distance $d(y, z) = 2$. Must there exist an $x \in X$ such that $d(x, z) = 1$ for some $z \in X$? Answer yes or no and justify your answer.

Answer: No


Question: Determine whether every uncountable subset $S$ of $\mathbb{R}$ contains a strictly decreasing sequence that converges to a real number.

Answer: Yes


Question: Let $V$ be a vector space, and consider linear transformations $S, T: V \to V$ such that $S \circ T = T \circ S$. If $v \in V$ is an eigenvector of $T$ with eigenvalue $\lambda$ and $\dim V_\lambda = 1$, where $V_\lambda$ is the eigenspace corresponding to $\lambda$, must $v$ also be an eigenvector of $S \circ T$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{x \to 0} \frac{e^{4x} - 4e^x + 3}{x^2} \) given that the function \( f(x) = \begin{cases} \frac{e^{4x} - 4e^x + 3}{x^2} & :x \neq 0 \\ B & :x=0\ \end{cases} \) is continuous at \( x=0 \).

Answer: 6


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

Answer: 0


Question:
Evaluate the limit:

$$ \lim_{n\rightarrow \infty}\frac{\sum\limits_{x = 0}^{n}\binom{n}{x}\left[1 + \mathrm{e}^{-(x+1)}\right]^{n + 1}}{\sum\limits_{x=0}^{n} \binom{n}{x}\left[1 + \mathrm{e}^{-x}\right]^{n + 1}}. $$

Answer: 0


Question: Consider a partially ordered set \( P \). If \( P \) contains antichains of arbitrarily large size, does it necessarily contain an infinite antichain? Answer 'yes' or 'no' and justify your answer.

Answer: No


Question: Consider the sequence of functions $\{g_n\}$ where $g_n:[1,\infty)\to \mathbb{R}$ is defined as $g_n(x)=\frac{x^2}{1+n+x^2}$. Determine if the limit $g=\lim_{n\to \infty} g_n(x)$ exists for each $x \in [1, \infty)$, and specify whether the convergence is pointwise or uniform on $[1, \infty)$. Provide a justification for your answer.

Answer: 0


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

Answer: C


Question: Evaluate the integral \( \int\limits_{t,t'}{\frac{dt\,dt'}{\|x(t)-y(t')\|}} \) where \( x(t) \) and \( y(t') \) are curves traversing the boundary of \([0,1]^2\) in \( \mathbb{R}^2 \) counterclockwise.

Answer: \infty


Question: Given that \(\sum \alpha\beta=-9\), \(\sum (\alpha\beta)^2=81\), and \(\sum (\alpha\beta)^3=-486\), find the value of \(\sum (\alpha\beta)^5\).

Answer: -26244


Question:
From a group of seven persons, seven committees are formed such that any two committees have exactly one member in common, and each person is in exactly three committees. Determine which of the following statements is true about the sizes of the committees:

(a) At least one committee must have more than three members.
(b) Each committee must have exactly three members.
(c) Each committee must have more than three members.
(d) Nothing can be said about the sizes of the committees.

Answer: B


Question: In circle \(\Gamma_1\) with radius 61, chord \(CD\) is perpendicular to diameter \(AB\) at point \(N\), where \(AN > NB\). Circle \(\Gamma_2\) is centered at \(C\) with radius 60, and intersects \(\Gamma_1\) at points \(P\) and \(Q\). The intersection of \(PQ\) and \(CD\) is point \(M\). Given that \(CN\) is a radius of \(\Gamma_2\), find the length of \(AM\).

Answer: 78


Question:
Given a sequence of real numbers \(\{a_n\}\) for \(n \ge 1\) satisfying the recurrence relation \(a_{n+2} = a_{n+1} - a_n\), and the conditions:
1. \(\sum_{n=1}^{999} a_n = 1003\)
2. \(\sum_{n=1}^{1003} a_n = -999\)

Find the value of \(\sum_{n=1}^{2002} a_n\).

Answer: 2002


Question: Determine whether a triangle is uniquely defined by its three heights. Answer 'yes' or 'no'.

Answer: yes


Question: In the integral form of the Fox H function, does the term with the product from $p+1$ to $n$ disappear when $n=p$?

Answer: Yes


Question: If $A$ is a symmetric, real matrix, does the singular value decomposition (SVD) of $A$ guarantee that $U = V$ in the decomposition $A = U \Sigma V^\top$?

Answer: No


Question: Evaluate the limit: $$\lim_{x \to 0^+} (1+\tan^2(\sqrt{x}))^{\frac{1}{2x}}$$

Answer: \sqrt{e}


Question: Given the set \(A=\left\{\left|z^{n}+\frac{1}{z^{n}}\right| \mid n \in \mathbb{N}, z \in \mathbb{C}, z^{4}+z^{3}+z^{2}+z+1=0\right\}\), find the sum of the squares of the elements of \(A\).

Answer: 7


Question: Given the process \(Y_t = Y_{t-1} + X_t - \phi^2 X_{t-2}\), where \(X_t = \phi X_{t-1} + Z_t + Z_{t-2}\) and \(Z_t\) is white noise with mean 0 and variance \(\sigma^2\), identify the parameters \((p,d,q)\) of the ARIMA process that describes \(Y_t\).

Answer: (0, 1, 3)


Question: Consider an integer $k > 1$. In the context of linear algebra, let $e$ denote an element in $\mathbb{Z}^k$ which consists of a 1 in the first coordinate and 0s in all other coordinates. Determine the size of the stabilizer of $e$ under the natural action of $\mathrm{SL}_k(\mathbb{Z})$. In other words, how many matrices in $\mathrm{SL}_k(\mathbb{Z})$ have $e$ as their first column?

Answer: \infty


Question: Evaluate the integral \( \displaystyle\int_\gamma \! \frac{1}{z-2} \, \mathrm{d}z \) where \( \gamma \) is a path such that \( \gamma(0) = 1 \) and \( \gamma(1) = 3 \) and \( \gamma(t) \in \mathbb{C}\setminus[2,\infty) \ \forall t \in [0,1].\)

Answer: -i\pi


Question: Determine the dimension of the moduli space of complex structures on \( \mathbb{R}^{4} \) with the standard complex structure at infinity.

Answer: 0


Question: Determine the number of solutions of the equation \(x \tan x = 1\) in the interval \([-2\pi, 2\pi]\).

Answer: 4


Question: Let $V$ be a vector space with $\dim(V)=n$. If a linear operator $T: V \rightarrow V$ has $n-2$ distinct eigenvalues, determine the maximum possible dimension of $\ker(T)$. \( \text{Express your answer as a single number.} \)

Answer: 3


Question: Calculate the limit: \[ \lim_{x\rightarrow2}(x-1)^{\frac{2x^2-8}{x^2-4x+4}} \]

Answer: e^8


Question: Evaluate the limit \( \lim_{{n\to\infty}}\sin(2\pi\sqrt{n^2+n^{1/2}}) \) for \( n \in \mathbb{N} \). If the limit exists, find its value.

Answer: 0


Question: Suppose $f:[a,b] \to [a,b]$ is a continuous function and $(v_n)$ is a sequence defined by $v_0 \in [a,b]$ and $v_{n+1}=f(v_n)$. If $(v_n)$ has a subsequence that converges to a limit $l$, and $\lim \limits_{n \to \infty}\left(v_{n+1}-v_n\right)=0$, does it follow that $(v_n)$ converges to $l$? Answer yes or no and justify your answer.

Answer: Yes


Question: Given the equation \( \frac{a_1}{k^2+2}+\cdots+\frac{a_5}{k^2+6}=\frac{1}{k^2+1} \) for all \( k = \{1,2,3,4,5\} \), find the value of \( \frac{a_1}{3}+\cdots+\frac{a_5}{7} \).

Answer: \dfrac{1}{2}


Question: Let $X_1, X_2, \ldots, X_n$ be independent random variables with the probability density function $f_X(x) = 1 - e^{-x}$ for $x > 0$. Define $M_n = \max(X_1, X_2, \ldots, X_n)$. Determine the function $G(z) = \lim_{n \to \infty} P\left(\frac{M_n - b_n}{a_n} \leq z\right)$. What is $G(z)$?

Answer: G(z) = e^{-e^{-z}}


Question: Solve the differential equation \( \frac{dP}{dt} = 0.05 \ln\left(\frac{4000}{P}\right)P \) with the initial condition \( P(0) = 750 \). Find \( \displaystyle \lim_{t \to \infty} P(t) \).

Answer: 4000


Question: Let $H \subset G$ be finite groups, and $H^{ab}$ and $G^{ab}$ be their abelianizations (i.e., quotients by commutator subgroups). Determine whether it is true that $|H^{ab}| \leq |G^{ab}|$.

Answer: False


Question: Let $x, y, z$ be non-negative real numbers such that $x + y + z = 1$. Determine the number of distinct ways to choose these numbers such that $x^2 + y^2 + z^2 = \min\{x, y, z\}$.

Answer: 1


Question: In a commutative ring $A$, if $a$ and $a+b$ are nilpotent elements, must $b$ also be nilpotent?

Answer: Yes


Question: Find the particular solution $p_n$ for the nonhomogeneous recurrence relation $a_n = 2a_{n-1} + 3a_{n-2} + 8n - 4$, given the initial conditions $a_0 = 1$ and $a_1 = 3$. The non-homogeneous term is $f(n) = 8n - 4$.

Answer: -2n - 3


Question: Simplify the expression \((C^{T} C)^{-1}C^{T}(D^{-1}C^{T})^{T}D^{T}D^{2}D^{-1}\), assuming that all matrices inverted in the expression exist.

Answer: D


Question: Evaluate the surface integral \(\iint_S \boldsymbol{F} \cdot d\boldsymbol{S}\), where the vector field \(\boldsymbol{F} = x^2 \boldsymbol{i} + 2z \boldsymbol{j} - y \boldsymbol{k}\) is defined over the curved surface \(S\) with the equation \(x^2 + y^2 = 25\), bounded by \(z = 0\) and \(z = 6\), and restricted to \(y \geq 3\).

Answer: 288


Question: Evaluate the limit: $$ \lim_{n\to\infty}\int_0^1x\sin^nx\,dx $$.

Answer: 0


Question: Determine an isomorphism for the quotient ring \( \mathbb{Z_8}[X]/(3X+5) \).

Answer: \mathbb{Z}_8


Question: Let $G$ be a locally compact abelian (LCA) group, and $H$ a topologically torsion group with a continuous group homomorphism $f: H \rightarrow G$. Is the topological closure of the image of $f$ in $G$ a topologically torsion group?

Answer: Yes


Question: Find the limit of the product as \( n \) approaches infinity: \[ \lim_{n \to \infty} \prod_{k=2}^{n} \left(1 - \frac{1}{k^2}\right) \] Express the result in a simplified form if possible.

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim\limits_{n \to \infty} \sqrt[4]{n^4+3n^3+1}-\sqrt[4]{n^4-2n^2+2}$$ using binomial expansion.

Answer: \dfrac{3}{4}


Question: Find the maximum possible value of the sum \(\sum_{1 \leq i < j \leq n} |x_i-x_j|\), where \(x_1,...,x_n\) are integers in the interval \([1,35]\).

Answer: 10404


Question: For every square matrix $A$, determine whether there always exists a non-singular matrix $B$ such that $AB = BA$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the arc length of the curve defined by the equation \( x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1 \).

Answer: 6


Question: Determine whether the sequence of random variables $X_n$ with the given probability density functions converges in probability to 0.

Answer: Yes


Question: Determine whether in a metric space $X$, every point of a closed subset $A$ is either an interior point or a limit point. Specifically, can $A$ be expressed as the union of its interior points and its limit points, $A = \text{int}(A) \cup E(A)$, where $E(A)$ denotes the set of limit points of $A$?

Answer: No


Question: Evaluate the limit \(\lim_{x\to 0} \frac{1-\cos{(1-\cos{(1-\cos x)})}}{x^8}\) using algebraic manipulations or substitutions, without employing Maclaurin series or L'Hopital's rule.

Answer: \dfrac{1}{128}


Question: Find the distance from the origin to the normal of the curve given by the parametric equations \(x = a\cos{(\theta)} + a\theta\sin{(\theta)}\) and \(y = a\sin(\theta) - a\theta\cos{(\theta)}\) at any point \(\theta\).

Answer: a


Question: Consider a sequence of functions \((f_n)_{n=1}^\infty\), where each \(f_n: D \to \mathbb{C}\), that converges uniformly to a function \(f: D \to \mathbb{C}\) on a domain \(D\). Given any subset \(S \subseteq D\), does the sequence of restricted functions \((f_n|_S)_{n=1}^\infty\), where \(f_n|_S: S \to \mathbb{C}\), converge uniformly to \(f|_S: S \to \mathbb{C}\) on \(S\)?

Answer: Yes


Question: Find the Lebesgue outer measure of the set \(E=(\mathbb{Q}\times \mathbb{R}) \cup (\mathbb{R}\times \mathbb{Q})\).

Answer: 0


Question: Given a regular expression $R$, design an algorithm $P$ that determines whether $L(R) = \{0,1\}^*$. If $L(R) = \{0,1\}^*$, the algorithm should return $1$; otherwise, it should return $0$. Ensure that your algorithm halts in a finite number of steps for any input regular expression $R$. Provide a brief explanation of your algorithm's logic.

Answer: P


Question: Determine whether the set of permutations in $S_n$ that fix elements modulo 4 forms a subgroup of $S_n$. Provide a justification for your answer.

Answer: Yes


Question: Can a group in which every element is its own inverse have an infinite number of elements?

Answer: Yes


Question: Given two random variables, $X$ and $Y$, where the probability density function (pdf) of $Y$ is $f_Y(y) = 2y$ for $0 \leq y \leq 1$, and conditioned on $Y=y$, $X$ follows a uniform distribution $X \sim U[0, y]$, find the variance of $Y$ given $X = 0.5$, i.e., $\text{Var}(Y \mid X=0.5)$. Compute the value of this variance.

Answer: \dfrac{1}{48}


Question: Determine whether the series \( \sum_{n=1}^{\infty} \frac{\cos \frac{n \pi}{3}}{n} \) is convergent. If it is convergent, find its value.

Answer: 0


Question: Calculate \(3^{202301} \mod 153\) without using Fermat's Little Theorem.

Answer: 63


Question: Determine the number of Sylow 5-subgroups in a group of order \(5^2 \cdot 7 \cdot 11\). Use Sylow's theorems, which state that the number of Sylow \(p\)-subgroups is congruent to 1 modulo \(p\) and divides the order of the group, to find your answer.

Answer: 1


Question: Simplify the expression \(\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}\).

Answer: -6


Question: Determine the number of integer solutions to the equation: \[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{y}} = \frac{1}{2016} \]

Answer: 82


Question: Suppose $f(x)$ is a differentiable function on $\mathbb{R}$ such that $f(x) \neq 0$ for all $x \in \mathbb{R}$. Given that $f(4)=\frac{1}{4}$, $f(8)=\frac{1}{2}$, and \( \int_4^8 \frac{(f'(x))^2}{(f(x))^4}dx=1 \), find the value of $f(6)$. \( \boxed{?} \)

Answer: \dfrac{1}{3}


Question:
Solve for $A$ in the equation:

$$\kappa^{-1} \sum_{n=1}^N r_n - \sum_{n=1}^N \big( x_n \cos(Ar_n) + y_n \sin(Ar_n) \big) = 0$$

where $\kappa, r_n, x_n, y_n \in \mathbb{R}$.

Answer: A


Question: Given the probability mass function $f_X(x)=0.1\delta(x)+0.9u(x)e^{-x}$, find the probability $P(X=0)$. Express your answer as a number.

Answer: 0.1


Question: In a cyclic quadrilateral $ABCD$, where $AB=11$ and $CD=19$, points $P$ and $Q$ lie on $AB$ and $CD$ respectively such that $AP=6$, $BP=5$, $DQ=7$, $CQ=12$, and $PQ=27$. If the line $PQ$ is extended to intersect the circle at points $R$ and $S$, determine the length of segment $RS$.

Answer: 31


Question: Evaluate the integral \( \int_{\mathbb{R}} (F(x+a) - F(x)) \, dx \) for any real number \( a \), where \( F \) is a distribution function.

Answer: a


Question: Determine the values of \( p \) for which the improper integral \( \int_0^{\infty} \frac{e^{-x}}{x^p} \, dx \) converges.

Answer: p < 1


Question: Consider a distribution function $F$ with associated density $f$. Define the sets $S_1=\{s: F(s+\varepsilon)> F(s-\varepsilon)\;\forall\;\varepsilon>0\}$ and $S_2=cl(\{s: f(s)>0\})$. Assuming $F$ is continuous, non-decreasing, and differentiable, determine if $S_1=S_2$. Provide a justification for your answer.

Answer: S_1 = S_2


Question: Determine the values of \( \alpha \) for which the integral \( \int_{B} \frac{1}{|x|^\alpha} \, dV \) exists, where \( B \) is the ball \( |x| \le 1 \) in \( \mathbb{R}^n \).

Answer: \alpha < n


Question: Calculate the total probability of winning a prize in a game where a fair coin is flipped $n$ times, with the probability of winning on a single flip being $\frac{1}{2}$. The probability of winning for even-numbered flips is given by $a_{2n} = \frac{1}{n+1}{2n\choose n}\left(\frac{1}{2}\right)^{2n}$. Find the sum of these probabilities for $n$ from $1$ to $\infty$. What is the total probability of winning?

Answer: 1


Question: Find the value of $a$ such that the system of partial differential equations \( \frac{\partial M}{\partial y} = (4x^2+2xy+6y)a(x+y)^{a-1} + (x+y)^a(2x+6) \) and \( \frac{\partial N}{\partial x} = (2x^2+9y+3x)a(x+y)^{a-1} + (x+y)^a(4x+3) \) satisfies \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \) at \( x = 3 \) and \( y = 1 \).

Answer: 1


Question: Compute the line integral of the vector field \( F \) over the parabolic arc \( C \) in the first quadrant, where \( C \) is the portion of the parabola \( z = y^2 \) from \((0,0,0)\) to \((1,1,1)\). Specifically, evaluate \( \int_C F \cdot \text{d}r \).

Answer: \dfrac{3}{2}


Question: Let $f \in L^1(\mathbb{R})$ be an odd function, i.e., $f(x) = -f(-x)$. Determine whether $\lim_{\xi \to \infty} \hat{f}(\xi) = 0$.

Answer: 0


Question: Evaluate the complex integral \(\int \limits_{\beta} (iy + x^2) \, dz\) where \(\beta\) is the path that starts at \(-i\), continues to \(2+5i\), and then goes to \(5i\). Determine a suitable parameterization for each line segment and compute the integral.

Answer: -12 + 2i


Question: Calculate the value of the infinite series \( \sum_{k=2}^{\infty}k\left(\zeta\left(k+1\right)-1\right) \).

Answer: 1


Question: Determine whether it is possible to find three fifth powers that can be added or subtracted to form a 40-digit number congruent to 10 modulo 11.

Answer: Yes


Question: Consider an inverse system of modules over an integral domain $R$ defined by the maps $A_{i+1} \to A_i, a \mapsto r_{i+1} a$, where $r_i \neq 0,1$. Determine if the inverse limit of this system is zero.

Answer: 0


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

Answer: -\dfrac{7}{6}


Question: Is there a finite ring whose rank is strictly less than both the rank of its additive group and the rank of its multiplicative monoid?

Answer: Yes


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

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


Question: Determine if the following statement is true or false: If $A > 0$ and $B$ is Hermitian with all eigenvalues of $AB$ less than $-1$, then $B < A^{-1}$.

Answer: True


Question: Provide an example of a countable set $S$ such that its set of limit points $S'$ is uncountable.

Answer: \mathbb{Q}


Question: Determine whether homeomorphisms, which are continuous functions with continuous inverse functions, are always bijective.

Answer: Yes


Question: Determine the asymptotic behavior of the function \(f(n) = 3^{2n-1} + 4^n + n^2 \log(n!)\) using big-O notation. Express your answer in the simplest form possible.

Answer: O(9^n)


Question: Is the free distance of a convolutional code equal to the minimum Hamming distance between the all-zero path and any other non-all-zero path in the trellis that originates and ends in the zero state?

Answer: Yes


Question: Find the limit of the expected value of the absolute difference between a Poisson process with rate \( \lambda \) and its mean as time approaches infinity.

Answer: \infty


Question: Determine whether there exists a one-to-one function $f: C \to \mathbb{R}$ such that the graph of $f$, denoted by $\Gamma_f$, is dense in $C \times \mathbb{R}$. Ignore continuity requirements.

Answer: Yes


Question: Find the smallest positive integer \( n \) for which the following limit holds true for two linearly independent solutions \( \phi_1(x) \) and \( \phi_2(x) \) of the differential equation \( 3x^2y''-(2x+x^2)y'+(x^2-3)y=0 \), given that \( \phi_1(0)=0 \): \[ \lim_{x \to 0}x^n\frac{\phi_2(x)}{\phi_1(x)}=0 \]

Answer: 3


Question: Is it possible for a model to be $|\mathcal M|^+$-saturated, where $|\mathcal M|^+$ denotes the successor cardinal of the cardinality of the model $\mathcal M$?

Answer: Yes


Question: Let $x \neq 0$ be a vector in $\mathbb{R}^n$ and let $a \in \mathbb{R}$ with $a > -1$. Calculate the determinant of the matrix $|x|^2 \cdot \mathrm{id} + a \, x \otimes x$. What is the determinant?

Answer: (1 + a) |x|^{2n}


Question: Evaluate the double integral \( \int_0^\infty \int _\frac{x}{\sqrt{4kt}} ^\infty y^2 e^{-y^2} dy\, dx \) using the property \( \int_0 ^\infty y^2 e^{-y^2} dy = \frac{\sqrt{\pi}}{4} \), where \( k \) and \( t \) are positive constants.

Answer: \sqrt{kt}


Question: Determine the radius of convergence of the power series \( \sum_{k=0}^{\infty} a_k x^{k^2} \), where \( a_k = (2k)! \) if \( k \) is a perfect square, and \( a_k = 0 \) otherwise.

Answer: 1


Question: Find the number of permutations $x \in S_n$ such that the cycle containing $1$ in the cycle decomposition of $x$ has length $k$, where $S_n$ is the group of permutations of $\{1,2,3, \ldots, n\}$ and $k$ is an integer between $1$ and $n$. \( \text{Express your answer in terms of } n \text{ and } k. \)

Answer: (n-1)!


Question: Find the degree of the field extension $[\mathbb{Q}(\alpha):\mathbb{Q}(\alpha^2 + 3)]$, where $\alpha$ is a root of the polynomial $p(x) = x^4 + 9x^2 + 15$. Express your answer as an integer.

Answer: 2


Question: Determine the order of the group formed by the permutations of the elements of the Klein four-group $K_4$ that are also group homomorphisms.

Answer: 6


Question: Determine whether the following statement is true or false: Given a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, if $f$ is differentiable at $x_0$ and $Df(x_0)=0$, then all directional derivatives at $x_0$ equal $0$. Justify your answer.

Answer: True


Question: Determine the limit of the sequence \((b_n)_{n \ge 1}\) defined by the initial condition \(b_1 > 0\) and the recursive relation \(b_{n+1} = \frac{(n+1)b_n}{n+2+b_n^2}\) for all \(n \ge 1\).

Answer: 0


Question: Determine whether a continuous projection in a topological vector space can be closed but not open.

Answer: Yes


Question: Find the permutation \( \sigma \) such that for a given permutation \( \tau \), the equation \( \tau^{-1}(134)\tau=(2\, 4 \, 3) \) holds.

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


Question: Let \(X\) and \(Y\) be random variables with a joint normal distribution such that \(\mathbb{E}X = \mathbb{E}Y = 0\), \(\operatorname{Var}(X) = 1\), \(\operatorname{Var}(Y) = 5\), and \(\operatorname{Cov}(X, Y) = -2\). Calculate \(\mathbb{E}(Y^2 \mid X)\).

Answer: 4X^2 + 1


Question: Is it possible to construct a field with an order that is not a power of a prime number? Provide a justification for your answer.

Answer: No


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{n} \left(\frac{k}{n^2}\right)^{\frac{k}{n^2}+2} \]

Answer: 0


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

Answer: e


Question: Evaluate the limit: $$\lim_{V\rightarrow {\bf{0}}^{n\times n}}\frac{|\text{tr}(V^2)|}{\|V\|_{op}}$$ where $\text{tr}(V^2)$ is the trace of the matrix $V^2$. Use this evaluation to determine if a function is Frechet differentiable.

Answer: 0


Question: Given that \( \frac{\sin \alpha + \sin \beta + \sin \gamma}{\sin(\alpha+\beta+\gamma)} = \frac{\cos \alpha + \cos \beta + \cos \gamma}{\cos(\alpha+\beta+\gamma)} = 2 \), find the value of \( \cos(\alpha+\beta) + \cos(\beta+\gamma) + \cos(\gamma+\alpha) \).

Answer: 2


Question: Determine the limit \( \lim_{n\to\infty}\left(\cos\frac{\pi}{n+1}+\cos\frac{2\pi}{n+2}+\ldots+\cos\frac{n\pi}{2n}\right)^{\frac{1}{n}} \) using the Squeeze Rule.

Answer: 1


Question: Given positive numbers $x, y, z$ such that $x^2 + y^2 + z^2 = 1$, find the minimum value of the expression $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$.

Answer: 3\sqrt{3}


Question: Evaluate the integral: \( \int_{0}^{+\infty} \mathrm{d}t \, f(t) \delta(t) \).

Answer: f(0)


Question: Given that $X$ is a Poisson random variable with parameter $\lambda$ and $\Lambda$ is an exponential random variable with parameter $1$, find the expected value of $e^{2\Lambda}$ given that $X=2$. Express your answer as a single number.

Answer: \infty


Question: Find the determinant of the matrix \( \begin{vmatrix} 1 & -\omega & \omega^2 \\ -\omega & \omega^2 & 1 \\ \omega^2 & 1 & -\omega \\ \end{vmatrix} \), where \( \omega \) is a cube root of unity and is equal to \( \frac{1}{2}(-1+i \sqrt{3}) \).

Answer: -4


Question: Determine the number of solutions for the differential equation \( \frac{dy}{dx} + |y| = 0 \), given the initial condition \( y(0) = 1 \).

Answer: 1


Question: Let $n$ be an even integer. In the expansion $$\sin n\theta=\sum_{r=0}^{n} b_r \sin^r\theta,$$ find the coefficient $b_2$. \(\boxed{\text{Your answer here}}\)

Answer: 0


Question: Determine whether the function \( f(x) = \begin{cases} n^2x, & 0 \le x \le \frac{1}{n} \\ 2n - n^2x, & \frac{1}{n} < x \le \frac{2}{n} \\ 0, & \text{else} \end{cases} \) converges pointwise.

Answer: 0


Question: Determine the Krull dimension of the quotient ring \( \frac{\mathbb{K}[X_1,X_2]}{(X_1X_2)} \), where \( \mathbb{K} \) is a field.

Answer: 1


Question: Find the particular solution of the differential equation \( y'' - 4y' + 4y = 16\sin(2t) \) using the Fourier series method.

Answer: 2\cos(2t)


Question: Find the values of \( k \) for which the matrix \( A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ k & 0 & 1 \end{pmatrix} \) is diagonalizable over \( \mathbb{R} \).

Answer: k > 0


Question: Does differentiability of a function $f$ everywhere on the interval $[0,1]$ guarantee its absolute continuity on that interval? Provide a justification for your answer.

Answer: No


Question:
Given the polynomials \( f(x) = x^3 + ax^2 + bx + c \) and \( g(x) = x^3 + bx^2 + cx + a \), where \( a, b, c \) are integers and \( c \neq 0 \), and the following conditions:

- \( f(-1) = 0 \)
- The roots of \( g(x) = 0 \) are the cubes of the roots of \( f(x) = 0 \)

Find the value of \( a^{2013} + b^{2013} + c^{2013} \).

Answer: 3


Question: Determine whether the property $AB = BA = I$ can hold for non-square matrices with full rank, where $I$ is the identity matrix. Provide a justification for your answer.

Answer: No


Question: In a right triangle ABC, with the right angle at C, the altitude CH and the median CM trisect the right angle. Given that the area of triangle CHM is K, determine the area of triangle ABC.

Answer: 4K


Question: Determine the values of \( p \) for which the series \[ \sum_{n=N}^\infty \frac{1}{n(\log(n))(\log\log(n))\dotsm(\log_{j}(n))(\log_{j+1}(n))^p} \] converges.

Answer: p > 1


Question: Determine whether a smooth function \( f \in L^2(\mathbb{R}) \) must satisfy \( |f(x)| \to 0 \) as \( |x| \to \infty \). Provide a justification for your answer.

Answer: No


Question: Let $(E,\mathcal{A},\mu)$ be a finite measure space and let $\mathcal{L}^1$ be the space of integrable functions on $E$. Suppose we have a sequence $\{f_n\} \subset \mathcal{L}^1$ that satisfies certain conditions, and assume that $f_n \to g$ pointwise almost everywhere. Can we conclude that $g \leq M$ almost everywhere for some constant $M$?

Answer: No


Question: Determine the positive integer $k$ for which the expression $\frac{103^{k/3}}{k!}$ reaches its maximum value.

Answer: 4


Question: Find the coefficient of $x^3y^4z^5$ in the expansion of $(xy+yz+zx)^6$.

Answer: 60


Question: Find all positive integer solutions (x, y) to the equation $7^x = 3^y - 2$.

Answer: (1, 2)


Question: Let $f(x)$ be a quadratic function such that $f(0)=1$ and the integral $\int \frac{f(x)}{x^2(x+1)^3} \, dx$ is a rational function. Determine the value of $f'(0)$. \( \boxed{} \)

Answer: 3


Question: Determine the order of the intersection of the cyclic subgroups generated by elements $a$ and $b$ in a group $G$, where $a$ has order 18 and $b$ has order 30, given that their intersection is non-trivial.

Answer: 6


Question: Parametrize the surface \( S = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = (2-z)^2, 0 \leq z \leq 1\} \) and calculate its area.

Answer: 3\sqrt{2}\pi


Question: Evaluate the integral \( \int_{C}\frac{\sin z}{(z-\pi)^3}dz \) over the unit circle centered at the origin.

Answer: 0


Question: Given an algebraically closed field $F$ and a polynomial $p \in F[X_1, ..., X_n]$ that is irreducible, determine if $p$ remains irreducible in $K[X_1, ..., X_n]$, where $K$ is a transcendental field extension of $F$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the limit \( \lim_{x\to \infty} \left[\frac{p(x)}{e^x}\right] \) always converges to 0, where \( p(x) \) is a polynomial.

Answer: 0


Question: Find the coefficient of \(x^{10}\) in the expression \(\frac{3x^4-x^2}{(1-x)^4}\) using combinatorial methods and generating functions.

Answer: 87


Question: Provide a counterexample to the statement: "If $H$ and $K$ are cyclic subgroups of a non-abelian group $G$, then the set $HK$ is not a subgroup of $G$."

Answer: S_3


Question: Let $(A_\alpha)_{\alpha \in I}$ be a family of countable sets such that for any $\alpha, \beta \in I$, either $A_\alpha \subset A_\beta$ or $A_\beta \subset A_\alpha$. If $I$ is an arbitrary index set, is $\bigcup_{\alpha \in I} A_\alpha$ necessarily countable? Answer yes or no.

Answer: No


Question: Let $P(x) = 2x^3 - 3x + 1$ be a polynomial with roots $p, q, r$. Calculate the value of $(q+r-5p)(p+q-5r)(r+p-5q)$. \( \)

Answer: 108


Question: Determine the number of non-abelian groups of order \( p^3 \) for a given prime \( p \).

Answer: 2


Question:
Given the piecewise function \( f(x) \) defined as:
\[
f(x) = \begin{cases}
ax(x-1) + b &; x < 1 \\
x + 2 &; 1 \le x \le 3 \\
px^2 + qx + 2 &; x > 3
\end{cases}
\]
where it is continuous for all \( x \) in \( \mathbb{R} \) except at \( x = 1 \), and \( |f(x)| \) is differentiable everywhere with \( f'(x) \) continuous at \( x = 3 \). If \( |a + p + b + q| = k \), find the value of \( k \).

Answer: 3


Question: Let $K$ be a compact Hausdorff space. Suppose that $f$ is a non-negative, continuous function on $K$ and $\mu$ is a continuous linear functional on $C(K)$. Is there a sequence $(f_{n})_{n}$ in $C(K)$ such that $f_{n}>0$ for each $n$ and $\mu(f_{n})\rightarrow \mu(f)$?

Answer: Yes


Question: Determine whether it is possible to construct an algebraic structure over a finite set such that for any elements \(a\) and \(b\) in the set, the equation \(x + a = x + b\) always has a solution. Provide a justification for your answer.

Answer: Yes


Question: Find the Taylor series of the function \( f(z) = \log(z^2-3z+2) \) at \( z = 0 \) and determine the radius of convergence for the series.

Answer: 1


Question: Determine the minimum number of distinct slopes formed by connecting each point from a set of $x_1$ points on the positive x-axis to each point from a set of $x_2$ points on the negative y-axis.

Answer: x_1 + x_2 - 1


Question: Determine whether there exists a finitely presentable torsion-free group whose abelianization has torsion.

Answer: Yes


Question: Let $V$ be the vector space of differentiable functions $f: [0,1] \to \mathbb{R}$, with the norm $\|f\| = \max\{|f'(x)| : x \in [0,1]\}$. Consider the map $S: V \to \mathbb{R}$ given by $S(f) = f(0)$. Determine the norm $\|S\|$.

Answer: \infty


Question: In a topological space $X$, determine whether any point $a$ is always included in a proper open subset of $X$. Answer with 'Yes' or 'No'.

Answer: No


Question: Let $L$ be a field and let $\zeta \in L$ be a primitive $n$-th root of unity. Let $\alpha$ and $\beta$ be automorphisms in $\text{Aut}(L)$. Determine if it is possible that $\alpha(\zeta) \neq \beta(\zeta)$. Provide a justification for your answer.

Answer: Yes


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

Answer: e^{-2}


Question: For how many distinct triangles $\triangle ABC$, with $AB = 2011$, are both $\cos(2\angle A + 3\angle C)$ and $\sin(2\angle B + \angle C)$ integers?

Answer: 5


Question: Is it possible to construct a continuous local martingale \((M(t))_{t \in [0,1]}\) such that \(M(0)=0\), \(M(t)\) is not always equal to 0, and \(M(t)\) is constant with positive probability?

Answer: Yes


Question: For every non-negative integer \( n \), a polynomial \( p_n(x) \) exists such that \( \int x^n e^x \,dx = p_n(x) e^x + C \). Determine \( \lfloor 1000 \cdot L \rfloor \), where \( L = \lim_{n \to \infty} \frac{|p_n(2)|}{n!} \).

Answer: 135


Question: Find the six-digit natural number $N$ with a digit sum of $43$, where exactly one of the following statements is false: (1) $N$ is a perfect square, (2) $N$ is a perfect cube, and (3) $N < 500000$. Determine the value of $N$. Given: The smallest six-digit number with a digit sum of $43$ is $169999$, and the largest number satisfying $N < 500000$ is $499993$. Checking for perfect squares in the interval $[169999, 499993]$ is computationally intensive.

Answer: 499849


Question: Given the equation with positive real numbers $x$ and $y$: \[\sqrt{x^2 + \sqrt[3]{x^4 y^2}} + \sqrt{y^2 + \sqrt[3]{x^2 y^4}} = 512,\] determine the value of $x^{2/3} + y^{2/3}$.

Answer: 64


Question: Let $f:[a,b]\to\mathbb{R}$ be a Riemann integrable function over its domain, and suppose $\lim_{x\to c\in(a,b)}f(x)=L$ for some $L\in\mathbb{R}$. Define the function $F:[a,b]\to\mathbb{R}$ by $F(x)=\int_a^x f(t)dt$. Prove that $F$ is differentiable at $c$ and that $F'(c)=L$. \(\text{True or False?}\)

Answer: A


Question:
Given $U_n = \int_{0}^{1}x^n(2-x)^n\,dx$ and $V_n = \int_{0}^{1}x^n(1-x)^n\,dx$, where $n \in \mathbb{N}$, determine the correct relationship between $U_n$ and $V_n$ from the following options:

(A) $U_n = 2^nV_n$

(B) $U_n = 2^{-n}V_n$

(C) $U_n = 2^{2n}V_n$

(D) $U_n = 2^{-2n}V_n$

Assume $V_n$ remains unchanged when $x$ is replaced by $1-x$, and $U_n = \int_{0}^{1}(1-x)^n(1+x)^n\,dx$. Find the correct relationship.

Answer: C


Question: Determine whether the following statement is true or false: If $A \in OPS^m(X)$, where $m \in \mathbb{R}$ and $X$ is a compact manifold, is an invertible pseudodifferential operator, then its inverse $A^{-1}$ is also a pseudodifferential operator, specifically $A^{-1} \in OPS^{-m}(X)$. Provide a justification for your answer.

Answer: True


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

Answer: 2


Question: Determine the number of integers in the range from 1 to $10^{10}$ that are not perfect squares, cubes, or fifth powers using the Principle of Inclusion-Exclusion.

Answer: 9999897804


Question: Suppose $V$ is an inner product space over $\mathbb{C}$. If $\|u\| \geq \|u + av\|$ for all $a \in \mathbb{C}$, does it imply that $u$ and $v$ are orthogonal? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Suppose that in the block matrix \( \pmatrix{A&B\\ C&D} \), the pairs of matrices \((A,C)\) and \((B,D)\) commute. Determine the expression for the determinant of this block matrix.

Answer: \det(AD - BC)


Question: Determine the number of pairs \((n, a)\) such that the congruence \(x^n \equiv a \pmod{31}\) has exactly 10 solutions.

Answer: 6


Question: Find the fundamental group of a cube with 12 points removed, where each point is located at the center of each face.

Answer: F_{11}


Question: Find the value of $\log_q(p)$ if $p$ and $q$ are solutions to the equation $5^{(\log_5x)^2} + x^{\log_5x} = 1250$. \( \boxed{} \)

Answer: -1


Question: Let \( G \) be a finite group, and let \( A \) be a symmetric subset containing the identity (i.e., \( A^{-1} = A \) and \( 1 \in A \)). The powers of \( A \) form a chain \( A \subsetneq A^2 \subsetneq A^3 \subsetneq \cdots \subsetneq A^d = \langle A \rangle \). Determine the optimal upper bound on \( d \) in terms of \( k \) such that \((k + 1)|A| > |G|\). For example, when \( 2|A| > |G| \), the optimal bound is \( d \leq 2 \), and when \( 3|A| > |G| \), it is \( d \leq 4 \). How does this bound change with \( k \)?

Answer: 2k


Question: Find the area of the region determined by the intersection of the curves \( y = \frac{2}{x^2+1} \) and \( y = |x| \).

Answer: \pi - 1


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left(\frac{x+2}{x-6}\right)^{\frac{x}{4}} \]

Answer: e^2


Question: Determine if the function \( f(x) = \int_{0}^{\infty} t^{x^2+1} e^{-2t} \, dt \) can be expressed as a power series in a neighborhood of zero. If it can, find the radius of convergence of this series.

Answer: \sqrt{2}


Question: Evaluate the integral \( \int_{-4}^{4} \cos{x} \delta(\sin{x}) \,{\rm d} x \).

Answer: -1


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

Answer: 1


Question: Evaluate the sum \( \sum_{k=1}^{2n} (-1)^{k+1} \frac{1}{\binom{2n}{k}} \).

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


Question: Evaluate the double integral \( \iint_D (x^2 - y^2)^{10} \, dx \, dy \), where \( D \) is the region bounded by \( |x| + |y| \leq 1 \). Use the substitution \( u = x - y \) and \( v = x + y \) to solve the integral.

Answer: \dfrac{2}{121}


Question: Find the smallest natural number \( n \) that satisfies the inequality: \[ n^{2^{n}} > (100^{100} \times n)^{100} \]

Answer: 15


Question: Evaluate the limit \( \lim_{(x,y) \to (0,0)} (1+x^2+y^2)^{\frac{1}{x^2 + y^2 - xy^2}} \).

Answer: e


Question: For the function $f : (-1, 1) \to \mathbb{R}$ satisfying $|f(x)| \le |x|^2$, determine whether $f'(0) = 0$.

Answer: 0


Question: Determine whether there exists an element of infinite order in the group of real 2x2 matrices with determinant 1 and inverse equal to its transpose.

Answer: Yes


Question: Find all pairs of prime numbers \(p\) and \(q\) such that the equation \(p + p^2 + \cdots + p^{10} - q = 2017\) holds true.

Answer: (2, 29)


Question: Evaluate the double integral \( \iint\limits_{R} x y \, dA \) where \( R = \{(x,y) : \frac{x^2}{36} + \frac{y^2}{16} \leq 1, x \geq 0, y \geq 0\} \).

Answer: 72


Question: Determine whether the following statement is true or false: A continuous linear functional $f$ on a finite-dimensional normed vector space $(X,||\cdot||)$ is injective if and only if there exists a constant $M>0$ such that for any given $x \in X$, $|f(x)|\ge M||x||$. Provide a justification for your answer.

Answer: A


Question: Let $U \subset \mathbb{R}^n$ be an open set. Suppose that $u$ is a function such that $\| u \|_{W^{k,p}(K)} \leq c$ for every relatively compact set $K \subset U$, where $c$ is a constant independent of $K$. Does this imply that $u \in W^{k,p}(U)$? Answer yes or no.

Answer: Yes


Question: Suppose you have a matrix $M$ over the reals with $m$ rows and $n$ columns, where $n > m$, and $M$ has full row rank. Is it always possible to add $n-m$ unique standard basis elements as rows to $M$ to form an $n \times n$ matrix of full rank? Justify your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \sum_{k=0}^n \frac{\binom{n}{k}}{k^2 2^n + n^2} \) using the approximation \( \binom{n}{k} \sim \sqrt{\frac{2}{n \pi}} \, 2^{n} \, e^{-\frac{(n-2k)^{2}}{2n}} \).

Answer: 0


Question: Determine the limit \( \lim_{n \to \infty} a_n \) for the coefficients \( a_n \) of the power series expansion of \((1-x)^{-\frac{1}{2}}\), where \( a_0=1, a_1=\frac{1}{2}, a_2= \frac{3}{8}, a_3=\frac{5}{16}, \ldots \).

Answer: 0


Question: Evaluate the function \( f(x) = \lim_{n\rightarrow \infty}\sum_{r=0}^{n}\left\{\frac{\tan \left(\frac{x}{2^{r+1}}\right)+\tan^3\left(\frac{x}{2^{r+1}}\right)}{1-\tan^2\left(\frac{x}{2^{r+1}}\right)}\right\} \).

Answer: \tan x


Question: Find the area enclosed by the curves \( y = \frac{x^2}{2} \) and \( x = \frac{y^2}{2} \) using a double integral.

Answer: \dfrac{4}{3}


Question: Given the equations $a^3 - 3ab^2 = 44$ and $b^3 - 3a^2b = 8$, where $a$ and $b$ are real numbers, find the value of $a^2 + b^2$. Express your answer as a single number.

Answer: 10\sqrt[3]{2}


Question: Determine whether the eigenvalues of the product $AB$ are equal to those of $BA$, where $A$ and $B$ are $n \times n$ square matrices.

Answer: Yes


Question: If \( f(z) = \frac{z}{(z-1)(z-2)(z-4)} \) has an antiderivative in the region \( \{ \mathrm{Re}(z)> 0\} \), determine the value of the integral \( \displaystyle\int_\gamma f(z)\,\mathrm{d}z \) for any simple closed curve \( \gamma \) lying entirely within this half-plane.

Answer: 0


Question: Find the limit of the sequence \(a_n = \sum_{r=0}^{n}{\frac{n \choose r}{n^r}\,\frac{1}{r+4}}\) as \(n\) approaches infinity.

Answer: 6 - 2e


Question: Evaluate the limit: \[ \lim_{(x,y) \to (0,0)}\frac{\sin(x^2+y^2)}{\ln(3x^2y^2+1)} \]

Answer: \infty


Question:
Assume $G$ is a group and $p, q$ are coprime positive integers. If for all $a, b \in G$, the following conditions hold:
\[a^p b^p = b^p a^p\]
and
\[a^q b^q = b^q a^q\]
Is $G$ necessarily an abelian group? Justify your answer.

Answer: Yes


Question: In a complete metric space $(X,d)$, consider a sequence $(x_n)_{n\in\mathbb{N}}$ such that $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ for all $n\in\mathbb{N}$. Does this condition ensure that the sequence $(x_n)_{n\in\mathbb{N}}$ is convergent?

Answer: No


Question: Determine the form of a matrix $A$ in $\mathbb{R}^3$ that commutes with every other matrix $X$ in the same space, i.e., $AX = XA$ for all matrices $X$. What is the form of matrix $A$?

Answer: \lambda I


Question: Let $\Omega$ be an open subset of $\mathbb{R}^n$ and let $\mu_1,\, \mu_2: \mathcal{B}(\Omega)\to [0,\,+\infty]$ be two Radon measures on the Borel algebra. Suppose that for every set $A$ in $\mathcal{B}(\Omega)$, if $\mu_1(A)=0$ then $\mu_2(A)=0$. Determine whether $\mu_2$ is absolutely continuous with respect to $\mu_1$. Provide a justification for your answer.

Answer: Yes


Question: Let \((X,d)\) be a complete metric space and \(f: X \rightarrow X\) be a map. If \(f^k\) is a contraction for some positive integer \(k\), does \(f\) necessarily have a fixed point? Provide a justification for your answer.

Answer: Yes


Question: Does there exist a metric space $(X, d)$ and a sequence $(x_n)$ in $X$ such that no subsequence of $(x_n)$ converges in $X$, yet the infimum of the set of distances between distinct elements of the sequence is zero? Specifically, does there exist a sequence with $\inf\{d(x_m, x_n) \mid m, n \in \mathbb{N}, m \neq n\} = 0$?

Answer: Yes


Question: Determine whether an antisymmetric matrix can have real eigenvalues other than 0.

Answer: No


Question: Evaluate the limit: \[ \lim_{x \to 10} (x^2 - 100)^{\frac{1}{\ln(x-10)}}. \]

Answer: e


Question:
Given that \( z \) is a root of the equation \( 11z^8 + 20iz^7 + 10iz - 22 = 0 \), determine the interval in which the expression \( S = |z|^2 + |z| + 1 \) belongs. Choose from the following options:

(A) \( S \leq 3 \)
(B) \( 3 < S < 7 \)
(C) \( 7 \leq S < 13 \)
(D) \( S \geq 13 \)

Answer: B


Question: Find the residue of the function \( f(z) = \frac{e^z}{z^6} \) at \( z = 0 \) using the Laurent series expansion.

Answer: \dfrac{1}{120}


Question: Evaluate the integral \( \int_1^4 \frac{x^2}{\sqrt{1-x^3}}\, dx. \)

Answer: -2i\sqrt{7}


Question:
Determine which of the following statements is true if and only if \( n \) is a prime number:

\[
\begin{alignat}{2}
&\text{(A)} &\quad n \mid (n-1)! + 1 \\
&\text{(B)} &\quad n \mid (n-1)! - 1 \\
&\text{(C)} &\quad n \mid (n+1)! - 1 \\
&\text{(D)} &\quad n \mid (n+1)! + 1 \\
\end{alignat}
\]

Select the correct divisibility condition that uniquely identifies a prime number.

Answer: A


Question: Determine the limit, if it exists, of the sequence \( a_n = \frac{1}{n} \sum_{i=2}^n \frac{1}{\ln i} \) for \( n \ge 2 \).

Answer: 0


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

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


Question: Let $g(x)$ be a one-to-one function such that $g(x)g(y)=g(xy)$ for all $x,y \in \mathbb{R} \setminus \{0\}$. Given that $g(1) = 1$ and $g'(1) = 3$, determine the function $g(x)$. Express your answer as a function of $x$.

Answer: x^3


Question: Suppose we have a polynomial of the form \(p(x) = \prod_{j=1}^n (x - e^{2 \pi i \alpha_j}) = x^n + A_{n-1}x^{n-1} + \ldots + A_1 x + A_0\) where \(\alpha_j \in \mathbb{C}\) and \(A_j \in \mathbb{C}\) for all \(j=1,\ldots,n\). If all roots \(a_j = e^{2 \pi i \alpha_j}\) are \(n\)-th roots of unity, meaning \(\alpha_j \in \mathbb{Q} \cap [0,1)\) and the set of roots is invariant under \(z \mapsto z^{-1}\), does this imply that the coefficients \(A_j \in \mathbb{R}\), i.e., \(p(x) \in \mathbb{R}[x]\)?

Answer: Yes


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function. Suppose there exists a bounded sequence $\{x_n\}$ such that $x_n \neq x_m$ for $n \neq m$ and $f(x_n) = 0$ for all $n$. Prove that there exists a point $x_0$ such that $f(x_0) = 0$ and $f'(x_0) = 0$.

Answer: x_0


Question: Determine the values of $c$ for which the function $g(z) = f(z) - z^2 = 2 (c-i) xy$ is analytic in the complex plane $\mathbb{C}$.

Answer: i


Question: Evaluate the limit: $$\lim_{{x\to+\infty}} \frac{1}{x} \int_{1}^{x} \frac{t^3}{1+t^3} \, dt$$

Answer: 1


Question: Given the function $f(x)$ defined on the interval $(-5, 5)$ by $e^{-x}f(x) = 2 + \int_{0}^{x} \sqrt{t^4 + 1} \, dt$, find the derivative of the inverse function $f^{-1}(x)$ at $x = 2$. Express your answer as a single value.

Answer: \dfrac{1}{3}


Question: A differentiable scalar field $g$ has a directional derivative of $-1$ at the point $(3, -1)$ in the direction of the vector toward $(1, 1)$, and a directional derivative of $+3$ in the direction of the vector toward $(4, -1)$. Calculate the directional derivative of $g$ at $(3, -1)$ in the direction of the vector toward $(0, 2)$. Express your answer as a single number.

Answer: -1


Question: In a simple group of order 60, there are 5 Sylow 2-subgroups. Determine the number of elements of order 2 in this group.

Answer: 15


Question: Determine the ideal $J(T)$ where $T \subset \mathbb{R}^n$ is the set of points with rational coordinates.

Answer: (0)


Question: Evaluate the limit \(\lim _{x\to +\infty }\left(\int _0^{x^2}e^{t^2}\,dt\right)^{\frac{1}{x^4}}\).

Answer: e


Question: Evaluate the limit: \[ \lim\limits_{(x,y)\to(0,0)}\dfrac{\log\left(\dfrac{1}{\sqrt{x^2+y^2}}\right)}{\dfrac{1}{\sqrt{x^2+y^2}}}. \]

Answer: 0


Question: Determine whether any monic irreducible polynomial of degree 4 in \(\mathbb{Q}[X]\) is separable.

Answer: Yes


Question: Determine the norm of the operator \( T: L^2(0,1) \to L^2(0,1) \) defined by \( Tu = e^t u \).

Answer: e


Question: In a right-angled triangle $\triangle ABC$ with $\angle B = 90^\circ$ and $\angle C = 15^\circ$, given that $|AC| = 7$, a point $D$ is chosen on $AC$. Perpendicular lines $DE$ and $DF$ are drawn to $AB$ and $AC$, respectively. Calculate the probability that $DE \cdot DF > 3$. Provide your answer as a decimal rounded to two decimal places.

Answer: 0.14


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

Answer: \dfrac{1}{e}


Question: Evaluate the limit: $$\lim\limits_{x\to0^+} \left(\frac1{\ln(2-x)}-\frac1x \right)$$ without using L'Hopital's rule or series expansions.

Answer: -\infty


Question: Compute $f''(0)$ given that $f'''(x)$ is continuous everywhere, and the limit \(\lim_{x \to 0}(1+x+\frac{f(x)}{x})^{1/x}=e^3\) holds. Use Taylor's theorem and the given limit to derive the expression for $f''(0)$. Provide your answer in the form of a single number.

Answer: 4


Question: Find the chromatic number of the graph obtained by removing $k$ vertex-disjoint edges from the complete bipartite graph $K_{m,n}$, where $k \leq \min\{m,n\}$. Express your answer in terms of $m$, $n$, and $k$. \( \boxed{} \)

Answer: 2


Question: Let $V$ be a vector space over a field $F$, and let $\varphi^1, \ldots, \varphi^k \in V^{\lor}$. Determine whether the following statement is true or false: $(\text{Span}(\varphi^1, \ldots, \varphi^k))^{\circ} = \bigcap_{i=1}^k \ker(\varphi^i)$.

Answer: True


Question: Solve for \( x \) in the equation: \[ 9^x - 2^{x + \frac{1}{2}} = 2^{x + \frac{7}{2}} - 3^{2x - 1}. \]

Answer: \dfrac{3}{2}


Question: Determine the number of points within the interval $[0, 6\pi]$ where the function $f(x)$, defined as $f(x) = \tan x$ for rational $x$ and $f(x) = x^2 + 1$ for irrational $x$, is continuous.

Answer: 6


Question:
Evaluate the determinant of the matrix $B^2$, where $B$ is given by:

$$B=\begin{pmatrix} 1 & 1 &1 \\ y_1 &y_2 &y_3 \\ y^{2}_1 & y^{2}_2 & y^{2}_3 \end{pmatrix}$$

and $y_1,y_2,y_3$ are the roots of the polynomial $y^3+py+q=0$. Provide your answer in terms of $p$ and $q$.

Answer: -4p^3 - 27q^2


Question: Evaluate the integral \( I = \int_\gamma \frac{1}{z^2} \, dz \), where \( \gamma \) is the upper half of the unit circle in the complex plane.

Answer: 2


Question: Determine the field over which all 2x2 orthogonal matrices are diagonalizable.

Answer: \mathbb{C}


Question: Given a group $G$ with subgroups $H$ and $K$, where $|H|=|K|=p^n$ for some prime $p$ and positive integer $n$, and $H$ and $K$ are not equal but contained in the same Sylow $p$-subgroup, does $p^{2n}$ necessarily divide $|G|$? Answer yes or no.

Answer: No


Question: Calculate the probability that the second largest number among three random numbers generated from a uniform distribution on the interval $[0,1]$ is less than a given value $x$. Express your answer in terms of $x$.

Answer: 3x^2 - 2x^3


Question: Evaluate \( \Gamma\left(-\frac{1}{2}\right) \) directly from the definition \( \Gamma(s) = \int_0^\infty e^{-t}t^{s-1}dt \), without using the known values \( \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} \) and \( \Gamma(s+1) = s\Gamma(s) \).

Answer: -2\sqrt{\pi}


Question:
Given a sequence of non-negative continuous functions $f_n(x)$ on $[0,1]$ for all $n \geq 1$, where $\lim_{n \to \infty} \int^1_0 f_n (x)dx=0$, determine which of the following statements is necessarily true:

A. $f_n$ converges to $0$ uniformly on $[0,1]$.
B. $f_n$ may not converge uniformly, but converges to $0$ pointwise.
C. $f_n$ converges pointwise, and the limit may be non-zero.
D. $f_n$ is not guaranteed to have a pointwise limit.

Answer: D


Question: Provide an example of a non-trivial function that satisfies the equation \( f(a+b) = f(a)f(b) \) for non-whole numbers.

Answer: e^x


Question: Evaluate the line integral \( \oint_{C} xy \, ds \) where \( C \) is the curve of intersection between the surfaces \( x^2 + y^2 = 4 \) and \( y + z = 8 \). The curve is traversed once in the counterclockwise direction when viewed from above.

Answer: 0


Question: Given that the function \( f(x) \) has \( n \) distinct real roots, determine the number of distinct real roots of the equation \( g'(x) = 0 \), where \( g(x) = f(x)f'(x)f''(x) \).

Answer: 3n - 4


Question: Evaluate the infinite product: $$\prod_{k=2}^{\infty} \frac{(k-1)(k+1)}{k^2}$$

Answer: \dfrac{1}{2}


Question: Calculate the area inside the circle given by the polar equation \( r = 3a\cos(\theta) \) and outside the cardioid given by \( r = a(1+\cos(\theta)) \).

Answer: \pi a^2


Question: Suppose we have a sequence of functions $f_n$ in $L^1[0,1]$ that converge pointwise almost everywhere to $f \in L^1[0,1]$. If $\lim_{n\to \infty}\int |f_n-f| = 0$, does this imply that $\int \vert f_n\vert \rightarrow \int \vert f\vert$? Answer yes or no and justify your answer.

Answer: Yes


Question: Determine whether there exists a bounded region in \(\mathbb{R}^2\) such that no point exists through which every line passing through it bisects the area of the region.

Answer: Yes


Question: Determine the number of points \( z \in D(0,1) \) for which the equation \( e^{z} + 3z = 0 \) holds true, using Rouche's Theorem.

Answer: 1


Question: Given the equation \( x \sin \pi x = \int_{0}^{x^2}f(t)\,dt \), where \( f \) is a continuous function, compute \( f(4) \).

Answer: \dfrac{\pi}{2}


Question: Solve the congruence equation: \( x^{37} \equiv 54 \mod 97 \).

Answer: 43


Question:
Let $P(x)$ be a separable polynomial of degree $n$ with distinct roots $x_1, x_2, \ldots, x_n$. If $Q(x)$ is a polynomial of degree at most $n-2$, find the value of the expression:

$$\sum_{i=1}^n\,\frac{Q\left(x_i\right)}{P'\left(x_i\right)}$$

Answer: 0


Question: Find the limit of the solution to the initial value problem \(x'(t)=Ax(t)+g(t)\) with \(x(0)=x_0\), given that all eigenvalues of the matrix \(A\) have negative real parts (\(\mathrm{Re}(\alpha_j)<0\)), and \(\lim_{t\to\infty} |g(t)|=g_0\).

Answer: -A^{-1} g_0


Question: Find the second directional derivative of the function \( f(x, y) = x^3 + 5x^2y + y^3 \) at the point \((2, 1)\) in the direction of the vector \( u = \left< \frac{3}{5}, \frac{4}{5} \right> \).

Answer: \dfrac{774}{25}


Question: Suppose $X_1, X_2, X_3$ are independent and identically distributed (iid) uniform random variables over the interval $(0,1)$. Evaluate the probability $P(X_1 > X_2 > X_3)$. Provide your answer as a single probability value.

Answer: \dfrac{1}{6}


Question: For a set of \( n \) distinct points in space, where \( n > 4 \), determine the number of equilateral triangles that can be formed when each point is connected to every other point with straight lines of equal length.

Answer: 0


Question: Determine whether the second cohomology groups of a non-abelian group and its abelianization are always different. Answer true or false.

Answer: False


Question: Does every uncountable subset of \(\mathbb{R}\) have a closed subset containing infinitely many elements?

Answer: Yes


Question: In a metric space $(X,d)$, consider a sequence $\{x_n\}_{n\in \mathbb{N}}$ such that $d(x,x_n) \to \infty$ for some point $x \in X$. Does this imply that the sequence $\{x_n\}_{n\in \mathbb{N}}$ cannot be contained in any compact subset of $X$? Answer yes or no and justify your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: For every non-degenerate bilinear form $f$ on a finite-dimensional $\mathbb{Z}_2$ vector space $V$, $f(x,x)=0$ for all $x\in V$. Provide a justification for your answer.

Answer: False


Question: Evaluate the integral \(\int_{|z|=2} \frac{(1-\cos z)}{(e^z-1)\sin z}\) using Cauchy's residue theorem.

Answer: 0


Question: Determine the cardinality of the set of functions that are either multiplicative or completely multiplicative, defined on the set of natural numbers and taking natural numbers as their values.

Answer: 2^{\aleph_0}


Question: In a metric space $M$, if two balls $B_r(x)$ and $B_s(y)$ have the same set of points for some $x, y \in M$ and $r, s \in \mathbb{R}$, does it necessarily follow that $x = y$ and $r = s$? Provide a justification for your answer.

Answer: No


Question: Determine whether it is possible to construct two spaces with different fundamental groups that share the same universal covering space.

Answer: Yes


Question: Find the limit of the product as \( n \) approaches infinity: \[ \lim_{n\to\infty}\prod_{k=1}^{n}\frac{(k+1)(2k)^2}{k(2k+1)^2} \]

Answer: \dfrac{\pi}{4}


Question: Calculate the absolute value of the line integral \( \int_{C}\vec{F}.\vec{dr} \), where \( C \) is the closed curve forming the boundary of the surface of the cone \( z=\sqrt{x^2+y^2} \) bounded by the planes \( z=0 \) and \( z=3 \), and the curl of the vector field \( \vec{F} \) is given by \( \vec{T}=\langle -y,x,0\rangle \).

Answer: 0


Question: Determine the probability distribution of the random variable \( Z = \frac{X+Y}{X} \), where \( X \) and \( Y \) are independent exponentially distributed random variables with parameter \( \beta = 1 \).

Answer: \frac{1}{z^2}


Question: Given a sphere with center at the origin and radius 'r', calculate the average of the reciprocal of the distance between a point located at a distance 'c' from the sphere's center (where 'c' > 'r') and all points inside the sphere.

Answer: \dfrac{1}{c}


Question: Find the smallest known value of $C$ such that there are infinitely many prime pairs differing by $C$. Provide your answer as a single number.

Answer: 246


Question: Consider two independent, identically distributed (IID) standard normal random variables $X$ and $Y$. Given that $E(X|X+Y>0) = \frac{1}{\sqrt{\pi}}$, calculate $E(XY|X+Y>0)$.

Answer: 0


Question: Let \(p\) be a prime number, and let \(\mathbb{F}_{p}\) be a finite field of order \(p\). Let \(G=GL_{n}(\mathbb{F}_{p})\) denote the general linear group and \(U_{n}\) denote the unitriangular group of \(n\times n\) upper triangular matrices with ones on the diagonal, over the finite field \(\mathbb{F}_{p}\). Let \(H\) be an abelian subgroup of order \(p^{m}\) in \(U_{n}\). Determine whether the subgroup \(H\) must be elementary abelian of rank \(m\) (i.e., \(H\simeq (\mathbb{F}_{p})^{m}\)).

Answer: No


Question:
Determine the asymptotic behavior of the function $xf(x)$ as $x$ approaches infinity, where $f(x)$ is defined by:

$$f(x)= \begin{cases} \frac{a}{\pi(ax^2+x^2)} & \text{if } x>0 \\ 0 & \text{otherwise} \end{cases}$$

Answer: \dfrac{a}{\pi(a + 1) x}


Question: Evaluate the surface integral \( \iint\limits_S \vec{v} \cdot \vec{n} \, dS \) for the vector field \( \vec{v} = \frac{\vec{r}}{|\vec{r}|^3} \), where \( \vec{r}=(x,y,z) \), and \( \vec{n} \) points outward from a closed and smooth surface \( S \) enclosing the origin.

Answer: 4\pi


Question: Find the area (in square units) of the region enclosed by the curve \(\sqrt{x} + \sqrt{y} = 1\), where \(x\) and \(y\) are non-negative, and the tangent to this curve at the point \(\left(\frac{1}{4}, \frac{1}{4}\right)\).

Answer: \dfrac{3 - 2\sqrt{2}}{6}


Question: Three projective planes $\pi_1$, $\pi_2$, and $\pi_3$ intersect such that $\pi_1 \cap \pi_2 \cap \pi_3$ is a point, $\dim\,(\pi_1 \cap \pi_2) = 0$, $\dim\,(\pi_1 \cap \pi_3) = 1$, and $\dim (\pi_2 \cap \pi_3) =1$. Determine the dimension of $\pi_1 \lor \pi_2\lor \pi_3$. \(\boxed{\ }\)

Answer: 4


Question: Determine the number of k-dimensional subspaces in an n-dimensional vector space over the finite field $\mathbb{F}_p$. Express your answer using the Gaussian binomial coefficient.

Answer: \dbinom{n}{k}_p


Question: Let $E$ be a field and $F$ be a subfield of $E$ which is isomorphic to $E$. Is $F$ necessarily equal to $E$? Provide a justification for your answer.

Answer: No


Question: Let $f$ be a function defined on $[0,1]$. If for any $c>0$, $f$ is Riemann integrable on $[c,1]$, and if $\lim_{c\to0^+}\int_c^1|f(x)|dx=0$, does it follow that $f$ is Riemann integrable on $[0,1]$? Answer yes or no.

Answer: No


Question: Evaluate the contour integral \( \oint_C \frac{z}{z-1} \, dz \) where \( C \) is a circle with radius 4 centered at the origin.

Answer: 2\pi i


Question: Is it possible to partition the set of natural numbers into finitely many sets such that each set contains an infinite arithmetic progression? Provide a yes or no answer.

Answer: Yes


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

Answer: 0


Question: Evaluate the polynomial \(x^{24} + 16x^{12} + 1\) at all 48th roots of unity. How many distinct values are obtained?

Answer: 4


Question: Given non-negative real numbers $a, b, c$ such that $a^2 + b^2 + c^2 = 1$, determine the maximum value of the expression $2\sqrt{6}ab + 8bc$. Provide your answer as a single number.

Answer: \sqrt{22}


Question: Find the greatest common factor (GCF) of the polynomials \( P(x) = x^{50} - 2x^2 - 1 \) and \( Q(x) = x^{48} - 3x^2 - 4 \).

Answer: x^2 + 1


Question: Find the smallest positive integer value of \(b\) such that there exist perfect squares \(a, b, c\) and a positive integer \(k\) satisfying \(a + 5k = b\) and \(b + 5k = c\).

Answer: 169


Question: Given the differential equations $f''(x) = -f(x)$ and $g'(x) = -f(x)$, where $g(x) = f'(x)$, and the function $F(x) = (f(x/2))^2 + (g(x/2))^2$, find the value of $F(10)$ if $F(5) = 5$. Provide your answer as a single number.

Answer: 5


Question: Evaluate the integral: $$\int_{1.96}^{\infty} e^{-\frac{x^2}{2}} \, dx$$

Answer: 0.0627


Question: Integrate the function \( \frac{e^x - 1}{x} \) and determine the constant \( c \) such that \( F(0) = 0 \).

Answer: 0


Question: Determine the intervals where the Fourier series of the function \(f(x) = \sin(x)\) for \(-2 < x \leq 2\) converges uniformly.

Answer: (-2, 2)


Question: Calculate the size of the stabilizer $|Stab_{S_7}((123)(456))|$ for the permutation $(123)(456)$ under the action of conjugation in the symmetric group $S_7$.

Answer: 18


Question: Determine whether the following statement is true or false: If the series \(\sum_{n\in\mathbb{N}}a_n\overline{b_n}\) converges for all sequences \((a_n)_{n\in\mathbb{N}}\) in the Hilbert space \(l_2(\mathbb{N,\mathbb{C}})\), then \((b_n)_{n\in\mathbb{N}}\) must also be in \(l_2(\mathbb{N,\mathbb{C}})\).

Answer: True


Question: Find the maximum value of the expression \((4a-3b)^2+(5b-4c)^2+(3c-5a)^2\) given that \(a^2+b^2+c^2=1\), where \(a\), \(b\), and \(c\) are real numbers.

Answer: 50


Question: Determine if the equation \( c^2\cos^6{x}+2ce\cos^4{x}+(e^2+a^2)\cos^2{x}-a^2=0 \) can be solved for some coefficients \(a\), \(c\), and \(e\).

Answer: Yes


Question: Let $z_1$ and $z_2$ be two complex numbers. If $\exp(tz_1) = \exp(tz_2)$ for every $t \in \mathbb{R}$, does this imply that $z_1 = z_2$? Provide a justification for your answer.

Answer: z_1 = z_2


Question: Suppose that $\{X_n\}$ are i.i.d. random variables with $\mathbb{E}[X_1] = 0$ and $\operatorname{Var}(X_1) = \sigma^2 \in (0, \infty)$. Define $\bar{X}(n) = \frac{1}{n}(X_1 + \cdots + X_n)$. Determine the probability that $|\bar{X}(n)^{-1}| \geq \epsilon$ for some positive number $\epsilon$ as $n$ approaches infinity.

Answer: 1


Question: Determine whether the decimal number $0.1234567891011121314\ldots$, formed by concatenating consecutive integers, is rational or irrational.

Answer: I


Question: Assume we have a symmetric Laplacian matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$ with $\text{rank}(\mathbf{A}) = n-1$. Construct a matrix $\mathbf{B} \in \mathbb{R}^{n+1\times n}$ by adding a new row to $\mathbf{A}$ such that the sum of its entries is non-zero. Will $\text{rank}(\mathbf{B})$ always be $n$?

Answer: Yes


Question: In triangle $ABC$, points $D$ and $E$ are on side $BC$. Perpendiculars from $D$ and $E$ to $BC$ intersect $BC$ at points $K$ and $L$, respectively. Given that $BK = R$ and $KL = 2R$, and that $DK$ and $EL$ are tangent to a circle centered at $B$, find the length of $BC$. Express your answer in terms of $R$.

Answer: 4R


Question: Evaluate \( \lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^p}{n^{p+1}} \).

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


Question: In a cyclic quadrilateral $PQRS$, the sides are given as $PQ = RS = x$, $QR = 2x$, and $PS = 1$. Determine the value of $x$.

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{x \to 0}{\frac{3^x-7^x+4^x-1}{x^3}} \] without using L'Hospital's rule.

Answer: \infty


Question: Determine the order of the Galois group of the residue field extension $k_i/\mathbf{F}_q$ in the context of the theorem relating the inertia index $f$ to the order of $q$ modulo $p$. Provide your answer as a single integer.

Answer: f


Question: Find the sum of all real solutions for \(x\) in the equation \((x^2 + 4x + 5)^{(x^2 + 4x + 5)^{(x^2 + 4x + 5)}} = 2018.\)

Answer: -4


Question: Evaluate the Taylor series of \(f(t)=\exp(t A)\, B \exp(-t A)\) up to the second-order term, given that \(f(0) = B\), \(f'(0) = [A,B]\), and \([A,[A,B]]=0\).

Answer: B + t [A, B]


Question: Determine the abscissa of convergence, \( \sigma_c \), for the Dirichlet series \( \sum_{n=1}^{\infty} \frac{n^{i \alpha n}}{n^s} \), where \( \alpha \in \mathbb{Z} \).

Answer: 0


Question: Compute the sum of the digits after applying the sum-of-digits operation four times to the number \((3^2 \cdot 5 \cdot 7)^{2023}\).

Answer: 9


Question: Determine if the function \(v(x,y)=\frac{(x^4+y^4)}{x^3+y^3}\) has a partial derivative with respect to \(y\) at the point \((0,0)\).

Answer: 1


Question: Evaluate the limit \( \lim_{n\to\infty} \left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}} \). Use logarithms and the Stolz-Cesàro theorem to find the value of this limit.

Answer: \sqrt{e}


Question: Calculate the entropy of the noisy observations $Y_{1:N}$ of a length-$L$ binary input sequence $X$ when transmitted through $N$ parallel binary symmetric channels (BSCs) with the same crossover probability $p$. Express your answer as $H(Y_{1:N})$.

Answer: N


Question: Find the integer $x$ such that $x^3 \equiv 2 \pmod{59}$.

Answer: 38


Question: Let $G$ be a non-abelian group of order $36$ and $H$ be a subgroup of $G$ with order 4. Determine whether $Z(H) = Z(G)$.

Answer: Z(H) \neq Z(G)


Question: Determine whether there exists an injective but not surjective function \( f: S \rightarrow S \) for an infinite set \( S \). If such a function exists, provide an example for \( S = \mathbb{R} \).

Answer: f(x) = e^x


Question:
Determine the asymptotic behavior of the expression:
\[ \frac{\int_{-n}^{n}ne^{-nx^{2}}dx}{\int_{-n}^{n}e^{-x^{2}}dx} \] as \( n \to \infty \).

Answer: \sqrt{n}


Question: Find the limit of the sequence \( \lim_{n \to \infty} \frac{n}{2^n}\sum_{k=1}^n \frac{2^k}{k} \).

Answer: 2


Question: Determine the genus $g$ of a closed Riemann surface if its tangent bundle $T$ is trivial.

Answer: 1


Question: Calculate the increase in size of a 10mm urchin over 60 days using the Johnson growth model. The differential equation is given by \( \frac{dS}{dt} = k \cdot S_t \cdot dT \cdot (\ln(S_\infty) - \ln(S_t))^2 \), where \( S_t \) is the size on day \( t \), \( k = 2.92 \), and \( S_\infty = 90 \).

Answer: 79.5


Question:
Let $x, y, z > 0$ such that $x + y + z = xy + yz + zx$. Determine the minimum value of the expression:
\[ P = \frac{x^3}{\sqrt{2(y^4+1)}} + \frac{y^3}{\sqrt{2(z^4+1)}} + \frac{z^3}{\sqrt{2(x^4+1)}}. \]

Answer: \dfrac{3}{2}


Question: Determine whether the series \(\sum_{j=n}^{\infty}\frac{1}{j^2}\) is of the order \(O\left(\frac{1}{n}\right)\) as \(n\) tends to infinity, and find the limit \(\lim_{n\to \infty}n\sum_{j=n}^{\infty}\frac{1}{j^2}\).

Answer: 1


Question: If $(g_n)$ is a sequence of functions such that $fg_n$ converges in $L^p$ for all $f \in L^p$, determine whether we can conclude that $(g_n)$ is uniformly bounded in $L^\infty$. Provide a justification for your answer.

Answer: Yes


Question: Consider the sequences $G_n$ and $P_n$ defined as follows: Let $G_0 = 1$, and for $n \geq 0$, $G_{n+1}$ is given by the linear combination of Fibonacci numbers $F_i$ such that $G_{n+1} = F_0G_n + F_1G_{n-1} + \cdots + F_nG_0$, with $F_n$ being the $n$th Fibonacci number where $F_0 = F_1 = 1$ and $F_{n+1} = F_n + F_{n-1}$. Define $P_n$ recursively by $P_0 = P_1 = 1$ and $P_{n+1} = 2P_n + P_{n-1}$ for $n \geq 2$. Determine whether $P_n = G_n$ for all $n$. Provide a proof or counterexample.

Answer: P_n \neq G_n


Question: A circular helix in $xyz$-space is defined by the parametric equations $x(\theta)= 5\cos(\theta)$, $y(\theta)=5\sin(\theta)$, and $z(\theta)=\theta$ for $\theta \in \mathbb{R}$. Let $L(\theta)$ represent the arc length of the helix from the point $P(\theta)=(x(\theta), y(\theta), z(\theta))$ to the point $(5, 0, 0)$, and let $D(\theta)$ denote the distance from $P(\theta)$ to the origin. Given that $L(\theta_0)=26$, find $D(\theta_0)$. Express your answer as a single number.

Answer: \sqrt{51}


Question:
Evaluate the Lebesgue integral \( \int_{[0,1]} f \, d\mu \), where \( d\mu \) is a Borel measure, \( f \) is defined as:
\[ f(x) = \begin{cases} x, & x \in C \\ 0, & x \in [0,1] \setminus C \end{cases} \]
and \( C \) is the Cantor set. Given that \( \mu(C) = 0 \), find the value of the integral.

Answer: 0


Question:
Let $X$ be a compact Hausdorff space, which is therefore normal. Consider a continuous function $f: X \to \mathbb{R}$, a point $x_0 \in X$, an open set $U$ containing $x_0$, and a positive real number $\epsilon$. Determine if there exists a continuous function $g: X \to \mathbb{R}$ such that:
1. $\|g - f\|_{\infty} < \epsilon$.
2. $g|_{U^c} = f|_{U^c}$.
3. $g|_{V} \equiv f(x_0)$ for some open neighborhood $V$ of $x_0$.

Can such a function $g$ be constructed using Urysohn's lemma?

Answer: Yes


Question: Given a cubic Bezier curve defined by the control points $P_0 = (0, 0)$, $P_1 = (0.4, 0)$, $P_2 = (0.2, 1)$, and $P_3 = (1, 1)$, find the time $t$ at which the velocity of a point on the curve is maximum.

Answer: 1


Question: Consider a measure space $(X, \mu)$ with $\mu(X) < +\infty$. Suppose there is a collection $\mathcal{G} \subseteq 2^X$ of subsets, each with positive measure, such that for every set $A \subseteq X$ with positive measure, there exists a subset $G \subseteq A$ that is in $\mathcal{G}$. Determine if it is possible to partition $X$ into countably many sets from $\mathcal{G}$ and one set with measure zero.

Answer: Yes


Question: Suppose $g$ is a real analytic function defined on an open set containing $\mathbb{R}$. If $g(\mathbb{Q}) = \{0\}$, where $\mathbb{Q}$ is the set of rational numbers, does it follow that $g(\mathbb{R}) = \{0\}$? Answer yes or no.

Answer: Yes


Question: Determine the values of $a \in \mathbb{R}$ for which the integral \( \int_0^{\infty} \frac{x^{a-1}}{1+x}\ dx \) converges.

Answer: (0, 1)


Question: Find the smallest integer $m$ such that the dihedral group $D_n$ is a subgroup of the symmetric group $S_m$. Provide your answer as a single integer.

Answer: n


Question:
Given the real numbers $x$ and $y$ that satisfy the following equations:

1. $(x+y)(x+1)(y+1) = 3$
2. $x^3 + y^3 = \frac{45}{8}$

Find the value of $xy$. \( \boxed{} \)

Answer: -\dfrac{1}{2}


Question: Find the value of the limit \( \lim_{x\rightarrow0}\int_{0}^{x^{2}}\frac{\cos(t^{2})}{x \sin x}dt \).

Answer: 1


Question: Given a polynomial $f(x)$ such that $f(0) > 0$ and $f(f(x)) = 4x + 1$ for all $x \in \mathbb{R}$, determine the value of $f(0)$. \( \boxed{} \)

Answer: \dfrac{1}{3}


Question: Find all natural numbers \(n\) such that \(\frac{n^3+7}{2^n}\) is a perfect square.

Answer: 1


Question: Let \(P \in \mathbb{R}[X]\) be a polynomial such that \(P(1) + P(2) + \dots + P(n) = n^5\) for all natural numbers \(n\). Compute \(P\left(\frac{3}{2}\right)\).

Answer: \dfrac{121}{16}


Question: Determine the index of the symmetric stable distribution with the characteristic function \( \phi(t) = e^{-\sqrt{|t|}} \).

Answer: \dfrac{1}{2}


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{1}{\sqrt{n}} \left( \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} \right) \) without using integrals.

Answer: 2


Question: Given a Markov chain with transition matrix \( M \), calculate the probability of transitioning from state 1 to state 3 in exactly \( n \) steps.

Answer: (M^n)_{1,3}


Question: Determine whether the statement \(a^n = \Omega(\ln(n!))\) is true or false for all positive integers \(n\) and positive real numbers \(a\).

Answer: False


Question: Determine the order type of the set \( \omega \times n \) with the lexicographic order, considering the first position first.

Answer: \omega


Question: Let $A=\{x_1,x_2,\ldots,x_{50}\}$ and $B=\{y_1,y_2,\ldots,y_{20}\}$ be two sets of real numbers. Determine the total number of surjective and non-decreasing functions $f:A\rightarrow B$.

Answer: \dbinom{49}{19}


Question: Determine if there exists any integer \( k \geq 4 \) such that the polynomial \( P_k(n) = \sum_{m=1}^n m^k \) is a perfect square of a polynomial in \( n \).

Answer: No


Question: Determine if the equation \( g(x) = \sqrt{3 + \sqrt{3}} \) has multiple solutions, where \( g(x) = \sqrt{x + \sqrt{2x + \sqrt{4x + \sqrt{8x}}}} \).

Answer: 2


Question: Let $u(x,t)$ be a bounded solution of the heat equation \( \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \) with the initial condition \( u(x,0)=\frac{e^{2x}-1}{e^{2x}+1} \) and boundary conditions \( u(0,t)=u(\pi,t)=0 \). Determine \( \lim_{t\to \infty}u(1,t) \).

Answer: 0


Question: A box initially contains 3 large donuts and 7 small donuts. Each hour, a donut is randomly selected: small donuts are consumed, while large ones are halved, with one half eaten and the other half returned as a small donut. What is the expected number of hours until only small donuts remain?

Answer: \dfrac{113}{12}


Question: Determine the fastest-growing function $g(x)$ such that the integral \( \int_N^{\infty}\frac{1}{g(x)}dx \) diverges, where \( N \) is a large number.

Answer: x \log x


Question: Determine whether the set \( \left\{\frac{\varphi(n)}{n} : n \in \mathbb{N} \setminus \mathbb{P}^k \right\} \) is dense in the interval \((0,1)\), where \(\mathbb{P}^k\) denotes the set of powers of prime numbers.

Answer: Yes


Question: Calculate the sum of the series \(\sum_{n=1}^{\infty }\frac{1}{(36n-13)(36n+23)}\) using the digamma function.

Answer: \dfrac{1}{828}


Question: Determine whether the eigenvalues of the product of any number of $n \times n$ matrices remain the same regardless of the order of multiplication. Answer with 'true' or 'false'.

Answer: false


Question: Evaluate the integral \( \int_c^d \frac{dx}{\sqrt{(x-c)(d-x)}} \) using the substitution \( x = c\cos^2t + d\sin^2t \).

Answer: \pi


Question: Express \( S(x) = \frac{a_0}{2} + \sum_{k=1}^\infty a_k \cos(2kx) + b_k \sin(2kx) \) in terms of \( f(x) = \frac{a_0}{2} + \sum_{k=1}^\infty a_k \cos(kx) + b_k \sin(kx) \).

Answer: S(x) = f(2x)


Question: Let $X = Y = \Bbb{N}$, where $\Bbb{N}$ is the set of natural numbers. Define $f: X \to Y$ by $f(x) = 2x$ and define $g: Y \to X$ by $g(y) = 3y$. Let $W = \bigcup_{n=0}^\infty (g(fg))^n(Y - D)$, where $D = f(X)$. $W$ is a subset of $X$. Find an element of $X$ that is not in $W$.

Answer: 2


Question: Evaluate the limit \( \lim_{h \to 0}\frac{f(2h, h)}{h} \) given that \( f \in C^{\infty}(\mathbb{R}^2) \) satisfies \( f(\sin 2t, \sin t) = 0 \) for all \( t \in \mathbb{R} \).

Answer: 0


Question: Determine whether the following statement is true or false: Every polynomial with coefficients in \( \mathbb{Z}_p \) has a solution in \( \mathbb{Q}_p \) if and only if it has a solution in \( \mathbb{Z}_p \), excluding constant polynomials.

Answer: False


Question: Compute the limit of \( \mathbb{P}\{S_n \leq n\} \) as \( n \to \infty \), where \( S_n = X_1 + \ldots + X_n \) and each \( X_i \sim \text{Poisson}(1) \) are independent and identically distributed random variables.

Answer: \dfrac{1}{2}


Question: Given the integral \( \int_{0}^{1} (2x^2-3f(x))f(x)\,dx=\frac{1}{15} \), find the value of \( \int_{0}^{1} f(x)\,dx \).

Answer: \dfrac{1}{9}


Question: Evaluate the sum \(\sum\limits_{k|(2p, 2q), k\nmid q} \varphi(k)\) where \(p\) and \(q\) are distinct odd primes.

Answer: 1


Question: Evaluate the sum \( \sum_{r=1}^{n} r\cdot (r-1)\cdot \binom{n}{r} \) using combinatorial methods.

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


Question: Suppose $f$ is a continuous, periodic function. If $f(x_0) > 0$ for some $x_0$, and the local maxima of $f$ do not decrease as $x \to \infty$, does $\int_{a}^{\infty} f(x)\,dx$ diverge? Answer yes or no.

Answer: Yes


Question: Determine whether there exists a solution to the equation \((a_0 + a_1\zeta + a_2\zeta^2 + a_3\zeta^3)^2 = 2\) in the field \(\mathbb{F}_3(\zeta)\), where \(\zeta\) is an imaginary solution of \(x^5 - 1 = 0\).

Answer: Yes


Question:
Determine whether it is possible to construct two disjoint subsets $A$ and $B$ of the real numbers $\mathbb{R}$ such that:

1. $A \cap B = \emptyset$ (Disjointness)
2. $A \cup B = \mathbb{R}$ (Exhaustiveness)
3. For any $x, y \in A$ with $x < y$, there exists $z \in B$ such that $x < z < y$, and vice versa for $B$.
4. $|A| = |B|$, ensuring the existence of a bijection $h: A \rightarrow B$.

Is it possible to construct such subsets $A$ and $B$ for $U = \mathbb{R}$?

Answer: Yes


Question: Calculate the sum of the series \( \sum\limits_{b=0}^\infty (b+k+1)2^{-b-k-1}\binom{b+k}{k} \).

Answer: 2(k + 1)


Question: Given that $a, b, c, d$ are real numbers satisfying the equation $a^2 + b^2 + c^2 + d^2 + a + 3b + 5c + 7d = 4$, determine the maximum value of $a + b + c + d$. Provide your answer as a single number.

Answer: 2


Question: Find a closed form expression for the series: \[ F(s) = \sum_{n>0} \frac{s^n}{n} \] for \(|s|<1\).

Answer: -\ln(1 - s)


Question: Let $P_1$, $P_2$, ..., $P_n$ be points on a circle of radius 1, centered at $O$. Define $G$ as the point such that $\overrightarrow{OG} = \frac{\overrightarrow{OP_1} + \overrightarrow{OP_2} + \dots + \overrightarrow{OP_n}}{n}$ and let $d = OG$. Express $\sum_{1 \le i < j \le n} (P_i P_j)^2$ in terms of $n$ and $d$. Provide your answer in terms of $n$ and $d$. \( \boxed{} \)

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


Question:
Evaluate the limit \( \lim_{n \to \infty} \int_0^1 f_n(x) \, dx \), where the function \( f_n \) is defined on the interval [0,1] as follows:

\[ f_n(x) = \begin{cases}
      0 & \text{if } x = 0 \\
      \sin\left(\frac{\pi}{2n}\right) & \text{if } x \in \left(0, \frac{1}{n}\right] \\
      \sin\left(\frac{2\pi}{2n}\right) & \text{if } x \in \left(\frac{1}{n}, \frac{2}{n}\right] \\
      \sin\left(\frac{3\pi}{2n}\right) & \text{if } x \in \left(\frac{2}{n}, \frac{3}{n}\right] \\
      & \vdots \\
      \sin\left(\frac{n\pi}{2n}\right) & \text{if } x \in \left(\frac{n-1}{n}, 1\right]
   \end{cases} \]

Discuss the connection between this limit and the Riemann integral.

Answer: \dfrac{2}{\pi}


Question: Find the sum of the fourth powers of the roots of the polynomial equation \( x^4 + x + 1 = 0 \).

Answer: -4


Question: Find all functions \( g: \mathbb{R} \to \mathbb{R} \) such that \( |g(x+y) - g(x-y)| \leq 2y^2 \) for all \( x, y \in \mathbb{R} \).

Answer: g(x) = c


Question: Evaluate the integral \( \frac{1}{2 \pi i} \oint_{C} \frac{z^{2} d z}{z^{2}+4} \) where \( C \) is the square with vertices at \( \pm2, \pm2+4i \), using the Cauchy integral theorem.

Answer: i


Question: Determine the number of elements in the quotient ring $\mathbb{Z}[i]/\langle 2-i \rangle$, where $\mathbb{Z}[i]$ is the ring of Gaussian integers and $\langle 2-i \rangle$ is the ideal generated by $2-i$. Provide your answer as a single integer.

Answer: 5


Question: Determine whether every infinite set contains a countably infinite subset.

Answer: Yes


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

Answer: 1


Question: Evaluate the limit: \[ \lim\limits_{x \to \infty} \left( \sqrt{\ln(x)} - \sqrt{\ln(x)-\ln(\ln(x))} \right) \]

Answer: 0


Question: Calculate the variance of the minimum of 6 randomly chosen variables from a set of 10 exponentially distributed random variables, each with a rate parameter of 2.

Answer: \dfrac{1}{144}


Question: Find the limit of the following function as \( x \) approaches infinity: \[ \lim_{x \to \infty} \frac{\frac{1}{2}\pi - \arctan(x)}{\ln\left(1 + \frac{1}{x^2}\right)} \]

Answer: \infty


Question: Evaluate the integral \( \int_0^{\infty}\frac{x}{x^4+1}dx \) using complex analysis techniques, such as the residue theorem.

Answer: \dfrac{\pi}{4}


Question: Determine whether the limit \( \lim_{{n\to\infty}} \frac{1+\sqrt[2]{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n} \) converges to 1.

Answer: 1


Question: Let $D$ be a finite central division algebra over its center $Z$, and $F$ be a maximal subfield of $D$. Is the dimension of $F$ over $Z$ an invariant?

Answer: Yes


Question: Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth function and $M$ be a smooth manifold of $\mathbb{R}^n$. Assume that $Df(x)v \neq 0$ for all $v$ being tangent to $M$ at $x$ and for all $x$ in $M$. Is $f$ locally injective on $M$?

Answer: Yes


Question: Let $p(x)$ be a polynomial with integer coefficients such that $p(m) - p(n)$ divides $m^2 - n^2$ for all integers $m$ and $n$. Given that $p(0) = 1$ and $p(1) = 2$, determine the largest possible value of $p(100)$. Provide your answer as a single integer.

Answer: 10001


Question: Calculate the line integral \( \int_\beta F \cdot n \, ds \) for the vector field \( F = \frac{xi + yj}{x^2+y^2} \), where \( \beta \) is a simple closed curve in a plane that encloses the origin. Use Green's theorem and parameterization to find the result.

Answer: 2\pi


Question: A fair die is rolled 15 times. Let $A$ be the number of times a five is rolled, and $B$ be the number of times a two is rolled. Calculate the covariance, $\operatorname{Cov}(A,B)$.

Answer: -\dfrac{5}{12}


Question: Determine whether there always exists an irreducible polynomial of degree $d$ over the finite field $\mathbb{F}_p$, where $p$ is a prime number and $d$ is a positive integer.

Answer: Yes


Question: Determine the values of \(\gamma > 0\) for which the sequence \(n^{\gamma} (\sqrt[n]{n} - 1)^2\) converges.

Answer: (0, 2)


Question: Evaluate the improper integral \( \int_{1}^{\infty}\frac{x^3}{e^x}dx \).

Answer: \dfrac{16}{e}


Question: Evaluate the limit: $$\lim\limits_{(x,y) \to (0,0)} \frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4}.$$ Use polar coordinates by setting $x = r\cos\theta$, $y = r\sin\theta$, and $r = \sqrt{x^2+y^2}$. Determine the limit from the expression $$\lim\limits_{r \to 0} \frac{e^{-\frac{1}{r^2}}}{r^4\cos^4\theta + r^4\sin^4\theta}.$$

Answer: 0


Question: Calculate the flux of the vector field \( \vec{F}=z\hat{i}+x\hat{j}+y\hat{k} \) through the surface of a sphere of radius \( a \) centered at the origin using Stokes' theorem.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1^{1/5} + 2^{1/5} + \cdots + n^{1/5}}{n \cdot n^{1/5}} \]

Answer: \dfrac{5}{6}


Question: Calculate the residue of the function \( f(z) = \frac{z - \sin z}{z} \) at \( z = 0 \).

Answer: 0


Question: Given the matrix \(A = \begin{bmatrix} 1 & 3 & 4\\ 3 & 6 & 9\\ 1 & 6 & 4 \end{bmatrix},\) where \(B\) is an invertible \(3\times 3\) matrix that satisfies \(A \cdot A^{T} \cdot A +3B^{-1} =0,\) determine the value of \(\det( \operatorname{adj} (A^{-1}(B^{-1}){2B^{T}})).\)

Answer: \dfrac{64}{81}


Question: For positive random variables $(X_1, Y_1)$ and $(X_2, Y_2)$, suppose that $(X_1, Y_1)$ and $(X_2, Y_2)$ have the same distribution and are independent. Also, suppose that $E[Y_1|X_1] = \theta X_1$. Let $Z=\frac{Y_1 + Y_2}{X_1+X_2}$. Find $E[Z]$. \( \theta \) is a constant.

Answer: \theta


Question: Calculate the sum of the infinite series: \[ S = 1 + \frac{1+2}{2!} + \frac{1+2+3}{3!} + \ldots. \]

Answer: \dfrac{3e}{2}


Question: A puzzle club meets in a schoolroom every Friday after school. Last Friday, a member said, "I’ve hidden a list of numbers in this envelope that add up to the number of this room." A girl asked, "If you told us the number of numbers in the envelope and their product, would that be enough to determine the room number?" He replied, "No." She then said, "Well, at least I’ve worked out their product." What is the number of the school room they meet in?

Answer: 4


Question: Determine if there exists a positive integer $n$ such that $\varphi(n)$ is a power of $3$, where $\varphi(n)$ denotes the number of positive integers less than $n$ that are relatively prime to $n$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of distinct similarity classes of $9 \times 9$ matrices with real entries that satisfy $A^2 = I$. Consider the minimal polynomial and the characteristic polynomial of $A$ in your solution.

Answer: 10


Question: Find the coordinates of the point on the curve \(xy(x + y) = 2a^3\), where \(a\) is a non-zero constant, such that the tangent at this point is parallel to the x-axis.

Answer: (a, -2a)


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

Answer: 2


Question: Given that \(-6 = \det\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}\), find the value of \(x\) where \(x = \det\begin{bmatrix} a & b & c \\ 2d & 2e & 2f \\ g+3a & h+3b & i+3c \\ \end{bmatrix}\).

Answer: -12


Question:
Determine the radius of convergence of the power series \( \sum_{n=0}^\infty a_n x^n \), where the coefficients \( a_n \) are defined by the recurrence relation:
\[
\begin{aligned}
a_{n+2} &= \frac{n(n+1) a_{n+1} - a_n}{(n+2)(n+1)}, \quad n\ge 1, \\
a_2 &= -\frac{a_0}{2}
\end{aligned}
\]
with arbitrary initial values \( a_0 \) and \( a_1 \).

Answer: 1


Question: Given an exponential distribution with the joint probability density function of the order statistics \(f_{x_{(1)}x_{(2)}x_{(3)}}(y_1,y_2,y_3) = 3!e^{-(y_1 +y_2 + y_3)}\), find the probability \(\Pr(X_1 < X_2 < X_3)\).

Answer: \dfrac{1}{6}


Question: Given a random sample $X_1, X_2, \ldots, X_n$, where $n > 2$, drawn from a binomial distribution $b(1, \theta)$, it is known that $Y_1 = X_1 + X_2 + \cdots + X_n$ is a complete sufficient statistic for $\theta$, and $Y_2 = \frac{X_1 + X_2}{2}$ is an unbiased estimator of $\theta$. Find the conditional expectation $E(Y_2 \mid Y_1 = y_1)$, where $y_1$ is a value of $Y_1$. Provide your answer in terms of $y_1$ and $n$. \(\theta\) is the parameter of the binomial distribution.

Answer: \dfrac{y_1}{n}


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

Answer: \dfrac{e}{2}


Question: Determine whether there exists an element of infinite order in the group \( \{ A \in GL_2(\mathbb{R}) : A^T = A^{-1} \} \), which consists of real invertible matrices whose transpose is their inverse.

Answer: Yes


Question: Determine whether the union $\bigcup_{n=1}^{\infty} I_n$ is open, closed, or neither in the complex plane, where $I_n = [0,a_n]$ with $a_n=\cos \left(\frac{\pi}{2n}\right) + i \sin \left(\frac{\pi}{2n}\right)$. Each $I_n$ represents a line segment connecting the origin (0) to the point $a_n$. Provide your answer as "open", "closed", or "neither".

Answer: neither


Question: Evaluate the limit: $$\lim_{n\to \infty}(\sin\sqrt{n+1} - \sin\sqrt{n}) .$$

Answer: 0


Question: Compute the derivative of \( \text{tr}(AXB) \) with respect to the matrix \( X \), where \( A, B \in \mathbb{R}^{2 \times 2} \) and \( X \in \mathbb{R}^{2 \times 2} \).

Answer: A^\top B^\top


Question: Assume $D \subseteq \mathbb{C}$ is a connected open set and $f: D \to \mathbb{C}$ is an analytic function. If there exists a continuous path $\gamma \subset D$ such that $f(z) = 0$ for all $z \in \gamma$, does it follow that $f(z) = 0$ for all $z \in D$? Answer yes or no.

Answer: No


Question: Determine the range of values of the constant \( k \) such that for all non-negative \( x \) and \( y \), the inequality \( \sqrt{\frac{2}{\pi}} \cdot (x+y)^2 \leq x^2 + y^2 + 2cxy + k \leq (x+y)^2 \) holds, where \( c = \sqrt{\frac{2}{\pi}} \).

Answer: 0


Question: Evaluate the integral \( \int_0^{\frac{\pi}{3}} \sin x\ \mathrm{d}x \) using Riemann sums. Use the identity \( \sum_{i = 1}^n \sin (i\theta) = \frac{\sin \frac{(n + 1)\theta}{2} \sin \frac{n\theta}{2}}{\sin \frac{\theta}{2}} \) and explain why setting \( \theta = \frac{\pi}{3n} \) is valid.

Answer: \dfrac{1}{2}


Question: Let $P$ be a torsion group where every element has an order that is a power of a prime $p$, i.e., a torsion $p$-group. Suppose $P$ acts on a finite set $X$. Is it true that every orbit of this action has a length that is a power of $p$?

Answer: Yes


Question: Determine the radius of convergence for the complex power series \(\sum_{n=1}^\infty \frac{n^2}{\ln n}z^n\) using the root or ratio test.

Answer: 1


Question: Let $m \geq 3$ be an integer and $\zeta = \exp(-2\pi i/m)$ be the $m$-th primitive root of unity. Consider the cyclotomic field $K = \mathbb{Q}(\zeta)$ and a cyclic subgroup $G = \langle\sigma\rangle \leq \text{Gal}(K/\mathbb{Q}) \cong \mathbb{Z}^\times$. When $\sigma(\zeta) = \zeta^{-1}$, the invariant subfield is $F = \mathbb{Q}(\zeta + \zeta^{-1})$. Given a prime number $p \in \mathbb{Z}_m^\times$ of order $d$ (i.e., the smallest $d > 0$ with $p^d \equiv 1 \pmod m$), suppose we have $G$ such that $\sigma(\zeta) = \zeta^p$. Determine if the invariant subfield $F = \mathbb{Q}(\xi)$, where $\xi = \zeta + \zeta^p + \dots + \zeta^{p^{d-1}}$, corresponds to $G$. Justify your answer.

Answer: \mathbb{Q}(\xi)


Question: Let $C = H \cap S$, where $S$ is a surface of general type canonically embedded in $\mathbb{P}^N$ for some $N > 0$, and $H$ is the divisor of a hyperplane in $\mathbb{P}^N$. Determine the degree of the divisor $H_{|C}$ on the curve $C$.

Answer: K_S^2


Question: Find the sixth derivative of the function \( \frac{\cos{(5x^2)}-1}{x^2} \) evaluated at \( x = 0 \).

Answer: 18750


Question:
Determine whether the following proof is valid:

Proof: Let $\{A_\lambda\}_{\lambda \in L}$ be an arbitrary collection of disjoint non-empty open subsets of $\mathbb{R}$. Since every non-empty open subset of $\mathbb{R}$ can be written as a countable union of disjoint open intervals, we can take the union $A = \bigcup\limits_{\lambda \in L}A_\lambda$ and decompose it $A = \bigcup\limits_{n \in \mathbb{N}} I_n$ in disjoint open intervals which forms a countable collection. We can also decompose each $A_\lambda$ as $\bigcup\limits_{m \in \mathbb{N}}J_{\lambda,m}$. For $\lambda \neq \mu \in L$, $A_\lambda \cap A_\mu = \emptyset$ and this is a new representation of $A$:

$$A = \bigcup_{n \in \mathbb{N}} I_n = \bigcup_{\substack{\lambda \in L \\ m \in \mathbb{N}}} J_{\lambda,m}$$

No matter how complicated the union over $L$ is, the $J_{\lambda,m}$ are disjoint open intervals. Thus, by the uniqueness, the two collections are exactly the same. As the final argument, we produce an injection $\varphi:\{A_\lambda\} \mapsto \{J_{\lambda,m}\}$ picking for each $A_\lambda$ some $J_{\lambda,m}$.

Is this proof valid?

Answer: Valid


Question:
Given the series:
\[ a = 1 + \frac{x^{3}}{3!} + \frac{x^{6}}{6!} + \cdots \]
\[ b = \frac{x}{1!} + \frac{x^{4}}{4!} + \frac{x^{7}}{7!} + \cdots \]
\[ c = \frac{x^{2}}{2!} + \frac{x^{5}}{5!} + \frac{x^{8}}{8!} + \cdots \]
Let \( u = a^2 + b^2 + c^2 \). Find the value of \( u(\ln 2) \).

Answer: \dfrac{5}{3}


Question: Solve the equation: \[ 5^{\frac{x}{2}} - 2^x = 1 \]

Answer: 2


Question: Let \( G \) be a finite abelian group. Determine whether it is true that for any integer \( n \) coprime to the order of \( G \), the set \( \{ng : g \in G \} = G \).

Answer: True


Question: Determine the Shnirelman density of the set \( A = \{ \lfloor n \ln (n) \rfloor : n = 1,2,3,... \} \). The Shnirelman density is defined as \( \inf \left\{ \frac{ | A \cap \{1,\ldots,n\}|}{n} : n \in \mathbb{N} \right\} \).

Answer: 0


Question: Evaluate the triple integral \( \iiint \frac{dx\,dy\,dz}{\sqrt{1 - \left(\frac{x^2}{9}+\frac{y^2}{16}+\frac{z^2}{25}\right)}} \) over the region inside the ellipsoid defined by \( \frac{x^2}{9}+\frac{y^2}{16}+\frac{z^2}{25}=1 \).

Answer: 60\pi^2


Question: Evaluate the integral \( \int_0^{\pi/2} (\cos^{10} x)(\sin 12x) \, dx \).

Answer: \dfrac{1}{11}


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{2x+x\cos(x)-3\sin(x)}{x^4\sin(x)}$$ without using L'Hôpital's rule.

Answer: \dfrac{1}{60}


Question: Evaluate the integral \( \oint_{S^1} dx \) where \( S^1 \) is a circle in the plane oriented counterclockwise.

Answer: 0


Question: Find the polynomial of minimal degree for which a primitive 3rd root of unity, \( \omega \), is a root.

Answer: x^2 + x + 1