Question: Determine the necessary and sufficient condition for the product measures \( \mu_1 \otimes \mu_2 \) and \( \mu_2 \otimes \mu_1 \) to be equal, given that \( \mu_1 \) and \( \mu_2 \) are measures on the measurable space \((E, \mathcal{P}(E))\) with \( E = \{0, 1\} \) and \( \mu_i(E) = 1 \) for \( i = 1, 2 \), where \( \mu_1(\{0\}) = p \) and \( \mu_2(\{0\}) = q \) with \( p, q \in [0, 1] \).

Answer: p = q


Question: Determine the value of \( k \) such that the number of monotonically increasing functions from the set \( \{1,2,3,4,5,6\} \) to itself, with the property that \( f(x) \ge x \) for all \( x \), is equal to \( \frac{2}{k} \cdot \binom{11}{5} \).

Answer: 7


Question: Let $f$ be a twice differentiable function with $f(0) = 0$, $f(1) = 1$, and $f'(0) = f'(1) = 0$. Prove that there exists some $x \in [0,1]$ such that $|f''(x)| \geq 4$.

Answer: 4


Question: If $F$ is a connected topological field, must $F$ also be path-connected?

Answer: Yes


Question:
Find the largest value of the series function:
\[ F(l, m) = \frac{(-1)^m}{2^{2l+m}}\sum_{j=0}^l C_l^jC_{m+l}^j(-3)^j, \]
where \( l, m \in \mathbb{N}_+ \) are positive integers and \( C_l^j \) is the binomial coefficient.

Answer: \dfrac{5}{8}


Question: Consider a totally ordered set \( \langle A, \leq_A \rangle \) with the property that for each \( a \in A \), the set \( \{ b \in A \mid b \leq_A a \} \) is finite. Additionally, \( \langle A, \leq_A \rangle \) does not have a maximal element. Determine if \( \langle A, \leq_A \rangle \) must have a minimum element strictly due to the property that \( \{ b \in A \mid b \leq_A a \} \) is finite for each \( a \in A \).

Answer: Yes


Question:
Given two sequences of random variables $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$, where:
1) $X_n \to 0$ almost surely as $n \to \infty$.
2) For all $n \in \mathbb{N}$, $X_n$ and $Y_n$ are equal in distribution: $X_n \stackrel{d}{=} Y_n$.

Does it follow that $Y_n \to 0$ almost surely as $n \to \infty$?

Answer: No


Question: Determine the value of \( r \) such that for a simple random walk defined by independent random variables \( Z_1, Z_2, \ldots, Z_n \), where each \( Z_j \) is either 1 or -1 with equal probability, the probability that the sum \( S_n = \sum_{j=1}^n Z_j \) satisfies \(|S_n| \leq r\) is 0.5.

Answer: 0.6745 \sqrt{n}


Question: Determine whether there exists an integer N such that the number of square-free numbers with exactly three prime divisors (3-primes) less than N is at least twice the number of square-free numbers with exactly two prime divisors (2-primes) less than N.

Answer: Yes


Question: Determine the maximum number of points that can be placed in \(\mathbb{R}^n\) such that no \(n+1\) points lie in the same \(\mathbb{R}^{n-1}\)-plane and the Euclidean distance between every pair of points is an integer.

Answer: n + 1


Question: Let $G$ and $H$ be countable abelian groups. Assume that for every prime number $p$, there is an isomorphism $G \otimes_{\mathbb{Z}} \mathbb{Z}[\frac{1}{p}] \cong H \otimes_{\mathbb{Z}} \mathbb{Z}[\frac{1}{p}]$. Does it follow that $G$ and $H$ are isomorphic as abelian groups?

Answer: No


Question: Determine the set of values for \(x\) such that the functions \(f(x) = \sin^{-1}(\sin(\tan x))\) and \(g(x) = \cos^{-1}(\sin(\sqrt{1-\tan^2 x}))\) are the same. Find the set of \(x\) values for which this holds true.

Answer: \emptyset


Question: Determine the values of \( n \in (1, \infty) \) for which the inequality \( x^n \leq n^x \) holds for all \( x \in (1, \infty) \).

Answer: e


Question: Consider the operator \( S: l^2 \to l^2 \) defined by \((Sa)_n = \left(\frac{3}{5}\right)^n a_n\) for all \( n \in \mathbb{N} \) and \( a_n \in \mathbb{R} \). Show that \( S \) is a bounded linear operator on \( l^2 \) and find its operator norm. Determine whether \( S \) is invertible.

Answer: \dfrac{3}{5}


Question: Let \( L^2([0, 1]) \) be the Hilbert space of all real-valued square-integrable functions on \([0, 1]\) with the usual inner product. Define the linear functional \( \phi \) on \( L^2([0, 1]) \) by \[ \phi(f) = \int_{\frac{1}{4}}^{\frac{3}{4}} 3\sqrt{2} f \, d\mu \] where \( \mu \) denotes the Lebesgue measure on \([0, 1]\). Find the norm \( \| \phi \| \) of the linear functional \( \phi \).

Answer: 3


Question: Determine the values of $p$ for which the series \( \sum_{n=2}^\infty \frac{1}{n(\ln n)^p} \) is convergent.

Answer: p > 1


Question: Is a bounded convex function on an open interval \((a, b)\) uniformly continuous?

Answer: Yes


Question: Let \( S \) be the set of all 12-digit positive integers where each digit is either 1, 4, or 7. What is the probability that a randomly selected number from \( S \) is divisible by 12?

Answer: \dfrac{1}{9}


Question:
Compute the product \(\mathbf{ab}\) in the Clifford Algebra \(\mathcal{Cl}_2\), where \(\mathbf{a}=e_2+e_{12}\) and \(\mathbf{b}=(1/2)(1+e_1)\). Use the basis \((1, e_1, e_2, e_{12})\) and the multiplication table provided:

\[
\begin{array}{cccc}
  & \mathbf{e_1} & \mathbf{e_2} & \mathbf{e_{12}} \\\\
  \mathbf{e_1}& 1 & e_{12} & e_2 \\\\
  \mathbf{e_2}& -e_{12} & 1 & -e_1 \\\\
  \mathbf{e_{12}}& -e_2 & e_1 & -1
\end{array}
\]

Answer: 0


Question: Let $X \sim \mathsf{Gamma}(2,3)$ and define $Y = 2X$. Find the probability density function (pdf) of $Y$ and evaluate it at $y = 13.5$. Provide your answer as a numerical value.

Answer: 0.0395


Question: Find the least possible degree of the polynomial \( f(x) \) in the expression \( \frac{d^{100}}{dx^{100}}\left(\frac{p(x)}{x^3-x}\right) = \frac{f(x)}{g(x)} \), where \( p(x) \) is a polynomial that does not have the factor \( x^3-x \).

Answer: 200


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

Answer: ab


Question: Evaluate the sum \( \sum_{n=1}^{\infty}\frac{1}{n!(n+2)} \).

Answer: \dfrac{1}{2}


Question: Find the average z-coordinate of all points on and within a hemisphere of radius 4 centered at the origin, with its base in the xy-plane.

Answer: \dfrac{3}{2}


Question: Use the epsilon-delta definition of limit to prove that \( \lim_{x \to 1} \frac{1}{2+\sqrt{x}} = \frac{1}{3} \).

Answer: \frac{1}{3}


Question: Prove that there are no non-trivial integer solutions to the equation \(a^2 + 3b^2 = 2c^2\).

Answer: There are no non-trivial integer solutions to the equation \(a^2 + 3b^2 = 2c^2\).


Question: Consider the Hilbert space \( l_2 \) with the inner product \( \langle x, y \rangle = \sum x_i y_i \). Let \( S = \{ x \in l_2 : \sum \frac{x_i}{n} = 0 \} \). Determine the number of interior points of the set \( S \).

Answer: 0


Question: Find the limit of \( \frac{3^{3n+3}+n^3 9^{9/n}}{9^n+n^9 3^{3/n}} \) as \( n \to \infty \).

Answer: \infty


Question: Suppose \(X\) and \(Y\) are independent Poisson random variables, each with mean 1. Find \(E[(X + Y)^2]\).

Answer: 6


Question:
Given the equation \( x^2 + \frac{x^2}{(x-1)^2} = 2010 \), find the value of \( \frac{x^2}{x-1} \). Choose from the following options:

(A) \( 1+\sqrt{2011} \)
(B) \( 1-\sqrt{2011} \)
(C) \( 1\pm\sqrt{2011} \)
(D) \( \sqrt{2011} \)

Answer: C


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

Answer: 6


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{\sin \left\{ \frac{2}{n} \right\}}{\left[ 2n \tan \frac{1}{n} \right] \left( \tan \frac{1}{n} \right)} + \frac{1}{n^2 + \cos n} \right)^{n^2}, \] where \([.]\) denotes the greatest integer function and \(\{.\}\) denotes the fractional part function.

Answer: 1


Question:
Let $\Omega = D(0,2) \setminus \{\frac{1}{2}\}$, where $D(0,2)$ is a disc, and let $f$ be holomorphic in $\Omega$. Given that $\frac{1}{2}$ is a simple pole for $f$ with residue $1$, calculate the limit:
\[ \lim_{n \to \infty} \frac{f^{(n)}(0)}{2^{n}n!} \]

Answer: -2


Question: Calculate the variance of the random variable $Z|Z|$, where $Z \sim N(0,1)$. Express your answer in terms of expectations: $\mathrm{Var} (Z|Z|) = \mathrm{E}(Z^4) - \mathrm{E}^2(Z|Z|)$. Compute this variance.

Answer: 3


Question: Simplify the expression \( \frac{1}{\pi i}\left[\frac{1}{2}\log[e^{-2ix}]-\log[e^{-ix}]\right] \) and determine its value as an integer multiple of \( \pi \), where \( x \) is a real number.

Answer: 0


Question: Evaluate the triple integral \( \int\int\int_{D} xy \, dV \), where \( D \) is the region defined by \( 0 < y < x \) and \( |z| < 1 - x \).

Answer: \dfrac{1}{20}


Question: Evaluate the limit without using L'Hôpital's Rule: \[ \lim_{ x \to 1}\frac{\sqrt[n]{x^n-1}}{\sqrt[n]{nx}-\sqrt[n]{n}-\sqrt[n]{nx-n}} \]

Answer: -1


Question: Prove that \( \sum_{i=0}^n {2n \choose 2i} = 2^{2n-1} \).

Answer: 2^{2n-1}


Question: Find the values of the real number $m$ such that the equations $x^3 - mx^2 - 4 = 0$ and $x^3 + mx + 2 = 0$ have a common root.

Answer: -3


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\ln\sin^2(ax)}{\ln\sin^2(bx)} \] without using L'Hopital's rule or Taylor series, where \( a \) and \( b \) are constants.

Answer: 1


Question: Find all natural numbers \( n > 0 \) such that \( n^3 - 6n^2 + 10n - 5 \) is a perfect square.

Answer: 1


Question: Determine the Galois group of the polynomial \( f(x) = x^4 + 4x - 1 \) over \( \mathbb{Q} \).

Answer: D_4


Question:
Use an appropriate change of coordinates to find the exact value of the integral:
\[ \int_{-\sqrt{3}}^{\sqrt{3}}\int_{-\sqrt{3-x^2}}^{\sqrt{3-x^2}}\int_{-3+x^2+y^2}^{3-x^2-y^2}x^2\,dz\,dy\,dx \]

Answer: \dfrac{9\pi}{2}


Question: Does the Central Limit Theorem fail to provide a good approximation for the Cauchy distribution due to its undefined mean and variance?

Answer: Yes


Question: Find the expression for $c$ given that $c=f(a+e^b)+g(a-e^b)$, where $f$ and $g$ are functions of $a+b^2$ and $a-b^2$ respectively. Determine $c$ such that when $b=0$, $c=0$ and $\frac{\partial c}{\partial b}=1+a$.

Answer: (a + 1)(e^b - 1)


Question: What is the maximal number of elements in a sigma-algebra generated by a collection of N subsets of a set X?

Answer: 2^{2^N}


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\frac{n!\cdot e^n}{\sqrt{n}\cdot n^n} \] without using Stirling's approximation.

Answer: \sqrt{2\pi}


Question: Let $(s_n)$ be a sequence such that $s_1, s_2, \ldots, s_{100}$ are positive and $s_n < 0$ for all $n > 100$. Assume $\lim_{n \to \infty} s_n = 0$. Determine $\lim_{n \to \infty} \frac{1}{s_n}$.

Answer: -\infty


Question: Identify the group generated by $a, b, c$ with the relations $a^2=1, b^2=1, c^2=1, ac=ca, bc=cb, abab=baba$. Verify if this group has order 16.

Answer: 16


Question: Evaluate the integral \( \int_{-4i}^{4i} \frac{1}{z^2} \, dz \) along any contour that does not pass through the origin.

Answer: \dfrac{i}{2}


Question: Determine the interval \( I \subset \mathbb{R} \) for which the function \( f(x) = \frac{\ln(t)}{1+t^{x}} \) is differentiable with respect to \( x \).

Answer: (-\infty, \infty)


Question: Given that \( \int_{0}^{\infty} \left(\frac{\sin x}{x}\right)^3 \, dx = A \), find the value of \( k \) such that \( \int_{0}^{\infty} \frac{x - \sin x}{x^3} \, dx = kA \).

Answer: \dfrac{2}{3}


Question: Determine whether $RP^3$ minus a point deformation retracts to a space homeomorphic to $RP^2$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the total number of ways to distribute 10 different red balls and 15 identical black balls into 4 boxes such that each box contains at least one red ball, and the fourth box contains at least 6 black balls.

Answer: 180074400


Question: Determine the smallest integer $N$ such that among the $\binom{N}{3}$ circles formed by every triple of $N$ points in general position in 3-D space, there exists at least one pair of linked circles. Note that circles sharing one of the $N$ points do not count as linked.

Answer: 6


Question:
Determine whether the following statement is true:

Suppose $0 \le a_n \le 1$ for all $n \ge 0$ and define $a_{-1} = 0$. Assume that $\sum\limits_{n=0}^{\infty} a_n s^n$ is finite for all $0 < s < 1$, but $\sum\limits_{n=0}^{\infty} a_n$ diverges. Is it true that
\[ \lim_{s \to 1^{-}} \sum_{n=0}^{\infty} (a_n - a_{n-1}) s^n = \lim_{m \to \infty} \frac{1}{m+1} \sum_{n=0}^{m} a_n \]
whenever the right-hand side limit exists?

Answer: True


Question: Is it possible to have a $3 \times 3$ matrix with all eigenvalues equal to zero that is not triangular?

Answer: Yes


Question: Determine if the solution B = 0 is correct for the equation \( \frac{\partial B^T \Lambda B}{\partial B} = 0 \), where \( \Lambda = \Sigma^{-1} \) and \( B = x - \mu \).

Answer: B = 0


Question: Determine whether the following statement is true: In a Banach space $V$, for a sequence of subsets $\{B_n\}_{n \in \mathbb{N}}$, is it true that \( \bigcap_{n=1}^\infty \overline{\operatorname{span} B_n} \subset \overline{\operatorname{span} \bigcap_{n=1}^\infty B_n} \)?

Answer: False


Question: Let \( \alpha \) be an element in an algebraic closure of \( GF(64) \) such that \( \alpha^4 = \alpha + 1 \). Determine the value of \( r \in \mathbb{N} \) for which the field extension \( GF(64)(\alpha) \) is isomorphic to \( GF(2^r) \).

Answer: 12


Question: Determine the value of $L(r,n)$ for all integers $r, n \geq 1$, where $L(r,n)$ is defined as the smallest integer such that for any sequence of length $L(r,n)$ and any $r$-coloring of the sequence, there exists a monochromatic increasing or decreasing subsequence of length $n+1$. Additionally, there exists a sequence of length $L(r,n)-1$ and an $r$-coloring of that sequence which has no monochromatic increasing or decreasing subsequence of length $n+1$. It is given that $L(1,n) = n^2 + 1$ for all $n$. Find $L(r,n)$.

Answer: r n^2 + 1


Question: Determine the number of transitive $G$-sets of the symmetric group $S_3$ up to isomorphism.

Answer: 4


Question:
Let the ellipse be defined by \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) and let \(AB\) be a chord of the ellipse. The elliptical angle of point \(A\) is \(\alpha\) and the elliptical angle of point \(B\) is \(\beta\). The chord \(AB\) intersects the major axis at point \(C\), and the distance from the center of the ellipse to \(C\) is \(d\). Determine the value of \(\tan \frac{\alpha}{2} \tan \frac{\beta}{2}\). Choose the correct option:

(A) \(\frac{d-a}{d+a}\)
(B) \(\frac{d+a}{d-a}\)
(C) \(\frac{d-2a}{d+2a}\)
(D) \(\frac{d+2a}{d-2a}\)

Answer: A


Question: Given that \( f(x) f(1/x) = f(x) + f(1/x) \) and \( f(2) > 1 \), find \( \lim_{x \to 1} f(x) \).

Answer: 2


Question: Calculate the limit: \[ \lim_{a\to 1}\int_{0}^{a}x\ln(1-x)\,dx, \quad a\in (0,1) \]

Answer: -\dfrac{3}{4}


Question: A function $f$ is defined on the whole real line with the property that $f(1+x)=f(2-x)$ for all $x$. The equation $f(x)=0$ has 8 distinct real roots. Find the sum of these roots.

Answer: 12


Question: Find the value of $x$ such that $x^x = 2^{2048}$. Express your answer in terms of powers of 2.

Answer: 2^8


Question: Evaluate the expression: $$\sqrt{\frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2} \cdot \left( \frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2} - \sqrt{41} \right) \cdot \left( \frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2} - \sqrt{29} \right) \cdot \left( \frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2} - \sqrt{10} \right)}$$

Answer: \dfrac{17}{2}


Question: Determine the number of roots of the equation \( \cot x - \tan x = a \) in the first quadrant, where \( a \in \mathbb{R} \).

Answer: 1


Question: Given that \((f(x))^3 - 3f(x) + x = 0\) and \(f(x) \neq 1\), find the value of \(|9f'(2)|\) if \(f(x)\) is a differentiable function.

Answer: 1


Question: Determine whether for every integer \( k \geq 2 \), there exists a natural number \( n \) such that the sum of divisors function \( \sigma(n) = k \cdot n \).

Answer: Yes


Question: Evaluate the complex function \( f(x) = x^{2/3} \) at \( x = -1 \).

Answer: -\frac{1}{2} + \frac{\sqrt{3}}{2}i


Question:
Martin wrote the following list of numbers on a whiteboard:
\[1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5},...,\frac{1}{2019}\]
Vincent must repeatedly choose two numbers, $a$ and $b$, from the board, erase them, and write $a+b+ab$ instead. This process continues until only one number remains on the board. What is the final number left on the board?

Answer: 2019


Question: Determine the number of zeroes of the polynomial \( f(z) = z^4 - 5z + 1 \) within the disk \( D(0; \frac{1}{4}) \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \left( 1- \left( 1- \frac{c}{n} \right)^{\frac{1}{2}} \right)^{\frac{n}{2}} \] where \( c < 0 \).

Answer: 0


Question:
Calculate the value of a European call option using the Black-Scholes formula when the strike price $K$ is 0. Use the following formula:

$$ C(s) = N(d_1)S - N(d_2)K $$

where

$$ d_1 = \frac{1}{\sigma \sqrt{T}} \left[ \ln{\frac{S}{K}} + \frac{\sigma^2}{2}T \right] $$
$$ d_2 = d_1 - \sigma \sqrt{T} $$

and

- $N$ is the cumulative distribution function of the standard normal distribution,
- $T$ is time to expiration,
- $S$ is the spot price of the underlying,
- $K$ is the strike price of the option,
- $\sigma$ is the volatility of the returns of the underlying.

Answer: S


Question: Determine the maximum value of the function \( F(y) = \int_{0}^{y} \sqrt{x^4 + (y-y^2)^2} \, dx \) for \( y \in [0, 1] \).

Answer: \dfrac{1}{3}


Question: Let \( f \in L^1(\mathbb{R}) \). Determine whether \( \lim_{n\rightarrow\infty}f(n^2 x)=0 \) almost everywhere for \( x \in \mathbb{R} \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{1}{x^2}\left(\frac{1}{2}\right)^{\left(\log x+ \log \left(\log \left(\frac{1}{x^2}\right)\right)\right)^2} \] as \( x \to 0 \).

Answer: 0


Question:
Given a decreasing sequence \(a_n\) of positive real numbers, for \(x>0\) define
\[
f(x)=\sum_n \min\left(\frac{x}{a_n}, \frac{a_n}{x}\right).
\]
Can \(a_n\) be chosen so that \(f(x)\to 0\) as \(x\to 0\)?

Answer: No


Question: Let $C$ be a convex compact subset of a finite-dimensional normed vector space, and let $f: C \to \mathbb{R}$ be a strictly convex and uniformly continuous function. Is it true that $f$ has exactly one minimizer?

Answer: Yes


Question: Find the steady-state solution of the differential equation \( \frac{\partial^2u}{\partial x^2}+\gamma^2(u-T)=\frac{1}{k}\frac{\partial u}{\partial t} \) for \( 0<x<a, \ t>0 \), with boundary conditions \( u(0,t)=T \), \( u(a,t)=T \) for \( 0<t \), and initial condition \( u(x,0)=T_1\frac{x}{a} \) for \( 0<x<a \).

Answer: T


Question: Determine the values of the parameter \(a\) for which the function \(f(x) = 8ax - a\sin 6x - 7x - \sin 5x\) is increasing and has no critical points for all real \(x\).

Answer: (6, \infty)


Question:
Given the AR(1) process \( \{X_t\} \) defined by:
\[ X_t = 2.62 - 0.84X_{t-1} + \epsilon_t, \quad \epsilon_t \sim WN(0, 2.27) \]
compute the variance of the average:
\[ \overline{X} = \frac{1}{3}\sum_{t=1}^{3} X_t \]

Answer: 0.90


Question: Given that the sequence of vectors $v_1, \cdots , v_n, u_1, \cdots, u_{m-1}$ is linearly independent and that $u_1, \cdots, u_m$ is also linearly independent where $u_m$ is in the $\text{span}$ of $v_1, \cdots, v_n$, let $V = \text{span}\{v_1, v_2, \cdots , v_n\}$ and $U =\text{span}\{u_1, \cdots, u_{m}\}$. Determine $\dim(U \cap V)$. Express $u_m$ as a linear combination of $v_1, \cdots, v_n$ and use this to find the dimension of the intersection.

Answer: 1


Question: Evaluate the integral \( \int_{\mathbb{R} } \frac{\sin^4(\frac{t}{2}) }{t^2} \, dt \). Use the fact that the Fourier transform of the function \( f(x) = (1-2|x|)\chi_{[-\frac{1}{2}, \frac{1}{2}]} (x) \) is given by \( \hat{f} (t) = \frac{8\sin^2(\frac{t}{4})}{t^2} \).

Answer: \dfrac{\pi}{4}


Question: Determine the number of roots of the equation \(z^5 - 12z^2 + 14 = 0\) that lie in the region \(\{z \in \mathbb{C} : 2 \leq |z| < \frac{5}{2}\}\).

Answer: 3


Question: Let $J = (uw - v^2, u^3 - vw, w^3 - u^5) \subseteq \mathbb{C}[u,v,w]$ and $I = (uw - v^2, u^3 - vw, w^2 - u^2v) \subseteq \mathbb{C}[u,v,w]$. Show that $J \subseteq I$.

Answer: J \subseteq I


Question: Calculate the limit: $$\lim_{n \to \infty} \prod_{k=2}^n \frac{k^3-1}{k^3+1}$$

Answer: \dfrac{2}{3}


Question: Given the implicit equation \( F(x, y) = x^2y + e^{x+y} = 0 \), where \( y = f(x) \) is defined for \( x > 0 \), compute \( \lim_{x \to 0^+} f(x) \).

Answer: -\infty


Question: Let $T_1$ be a random variable with probability density function (pdf) given by \( f(t) = \frac{6+2t}{7} \), and let \( T_2 \sim \text{Exp}\left(\frac{1}{3}\right) \). Given that $T_1$ and $T_2$ are independent, calculate \( P(T_1 + T_2 > 1) \).

Answer: \dfrac{6}{7}


Question: Use Stokes' Theorem to evaluate the line integral \( \int_C \mathbf{F} \cdot d \mathbf{r} \), where \( \mathbf{F} = (2x+3y, 1y+5z, 9z+1x) \) and \( C \) is the triangle with vertices \((4,0,0)\), \((0,6,0)\), and \((0,0,24)\), oriented in the specified order.

Answer: -444


Question: A fair 20-sided die, numbered from 1 to 20, is rolled 20 times. The results of the rolls are summed. Determine the number below which 99% of the sums will fall.

Answer: 270


Question: Find an irreducible polynomial of degree 2 over a finite field of characteristic 2.

Answer: x^2 + x + 1


Question: Given that \( \frac{a_0}{1}+\frac{a_1}{2}+\dots+\frac{a_n}{n+1}=0 \), prove that there exists some \( x \) in the interval \([0,1]\) such that \( a_0+a_1x+\dots+a_nx^n=0 \).

Answer: 0


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

Answer: 1


Question: Maximize the function \( f(x, y, z) = \min(x, 5y + 2z) \) subject to the constraint \( x + 15y + 7z = 44 \) and \( x, y, z \geq 0 \).

Answer: 11


Question: Determine the values of $b$ for which the series \( \sum_{k=1}^\infty \frac{k!}{(b+1)(b+2)...(b+k)} \) converges.

Answer: b > 1


Question: Determine which function is asymptotically greater as $n$ approaches infinity: $f(n) = 3^{2^n}$ or $g(n) = n! \times n^3$.

Answer: f(n)


Question: In a linear programming problem, consider a variable $x_k$ that does not have a non-negativity constraint. We can express it as $x_k = u_k - v_k$, where both $u_k$ and $v_k$ are non-negative. Can there exist a basic feasible solution where both $u_k$ and $v_k$ are basic variables? Answer yes or no.

Answer: No


Question: Given a polynomial \(P(x)\) of degree 4, it is known that \(P(x) \equiv x-1 \pmod{2x^2+4}\), \(P(x) \equiv x-1 \pmod{x^2-x-2}\), and \(P(x) \equiv 141 \pmod{x+2}\). Calculate the sum of the coefficients of \(P(x)\).

Answer: -36


Question: Is it true that for a function $f \in L^p(\mu)$, the inequality \(|\int_n^{n+1} f(x) \, dx| \leq \int_n^{n+1} |f(x)| \, dx\) holds?

Answer: Yes


Question: Let \(X\) and \(Y\) be independent random variables following \(N(0,1)\). Calculate the probability \(P(X \geq 0, X+Y \geq 0)\).

Answer: \dfrac{3}{8}


Question: Suppose $X$ is a continuum of dimension one and $Y$ is a subcontinuum of $X$. If the first Cech cohomology group $\check{H}^1(X;\mathbb{Z})$ is trivial, must $\check{H}^1(Y;\mathbb{Z})$ also be trivial? Provide a justification for your answer.

Answer: Yes


Question: In triangle $\triangle ABC$, point $D$ is located inside the triangle. Given that $\angle DBA = 10^\circ$, $\angle DBC = 20^\circ$, $\angle DAB = 30^\circ$, and $\angle DAC = 50^\circ$, find the measure of $\angle DCA$. Express your answer in degrees.

Answer: 30


Question: Let $X_1, X_2, \ldots$ be a sequence of independent uniform random variables on $(0,1)$. Define $N$ as the minimum index $n \geq 2$ such that $X_n < X_{n-1}$. Calculate the expected value $E(N)$. Provide your answer as a single number.

Answer: e


Question: Consider a set $S \subset \mathbb{R}^n$ that is not of measure zero. Is it always true that there exists a closed rectangle $C$ such that $C \subseteq S$?

Answer: No


Question:
Consider the function \( f(x) = \frac{x^n}{x-1} \) where \( n \) is a positive integer. Let \( G \) represent the greatest possible number of local extrema for \( f(x) \), and let \( L \) represent the least possible number of local extrema. Which statement is true?

(a) \( G = n \) and \( L = n-1 \)
(b) \( G = n \) and \( L = 1 \)
(c) \( G = n-1 \) and \( L = 0 \)
(d) \( G = 2 \) and \( L = 1 \)
(e) \( G = 2 \) and \( L = 0 \)

Answer: E


Question:
Suppose $a : \mathbb{R}_+ \to \{-1,1\}$ is a measurable function. Let $X_0 = \frac{1}{2}$. A particle moves on the $X$-axis according to the equation:
\[ X_t = X_0 + \int_0^t a_s \, ds \]
where the integral is a Lebesgue integral. Fix $T = \frac{1}{2}$, so $X_t \in [0,1]$ for all $t \le T$. Let $S \subset [0,1]$ be a set such that $\ell(S) = 1$, where $\ell(\cdot)$ denotes the Lebesgue measure. Define:
\[ G := \{t \le T : X_t \in S\} \]
Is it true that $\ell(G) = \ell([0,T]) = \frac{1}{2}$?

Answer: \dfrac{1}{2}


Question: Calculate the asymptotic relative efficiency (ARE) for the Poisson process given by \(\frac{\lambda^t \exp(-\lambda)}{t!} =  P(X=t)\). For \(X = t = 0\), the best unbiased estimator of \(e^{-\lambda}\) is \(((n-1)/n)^y\), where \(y = \Sigma{X_i}\), the complete sufficient statistic for \(\lambda\). For \(X = t = 1\), the best unbiased estimator of \(\lambda e^{-\lambda}\) is \(\frac{y}{n}\left(\frac{n-1}{n}\right)^{y-1}\). The ARE for \(t=0\) is given by: \[\left[\frac{e^{-\lambda}}{\left(\frac{n-1}{n}\right)^{n\lambda}\log\left( \frac{n-1}{n}\right)^n}\right]^2\]. Calculate this expression.

Answer: 1


Question: Given that $f(x)$ and $g(x)$ are non-zero polynomials with real coefficients, and they satisfy the equation $f(g(x)) = f(x) \times g(x)$, find the polynomial $g(x)$ if $g(2) = 37$.

Answer: x^2 + 33x - 33


Question:
Let $f: \mathbb{R} \to \mathbb{R}$ be a real function, and let $0 \leq n \leq +\infty$. Assume that for every $a \in \mathbb{R}$ and for $k=n$ (or for any $k \geq 0$ if $n=+\infty$), there exist real numbers $c_0(a), \ldots, c_k(a)$ such that:
\[ f(x) = c_0(a) + c_1(a)(x-a) + \frac{1}{2}c_2(a)(x-a)^2 + \cdots + \frac{1}{k!}c_k(a)(x-a)^k + o((x-a)^k) \]
where $o((x-a)^k)$ means $(x-a)^k \varepsilon_{a,k}(x)$ for some function $\varepsilon_{a,k}$ tending to $0$ as $x \to a$. Additionally, assume that for each $0 \leq k \leq n$, the function $c_k(a)$ is continuous in $a$. Can you conclude that $f$ is $C^n$?

Answer: Yes


Question: Given that \(m^2+n^2=1\), \(p^2+q^2=1\), and \(mp+nq=0\), find the value of \(mn+pq\).

Answer: 0


Question: Let \( p(x) = ax^3 + bx^2 + cx + d \) be a cubic function. There exist real numbers \( r \) and \( s \) such that \( 0 < r < s < 1 \), and the average value of \( p(x) \) on the interval \([0, 1]\) is equal to the average of \( p(r) \) and \( p(s) \). Find the product \( rs \) expressed as a fraction.

Answer: \dfrac{1}{6}


Question: In the derived category $D^b(X)$ of constructible sheaves on a reasonable space $X$, consider an open inclusion $j: U \to X$ and the closed complement $i: Y \to X$. Let $M, N \in D^b(X)$ and $f \in \text{Hom}_{D^b(X)}(M, N)$. If $i^*f = 0$ and $j^*f = 0$, is it true that $f = 0$? Justify your answer.

Answer: Yes


Question: Let $G$ be a torsion-free divisible abelian group that is also a $\mathbb{Q}$-vector space with finite dimension. Can $G$ be expressed as a finite direct sum of copies of $\mathbb{Q}$?

Answer: Yes


Question: Let $f$ be a function in $L^1(\mathbb{R})$. Prove that \( \lim\limits_{k \rightarrow \infty} \int\limits_k^\infty f(s)\,ds = 0 \).

Answer: 0


Question: Given that \( a \neq p \), \( b \neq q \), \( c \neq r \), and the determinant \( \left|\begin{array}{ccc} p & b & c \\ a & q & c \\ a & b & r \end{array}\right| = 0 \), find the value of \( \frac{p}{p-a} + \frac{q}{q-b} + \frac{r}{r-c} \).\[\text{(a) } -1 \quad \text{(b) } 1 \quad \text{(c) } -2 \quad \text{(d) } 2\]

Answer: D


Question:
Given that $X \sim U(0,1)$ and $Y|X=x \sim U(0,x)$, find the unconditional distribution of the random variable $Y$. Use the relationship:
\[ f_Y(y) = \int_{0}^{1} f_Y(y|x) f_X(x) \,dx \]
where $f_X(x) = 1$ for $0 \leq x \leq 1$ and $f_Y(y|x) = 1/x$ for $0 \leq y \leq x$. Determine the correct limits of integration and find $f_Y(y)$. What is the distribution of $Y$?

Answer: -\ln(y)


Question: Suppose that $y \in F_p((x))$ is transcendental over $F_p(x)$. Denote $L:=F_p(x, y)$ and let $L^p$ be the field of $p$th powers of $L$. Prove that $[L : L^p]=p^2$.

Answer: p^2


Question: Let $f: X \to Y$ be a finite, surjective morphism of smooth, quasi-projective varieties over a field $k$ of characteristic zero. Let $p \in X$. If $\dim X > 0$, does there necessarily exist a smooth curve $C$ on $X$ such that $f(p)$ is a smooth point of the closure of $f(C)$ in $Y$?

Answer: Yes


Question:
Let $U$ and $W$ be subspaces of polynomials over $\mathbb{R}$ defined as follows:

\[ U = \text{Span}(t^3 + 4t^2 - t + 3, t^3 + 5t^2 + 5, 3t^3 + 10t^2 -5t + 5) \]
\[ W = \text{Span}(t^3 + 4t^2 + 6, t^3 + 2t^2 - t + 5, 2t^3 + 2t^2 -3t + 9) \]

Determine the dimension of the intersection $\dim(U \cap W)$. Provide your answer as a single integer.

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to\infty}e^{-((\log(\frac{x+a}{\sigma}))^\nu-(\log(\frac{x}{\sigma}))^\nu)}$$ where $\nu, \sigma>0$.

Answer: 1


Question: Find the value of \( a \) such that the sum of solutions to the equation \( \sin(\sqrt{ax-x^2})=0 \) is equal to 100.

Answer: 25


Question:
Consider a sequence of rooted trees generated by the following operations:

1. Contract an edge: Choose an edge $E$ and join the two vertices adjacent to $E$.
2. Grow a leaf: Choose any vertex and connect it to a new leaf.

Starting with any rooted tree, these operations produce an infinite sequence of trees. Does this sequence always contain a tree $T$ such that there are infinitely many trees in the sequence with $T$ as a subgraph?

Answer: Yes


Question: Is the volume of a convex, solid $n$-dimensional body (containing the origin) equal to $cE[R^n]$, where $c$ is the volume of a unit $n$-ball and $R$ is the distance from the origin to a point on the body located at a random angle, distributed according to the rotation-invariant distribution?

Answer: Yes


Question: Does there exist a function \(a : [0,1] \to (0,\infty)\) and a family \(\{D_x : x \in [0,1]\}\) of countable, dense subsets of \([0,1]\) such that \(\bigcup_{x \in [0,1]} D_x = [0,1]\) and \(\sum_{r \in D_x} a(r) < \infty\) for all \(x \in [0,1]\)?

Answer: Yes


Question: Let $R$ be a ring and $M$ a free $R$-module with basis $B$. If $P$ is a prime ideal of $R$, is $B_P$ a basis for the $R_P$-module $M_P$?

Answer: Yes


Question: A hexagon-shaped room has walls that need to be painted such that no two adjacent walls are the same color. Given 10 different colors of paint, determine the number of distinct ways to paint the walls of the hexagon, considering rotations of the hexagon as distinct colorings.

Answer: 531450


Question: Determine whether the following proposition is true or false: If \( \lim_{t\to \infty}x(t) \) is convergent, then \( \lim_{t\to \infty}x'(t)=0 \), where \( ' = \frac{d}{dt} \).

Answer: False


Question: Evaluate the limit: \[ \lim_{n \to +\infty} \int_{1}^{+ \infty} \frac{\cos^{n}x}{x^2} \, dx \]

Answer: 0


Question: For the function \( f(x) = \int_0^{\ln x} \frac{1}{\sqrt{4+\mathrm{e}^{t}}} \, \mathrm{d}t \) where \( x > 0 \), find the value of \((f^{-1})'(0)\).

Answer: \sqrt{5}


Question: Evaluate the partial derivative \( \frac{\partial f(t,v)}{\partial v} \) for the function \( f(t, v) = 25(\cos t - v + t/5) \) at the point \( t = 0, v = 1 \).

Answer: -25


Question: If $f(f(x)) = x$ for a function $f(x)$, does it imply that $f(x)$ is its own inverse? Determine whether this statement is never, sometimes, or always true.

Answer: A


Question: Let $X$ and $Y$ be independent random variables uniformly distributed on $(0,1)$. Calculate the probability $P(X^2 + Y^2 \leq 1 \mid X \leq Y)$. Express your answer in terms of known constants.

Answer: \dfrac{\pi}{4}


Question: Given an uncountable subset $S$ of the unit interval $[0, 1]$, is there an accumulation point of $S$ that also lies in $S$?

Answer: Yes


Question: Consider a homomorphism \( \varphi \) from a cyclic group \( G = \mathbb{Z}/d\mathbb{Z} \) to an arbitrary group \( H \). If \( \varphi \) is defined by choosing an element \( h \in H \) such that \( \varphi(g_G) = h \) for a generator \( g_G \) of \( G \), determine whether the condition \( h^d = e_H \) is necessary and sufficient for \( \varphi \) to be a homomorphism.

Answer: Yes


Question: Given the function \( A(y) = \frac{y \sqrt{1 - y^2}}{2} + \int_{y}^{1} \sqrt{1 - t^2} \, dt \) on \([-1, 1]\), with \( A'(y) = \frac{-1}{2\sqrt{1 - y^2}} \), and the definitions \( A(\cos x) = \frac{x}{2} \) and \( \sin x = \sqrt{1 - \cos^2 x} \), find the derivative \( \cos'(x) \).

Answer: -\sin x


Question: Determine if the limit \( \lim_{\bar{x} \to \bar{0}} \frac{\ln(1+\vert \bar{x} \vert^2)}{\vert \bar{x} \vert^2 + \sin(x_1x_2x_3)} \) exists, where \( \bar{x} = (x_1,x_2,x_3) \) and \( \vert \bar{x} \vert = \sqrt{x_1^2 + x_2^2 + x_3^2} \). If it exists, find its value.

Answer: 1


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\left( \frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot (2n)}\right)^3 \]

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to\infty}\left(\sqrt{9x^6+7x^3+5x}-\sqrt{9x^6+6x^2+3x}\right)$$

Answer: \dfrac{7}{6}


Question:
Consider a dice game played with a fair $n$-sided die. On the first roll, a roll of $n$ wins, while a roll of $1$ loses. For any other result, the player rolls again. On the second roll, a roll of $n$ wins, while a roll of $1$ or $2$ loses. The game continues similarly, where on roll $k$, the player wins with a roll of $n$ and loses with a roll of $k$ or below. The game must be won in no more than $n - 1$ turns. For a given $n$, the probability of winning is given by:

$$
\mathrm{P}(\text{win}) = \frac{1}{n} + \sum_{i = 2}^{n - 1}\frac{(n - 2)!}{(n - i - 1)!\, n^{i}}
$$

Determine whether:

$$
\lim_{n \to \infty}\mathrm{P}(\text{win}) = 0
$$

or if $\mathrm{P}(\text{win})$ converges to some other nonzero probability as $n \to \infty$. Justify your answer.

Answer: 0


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

Answer: 1


Question: Evaluate the integral \( \int_{\gamma} \sin(z^3) + e^{z^2} \, dz \), where the path \( \gamma: [0, \pi] \to \mathbb{C} \) is defined by \( \gamma(t) = \sin t + i \arctan(t^4 - \pi t^3) \).

Answer: 0


Question: Given that \( f(1) = 4 \) and \( f'(x) = \sum_{k=0}^{\infty} \frac{(-1)^{k} (x-1)^{k}}{(k+1)!} \), find \( f''(1) \).

Answer: -\dfrac{1}{2}


Question: Evaluate the integral \( \int_0^{\infty} x^{-3/2} (1 - e^{-x})\, dx \) using Beta and Gamma functions.

Answer: 2\sqrt{\pi}


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

Answer: -1


Question: Determine the range of values for \( m \) such that the function \( f(x) = mx^2 + 1 - \ln(1+x) \) has two extreme local points.

Answer: (-\infty, -2)


Question: Evaluate the limit: $$\lim_{n\to+\infty}n\int_{0}^{1}\left(f\left(\frac{x^{2}}{n}\right)-1\right)\,dx$$ where $f(x) = (x^2 + 1)e^x$. Determine the value of this limit.

Answer: \dfrac{1}{3}


Question: Consider a strict short exact sequence of complete nuclear spaces: \(0 \to V \to W \to L \to 0\), where all maps are continuous, \(V \to W\) is a closed embedding, the topology on \(V\) is induced from \(W\), and \(W \to L\) is open. Let \(U\) be a complete nuclear space. Is the sequence obtained by the completed tensor product with \(U\), \(0 \to V \hat{\otimes} U \to W \hat{\otimes} U \to L \hat{\otimes} U \to 0\), also a strict short exact sequence?

Answer: Yes


Question: Let $S_n = \sum_{i=1}^n X_i$ be a symmetric random walk. Compute the conditional expectation $E(X_i \mid \mathcal{G}_{n+1})$ for $1 \leq i \leq n$, where $\mathcal{G}_{n} = \sigma(S_{n},S_{n+1},\dots)$. Assume $S_n$ is a martingale.

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


Question: Determine the radius of convergence for the series \( g(x) = \sum_{n=0}^{\infty} \lceil \frac{n+1}{2} \rceil x^n \).

Answer: 1


Question: Determine whether the limit \( \lim_{n\to\infty} \left( \sum_{m=1}^n \frac{1}{\sqrt{m}} - \sqrt{n} \right) \) is convergent.

Answer: \infty


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

Answer: q > -2


Question: Calculate the fundamental group of the space defined by the equation: \[ X=\{(x,y,z)\in\mathbb{R}^3:(x^2+y^2+z^2-100)[(x-10)^2+y^2+z^2-1]=0\}. \]

Answer: 0


Question: Suppose $u_n$ converges weakly to $u$ in $L^2(\Omega)$. Does $|u_n|$ converge weakly to $|u|$ in $L^2(\Omega)$?

Answer: No


Question: Find the degree of the field extension \([F:\mathbb{Z}_{11}]\) where \(F\) is the splitting field of a cubic polynomial over \(\mathbb{Z}_{11}\).

Answer: 3


Question: Evaluate the limit \( \lim_{r \to 0^+} r^{n-1}(\log(1/r))^n \) for \( n > 1 \).

Answer: 0


Question:
Find the solution of the following higher derivative using Leibniz's formula:
\[ \frac{d^{p+q}}{dx^{p+q}}x^p(1+x)^q \]

Answer: (p + q)!


Question: Determine the number of solutions to the equation \( \sin x = \frac{x}{10} \).

Answer: 7


Question: Evaluate the Lebesgue integral of the function \( f(x) = \begin{cases}\frac{1}{\sqrt[]{x}} & x\in(0,1] \\ 0 &x\notin (0,1]\end{cases} \) over the interval \((0,1]\). Show that \( \int_{(0,1]} f(x) \, dx = 2 \).

Answer: 2


Question: Determine the radius of convergence for the series \( S=\sum_{n=1}^{\infty}\frac{(-2)^{n}+3^{n}}{n}x^{n} \) using the root test.

Answer: \dfrac{1}{3}


Question:
If $x_1, x_2, \ldots, x_n$ are the roots of the equation $1 + x + x^2 + \ldots + x^n = 0$, find the value of the expression:
\[ \frac{1}{x_1 - 1} + \frac{1}{x_2 - 1} + \ldots + \frac{1}{x_n - 1}. \]

Answer: -\dfrac{n}{2}


Question: Evaluate the limit \( \lim\limits_{n \rightarrow \infty} \int_{0}^{1} \left(1+ \frac{x}{n}\right)^n dx \).

Answer: e - 1


Question: Determine the radius of convergence for the Taylor series of the function \( f(z) = \frac{10z}{z^2+z-6} \) about the point \( z = i \).

Answer: \sqrt{5}


Question: Find all pairs of positive integers \((x, y)\) such that \(x^y = (x+y)^2\), where at least one of \(x\) or \(y\) is a prime number.

Answer: (2, 6)


Question: Determine the largest constant \( c > 0 \) such that the inequality \(-3x^2 + 2|x||y| - y^2 \leq -c(x^2 + y^2)\) holds for all real numbers \( x \) and \( y \).

Answer: 2 - \sqrt{2}


Question: Does there exist a connected topological space on which the permutation group $S_3$ acts without fixed points?

Answer: Yes


Question: Evaluate the limit: \[ \lim\limits_{x \rightarrow 0^+} \left ( 2 \sin \left ( \frac{1}{x} \right ) + \sqrt{x} \sin \left ( \frac{1}{x} \right ) \right )^x. \]

Answer: 1


Question: Is it possible to construct matrices $A$ and $B$ such that $AB$ is invertible, where $A$ is an $n \times m$ matrix, $B$ is an $m \times n$ matrix, and $n < m$?

Answer: Yes


Question: Given a Markov chain with a steady state probability of state "a" denoted as \( s_a \), calculate the expected number of times the chain will be in state "a" over 100 time steps.

Answer: 100 s_a


Question: In a standard deck of 52 playing cards, there are 26 red cards and 26 black cards. If the deck is well shuffled, what is the average length of a run of cards of the same color?

Answer: \dfrac{52}{27}


Question: Find the Hausdorff dimension of the set \( F \), where \( F \) consists of numbers \( x \in [0,1] \) with base 3 expansions \( 0.a_1a_2\ldots \) such that there exists an integer \( k \) for which \( a_i \neq 1 \) for all \( i \geq k \).

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


Question: A polynomial function \( f(x) \) of degree 5 with leading coefficient 1 increases on the intervals \((-\infty, 1)\) and \((3, \infty)\), and decreases on the interval \((1, 3)\). Given that \( f'(2) = 0 \) and \( f(0) = 4 \), find the value of \( f'(6) \).

Answer: 1200


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x} \]

Answer: e^3


Question: Given a coin with an unknown bias, you observe $N$ heads and $0$ tails. Using Maximum Likelihood Estimation (MLE), determine the expected probability that the next flip will result in a head.

Answer: 1


Question: Calculate the limit: \( \lim_{x \to 0}\frac{\sqrt[3]{\cos x}-\sqrt{\cos x}}{x^2} \)

Answer: \dfrac{1}{12}


Question: Determine the number of solutions to the differential equation \( x'(t) = x(t) - e^{-t^2} \) that satisfy the condition \( \lim_{t \rightarrow \pm \infty} x(t) = 0 \).

Answer: 1


Question: Determine whether there exists a monotonic function \( f: \mathbb{N} \to \mathbb{R} \) such that the series \( \sum_{n=1}^{\infty} f(n) \) diverges, but the series \( \sum_{p \text{ prime}} f(p) \) converges.

Answer: Yes


Question: Determine the number of times L'Hopital's rule must be applied to evaluate the limit \( \lim _{x\to 0}\left(\frac{e^{x^2}+2\cos \left(x\right)-3}{x\sin \left(x^3\right)}\right) \).

Answer: 4


Question:
Evaluate the limit:
\[ \lim_{n \to \infty} \left( \sum_{k=10}^{n+9} \frac{2^{11(k-9)/n}}{\log_2 e^{n/11}} - \sum_{k=0}^{n-1} \frac{58}{\pi\sqrt{(n-k)(n+k)}} \right) \]

Answer: 2018


Question:
Let $A$ be a nonempty subset of a topological space $X$. Determine which of the following statements is true:

1. If $A$ is connected, then its closure $\overline{A}$ is not necessarily connected.
2. If $A$ is path-connected, then its closure $\overline{A}$ is path-connected.
3. If $A$ is connected, then its interior $A^o$ is not necessarily connected.
4. If $A$ is path-connected, then its interior $A^o$ is connected.

Select the correct statement.

Answer: 3


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

Answer: -\dfrac{1}{2}


Question:
Determine the type of singularity at the point $z=0$ for the function $f(z)=\sin\left(\cos\left(\frac{1}{z}\right)\right)$. Choose one of the following options:
(a) Removable singularity
(b) Pole
(c) Essential singularity
(d) Non-isolated singularity

Answer: C


Question:
Find the range of values for \( \phi \) that satisfy the inequality:
\[ -1.96 \leq \frac{6.4-10\phi}{\sqrt{1-\phi^2}} \leq 1.96 \]

Answer: [0.467, 0.766]


Question: Determine whether there exists a holomorphic function \( f \) on the domain \( D(0,1) \) such that for all \( z \in D(0,1/2) \), \( f(2z) = f(z) \). If such a function exists, describe its form.

Answer: f(z) = c


Question: Evaluate the infinite sum \( \sum_{n=1}^\infty \frac{3n^2-4n+2}{n!} \).

Answer: 4e - 2


Question: Find the minimum value of the expression \[ \frac{xy}{x^5+xy+y^5} + \frac{yz}{y^5+yz+z^5} + \frac{xz}{x^5+xz+z^5} \] given that \( x, y, z > 0 \) and \( x + y + z = 1 \).

Answer: 0


Question:
Find the number of solutions $h_n$ for the equation $2x_1 + 4x_2 = n$, where $x_i \in \mathbb{N}$. The generating function for the sequence $h_n$ is given by:
\[ \frac{1}{1-x^2} \cdot \frac{1}{1-x^4} \]
Calculate $h_{2000}$.

Answer: 501


Question: Let $u(x,y)=2f(y)\cos(x-2y)$ be a solution of the Initial Value Problem $2u_x+u_y=u$ with the initial condition $u(x,0)=\cos(x)$. Find the value of $f(1)$. Use Lagrange's Auxiliary Equations $\frac{\operatorname{dx}}{2}=\frac{\operatorname{dy}}{1}=\frac{\operatorname{du}}{u}$ to assist in solving the problem.

Answer: \dfrac{e}{2}


Question: Evaluate the limit: \[ \lim_{x\to \infty}\left(\frac{2\arctan(x)}{\pi}\right)^x. \]

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


Question: Calculate the residue of the function \( f(z) = z \sin\left(\frac{1}{1-z}\right) \) at \( z = 1 \).

Answer: -1


Question: Let $X$ and $Y$ be Hilbert spaces with dot products $(\cdot,\cdot)_X$ and $(\cdot,\cdot)_Y$, respectively. Consider two functions $f, g: X \to Y$. Prove that $f = g$ if and only if for all $x \in X$ and $y \in Y$, the equality $(f(x), y)_Y = (g(x), y)_Y$ holds.

Answer: f = g


Question: Given a holomorphic function \( f: D \to \mathbb{C} \) on an open connected domain \( D \), where the k-th derivative of \( f \) satisfies \( \left|\frac{d^k f}{dz^k}\right| > 1 \) for all \( z \in D \), determine a bound on the number of zeros of \( f \) in terms of \( k \).

Answer: k


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{\sin(6x^2)}{\ln \cos(2x^2-x)}$$

Answer: -12


Question: Let $p(x)$ be a polynomial of degree four with extrema at $x=1$ and $x=2$. Given that \( \lim \limits_{x \to 0}\left(1+\frac{p(x)}{x^2}\right)=2 \), find the value of $p(2)$. \( \boxed{} \)

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{1}{x^2}\int_0^{\sin x}\ln(1+t)\,dt \]

Answer: \dfrac{1}{2}


Question:
Consider the partial differential equation \( a(x,y) u_{xx} + b(x,y) u_{xy} + c(x,y) u_{yy} = 0 \), where \( a \), \( b \), and \( c \) are continuous functions on \( \mathbb{R}^2 \). The equation is classified as elliptic, parabolic, or hyperbolic in regions \( V_1 \), \( V_2 \), and \( V_3 \) respectively, such that \( V_1 \cup V_2 \cup V_3 = \mathbb{R}^2 \). Determine which of the following statements is true:

1. \( V_1 \), \( V_2 \), \( V_3 \) are open sets in \( \mathbb{R}^2 \).
2. \( V_1 \), \( V_3 \) are open sets in \( \mathbb{R}^2 \).
3. \( V_1 \), \( V_2 \) are open sets in \( \mathbb{R}^2 \).
4. \( V_2 \), \( V_3 \) are open sets in \( \mathbb{R}^2 \).

Use the condition \( b^2 - 4ac \) to classify the regions: \(< 0\) is elliptic, \(> 0\) is hyperbolic, and \(= 0\) is parabolic.

Answer: 2


Question: Determine whether the limit \( \lim_{t\to 0}\ln\left(1+\frac{1}{t}\right)^t \) exists.

Answer: 0


Question: Given that $A$ is an $m \times n$ matrix and $B$ is an $n \times m$ matrix, and that the product $AB$ is non-singular, determine the null space of the matrix $B$. Provide a justification for your answer.

Answer: \{\mathbf{0}\}


Question: Find the probability \( P(X^2 + Y^2 < t) \) where \( X \) and \( Y \) are independent standard normal random variables \( N(0,1) \).

Answer: 1 - e^{-t/2}


Question: Determine all positive integers \( n \) such that \( \gcd(n, 5) = 1 \) and \( n^4 + 4^n \) is a prime number.

Answer: 1


Question: Determine the Big-O notation for the function \((n^n + n2^n + 5^n)(n! + 2^n)\).

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


Question: Evaluate the limit \( \lim_{x \rightarrow 0} x^2 \left( 1 + 2 + 3 + \cdots + \left\lfloor \frac{1}{|x|} \right\rfloor \right) \), where \( \lfloor x \rfloor \) denotes the floor function of \( x \).

Answer: \dfrac{1}{2}


Question: Find the last three digits of \(2003^{2002^{2001}}\).

Answer: 241


Question: Is it true that if $D$ is a Riemannian $2$-disk with constant Gaussian curvature equal to $1$ and a boundary with constant geodesic curvature, then $D$ is isometric to a geodesic ball on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$? Provide a proof or counterexample.

Answer: Yes


Question: Find the derivative of the solution to the differential equation \( \ddot{x} = \dot{x}^2 + x^3 \) with respect to the parameter \( A \), given the initial conditions \( x(0)=0 \) and \( \dot{x}(0) = A \), evaluated at \( A = 0 \).

Answer: t


Question: If the automorphism group of a group \( G \), denoted \( \text{Aut}(G) \), is cyclic, does it necessarily follow that \( G \) is cyclic? Provide a justification for your answer.

Answer: Yes


Question: Let $U \subset \mathbb{R}^n$ be a vector space with $\dim(U)=d$. A standard normal distribution on $U$ is defined as the law of a random vector $X=(X_1, \ldots, X_n)$ taking values in $U$, where the coordinates of $X$ in any orthonormal basis of $U$ are $d$ independent standard normal distributions $\mathcal{N}(0, 1)$. Consider a random vector $Y=(Y_1, \ldots, Y_n)$ following a standard normal distribution on $\mathbb{R}^n$. Is the conditional distribution of $Y$ given $Y \in U$ the standard normal distribution on $U$?

Answer: Yes


Question: Given that $\mathcal{L}^{−1}\{[Iz - B]^{-1}\}(z) = e^{Bt}$, use the convolution theorem to determine $\Phi(t)$ if $\hat{\Phi}_z = A[Iz - B]^{-1}$. Is $\Phi(t) = Ae^{Bt}$?

Answer: \Phi(t) = A e^{Bt}


Question: Calculate the work required to fill a hemispherical tank with water through a hole in its base. The tank has a radius of 2 feet and is positioned with its base circular. The water source is at the base, and the weight-density of water is 62.4 pounds per cubic foot. Provide the work done in foot-pounds.

Answer: 249.6\pi


Question: Determine whether the limit \( \lim _{x\to 0} \frac{1}{x} \int_0^x \left|\cos \frac{1}{t} \right| dt \) exists. If it does, find its value.

Answer: \dfrac{2}{\pi}


Question: Evaluate the limit: \[ \lim_{t \to(\pi/2)^-} \log\left(\frac{2 t}{\pi}\right) \log(\cos(t)) \]

Answer: 0


Question: How many labeled trees with 10 vertices have exactly one vertex of degree 4?

Answer: 29841840


Question:
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), the following equations hold:
\[ f(3x) = f\left(\frac{x+y}{(x+y)^2+1}\right) + f\left(\frac{x-y}{(x-y)^2+1}\right) \]
and
\[ f\left(x^2-y^2\right) = (x+y)f(x-y) + (x-y)f(x+y) \]

Answer: 0


Question: Let \( A \in M_{m}( \mathbb{C}) \) be a matrix such that there exists an integer \( r > 0 \) for which \( A^{r} = I \), where \( I \) is the identity matrix. Suppose \( \lambda \) is the only eigenvalue of \( A \). Prove that \( A = \lambda I \).

Answer: A = \lambda I


Question: Consider a compact manifold $X$ with a Morse function $f$ that has a saddle point at $x_0$. Suppose we perturb $f$ by adding a function of the form $\epsilon \rho Q$, where $Q$ is a quadratic polynomial, $\epsilon$ is a small number, and $\rho$ is a bump function that is identically 1 in a neighborhood of $x_0$. For sufficiently small $\epsilon$, does the perturbed function remain a Morse function with the same critical points as $f$?

Answer: No


Question: Let $H$ be a complex Hilbert space and $B: H \times H \to \mathbb{R}$ be a continuous sesquilinear form. Consider the quadratic form $Q(\psi) = B(\psi, \psi)$. Suppose there exists a continuous one-parameter group of unitary operators $\{e^{i\theta T}\}_{\theta \in \mathbb{R}}$ such that \( Q\left( e^{i\theta T}\psi - \psi \right) = O\left(\lvert \theta\rvert^3\right) \) as $\theta \to 0$, where $\psi$ is in the domain of the self-adjoint generator $T$. Is it true that $Q(T\psi) = 0$?

Answer: Yes


Question: Prove that \( \left|\frac{e^{-ixt}-1}{t}\right| \le |x| \) for all real numbers \( x \neq 0 \) and \( t \neq 0 \).

Answer: \left|\frac{e^{-ixt}-1}{t}\right| \le |x|


Question: Evaluate the integral: $$\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} xe^{\displaystyle\frac{-x^{2}}{2}}dx$$

Answer: 0


Question:
Find the function $f(x)$ that satisfies the recurrence relation $f(x+2) = f(x+1) + 2f(x)$ for all positive integers $x$, given the initial conditions $f(1) = 4$ and $f(2) = 2$. Express your answer as a closed form function of $x$.

\[ f(x) = \]

Answer: 2^x - 2(-1)^x


Question: Given that the number $110179$ can be expressed as a product of two distinct prime numbers $p$ and $q$, and that the number of integers less than $110179$ and relatively prime to it is $109480$, find the value of $p+q$. Also, determine the values of $p$ and $q$.

Answer: 700


Question: Determine whether the function $F(X) = e^X$ is a solution to the stochastic differential equation $dX = \mu dt + \sigma dW$.

Answer: No


Question: Evaluate the integral \( \int_\gamma x \cdot n(x) \, ds(x) \), where \( x = (x, y) \in \mathbb{R}^2 \), \( n(x) \) is the unit outward normal to the ellipse \( \gamma \) given by \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \).

Answer: 12\pi


Question: In triangle $ABC$, where $BC=5$, $AC=6$, and $AB=7$, points $P_1, P_2, P_3, P_4$ are placed on $BC$ such that $BP_1 = P_1P_2 = P_2P_3 = P_3P_4 = P_4C = 1$. Calculate the value of $AP_1^2 + AP_2^2 + AP_3^2 + AP_4^2$. Provide your answer as a single number.

Answer: 150


Question: Find an ideal \( I \subseteq \mathbb{Z}[x] \) such that \( \mathbb{Z}[x]/I \cong \mathbb{Z}[i] \), or prove that no such ideal exists.

Answer: (x^2 + 1)


Question: Calculate the limit: \[ \lim _{x\rightarrow 0^{+}}\left( \dfrac{1+x}{2+x}\right) ^{\dfrac{1-\sqrt{x}}{1-x}} \]

Answer: \dfrac{1}{2}


Question:
Given a Markov chain \( X_n \) with state space \( S = \{1, 2, 3, 4\} \) and transition matrix \( P \) as follows:
\[
P = \begin{pmatrix}
  \frac{1}{4} & 0 & \frac{1}{2} & \frac{1}{4} \\
  \frac{1}{4} & \frac{1}{4} & \frac{1}{2} & 0 \\
  0 & \frac{2}{3} & \frac{1}{3} & 0 \\
  0 & \frac{1}{2} & 0 & \frac{1}{2}
\end{pmatrix}
\]
Calculate \( EX_{\tau} \), where \( \tau = \inf \{ n : X_n = 2 \text{ or } X_n = 3 \} \), given that the initial state is \( X_0 = 1 \).

Answer: \dfrac{8}{3}


Question:
Let \( u \in H^1(U) = W^{1,2}(U) \), where \( U = B(0,1) \) is the unit ball in \( \mathbb{R}^n \). Show that there exists a constant \( C(\alpha, n) \) such that
\[
  \int_U u^2 \, dx \le C \int_U |Du|^2 \, dx
\]
provided that
\[
  |\{x \in U: u(x)=0\}|\ge \alpha.
\]

Answer: C(\alpha, n)


Question: Evaluate \( \lim_{n \rightarrow \infty} \int_{0}^{x} \frac{(-\ln{z})^n}{n!} \, dz \) for \( x \in (0,1) \).

Answer: 1


Question: Evaluate the integral \( \iint (3x + 2y)(x + 4y) \, dx \, dy \) over the region \( R \) bounded by the lines \( y = -\frac{3}{2}x + 1, \ y = -\frac{3}{2}x + 3 \) and \( y = -\frac{1}{4}x, \ y = -\frac{1}{4}x + 1 \). Use the change of variables \( u = 3x + 2y, \ v = x + 4y \) to find expressions for \( x \) and \( y \) in terms of \( u \) and \( v \), and compute the Jacobian \( \frac{\partial(x, y)}{\partial(u, v)} \). Then, evaluate the transformed integral \( \iint uv J(u,v) \, du \, dv \) with the appropriate boundaries.

Answer: \dfrac{64}{5}


Question:
Consider the fiber products of the schemes $Y_1(5)^
\circ$ and $Y_1(7)^
\circ$ over the moduli stack of elliptic curves $\mathcal{M}_{1,1}^\circ$ and the $j$-line $\mathbb{A}_j^{1\circ}$, respectively:
\[ A := Y_1(5)^\circ\times_{\mathcal{M}_{1,1}^\circ}Y_1(7)^\circ, \quad B := Y_1(5)^\circ\times_{\mathbb{A}_j^{1\circ}}Y_1(7)^\circ. \]
Both $A$ and $B$ are schemes, and $Y_1(7)^\circ$ is finite étale over both $\mathcal{M}_{1,1}^\circ$ and $\mathbb{A}_j^{1\circ}$ with degree 24. The universal property of fiber products provides a map $A\rightarrow B$ that is finite étale. By comparing degrees, determine if this map is an isomorphism, i.e., is $A \cong B$? If so, explain the implications for elliptic curves $E_1/K$ and $E_2/K$ with points of order 5 and 7, respectively, such that $j(E_1) = j(E_2) \neq 0,1728$. What does this imply about the twists of elliptic curves with $K$-rational points of order $\ge 4$?

Answer: A \cong B


Question: Does the Jordan outer measure \(m^{*,(J)}(E) = \inf_{B \supset E, B\text{ elementary}} m(B)\) satisfy finite subadditivity? Specifically, does it hold that \(m^{*,(J)}(E \cup F) \leq m^{*,(J)}(E) + m^{*,(J)}(F)\) for any sets \(E\) and \(F\), where an elementary set is a finite union of boxes?

Answer: Yes


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

Answer: 1


Question: Evaluate the double integral \( \iint (x^2 + y^2) \, dx \, dy \) over the region enclosed by the curves \( y = 8x \), \( x + y = 6 \), \( y = 4 \), and \( y = 0 \).

Answer: \dfrac{1301}{8}


Question: Determine the optimal value of $x \in [0,1]$ that maximizes the function \( z = 8xy - 3x - 3y + 1 \) given that $y \in [0,1]$ is chosen by an opponent who aims to minimize $z$. Neither player knows the other's choice.

Answer: \dfrac{3}{8}


Question: Is it true that for any continuous function \( f: [a, b] \to \mathbb{R} \) and any partition \( P \) of \( [a, b] \), the inequality \( \mathcal{W}(f, P) \leq V_f[a, b] \) holds, where \( V_f[a, b] \) is the total variation of \( f \) on \( [a, b] \) and \( \mathcal{W}(f, P) \) is defined as \( \sum_{k=1}^n (M_k - m_k) \) with \( M_k = \sup \{f(x): x \in [x_{k-1}, x_k]\} \) and \( m_k = \inf \{f(x): x \in [x_{k-1}, x_k]\} \) for a partition \( P = \{x_0, x_1, \ldots, x_n\} \)?

Answer: Yes


Question: Let \( f(x) \) be a fourth differentiable function such that \( f(2x^2-1) = 2xf(x) \) for all \( x \in \mathbb{R} \). Find \( f^{(4)}(0) \).

Answer: 0


Question: Given two simple curves $M, N \subset \mathbb{R}^n$ with the same finite length $L[M] = L[N] < \infty$, determine whether there exists a smooth bijection $F: M \to N$ such that for every parametrized curve $\gamma$ in $M$, the length condition $L[\gamma] = L[F \circ \gamma]$ holds.

Answer: Yes


Question: Evaluate the limit: \[ \lim\limits_{(x,y)\rightarrow (0,0)} \dfrac{x^2y^2}{(x^2+y^4)\sqrt{x^2+y^2}} \]

Answer: 0


Question: Compute \( \mathbb{E}[X_{1} \mid X_{1} + X_{2} + \cdots + X_{n} = x] \) where \( X_{1}, \ldots, X_{n} \) are i.i.d random variables.

Answer: \dfrac{x}{n}


Question: Given any function \( f: \mathbb{N}_+ \to \mathbb{N}_+ \), is there an irrational number \( \alpha \in [0,1] \setminus \mathbb{Q} \) such that \( f \leq^* \text{appr}_\alpha \)?

Answer: Yes


Question: Evaluate the line integral \( \oint_{C}\left[\left(2x - y^{3}\right)\mathrm{d}x - xy\,\mathrm{d}y\right] \), where \( C \) is the boundary of the annular region enclosed by the circles \( x^{2} + y^{2} = 1 \) and \( x^{2} + y^{2} = 9 \).

Answer: 60\pi


Question: Find the inverse of the generator $g = 0010$ in the finite field $\mathbb{F}_{2^4}$, given the irreducible polynomial $f(x) = x^4 + x + 1$. Express your answer in binary form.

Answer: 1001


Question: Evaluate \( \lim_{x \to \infty} e^x \int_x^\infty \frac{dt}{te^t} \).

Answer: 0


Question: Compute \(3^{100} \pmod{9797}\) using Euler's Theorem and the Chinese Remainder Theorem. (Hint: Factor 9797 into its prime components and apply the theorems accordingly.)

Answer: 2021


Question:
Evaluate the limit \( L \) using the ratio test for the series \( \sum_{n=1}^{\infty}\frac{(-1)^n}{(\ln n)^n} \). Determine \( L \) by calculating:
\[
L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty}\left[\frac{\ln(n)}{\ln(n+1)}\right]^n\frac{1}{\ln(n+1)}
\]
Find the value of \( L \).

Answer: 0


Question: Let \(z_1\) and \(z_2\) be two complex numbers such that \(z_1^2 - 4z_2 = 16 + i20\). If \(a\) and \(b\) are roots of the equation \(x^2 + z_1 x + z_2 + M = 0\) (where \(M\) is a complex number) and \(|(a-b)^2| = 28\), find the maximum value of \(|M|\).

Answer: 7 + \sqrt{41}


Question: Determine the value of \( k \) such that the solutions to the differential equation \( y'' + 2y = \cos(kt) \) are unbounded.

Answer: \sqrt{2}


Question: Given that $p$, $q$, $r$, and $s$ are the sides of a quadrilateral, find the minimum value of the expression \( \frac{p^2 + q^2 + r^2}{s^2} \).

Answer: \dfrac{1}{3}


Question: Determine if the kernel \( K(x,y) = \frac{e^{-\frac{xy}{x+y}}}{x+y} \) is invertible for \( x \in [0,1] \) and \( y \in [0, \infty) \). Specifically, if \( \int_0^1 K(x,y) f(x) \, dx = 0 \) for all \( y \in [0, \infty) \), can we conclude that \( f(x) = 0 \) for \( x \in [0,1] \)?

Answer: f(x) = 0


Question: Find all functions \( f : \Bbb Z \rightarrow \Bbb Z \) such that for all integers \( x \) and \( y \), the equation \( f(x+f(f(y)))=y+f(f(x)) \) holds.

Answer: f(x) = x


Question:
Determine the smallest possible sample size \( k \) such that for a sample \( Y_1, \ldots, Y_k \) from a normal distribution \( N(\mu, \sigma^2) \) with unknown parameters \( \mu \) and \( \sigma^2 \), the following probability condition holds:

\[ P\left( |\bar{Y}_k - \mu| \leq \frac{\sigma}{5} \text{ and } |s_k - \sigma| \leq \frac{\sigma}{5} \right) \geq \frac{1}{2} \]

where \( \bar{Y}_k \) is the sample mean and \( s_k \) is the sample standard deviation. Provide the smallest \( k \) that satisfies this condition.

Answer: 21


Question: Determine the maximal possible dimension of an abelian Lie subalgebra of the Heisenberg Lie algebra \( \mathfrak{h}_n \) of dimension \( 2n+1 \).

Answer: n + 1


Question: Determine the number of roots, counted with multiplicity, of the polynomial \(2z^5 - 15z^2 + z + 2\) inside the annulus \(1 \leq |z| \leq 2\).

Answer: 3


Question: Find the maximal volume of a rectangular box with a fixed surface area of 150 square units using the method of Lagrange multipliers.

Answer: 125


Question: In a two-player game, each player draws a number from a continuous uniform distribution over the interval [0, 100]. After observing their own number, each player independently chooses either to SWAP or RETAIN their number. If both players choose SWAP, they exchange their numbers; otherwise, they retain their original numbers. What is the probability that the players will exchange their initially drawn numbers?

Answer: \dfrac{1}{4}


Question: Find the minimum value of the sum of distances $PX + PY$, where $P$ is a variable point on the line $2x + y + z = 1$. The fixed points are $X(1,1,1)$ and $Y(2,2,4)$. What is the minimum value of $PX + PY$?

Answer: \sqrt{29}


Question:
Given complex numbers \(z_1\) and \(z_2\) such that \(|z_1| = |z_1 + z_2| = 3\) and \(|z_1 - z_2| = 3\sqrt{3}\), find the value of \(\log_{3}|(z_{1}\overline{z_{2}})^{2000}+(
\overline{z_{1}}z_{2})^{2000}|\).

Answer: 4000


Question: Let $G$ be a group, and let $A$ and $B$ be subgroups of $G$ such that $A \cap B$ has finite index in both $A$ and $B$. If the set $A \cup B$ is normalized by $G$, does $A \cap B$ have a subgroup of finite index that is normal in both $A$ and $B$?

Answer: Yes


Question: Two independent events A and B follow an exponential distribution with parameter \( \lambda \): \( f(t) = \lambda e^{-\lambda t} \) for \( t \geq 0 \). If \( X \) is the time when A occurs and \( Y \) is the time when B occurs, calculate the probability \( P[X \geq 2Y] \), meaning that A happens at least after double the time that B occurred.

Answer: \dfrac{1}{3}


Question: A deck consists of 25 cards: 10 black cards, and 5 sets of 3 identical cards each in green, red, yellow, blue, and pink. How many different hands of 10 cards can be drawn from this deck?

Answer: 99404


Question: Let \( \{a_n\} \) be a sequence with \( a_n \ge 0 \). Assume the series \( \sum_{n=1}^{\infty} n^q a_n \) converges for any arbitrarily large \( q \). Given \( p \ge 1 \), determine if there exists a sufficiently large \( q \) such that the series \( \sum_{n=1}^{\infty} n^p \sqrt{a_n} \) also converges.

Answer: Yes


Question: Determine the interval of convergence for the power series \( F(x) = \sum_{n=0}^\infty g(1+1/n)x^n \), where \( g \) is a continuous function on \([0, 2]\) and \( g(t) = 2 \) as \( t \to 1 \).

Answer: (-1, 1)


Question: Let $a, b, c, d$ be four distinct non-zero integers such that $a + b + c + d = 0$. It is known that the number \( M = (bc - ad)(ac - bd)(ab - cd) \) lies strictly between 96100 and 98000. Determine the value of \( M \).

Answer: 97344


Question: Let $X$ and $Y$ be integral separated schemes of finite type over $\mathbb{C}$, with $Y$ being normal. Consider a surjective morphism of schemes $f: X \rightarrow Y$. Can the non-flat locus of $f$ be non-empty and have codimension at least 2 in $X$?

Answer: Yes


Question: Given that \(x + x^{-1} = \frac{1}{2}(1 + \sqrt{5})\), find the value of \(x^{2000} + x^{-2000}\).

Answer: 2


Question: Let \((B_t)_{t \geq 0}\) be a standard Brownian Motion. Define the stopping time \(T:=\inf\{t \geq 0: B_t=at-b\}\) for some positive constants \(a\) and \(b\). Calculate \(\mathbb{E}[T]\).

Answer: \dfrac{b}{a}


Question: Which is larger: a googolplex or \(10^{100!}\)? A googolplex is defined as \(10^{10^{100}}\).

Answer: 10^{100!}


Question: Find the value of the 7th derivative of the function \( y(x) = x^3 \cdot \tan(2x) \) at \( x = 0 \), denoted as \( y^{(7)}(0) \).

Answer: 0


Question: Find the remainder when \(3^{3^{3^{\cdots}}}\) (with 2020 copies of 3) is divided by 46.

Answer: 41


Question: Determine the cardinality of the set of all circles in the plane whose centers have rational coordinates and whose radii are the square roots of prime numbers.

Answer: \aleph_0


Question: Find the norm of the bounded linear operator \( S: l^{\infty} \to l^1 \) defined by \( S(x_1, x_2, x_3,...) = \left(\frac{x_1}{2}, \frac{x_1+x_2}{2^2}, \frac{x_1+x_2+x_3}{2^3},...\right) \).

Answer: 2


Question: Calculate the area enclosed by the region defined by the inequality \(x^{\frac{2}{3}} + y^{\frac{2}{3}} \leq 4\).

Answer: 24\pi


Question: A triangle has two of its vertices at the points (3.052, 0, 1.456) and (0, 3.052, 1.456). The third vertex lies on the curve defined by all points of the form (3.052, 3.052, a^2 + 1.456), where a is a real number. Express the area of the triangle as a function of a, and determine the value of a for which this area function attains its minimum value.

Answer: 0


Question: Let $R$ and $S$ be commutative rings, and let $f: R \to S$ be a surjective ring homomorphism. Suppose $I$ and $J$ are coprime ideals in $S$. Are the ideals $f^{-1}(I)$ and $f^{-1}(J)$ coprime in $R$?

Answer: Yes


Question:
Suppose $\mu_1$ and $\mu_2$ are finite and non-negative measures. Is it always true that for any measurable set $A$ and integrable function $f$, the following holds?
\[ \int_{x \in A} f(x) \, d\mu_1 (x) + \int_{x \in A} f(x) \, d\mu_2 (x) = \int_{x \in A} f(x) \, d(\mu_1 + \mu_2)(x) \]

Answer: Yes


Question: Use Green's Theorem to evaluate the integral \( \int \int_R {((\nabla \times F) \cdot k)} \, dx \, dy \) over the region in the first octant bounded by the curves \( y = x^2 \) and \( y = x \), where \( F = (xy^2, y+x) \).

Answer: \dfrac{1}{12}


Question: Determine whether the series \( u = \sum_{n=1}^{\infty}\frac{1}{2^{n}}e_n \) converges in the Banach space \( X \), where \( \{e_{n}\}_{n=1}^{\infty} \) is a maximal set of linearly independent vectors in \( X \). Specifically, are the partial sums \( \sum_{n=k}^{m}\frac{1}{2^{n}}e_n \) a Cauchy sequence?

Answer: Yes


Question: Let $f(x)$ be the unique polynomial that satisfies $f(n) = \sum_{i=1}^{n} i^{101}$ for all positive integers $n$. The leading coefficient of $f(n)$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are positive coprime integers. Find the value of $a + b$. \( \text{Express your answer as an integer.} \)

Answer: 103


Question:
Find the limit of the following expression as \( s \to 1 \):
\[ \lim_{s \to 1} \frac{D(s)}{\zeta(s)} \]
where \( D(s) = e^{i \theta} + \frac{e^{i \theta/2}}{2^s} + \frac{e^{i \theta / 3}}{3^s} + \dots \) is a Dirichlet series.

Answer: 1


Question: Determine the condition for repeated roots in the fourth-order polynomial equation: \( x^4 + ax^3 + bx^2 + cx + d = 0 \).

Answer: \Delta = 0


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} n(x^{1/n}-1) \]

Answer: \ln x


Question: Given a matrix $A \in \mathbb{R}^{N \times N}$ where there is a row (say the $i$-th row) such that $a_{ij} = \delta_{ij}b$ for all $j \in \{1, \ldots, N\}$, where $\delta_{ij} = \begin{cases} 0 & i \neq j \\ 1 & i = j \end{cases}$, use the Gershgorin circle theorem to determine if this matrix has at least one eigenvalue equal to $b$. Assume $A$ is not necessarily diagonal.

Answer: Yes


Question: Find all entire functions \( f(z) \) such that \( f(x) = e^x \) for all real numbers \( x \).

Answer: e^z


Question: Determine whether there exists a convex function \( \Phi : \mathbb{R}_{++} \to \mathbb{R} \) such that the function \( g(t) = \alpha t \cdot \Phi\left(\frac{1}{t} + 1\right) + 2(1-\alpha)t \cdot \Phi\left(\frac{1}{t}+\frac{1}{2}\right) \) is constant for \( t \in [1,2] \), where \( \alpha \in (0,1) \) is fixed.

Answer: Yes


Question: Find an integer \( n_1 \) such that the \( n_1 \)-th order Taylor polynomial for \( \sin(x) \) about \( x=0 \) approximates \( \sin(x) \) with an error of less than \( 5 \cdot 10^{-10} \) for all \( x \) in the interval \( [0, \frac{1}{2}\pi] \). You may assume \( 2 < \pi < 4 \). Show that your chosen \( n_1 \) satisfies the error requirement without using a calculator.

Answer: 15


Question: Let $R$ be a Noetherian local domain which is not a UFD, and let $P$ be a height one prime ideal of $R$. Determine if there exists an element $x \in P$ such that $P$ is the only minimal prime ideal containing $x$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of roots of the polynomial \( P(z) = z^5 + 2z^3 + 3 \) that lie within the closed unit disk \( \{z : |z| \le 1\} \).

Answer: 1


Question: Let \(a\) and \(b\) be integers such that \(\gcd(a, p) = \gcd(b, p) = 1\) and \(v_p(a-b) = n\), where \(p\) is an odd prime. Prove that for every natural number \(n\), the p-adic valuation of \(a^p - b^p\) is \(n+1\).

Answer: n + 1


Question: Given the surface $S_g = \#_g T^2$ where $g$ is a non-negative integer, determine if it is always possible to construct a covering space $S_h$ of $S_g$ such that $S_h$ is an $n$-sheeted covering and $h = n(g-1) + 1$ for some $n \ge 1$. Assume that the condition on the Euler characteristic is met.

Answer: Yes


Question: Compute the Fisher information $I_n(\theta)$ for the i.i.d. random variables $X_1, X_2, \ldots, X_n$ with density $\frac{1}{2} \exp(-|x- \theta|)$, where $\theta \in \mathbb{R}$ is unknown. Verify if $I_n(\theta)$ can be zero.

Answer: n


Question: Determine the limit of the sequence \( f_n(a) = a^{f_{n-1}(a)} \) as \( n \) approaches infinity, where \( f_0(a) = a \) and \( a = 1.1 \).

Answer: 1.1118


Question: Consider the set of dyadic rationals in the interval \([0, 1]\). If \(\{d_n\}\) is a sequence containing all distinct dyadic rationals in \([0, 1]\), does this sequence necessarily have a subsequence that converges to \(\frac{1}{2}\)?

Answer: Yes


Question: Calculate the limit: \[ \lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} (\alpha(k) - \beta(k)) \] where \(\alpha(n)\) is the number of even divisors of \(n\) and \(\beta(n)\) is the number of odd divisors of \(n\).

Answer: -\log 2


Question:
Consider the function defined as follows: \[ D(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q}, \\ 0 & \text{if } x \in \mathbb{R} \setminus \mathbb{Q} \end{cases} \] and \[ f(x) = x \cdot D(x). \] Determine which of the following statements is true about the differentiability of $f$ at $x=0$:

- $f$ is not differentiable at $x=0$
- $f'(0)=0$
- $f'(0)=\frac{1}{2}$
- $f'(0)=1$

Select the correct statement.

Answer: A


Question: Determine the number of non-equivalent colorings of the vertices of a regular hexagon using the colors red, white, and blue.

Answer: 92


Question: Given that \( \lim \limits_{x \to 1} \frac{f(x)+2}{x-1} = 3 \), where \( f(x) \) is a polynomial, compute the following limit: \[ \lim \limits_{x \to -1} \frac{(f(-x))^2-4}{x^2-1} \]

Answer: -6


Question: Determine if the function \( f(x) = \begin{cases} \frac{1}{10^n}, & \text{if } x \in (2^{-(n+1)}, 2^{-n}) \\ 0, & \text{if } x = 0 \end{cases} \) is integrable over the interval \([0, 1]\). If it is integrable, find the value of \( \int_0^1 f(x) \, dx \).

Answer: \dfrac{10}{19}


Question: Let \(G\) be a locally compact group and \(f\) belong to \(C_c(G)\), the space of continuous functions with compact support. Define an operator \(T_f\) on \(L^2(G)\) by \(T_f(g) = f * g\) (the convolution product). If \(T_f\) is positive and invertible, determine if \(\|T_f\|\) can belong to the point spectrum of \(T_f\).

Answer: Yes


Question:
Given that \( \left( \frac{r}{\sqrt{r^2-1}}, \frac{r}{\sqrt{r^2+1}} \right) \) are the roots of the equation \( x^2 - bx + 3 = 0 \), determine the value of \( b \). Choose the correct answer from the following options:

1) \( \pm 2\sqrt{6} \)
2) \( \pm 2\sqrt{3} \)
3) \( 2\sqrt{6} \)
4) \( 2\sqrt{3} \)

Answer: 1


Question: Determine the number of points in the interval \([0, 2\pi]\) where the function \( g(x) = \left[\cos x\right] \) is discontinuous. Here, \([\cdot]\) denotes the greatest integer function.

Answer: 4


Question: Determine the number of positive integers \(x\) such that \(x \leq 1000\) and \(x!\) is divisible by the sum of the first \(x\) positive integers, \(1 + 2 + 3 + \ldots + x\).

Answer: 833


Question: Find the expected squared distance to the origin of a $d$-dimensional spherical Gaussian centered at the origin with variance $\sigma^2$. Provide your answer in terms of $d$ and $\sigma^2$.

Answer: d \sigma^2


Question: Evaluate the surface integral \( \int_S \mathbf{F} \cdot \text{d}\mathbf{s} \) over a square plane, where the unit normal vector to the plane is \( \mathbf{a_n}=\frac{1}{\sqrt{2}}(\mathbf{a_y}+\mathbf{a_z}) \) and the dot product of the vector field \( \mathbf{F} \) and the plane's normal vector is \( \mathbf{F} \cdot \mathbf{a_n}=\frac{5}{\sqrt{2}} \). Determine the correct limits of integration and find the value of the integral, given that the answer is \( 20 \).

Answer: 20


Question: Evaluate the line integral \( \int_C y^3 \, dx - x^3 \, dy \) where \( C \) is the boundary of the circle \( x^2 + y^2 = 4 \) using Green's Theorem.

Answer: -24\pi


Question: Calculate the norm of the functional \( P(f) = f(1) + f(-1) - 2f(0) \) on the space of continuous functions \( C[-1,1] \).

Answer: 4


Question: Given the function \( f(x) = \ln(x) + \arctan(x) \), find \( (f^{-1})'\left(\frac{\pi}{4}\right) \).

Answer: \dfrac{2}{3}


Question: In triangle $ABC$, the medians from vertices $A$ and $B$ intersect at right angles at point $P$. Given that $AB = 12$, $BC = 13$, and $AC = 11$, find the length of $AP$. Express your answer as a simplified fraction.

Answer: \dfrac{19}{3}


Question: Let $a$ and $b$ be positive integers, and let $c$ be a non-negative integer. Determine whether the expression \( \frac{a^2 + b^2 - abc}{ab + 1} \) is always a perfect square if it is a positive integer.

Answer: Yes


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

Answer: 0


Question:
Determine the nature of the solution set for the equation $x^{10} = 10^x$. Choose the correct option:

(A) One negative and one positive number
(B) One negative and two positive numbers
(C) Two positive numbers
(D) Two negative and one positive number

Answer: B


Question: If $f(z) = u + iv$ is a complex function where both $u$ and $v$ are harmonic functions, is $f(z)$ necessarily analytic? Provide a justification for your answer.

Answer: No


Question: Find all functions \( f: \bar{\Omega} \rightarrow \mathbb{C} \) that satisfy the following conditions: \( f \) is continuous, \( f \) is holomorphic on \( \Omega \), and \( f(z) = e^z \) for all \( z \in \partial\Omega \).

Answer: e^z


Question: Determine the values of the positive parameter $a$ for which the integral \( \int_0^\infty \frac{\sin x}{x^{a}+x^{2a}}\, \mathrm{d}x \) is convergent.

Answer: (0, 2)


Question: In triangle ABC, H is the orthocenter. It is given that AM = MH and BN = NC. Calculate the measure of angle \( \angle BNM \).

Answer: 90^\circ


Question:
Determine the existence of positive real solutions for the following equations:

1. \( x^{x^{2018}} = 2018 \)
2. \( x^{x^{x^{\ldots}}} = 2018 \)

Choose the correct statement:

1) One equation has a positive real solution, the other does not.
2) Both equations have positive real solutions, and they are the same.
3) Both equations have positive real solutions, but they are different.

Answer: 1


Question:
Determine which of the following functions is the solution to the integral equation \( u(x) = x + \int_0^x (t-x)u(t)dt \):

(a) \( \sin(x) \)

(b) \( x\cos(x) \)

(c) \( \ln(1+x) \)

(d) \( xe^{-x} \)

(e) \( xe^x \)

Answer: a


Question: Find the value of \( n \) such that \( \lim_{x\rightarrow 0}\frac{x^n\sin^{n}(x)}{x^{n}-(\sin x)^{n}} \) is a finite non-zero number.

Answer: 2


Question: Given the equations \( xy + y = 3x^2 + x + 5xy \) and \( (8x^2 + xy + y^2)g(x,y) = (2x^2 + 3xy + 5y^2)h(x,y) \), where \( h(0, 0) = 10 \), find the value of \( g(0, 0) \).

Answer: 10


Question: Evaluate the limit \( \lim_{x\rightarrow 0}(1+\arctan(\frac{x}{2}))^{\frac{2}{x}} \) without using L'Hôpital's rule.

Answer: e


Question: Determine the maximum dimension of a subspace M of the space of 6x6 skew-symmetric matrices over the real numbers, where all non-zero elements of M have rank 2.

Answer: 5


Question: Transform the expression \( \sum_{k=0}^{n-1} k \cdot 2^{n - 1 - k} \) into the equivalent expression \( 2^n - n - 1 \).

Answer: 2^n - n - 1


Question: Determine the values of \( \lambda \) for which the function \( N_\lambda(x,y) = \sqrt{x^2 + y^2 + 2\lambda xy} \) defines a norm on \( \mathbb{R}^2 \).

Answer: (-1, 1)


Question: Let $X$ be a compact Hausdorff space and let $C(X,I)$ be the set of all continuous functions from $X$ into the closed interval $[0,1]$. If we equip $C(X,I)$ with the topology of uniform convergence, is $C(X,I)$ compact? Provide a justification for your answer.

Answer: No


Question: Suppose \(X_1, \ldots, X_n\) are a random sample from a normal distribution with mean \(\theta\) and variance \(\sigma^2\), where \(\sigma^2\) is fixed and \(\theta > 0\) is a parameter. Find the maximum likelihood estimator of \(\sqrt{\theta}\).

Answer: \sqrt{\overline{X}}


Question: $A, B, C,$ and $D$ are collinear points such that $AB = BC = CD$. Point $P$ lies on a circle with $BC$ as its diameter. Calculate the value of $\tan\angle APB \cdot \tan\angle CPD$. Provide your answer as a single expression.

Answer: \dfrac{1}{4}


Question: Is it true that any orthogonal projector onto an eigenspace of a self-adjoint linear operator on a Euclidean finite-dimensional space can be represented as a polynomial of the operator?

Answer: Yes


Question: Calculate the sum of the series \( \sum\limits_{n=1}^8 3(3n-2)(-2)^{n-1} \).

Answer: -5889


Question: Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function that is differentiable everywhere except possibly at \( x = 0 \). If \( f'(x) \to 0 \) as \( x \to 0 \), is \( f \) differentiable at \( x = 0 \)? Provide a justification for your answer.

Answer: Yes


Question: Let $O$ be an open set in $\mathbb{R}^m$. Does there exist a real-valued function $f(x)$ defined on $O$ and an open subset $V$ of $O$ such that $f$ is not locally bounded at any point of $V$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{n\rightarrow\infty} n^2C_n \) where \( C_n=\int_{\frac{1}{n+1}}^{\frac{1}{n}}\frac{\tan^{-1}nx}{\sin^{-1}nx}dx \).

Answer: \dfrac{1}{2}


Question:
Determine which of the following substitutions can transform the differential equation \(y' = \frac{y}{x^2} + x^3y^3\) into a first-order linear equation in terms of \(v(x)\):

1. \(v(x) = x^3y^3\)
2. \(v(x) = y^2\)
3. \(v(x) = y^{-2}\)
4. \(v(x) = \frac{y}{x}\)

Provide the correct substitution.

Answer: 3


Question: Given that $e^{f(x)} = \ln x$, and $g(x)$ is the inverse of $f(x)$, find the derivative of $g(x)$. Use the formula for the derivative of an inverse function: $$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}.$$

Answer: e^{x + e^x}


Question: In the sequence \( \{a_n\} \), where \( a_1 = 2 \) and \( 2a_{n+1} = a_n^2 + 1 \), define \( b_n = \frac{2a_n - 1}{a_n + 1} \). Determine the maximum integer \( t \) such that \( b_1 + b_2 + \cdots + b_{2019} > t \).

Answer: 4035


Question: Find the maximum value of the expression \((a-d)(\beta-\gamma)+b(-\alpha+\gamma+\delta)+c(\alpha-\beta-\delta)\) given that \(0 \le a, b, c, d \le 1\) and \(a+b+c+d=1\), and \(0 \le \alpha, \beta, \gamma, \delta \le 1\) with \(\alpha+\beta+\gamma+\delta=1\).

Answer: 1


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

Answer: 0


Question: Find the remainder when \(1^{2016} + 2^{2016} + \cdots + 2016^{2016}\) is divided by 2016.

Answer: 48


Question: Consider topological spaces $X$ and $Y$ with maps $f: X \to Y$ and $g: Y \to X$ such that $g \circ f \simeq 1_X$. If $X \not\simeq Y$, is it true that the homotopy dimension of $X$ is strictly smaller than the homotopy dimension of $Y$?

Answer: No


Question: Given an n x n matrix, determine the number of lists that can be formed by selecting one element from each row, such that no two lists differ by only one element. Calculate the total number of such lists.

Answer: n^{n-1}


Question: Consider a continuous function \( f : \mathbb{R} \to \mathbb{R} \) such that \( f(i) = 0 \) for all integers \( i \). Can the image of this function be not closed? Provide a justification for your answer.

Answer: Yes


Question: Given the function \( f(x) = 8 - x^4 \) defined on the interval \([0, 1]\), and the partition \( P_n = \{0, \frac{1}{n}, \frac{2}{n}, \ldots, 1\} \) of \([0, 1]\) into \( n \) subintervals of equal width, find the value of \( \sup\{L(f, P_n) : n \in \mathbb{N}\} \). Use the formula \( \sum_{k=1}^{n} k^4 = \frac{1}{5} n^5 + \frac{1}{2} n^4 + \frac{1}{3} n^3 - \frac{1}{30}n \) to assist in your calculations.

Answer: \dfrac{39}{5}


Question: Consider a shrinking Ricci soliton $(M,g,X)$. Determine if it is possible for the Ricci curvature $Ric$ to satisfy the inequality $$Ric_x(v) \leq \frac{C}{r} \quad \forall v \in T_xM \text{ and } \forall x \in B(2r),$$ where $B(2r)$ is the geodesic ball with radius $r$ centered at a fixed point $o \in M$, and $C > 0$ is a constant.

Answer: Yes


Question: Given the functional equation \( f\left(\frac{x+y}{3}\right) = \frac{2+f(x)+f(y)}{3} \) for all real \( x \) and \( y \), and the condition \( f'(2) = 2 \), determine the function \( f(x) \).

Answer: 2x + 2


Question: Solve the differential equation \( y'+6y(t)+9\int_0^t y(\tau)d\tau=1 \) with the initial condition \( y(0)=0 \).

Answer: t e^{-3t}


Question: Evaluate the line integral \( \int_C \frac{\mathrm{Re}(z)}{z} \, \mathrm{d}z \), where \( C \) is the shortest path from \( 1 + i \) to \( 5 + 5i \).

Answer: 4


Question: Let X and Y be independent random variables that are both uniformly distributed on the interval [0,1]. Calculate the expected value \( E[Y\,|\,X<Y^2] \).

Answer: \dfrac{3}{4}


Question: Find the partial derivative of the expression $x^TA^TAx$ with respect to the matrix $A$, where $x \in \mathbb{R}^n$ is a vector and $A \in \mathbb{R}^{m \times n}$ is a matrix.

Answer: 2 A x x^T


Question: Is it necessary for a Riemann integrable function on the interval $[a, b]$ to have left and right hand limits at every point within the interval?

Answer: No


Question: Let $p$ be a prime number and $d$ be a divisor of $p-1$. Prove that the congruence $x^d \equiv 1 \pmod{p}$ has exactly $d$ incongruent solutions.

Answer: d


Question: Determine the number of distinct groups of order 12.

Answer: 5


Question: Let $H$ and $K$ be normal subgroups of a group $G$ such that $H \cap K = \{1_G\}$. Prove that $xy = yx$ for all $x \in H$ and $y \in K$. Provide a detailed proof.

Answer: xy = yx


Question: Calculate the integral \( \int_{\gamma}\frac{1}{z}dz \) using the definition of integral along a curve, where \( \gamma \) is the contour of a circle that does not contain the origin.

Answer: 0


Question: Evaluate the integral \( \int\limits_{|x|\geq1}\frac{1}{|x|^n}dx \) over \( x \in \mathbb{R}^n \).

Answer: \infty


Question: Find \( \frac{dy}{dz} \) when \( y = \cot^{-1}(f(x)) \) and \( z = \cos^{-1}(\sqrt{1-x^4}) \), where \( f(x) = \frac{2\sqrt{1+x^2}-5\sqrt{1-x^2}}{5\sqrt{1+x^2}+2\sqrt{1-x^2}} \).

Answer: -\dfrac{1}{2}


Question: Determine whether the following statement is true for all functions $f(x)$ that have a series expansion at $x=0$: \[ \frac{d O(1)}{d x} = O(1) \] Here, $O$ denotes the big-O notation, and the expansion is around the point $x=0$.

Answer: True


Question: Determine the maximal \( \delta > 0 \) such that \( |f(x)| < \frac{1}{2} \) for all \( x \in (1 - \delta, 1 + \delta) \), where \( f(x) = e \cdot \ln(x) - x + 1 \). If a maximal \( \delta \) does not exist, find any such \( \delta \).

Answer: 0.238


Question: Determine whether the sequence \( n!^{\frac{1}{n}} \) converges as \( n \to \infty \).

Answer: \infty


Question: Let $C \subset [0,1]^2$ be the union of all sets of the form $\{p/q\} \times [0,1/q]$, where $p/q$ is a rational number in $[0,1]$ written in lowest terms. Prove that the set $C$ has content 0.

Answer: 0


Question: Determine the degree of the differential equation: \((y''')^{\frac{4}{3}}+(y')^{\frac{1}{5}}+ y = 0\).

Answer: 20


Question: Evaluate the limit: \[ L = \lim_{x \to 0} \frac{\cos(\sin x) - (1 + x^2)^{-\frac{1}{2}}}{x^4} \]

Answer: -\dfrac{1}{6}


Question: Evaluate the limit: $$\lim_{x \to 0^+} (\ln{x} + \cot{x})$$

Answer: \infty


Question: Evaluate the integral: \[ \int_{0}^{1}{\sqrt[3]{2x^3-3x^2-x+1}}\,\mathrm{d}x. \]

Answer: 0


Question: Is it always true that for any Hermitian matrix \( A = A^{\dagger} \), there exists a unitary matrix \( P \) such that \( PAP^{\dagger} = D \), where \( D \) is a diagonal matrix?

Answer: Yes


Question: Is there an injective map from the set of continuous real-valued functions on $[0, 1]$, denoted as $C[0, 1]$, to the set of all sequences of real numbers, $\\mathbb{R}^{\mathbb{N}}$?

Answer: Yes


Question: Determine whether the following conjecture is true or false: For every undirected graph $G=(V,E)$, there exist three pairwise disjoint sets of vertices $V_1,V_2,V_3$ such that for every $i\in\{1,2,3\}$ and for every cycle $C$ in $G$, $C\cap V_i\neq \emptyset$. Each $V_i$ must intersect all cycles in $G$, and the $V_i$ are pairwise disjoint.

Answer: False


Question: Calculate the limit: \[ \lim\limits_{(x,y)\to(0,0)}{\frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4}} \]

Answer: 0


Question:
Determine if there exist positive integers \(m, k, n\) such that \(k\) is a non-multiple of 3 and the following condition holds:
\[
\frac{2^n k -\sum_{i=0}^{n-1} 2^{n-1-i} \times 3^i}{3^n}= m < k
\]

Answer: Yes


Question:
Find the volume of the region in \( \mathbb{R}^3 \) that satisfies the following conditions:
\[ 0 < \sqrt{x^2+y^2} \leq x^2+y^2 \leq z\sqrt{x^2+y^2} \leq 2 \sqrt{x^2+y^2} + y. \]

Answer: \dfrac{7\pi}{3}


Question: Let $f: A \to \mathbb{C}$ be an analytic function on a convex region $A$, such that $f(z) \neq 0$ for all $z \in A$. Is it always true that there exists an analytic function $g(z)$ on $A$ such that $f(z) = e^{g(z)}$ for all $z \in A$?

Answer: Yes


Question: Determine the best bound in the form $O(A^k)$ for the expression \( \min\big\{A^2, \frac{1}{B^2}\big\}(1+BA) \), where \( A \in \mathbb{N} \) and \( 0 \leq B \leq 1 \).

Answer: O(A^2)


Question: Determine whether the following statement is true: For a sequence of linear operators \(T_n\), is it always the case that \(\left\|\lim_{n} T_{n}\right\| = \lim_{n} \left\|T_{n}\right\|\)?

Answer: False


Question: Find the value of the real number \( \alpha \) such that \( \exp(2B(t) - \alpha t) \) is a martingale.

Answer: 2


Question: Given that $f(x)$ is an injective function defined by the equation \( f\left(x + \frac{1}{f(y)}\right) = \frac{f(x)f(y)}{f(x) + f(y)} \) for \( x, y \geq 0 \), and \( f(0) = 1 \), find the value of \( f(2012) \).

Answer: \dfrac{1}{2013}


Question: Prove in Peano Arithmetic (PA) that for all natural numbers \(v_0\) and \(v_1\), if \(v_0 < v_1\), then there exists a natural number \(v_2\) such that \(v_0 + v_2 = v_1\).

Answer: \forall v_0 \forall v_1 (v_0 < v_1 \rightarrow \exists v_2 (v_0 + v_2 = v_1))


Question: Let $f(x+y)=f(x)f(y)$ for all $x, y \in \Bbb R$. Suppose that $f(3)=3$. Find the value of $f'(3)$. \( \text{(Express your answer as a number.)} \)

Answer: \ln 3


Question: Find the maximum value of \(|\ln(x) - f(x)|\) for \(0.3 \le x \le 1.7\), where \(f(x)\) is the sum of the first three nonzero terms of the Taylor series for \(\ln(x)\) centered at \(x=1\).

Answer: 0.1446


Question: Consider a function \( m \in L^2(0,1) \) such that \( \int_0^1 m(x)^2 \, dx < \infty \) and \( m \not\in L^\infty(0,1) \). Determine whether \( \int_{|m(x)|>A} m(x)^2 \, dx \to 0 \) as \( A \to \infty \).

Answer: Yes


Question: Let $k$ be a field, $K/k$ a separable quadratic extension, and $D/K$ a central division algebra of dimension $r^2$ over $K$ with an involution $\sigma$ of the second kind (i.e., $\sigma$ acts non-trivially on $K$ and trivially on $k$). Does there exist a field extension $F/k$ such that $L := K \otimes_k F$ is a field, and $D \otimes_K L$ splits (i.e., is isomorphic to the matrix algebra $M_r(L)$ over $L$)?

Answer: Yes


Question: Find the limit: \[ \lim_{x \to +\infty}(x^{\frac{1}{x}}-1)^{\frac{1}{\ln x}}. \]

Answer: \dfrac{1}{e}


Question:
Calculate the following limit:
\[ \lim_{n \to \infty} \inf_{x \in \mathbb{R}} \left( \sum_{k=0}^{2n} x^k \right) \]
Simplify the expression to:
\[ \lim_{n \to \infty} \inf_{x \in \mathbb{R}} \left( \frac{x^{2n+1} - 1}{x - 1} \right) \]

Answer: \dfrac{1}{2}


Question:
Determine the values of \( z \in \mathbb{C} \) for which \( A_2(z) = 0 \), where \( e^{jz} = A_0(z) + j A_1(z) + j^2 A_2(z) \) and:
\[
\begin{cases}
A_0(z) = \displaystyle \sum_{n \geq 0} \frac{x^{3n}}{(3n)!} \\
A_1(z) = \displaystyle \sum_{n \geq 0} \frac{x^{3n+1}}{(3n+1)!} \\
A_2(z) = \displaystyle \sum_{n \geq 0} \frac{x^{3n+2}}{(3n+2)!}
\end{cases}
\]
with \( j = e^{i \frac{2\pi}{3}} \).

Answer: 0


Question: Calculate the limit: \[ \lim_{n\to\infty} \left(\frac{1}{n}(n!)^{1/n}\right) \]

Answer: \dfrac{1}{e}


Question: Given the joint probability density function \( f(x,y) = \begin{cases} x y & \text{if } 0 \leq x \leq 2 \text{ and } 0 \leq y \leq 1, \\ 0 & \text{elsewhere.} \end{cases} \), find the probability \( P(X/2 < Y < X) \).

Answer: \dfrac{3}{8}


Question: Let $X$ be a normed linear space and $C$ be a compact convex subset of $X$. Is it true that for every open set $V$ in $X$ containing $C$, there exists a convex open set $U$ such that $C \subseteq U \subseteq V$?

Answer: Yes


Question: Evaluate the double integral \( \iint_D \sin\left(\frac{y-x}{y+x}\right) \, dy \, dx \) over the trapezoid \( D \) with vertices \((0,1), (0,2), (2,0), (1,0)\).

Answer: 0


Question: Calculate the multiplicity of the intersection of the two plane curves \(F=2X_0^2X_2-4X_0X_1^2+X_0X_1X_2+X_1^2X_2\) and \(G=4X_0^2X_2-4X_0X_1^2+X_0X_1X_2-X_1^2X_2\) at the point \((1:0:0)\).

Answer: 2


Question: Let \( \varphi : \mathbb{R}[x] \to \mathbb{C} \) be defined by \( f(x) \mapsto f(2+i) \). Find the generator for the kernel \( \ker \varphi \).

Answer: x^2 - 4x + 5


Question:
Let $X$ be a normal Hausdorff space. Let $A_1$, $A_2$, and $A_3$ be closed subsets of $X$ which are pairwise disjoint. Which of the following statements is true regarding the existence of a continuous real-valued function $f$ on $X$ such that $f(x) = a_i$ if $x$ belongs to $A_i$, $i=1,2,3$?

(A) This is possible iff each $a_i$ is either 0 or 1.

(B) This is possible iff at least two of the numbers $a_1$, $a_2$, $a_3$ are equal.

(C) This is possible for all real values of $a_1$, $a_2$, $a_3$.

(D) This is possible only if one among the sets $A_1$, $A_2$, $A_3$ is empty.

Answer: C


Question: Let $f_n, f: X \to \mathbb{R}$ be continuous functions such that $\frac{1}{n}f_{n}(x) \to f(x)$ as $n \to \infty$. Determine if it is true that $\left|\frac{1}{n-K}f_n(x)-\frac{n-H}{n}f(x)\right| \to 0$ as $n \to \infty$ for some integers $H, K \in \mathbb{N}$. Justify your answer.

Answer: True


Question:
Consider the Schrödinger operator \(H = -\frac{d^2}{dx^2} + V(x)\) where \(V: \mathbb{R} \rightarrow \mathbb{R}^{+ *}\) is a real positive function such that \(\lim_{x \to \pm\infty} V(x) = +\infty\). The operator \(H\) has a pure discrete spectrum \((\lambda_i)_{i \geq 0}\) with \(\lim_{i \to +\infty} \lambda_i = + \infty\). Each eigenvalue \(\lambda_i\) has an associated eigenfunction \(\phi_i \in L^2(\mathbb{R})\) satisfying:

\[-\phi_i''(x) + V(x)\phi_i(x) = \lambda_i \phi_i(x), \quad \forall x \in \mathbb{R}\]

and \(||\phi_i||_{L^2(\mathbb{R})} = 1\). Determine if there exists a constant \(M > 0\) such that for all \(i \geq 0\), the eigenfunctions satisfy \(|| \phi_i ||_\infty < M\).

Answer: Yes


Question:
Calculate the value of \( S = \frac{(b+c)bc}{P'(a)} + \frac{(c+a)ca}{P'(b)} + \frac{(a+b)ab}{P'(c)} \), where \( P(x) = x^3 - mx^2 - nx - p \) and \( a, b, c \) are the roots of \( P(x) \). Using Vieta's formulas, we have:
\[ a + b + c = m, \quad ab + bc + ac = -n, \quad abc = p. \]
The possible values for \( S \) are:
\( a) \; S = m \)
\( b) \; S = m+n \)
\( c) \; S = m+n+p \)
Determine the correct value of \( S \).

Answer: a


Question: Find all pairs of prime numbers \((p, q)\) such that \(p^3 - q^5 = (p+q)^2\).

Answer: (7, 3)


Question: Find the residue of the function \( \sin\left(\frac{1}{z}\right) \) at its poles.

Answer: 1


Question: Determine the third homotopy group of $SO(5)$, denoted as $\pi_3(SO(5))$. Provide your answer in standard group notation.

Answer: \mathbb{Z}


Question: A beacon light is located 4 miles offshore, perpendicular to a straight shoreline, and rotates at 3 revolutions per minute. Determine the speed at which the spot of light moves along the shoreline when the beam forms a 60° angle with the shoreline.

Answer: 32\pi


Question: Determine the real parameter \( m \) such that the substitution \( y = z^m \) transforms the differential equation \( 2x^4yy' + y^4 = 4x^6 \) into a first-order homogeneous differential equation.

Answer: \dfrac{3}{2}


Question: Find all solutions for the equation \( \tan(\alpha) = \cos(\alpha + 33.44^\circ) \) within the interval \( 0 < \alpha < 90^\circ \).

Answer: 26.56^\circ


Question: Find the maximum value of the function \( f(x, y) = xy \) subject to the constraint \( 4x^2 + 9y^2 = 32 \).

Answer: \dfrac{8}{3}


Question: Find the polynomial \( P(z) \) such that \( P(x - \frac{1}{x}) = x^5 - \frac{1}{x^5} \).

Answer: z^5 + 5z^3 + 5z


Question: Find the maximum value of the function \( f(x) = 12x^2\int_0^1 yf(y) \, dy + 20x\int_0^1 y^2 f(y) \, dy + 4x \).

Answer: \dfrac{1}{8}


Question: Evaluate the limit: $$ \lim_{n \to \infty }\int_{0}^{\pi} x^n\sin x \,\mathrm{d}x $$

Answer: \infty


Question: Is there a linear time algorithm to determine if the complement of a given undirected graph is bipartite?

Answer: Yes


Question: Evaluate the double integral \( \int_0^{1/2} \int_x^{1-x}  (x+y)^9(x-y)^9 \, dy \, dx \).

Answer: -\dfrac{1}{400}


Question: List all monic irreducible polynomials of degree 2 in the finite field $F_3$. How many such polynomials are there?

Answer: 3


Question: Calculate \( \lim_{n\rightarrow \infty }\int_{0}^{\pi /2}\sqrt[n]{\sin^nx+\cos^nx}\,dx \).

Answer: \sqrt{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty} \sqrt[n]{n^e+e^n} = e \]

Answer: e


Question: If the Lebesgue integral of a strictly positive function over a set \( A \) is zero, does it imply that the Lebesgue measure of \( A \) is zero?

Answer: Yes


Question: Find the minimum value of the function \( \sqrt{2x} - \left\lceil \frac{\sqrt{1+8x}}{2} \right\rceil \) for positive integer values of \( x \).

Answer: -1


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

Let $E^n$ be an $n$-dimensional vector space with a dot product $\langle\cdot,\cdot\rangle$. Suppose $f:E^n\to E^n$ is an isomorphism such that for all $x,y\in E^n$, if $||x||=||y||$, then $||f(x)||=||f(y)||$. Then there exists $\kappa>0$ such that $\langle f(x),f(y)\rangle=\kappa \langle x,y\rangle$ for any $x,y\in E^n$. Is this statement true?

Answer: True


Question: Determine the maximal possible dimension of a subspace \( U \subseteq V \), where \( V \) is the vector space of functions from \([0,1]\) to \( \mathbb{R} \), and \( U \) consists of monotone functions.

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\sqrt{1 + \tan x} - \sqrt{1 + \sin x}}{x^3} \]

Answer: \dfrac{1}{4}


Question: Given the function \( f(x) = \frac{x^3 + 1}{3} \) with fixed points \( \alpha, \beta, \gamma \) in the intervals \((-2,-1), (0, 1)\), and \((1, 2)\) respectively, a sequence is defined as \(x_1 = \gamma - 0.01\) and \(x_{n+1} = f(x_n)\) for all \(n = 1, 2, 3, \ldots\). Given that the sequence converges, find \( \lim_{n\rightarrow\infty}x_n \).

Answer: \beta


Question: Determine the interval of values for $x$ such that \((x^2-5x+4)(y^2+y+1)<2y\) holds for all real values of $y$. What is the interval for $x$?

Answer: (2, 3)


Question: If a matrix has linearly independent columns, does it automatically have a left inverse? Answer yes or no.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{\int_0^{\sin x} \sqrt{\tan t} \,\mathrm dt}{\int_0^{\tan x} \sqrt{\sin t} \,\mathrm dt}. \]

Answer: 1


Question: Let \( f(z) = \sum_{n=0}^{\infty} a_n z^n \) be a function given by a power series with radius of convergence \( R \). Define \( g(z) = \frac{1}{3}(f(z) + f(w \cdot z) + f(w^2 \cdot z)) \), where \( w = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) \). Find the power series for \( g(z) \) and estimate its radius of convergence.

Answer: R


Question: Let $X$ be a normal random variable with mean $2$ and variance $4$. Find the value of $a$ that maximizes $g(a) = P(a \leq X \leq a+2)$. Provide your answer as a single value of $a$.

Answer: 1


Question: Consider \( \mathbb{R}^n \) with the usual metric. Let \( U \subset \mathbb{R}^n \) be an open set and \( K \subset U \) a compact set. Is it true that there exists an open set \( U^{'} \) in \( \mathbb{R}^n \) such that \( \overline{U^{'}} \subset U \) and \( K \subset U^{'} \)?

Answer: Yes


Question: Embed \(\mathbb{R}^2\) in the projective plane \(\mathbb{RP}^2\) by the map \((x,y)\rightarrow [1,x,y]\). Find the point of intersection in \(\mathbb{RP}^2\) of the projective lines corresponding to the parallel lines \(y = mx\) and \(y = mx+c\) in \(\mathbb{R}^2\).

Answer: [0, 1, m]


Question: Let $E$ be a smooth vector bundle over a manifold $M$ with dimension greater than 1, equipped with a metric. Consider a metric connection $\nabla$ on $E$. Suppose there exists a non-zero section $\sigma \in \Gamma(E)$ such that $\sigma$ lies in $\ker R(X,Y)$ for all $X,Y \in \Gamma(TM)$. Does $\nabla$ admit a parallel section locally?

Answer: Yes


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

Answer: \infty


Question: Evaluate the triple integral \( \iiint_D f(x,y,z) \, dxdydz \) where \( f(x,y,z) = \sqrt{x^2+y^2} \) and \( D \) is the region inside the cone defined by \( z^2 = x^2 + y^2 \), bounded by \( z = 0 \) and \( z = 1 \).

Answer: \dfrac{\pi}{6}


Question: Find a function \( f(n) \) such that the limit \[ \lim_{n\to\infty}f(n)\left(1-\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} \,\mathrm{d}x_1\cdot\mathrm{d}x_2\cdots\mathrm{d}x_n\right) \] is finite. Determine the precise value of this limit.

Answer: 1


Question:
Evaluate the integral \( \iint_{\mathcal{D}} x^{3}+y^{3}\,\mathrm{d}y\,\mathrm{d}x \) over the domain \( \mathcal{D} = \lbrace (x,y)\in\mathbb{R}^{2}:1<xy<2,x^{2}<y<2x^{2}\rbrace. \) Choose the correct answer from the following options:

(i) \( \frac{3}{4} \),
(ii) \( 1 \),
(iii) \( \frac{36}{37} \),
(iv) \( \frac{37}{36} \).

Answer: iv


Question: Compute the closed form of the series \( \sum_{k=0}^\infty k(k-1)\cdots(k-r+1) \frac{ \alpha^k e^{- \alpha}}{k!} \) for \(r=2,3,\ldots\) and \(\alpha > 0\).

Answer: \alpha^r


Question: A variable line \(ax+by+c=0\), where \(a, b, c\) are in arithmetic progression, is normal to the circle \((x-\alpha)^2+(y-\beta)^2=\gamma\). This circle is orthogonal to the circle \(x^2+y^2-4x-4y-1=0\). Find the value of \(\alpha + \beta + \gamma\).

Answer: 7