Question:
Determine whether the following limit converges to 0.5:
\[ \lim_{n\to\infty}\frac{\sum_{r=\lceil\frac{n}{2}\rceil}^n\binom{n}{r}}{\sum_{r=0}^{n}\binom{n}{r}} = 0.5 \]

Answer: 0.5


Question: Let $D$ be the open ball of center $x_{0} \in \mathbb{R}^{k}$ with $k \geq 1$ and radius $R > 0$. Let $f$, $g$, and $h$ be three positive real analytic functions on $D$ such that $f(x) = g(x) + h(x)$ for all $x \in D$. Suppose there exists a positive number $r < R$ such that the Taylor series of $f$ converges for $|x - x_{0}| < r$. Can we conclude that the Taylor series of $g$ and $h$ also converge for $|x - x_{0}| < r$?

Answer: Yes


Question:
Determine whether the following inequality holds for all \( |x| \neq 0 \) in \( \mathbb{R}^3 \) with \( \lambda > 0 \):
\[
\frac{|e^{i\sqrt{\lambda+i\varepsilon}|x|}-e^{i\sqrt{\lambda}|x|}|^2}{|x|^2} \leq \frac{C}{|x|^2}
\]
where \( \sqrt{\lambda+i\varepsilon} = \sqrt{\lambda_\varepsilon} + i\frac{\varepsilon}{2\sqrt{\lambda_\varepsilon}} \) and \( \lambda_\varepsilon = \frac{\sqrt{\lambda^2+\varepsilon^2}+\lambda}{2} \).

Answer: Yes


Question: Find all continuous and non-negative functions \( f(t) \) such that \( f(t) \leq \int_0^t f(s) \, ds \) for \( 0 \leq t \leq 1 \).

Answer: 0


Question:
Let \( f: \mathbb{C} \setminus \{0\} \to \mathbb{C} \) be a function such that \( f(z) = f\left(\frac{z}{|z|}\right) \) and its restriction to the unit circle is continuous. Determine which of the following statements is true:

1. \( \lim_{z \to 0} f(z) \) exists.
2. \( f \) is analytic but not necessarily a constant function.
3. \( f \) is continuous but not necessarily analytic.
4. \( f \) is a constant function.

Answer: C


Question: Given the equation \( \int_0^x f(t) \, dt = x + \int_0^1 f(t) \, dt \), find the value of \( f(1) \).

Answer: 1


Question: Given a non-singular matrix $A \in \mathbb{R}^{n\times n}$ with spectral radius $\rho(A)$, determine if there exists a constant $a > 0$ such that for all $k \in \mathbb{N}_0$, the inequality \( \|A^k\| \leq a k^{n-1} \rho^k(A) \) holds, where \( \|\cdot\| \) is an induced matrix norm.

Answer: Yes


Question: Determine whether the subring \( \mathbb{Z}_3 + \mathbb{Z}_3 \mathbf{i} + \mathbb{Z}_3 \mathbf{j} + \mathbb{Z}_3 \mathbf{k} \) of the quaternion division ring over \( \mathbb{Q}_3 \) is a noncommutative local ring. Specifically, does it have a unique maximal left ideal (or equivalently, a unique maximal right ideal)?

Answer: Yes


Question: Consider a topological space $X$ and a topological group $T$ that acts continuously on $X$ on the right. Let $(X, \mathbb{R})$ be a topological group, and fix $x \in X$. For an open neighborhood $U$ of $x$, define $A_x(U)$ as the set of all $t \in \mathbb{R}$ such that $x.t \in U$. We say $x$ is an almost periodic point if for every neighborhood $U$ of $x$, there exists a compact set $K_x(U)$ such that $\mathbb{R} = K_x(U) + A_x(U)$. Does this condition imply that $A_x(U)$ is relatively dense? Justify your answer.

Answer: Yes


Question: Find the maximum perimeter of a rectangle inscribed in the ellipse \(\frac{x^2}{20} + \frac{y^2}{12} = 1\).

Answer: 16\sqrt{2}


Question: Let \( f(x) \) be a continuous function satisfying \( f(x^2+1) = \frac{x^2}{f(2^x)-1} \) and \( f(x) > 0 \) for all \( x \in \mathbb{R} \). Find \( \lim_{x\rightarrow 1}f(x) \).

Answer: 1


Question: For any positive number \( \alpha \), is there a zero-dimensional ideal \( I \) in \( K[x_1, \dots, x_n] \) such that the difference \( \dim_K(R/I) - |V(I)| \geq \alpha \)? Here, \( V(I) \) is the variety defined as \( \{ (\alpha_1, \dots, \alpha_n) \in K^n : f((\alpha_1, \dots, \alpha_n)) = 0 \ \forall f \in I \} \).

Answer: Yes


Question: Calculate the Frobenius norm \(||AA^+ - I||_F\), where \(A^+\) is the Moore–Penrose inverse of matrix \(A\). The matrix \(A\) has dimensions \(n \times m\) and rank \(r\). The Frobenius norm is defined as \(||A||_F = \sqrt{\text{Tr}(AA^T)}\).

Answer: \sqrt{n - r}


Question: Find the maximum area of a triangle inscribed in the ellipse given by the equation \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \).

Answer: 15\sqrt{3}


Question: Evaluate the integral \( \int_{-\infty}^{\infty} e^{-e^{-x} - x} \, dx \) and confirm that it equals 1.

Answer: 1


Question: Determine the values of $t \in \mathbb{R}$ for which the sequence $b_n = \sqrt{n^2 + n^t} - n$ converges as $n \to \infty$. Specifically, analyze the behavior of the sequence for $t \in (1, 2)$.

Answer: (-\infty, 1]


Question: Given the vector function \( r(t) = (3t + 4\sin(t) + 4\cos(t)) \), use the curvature formula \( \frac{|r' \times r''|}{|r'|^3} \) to find the curvature. Verify your calculations to ensure the result is \( \frac{4}{25} \).

Answer: \dfrac{4}{25}


Question: Represent the function \( \frac{x^2}{(8+x)^3} \) as a power series and determine its radius of convergence.

Answer: 8


Question: Given the matrix \( A = \begin{bmatrix}1 & -2 & -1 \\ 1 & 2 & 1 \\ -1 & 0 & 1\end{bmatrix} \), find the trace of the matrix \( A^{2014} \). The eigenvalues of \( A \) are \( 2 \), \( 1 + i \), and \( 1 - i \).

Answer: 2^{2014}


Question:
Consider the sequence \(x_n\) defined for \(n \geq 0\) with \(0 < x_0 < 1\) and the recurrence relation:
\[ x_{n+1} = x_n - x_n^2 + x_n^3 - x_n^4 + x_n^5 - x_n^6. \]
Calculate the limit:
\[ \lim_{n \to \infty} n x_n. \]

Answer: 1


Question: Let $f: G \to H$ be a surjective homomorphism from a compact Lie group $G$ to an Abelian Lie group $H$. Is the restriction of $f$ to the maximal torus $T$ of $G$ also surjective?

Answer: No


Question: Determine whether the Lebesgue measure of the set \( \{t \in \mathbb{R} \mid \sum_{i=1}^N \beta_i e^{\alpha_i t} = 0 \} \) is zero, given that \( \{\beta_i\}_{i=1}^N \) and \( \{\alpha_i\}_{i=1}^N \) are non-zero complex numbers.

Answer: 0


Question: Find the constant term in the expansion of \(\left(1 + x + 2y^{2} - \frac{1}{x^{2}y}\right)^{15}\).

Answer: 4054051


Question:
Suppose $X \rightarrow \operatorname{Spec}(A)$ is a morphism between a scheme $X$ and an affine scheme $\operatorname{Spec}(A)$ that satisfies the following conditions:
1. The morphism is a bijection topologically.
2. Each scheme-theoretic fiber is a single reduced point.
3. Both $X$ and $A$ are reduced.

Is $X$ an affine scheme?

Answer: Yes


Question: Solve the equation $f(x) = f^{-1}(x)$ for the function $f(x) = x^{4}-5x^{2}+x+4$ where $x \leq -1.6$. It is given that $f^{-1}(x)$ exists.

Answer: -2


Question: Show that the solution to the optimization problem \( \min\limits_x \|x\|_2^2 \) subject to \( Ax = b \) is given by \( x^* = A^T(AA^T)^{-1}b \), where \( A \in \mathbb{R}^{m \times n} \) with \( m < n \).

Answer: x^* = A^T(AA^T)^{-1}b


Question: Determine the value of $z$ such that the series \( \sum_{n=0}^\infty \frac{1}{n^2}\left(z^n+\frac{1}{z^n}\right) \) converges.

Answer: |z| = 1


Question: Solve the equation \(3^{\log_a x} + 3x^{\log_a 3} = 2\) for \(x\).

Answer: a^{-\log_3 2}


Question: Evaluate the line integral \( \oint \frac{\partial F}{\partial n} \, ds \) around the circle \( x^2 + y^2 = 1 \), where \( F(x,y) = x^2 - y^2 \) and \( \frac{\partial F}{\partial n} \) is the directional derivative of \( F \) along the outer normal. Use first principles to solve the problem.

Answer: 0


Question: Evaluate the integral \( \displaystyle\iiint\limits_{\mathbb{R}^3}{\frac{5\cdot e^{-4\cdot (z^2+y^2+x^2)}}{\sqrt{z^2+y^2+x^2}}}\,\mathrm{d}V \) using spherical coordinates.

Answer: \dfrac{5\pi}{2}


Question: Use Hensel's lemma to find all roots of the polynomial \( f(x) = x^3 + 4x + 79 \) in \( \mathbb{Z}/(125) \). Given that \( 2 \) is the unique root of \( f(x) \) in \( \mathbb{Z}/(5) \), determine the roots modulo 125.

Answer: 57


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{\sqrt[n]{\prod_{i=0}^{n-1} (a+ib)}}{\sum_{i=0}^{n-1} (a+ib)} \) where \( a > 0 \) and \( b > 0 \).

Answer: 0


Question: Find the volume of the solid that lies under the surface \( z = 4 - x^2 - y^2 \) and above the region \((x-1)^2 + y^2 \leq 1\) using cylindrical coordinates and a triple integral.

Answer: \dfrac{5\pi}{2}


Question: Determine whether the statement is true or false: A group of order 12 always has a normal 2-Sylow subgroup.

Answer: False


Question: Determine the number of functions \( \phi: G \times G \rightarrow G \) that define a group structure on the set \( G = \{a_1, a_2, a_3, a_4, a_5, a_6\} \) with \( a_1 \) as the neutral element. Consider the groups \( \mathbb{Z}_6 \) and \( S_3 \) up to isomorphism.

Answer: 80


Question: Three friends each tell the truth with a probability of \( \frac{2}{3} \) and lie with a probability of \( \frac{1}{3} \). You ask them independently if it is raining in your city, and they all say yes. Assuming no prior probability of rain, what is the probability that it is actually raining?

Answer: \dfrac{8}{9}


Question: Let $A, B, C$ be finite subsets of the natural numbers. Define $d(A, B) = \sum_{x \in A \mathbin{\Delta} B} 2^{-x}$, where $A \mathbin{\Delta} B = (A \cup B) \setminus (A \cap B)$ is the symmetric difference between $A$ and $B$. Does the function $d$ satisfy the triangle inequality, i.e., is $d(A, B) \leq d(A, C) + d(C, B)$ for all such subsets $A, B, C$?

Answer: Yes


Question: Evaluate the integral \( \oint_{|z|=1} \left(\frac{1}{z}+\frac{2}{z^2}\right) dz \).

Answer: 2\pi i


Question: Can axis/angle notation represent all possible orientations of a rotation matrix in 3D space? Given a pair \((\mathbf{u}, \theta)\), where \(\mathbf{u}\) is a vector representing the axis of rotation and \(\theta\) is the rotation angle, determine if this pair can represent all possible rotations in the Euclidean space \(\mathbb{R}^3\).

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty} |\lambda_{i}|^{n} n^{\ell} \] where \(|\lambda_{i}|<1\) and \(\ell\) is a positive integer.

Answer: 0


Question: Find all entire functions \( f(z) \) such that \( f(z) = f\left(\frac{1}{z}\right) \) for all \( z \neq 0 \).

Answer: f(z) = c


Question: Find the probability density function (pdf) of the time of the first arrival, \( T_1 \), given that there are exactly two arrivals by time 1 in a Poisson process \((N_t, t \geq 0)\).

Answer: 2(1 - t)


Question:
Solve the equation for $x$:
\[ \frac{1}{1}\left(\frac{x}{2008}+\frac{1}{2}\right)+\frac{1}{2} \left(\frac{x}{2008}+\frac{2}{3}\right) + \frac{1}{3}\left(\frac{x}{2008}+\frac{3}{4}\right)+\cdots+\frac{1}{2008}\left(\frac{x}{2008}+\frac{2008}{2009}\right)=\frac{x}{2009} \]
Provide a closed form for $x$.

Answer: -2008


Question: Determine the dimension of the vector space \( k[x,y,z] / (x, y^2, z^2) \) over the field \( k \).

Answer: 4


Question: Calculate the probability that a standard normal variable $X_1$ is less than the maximum of eight independent and identically distributed standard normal variables $Y_1, \dots, Y_8$. Specifically, find $P(X_1 < Y_{(8)})$, where $Y_{(8)} = \max(Y_1, \dots, Y_8)$. Assume $X_1, Y_1, \dots, Y_8$ are all standard normal variables.

Answer: \dfrac{8}{9}


Question: For $n \geq 1,$ let $G_n$ be the geometric mean of the set $\{\sin (\frac{\pi}{2} \frac{k}{n}) : 1 \leq k \leq n\}.$ Determine $\lim_{n \rightarrow \infty} G_n.$

Answer: \dfrac{1}{2}


Question: Determine the range of values of $p$ for which the series \( \sum_{k = 1}^\infty \frac{\exp(-p k)}{k^{1-p}} \) converges, where \( p \geq 0 \).

Answer: (0, \infty)


Question: Determine the number of Q-field homomorphisms from the algebraic closure of the rational numbers to the complex numbers that keep the rational numbers fixed.

Answer: 2^{\aleph_0}


Question: Find \( \lim_{n \rightarrow \infty} \frac{1}{\sqrt{n}}\left(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}\right) \).

Answer: 2


Question:
Consider the duality product defined between $L^2(\Omega)\times H^{-1}(\Omega)$ and $L^2(\Omega)\times H^1_0(\Omega)$ as follows: for all $(\phi_0,\phi_1)\in L^2(\Omega)\times H^{-1}(\Omega)$ and $(f_0,f_1)\in L^2(\Omega)\times H^1_0(\Omega)$, \[<(
\phi_0,\phi_1),(f_0,f_1)>=<\phi_1,f_1>_{H^1_0,H^{-1}}-\int_{\Omega}\phi_0 f_0 \, dx.\] Determine if the following inequality holds: \[<(
\phi_0,\phi_1),(f_0,f_1)>\leq ||(\phi_0,\phi_1)||_1||(f_0,f_1)||_2,\] where $||.||_1 $ and $||.||_2 $ are the norms in the respective product spaces.

Answer: Yes


Question: Let \( \mathbb{G} \) be a discrete amenable semigroup, and \( \{ F_{n} \} \) a Folner sequence. For a subset \( S \subset \mathbb{G} \), define the upper density as \( D^{\ast}(S) = \limsup_{n \rightarrow \infty} \frac{|S \cap F_{n}|}{|F_{n}|} \). Suppose there exists \( m \geq 0 \) such that for every \( g \in \mathbb{G} \), we have \( gF_{m} \cap S \neq \varnothing \). Is it true that \( D^{\ast}(S) > 0 \)?

Answer: Yes


Question: Calculate the approximation of the integral \( I = \int_{-1}^{1} e^{-x^{2}} \, dx \) by expanding the integrand into a power series. Determine how many terms of the series are needed so that the error is smaller than \( \varepsilon = 10^{-3} \).

Answer: 6


Question:
Evaluate the limit of the following expression as \( n \to \infty \):
\[ M_{Y_n}(t) = e^{-t\sqrt{n}} \left(\frac{1}{1 - t/\sqrt{n}}\right)^n. \]

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


Question:
Find the non-negative integer value of $a$ such that the following limit holds:
\[ \lim_{x \to 1 }\left\{ \frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1} \right\}^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4} \]

Answer: 0


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( f(x) = f(x^2 + \frac{x}{3} + \frac{1}{9}) \) for all \( x \in \mathbb{R} \).

Answer: f(x) = c


Question: Calculate the area bounded by the equation \((3x + 4y + 7)^2 + (2x + y + 3)^2 = 1\) using a double integral.

Answer: \dfrac{\pi}{5}


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables. Calculate the expected value: \[ E\left[ \frac{X_1 + X_2 + \cdots + X_k}{X_1 + X_2 + \cdots + X_n} \right] \] where $1 \leq k \leq n$. Assume $X_i$ are non-negative and have finite expectation.

Answer: \dfrac{k}{n}


Question: Solve the equation \(\log_{1-2x}(6x^2-5x+1) - \log_{1-3x}(4x^2-4x+1) = 2\) for \(x\), given that \(x\) must satisfy the conditions \(1-2x > 0\), \(6x^2-5x+1 > 0\), \(1-3x > 0\), and \(x \neq \frac{1}{2}\).

Answer: \dfrac{1}{4}


Question: Evaluate the line integral \( \int_{\gamma} x \, dz \), where \( \gamma \) is the path around the circumference of the unit square in the complex plane, traversed counterclockwise. Express your answer as a single complex number.

Answer: i


Question: Is it true that if a function $f: I \to \mathbb{R}$ is differentiable and its derivative $f'$ is one-to-one, then $f'$ is continuous? Here, $I$ is an interval and $\mathbb{R}$ represents the real numbers.

Answer: Yes


Question: Let $D=(d_{ij})_{i,j=1}^{n}$ be a diagonal matrix and $A=(a_{ij})_{i,j=1}^{n}$ a skew-symmetric matrix, where $d_{ij},a_{ij} \in \mathbb{R}$ for all $i,j$. Suppose the following equation holds: $$ AA=D. $$ Does $d_{jj} \neq d_{kk}$ imply $a_{jk}=a_{kj}=0$?

Answer: Yes


Question: Evaluate the integral \( \int_{|z|=1} \frac{2+\sin(z)}{z} \, dz \) where the contour is the unit circle traversed once counterclockwise.

Answer: 4\pi i


Question: Determine the points of continuity for the function \( f:(0,1)\rightarrow \mathbb{R} \) defined by \( f(x)= x^{2} \) if \( x \) is rational and \( f(x) = 2-x^{2} \) if \( x \) is irrational.

Answer: \emptyset


Question:
Determine the convergence rate of the iterative method defined by:
\[ x_{k+1} = x_k - \frac{f(x_k)}{g(x_k)} , \quad g(x_k) = \frac{f(x_k + f(x_k)) - f(x_k)}{f(x_k)} \]
where \( f(x) \in C^2[a,b] \) and there exists a point \( x_0 \) such that \( f(x_0) = 0 \) and \( f'(x_0) \neq 0 \).

Answer: 2


Question:
Let Y be an exponential random variable with mean \( \frac{1}{\theta} \), where \( \theta > 0 \). The conditional distribution of X given Y is a Poisson distribution with mean Y. What is the variance of X? Choose the correct answer from the following options:

(A) \( \dfrac{1}{\theta^2} \)
(B) \( \dfrac{\theta+1}{\theta} \)
(C) \( \dfrac{\theta^2+1}{\theta^2} \)
(D) \( \dfrac{\theta+1}{\theta^2} \)

Answer: D


Question: Evaluate the limit: \[ \lim_{n \to +\infty} \lim_{x \to 0^+} f^{(n)}(x) \] where \( f(x) = e^{\sqrt{x}} + e^{-\sqrt{x}} \).

Answer: 0


Question: Calculate the second partial derivative of the function \( f(x,y) = x^3 + 5x^2y + y^3 \) and evaluate it at the point \((2,1)\).

Answer: 20


Question: Calculate the limit: \[ \lim_{n\to+\infty} \frac{2^n \cdot (\sqrt{4^n+2n} - 2^n)}{n+1} \]

Answer: 1


Question: Find the area of the region defined by the inequality $0 \leq x \leq x^2 + y^2 \leq 1$ using double integration.

Answer: \dfrac{\pi}{4}


Question: Find the limit of the sequence: \[ \lim_{{n}\to{\infty}} (n+1)!(e-1-{1\over2}-\cdots-{1\over n!}) \]

Answer: 1


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

Answer: 0


Question: Evaluate the sum \( \sum_{0 \leq k \leq 100, \ \gcd(k, 100) = 1} f \left( \frac{k}{100} \right) \) where \( f(x) = \frac{9^x}{3+9^x} \).

Answer: 20


Question: Determine whether the following statement is true or false: For a sequence \( x_n \) in \( l_1 \), \[ \sum \frac{|x_i|^2}{2^i} \leq (\sup |x_i|)^2. \]

Answer: True


Question: Find the remainder when \(6^{936}\) is divided by 2014.

Answer: 1008


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{d x}{\left(x^{3}+\frac{1}{x^{3}}\right)^{2}} \).

Answer: \dfrac{\pi}{18}


Question: Find the minimum value of \( f(a,b,c) = \int_0^1 (a+bx+cx^2)^2 \, dx \) subject to the constraint \( a+b+c=1 \).

Answer: \dfrac{1}{9}


Question: Given real numbers $a, b, c, d, e$ such that $a + b + c + d + e = 8$ and $a^2 + b^2 + c^2 + d^2 + e^2 = 16$, determine the maximum value of $e$. Express your answer as a single number.

Answer: \dfrac{16}{5}


Question: Suppose that $p, q,$ and $r$ are primes such that $p > q > r$, and $kp + 1$ divides $qr$ for some $k \neq 0$. Prove that $1 + kp = qr$.

Answer: 1 + kp = qr


Question: Find all pairs of prime numbers \((p, q)\) such that \(p^2 + 3pq + q^2 = 5^t\) for some integer \(t\).

Answer: (5, 5)


Question: Find all functions \( f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} \) such that \( f(f(m) + f(n)) = m + n \) for all non-negative integers \( m \) and \( n \).

Answer: f(n) = n


Question: Suppose $(X, d)$ is a metric space with countably many elements, and every finite subset of $X$ can be covered with three subsets of diameter at most $1$. Can the entire space $X$ be covered with three subsets of diameter at most $1$?

Answer: Yes


Question: Determine the number of values of \( a \) for which the polynomials \( x^3 + ax + 1 = 0 \) and \( x^4 + ax^2 + 1 = 0 \) have at least one common root.

Answer: 1


Question: Let \( f:[0, 1] \to \mathbb{R} \) be an increasing function with \( f(0) = 0 \) and, for all \( n \in \mathbb{Z}^{+} \), \( f(x) = \frac{1}{n} \) on \( \left(\frac{1}{n+1},\frac{1}{n}\right] \). Find \( \int^{1}_{0} f(x) \, dx \) to one decimal place.

Answer: 0.6


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{x^2}{x^6 + 9} \, dx \).

Answer: \dfrac{\pi}{9}


Question:
Evaluate the limit: \[ \lim_{x\to \infty} \left((x+2) \tan^{-1} (x+2) - x\tan^{-1} x \right) \] Choose the correct answer from the following options:
(a) \( \frac{\pi}{2} \)
(b) Doesn't exist
(c) \( \frac{\pi}{4} \)
(d) None of the above.

Answer: D


Question: Evaluate the integral \( \int_\gamma \left(\frac{z}{z-1}\right)^n dz \), where the contour \( \gamma(t) = 1 + e^{it} \) for \( t \in [0, 2\pi] \).

Answer: 2\pi i n


Question: Let \(A_n\) be a sequence of \(d\times d\) symmetric matrices, and let \(A\) be a \(d\times d\) symmetric positive definite matrix with real entries. Assume that each element of \(A_n\) converges to the corresponding element of \(A\) as \(n\to \infty\). Can we conclude that there exists an \(\epsilon > 0\) such that the smallest eigenvalue of \(A_n\) is larger than \(\epsilon\) for all sufficiently large \(n\)?

Answer: Yes


Question: Find the last two digits of \(3^{3^{100}}\).

Answer: 03


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

Answer: \dfrac{3e}{2}


Question: Find the value of the positive constant \( A \) such that the graph of the function \( f(x) = A\sin x \) intersects the graph of its derivative perpendicularly.

Answer: \sqrt{2}


Question: Does there exist a non-constant holomorphic mapping $f: U \to V$ where $U \subset Y$ and $V \subset \Omega$ are open neighborhoods of $y_0$ and $z_0$ respectively, such that $f(y_0) = z_0$ and $f(y) \neq z_0$ for all $y \neq y_0$?

Answer: Yes


Question: Consider a set $S$ of $2n$ elements and let $F$ be the family of all $k$-size subsets of $S$, where $1 < k \le n$. Is it possible to pair up the sets in $F$ such that each pair has a nonempty intersection? If $|F|$ is odd, one set may be paired with itself.

Answer: Yes


Question: Calculate the conditional expectation $E[X_{1}|\bar{X}]$ where $X_{1}, X_{2}, \ldots, X_{n}$ are i.i.d. random variables following a normal distribution $N(\mu, \theta)$. Assume $\bar{X}$ is the sample mean of $X_{1}, X_{2}, \ldots, X_{n}$. Provide the expression for $E[X_{1}|\bar{X}]$.

Answer: \bar{X}


Question: Calculate \( M = \left\lfloor \sum_{n=1}^{1000} \frac{1}{n^{\frac{2}{3}}} \right\rfloor \). Find the value of \( M \).

Answer: 27


Question: Let $V$ be an $n$-dimensional real vector space and \( \eta: V \times V \to \mathbb{R} \) a nondegenerate symmetric bilinear form. Define \( r \) as the maximum dimension among all subspaces on which the restriction of \( \eta \) is positive definite. Can there be two different \( r \)-dimensional subspaces of \( V \) such that the restriction of \( \eta \) to them is positive definite?

Answer: Yes


Question: Given the equation $x^{13} \equiv 17 \pmod{47}$, find the integer $x$ such that $0 \leq x < 47$.

Answer: 16


Question: Let \( f: \mathbb{R}^n \rightarrow \mathbb{R} \) be a \( C^2 \) function and \( x^* \) be a point such that \( \nabla^2 f(x^*) \) is positive definite. Is it always true that there exists a neighborhood around \( x^* \) such that for all points \( x \) in that neighborhood, \( \nabla^2 f(x) \) is positive definite?

Answer: Yes


Question: Solve the equation \( \sqrt{3x^2+6x+7} + \sqrt{5x^2+10x+14} = 4 - 2x - x^2 \).

Answer: -1


Question: Evaluate the limit: \[ \lim_{x\to \frac{1}{\sqrt{2}^+}} \frac{\cos ^{-1} \left( 2x\sqrt{1-x^2}\right)}{x-\frac{1}{\sqrt{2}}} \]

Answer: 2\sqrt{2}


Question: Calculate the expected residual service time for an arbitrary job in a queueing system with two types of arrivals. Type A arrivals occur at a rate of \(\lambda_A = 1/20\) per second and have a service time of exactly 8 seconds. Type B arrivals occur at a rate of \(\lambda_B = 1/30\) per second and have a service time of exactly 12 seconds. Use the formula for expected residual time: \[E(R) = \frac{E(B^2)}{2E(B)}\] where \(E(B)\) is the expected service time of an arbitrary job. Provide the expected residual time \(E(R)\) as a single value.

Answer: 5


Question: Let $X \neq \{0\}$ be a reflexive Banach space and let $f \in X^*$, where $X^*$ is the dual space of $X$. Does there exist an $x \in X$ with $\|x\| = 1$ such that $f(x) = \|f\|$, where $\|f\|$ is defined as $\sup\{|f(x)| : x \in X, \|x\| = 1\}$?

Answer: Yes


Question: A wire is shaped as a circle defined by the equation \(x^2 + y^2 = a^2\). The density at any point \((x, y)\) on the wire is given by \(|x| + |y|\). Calculate the moment of inertia of the wire about a diameter.

Answer: 4a^4


Question: Find the completion of the metric space \((\ell^1(\mathbb{N}), d_*)\), where \(\ell^1(\mathbb{N}) = \{(a_n): a_n \in \mathbb{R}, \sum |a_n| < \infty \}\) and the distance function \(d_*\) is defined by \(d_*(a,b) = \sup |a_n - b_n|\) for \(a, b \in \ell^1(\mathbb{N})\).

Answer: c_0


Question: Compute the limit: \[ \lim_{x \to \infty} \int_{0}^{\pi} \frac{\sin(t)}{1+\cos^{2}(xt)} \, \mathrm{d}t. \]

Answer: \sqrt{2}


Question: Consider a compact, zero-dimensional, first countable space $Z$ with density $d(Z) = \mathfrak{c}$. Suppose we pick a point $p$ in $Z$ such that $p$ is not isolated. Let $X = Z \setminus \{p\}$. Does $X$ satisfy the following properties: first countable, zero-dimensional, locally compact, Lindelöf, non-compact, and all non-empty open sets in $X$ have $\pi$-weight $\mathfrak{c}$? Justify your answer.

Answer: Yes


Question: Consider smooth vector fields $X_i$, $1 \le i \le n$, on an open set $U \subset \mathbb{R}^n$, which are linearly independent at each point. Define $[X_i, X_j] = \sum_{k=1}^n c_{ij}^kX_k$ with constant coefficients $c_{ij}^k$. Let $\overline{X}_i$, $1 \le i \le n$, be another copy on $\overline{U} \subset \mathbb{R}^n$. Define $Y_i = X_i + \overline{X}_i$ on $U \times \overline{U}$. Do the vector fields $Y_i$ span an integrable distribution $E \subseteq T(U \times \overline{U})$?

Answer: Yes


Question: Calculate the fundamental group \(\pi_1(Y)\) of the space \(Y = \{(x_1,x_2,x_3):x_1^2+x_2^2+x_3^2=1\} \cup \{(x_1,0,0):-1\leq x_1\leq 1\} \cup \{(0,x_2,0): -1\leq x_2 \leq 1\}\).

Answer: 0


Question: Evaluate the limit: $$\lim _{n\to \infty \:}\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2n-1}-\frac{1}{2n}\right)$$

Answer: \ln 2


Question: Consider a collection of 20 marbles drawn from an infinite pool of pink and blue marbles, with replacement. The probability of drawing a pink marble is 0.4, and the probability of drawing a blue marble is 0.6. You draw 20 marbles and check if there are at least 12 pink marbles in the collection. If true, you note this result. Then, you randomly discard one marble from the collection and draw a new marble to replace it. You repeat this process of inspecting, noting the result, discarding, and replenishing a total of 100 times. What is the probability that you will encounter at least one instance where the collection has at least 12 pink marbles during these 100 cycles?

Answer: 0.997


Question: Calculate the expected value of the estimator \( \hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^{n} (\ln(x_i) - \mu)^2 \) where \( x_i \sim \mathrm{Lognorm}(\mu, \sigma^2) \).

Answer: \sigma^2


Question: Evaluate the integral \( \int \int (x-y)e^{x^2-y^2} \, dx \, dy \) over the region \( R \), which is bounded by the lines \( x+y=1 \) and \( x+y=3 \) and the curves \( x^2 - y^2 =1 \) and \( x^2 - y^2 = -1 \), using the change of variables \( u=x-y \) and \( v=x+y \). Find the Jacobian of the transformation and determine the new integration limits for \( u \) and \( v \).

Answer: \dfrac{2}{3e}


Question: Determine if the inhomogeneous heat equation \( u_t = u_{xx} + h(x,t) \) for \( -\infty < x < \infty , t > 0 \) with initial condition \( u(x,0) = 0 \) can be transformed into the integral equation \[ \int_0^t \int_{-\infty}^{\infty} {\frac{1}{2\sqrt{\pi(t-s)}} \exp\left(\frac{-(x-y)^2}{4(t-s)}\right) \times h(y,s)\, dy\, ds} \] using the Fourier transform.

Answer: Yes


Question: The graph of the quadratic function \(f(x) = ax^2 + bx + c\) contains the points \((m, 0)\) and \((n, 2016^2)\). Given that \(a, b, c, m,\) and \(n\) are integers with \(m < n\), how many distinct values of \(n - m\) are possible?

Answer: 165


Question:
Let \( \{f_{n}\}_{n \in \mathbb{N}} \) be a sequence of measurable functions on a measure space \( M \) such that \( f_{n} \rightarrow f \) almost surely, where \( f \) is also a measurable function. Suppose there exists a nonnegative measurable function \( g \) satisfying the following conditions:

1. \( |f_{n}| \leq g \) almost surely for all \( n \in \mathbb{N} \).
2. \( \int_{M} g^{p} \, d\mu < \infty \) for some \( p > 0 \).

Prove that \( \int_{M} |f_{n} - f|^{p} \, d\mu \rightarrow 0 \) as \( n \rightarrow \infty \).

Answer: 0


Question:
Let $A$, $B$, and $C$ be three different real $3 \times 3$ matrices with the following properties:

- $A$ has the complex eigenvalue $\lambda=3-5i$.
- $B$ has eigenvalues $\lambda=0$, $\lambda=5$, $\lambda=-5$.
- $C = M M^T$ for some real $3 \times 2$ matrix $M$.

Which of the matrices are necessarily diagonalizable? In the case of complex eigenvectors, diagonalization is over $\mathbb{C}$.

(A) Only $B$
(B) Only $A$ and $B$
(C) Only $B$ and $C$
(D) All three of them
(E) None of them

Answer: D


Question:
Find the last three digits of the product:
\[ 3 \times 7 \times 11 \times 15 \times 19 \times 23 \times \ldots \times 115 \times 119 \]

Answer: 125


Question: Given that $f(6) = 144$ and the recurrence relation $f(n+3) = f(n+2) \cdot (f(n+1) + f(n))$ for $n = 1, 2, 3, 4$, find the value of $f(7)$. Note that the solution is not unique, but all solutions are positive integers.

Answer: 3456


Question: Let \( f \) be a three times continuously differentiable real-valued function on \((0,5)\) such that its third derivative \( f'''(x) = \frac{1}{100} \) for all \( x \in (0,5) \). If \( P(x) \) is a polynomial of degree \( \leq 2 \) such that \( P(1) = f(1) \), \( P(2) = f(2) \), and \( P(3) = f(3) \), find the value of \(|f(4) - P(4)|\).

Answer: \dfrac{1}{100}


Question: Evaluate the limit: \[ \lim_{z\rightarrow 0}\frac{12z^2+6\sin^2z-18(\cos z \sin z)z} {\sin^4z} \]

Answer: 10


Question: Evaluate the limit: \[ \lim_{x\rightarrow-\infty} \left( \sqrt{x^2+2x+2} + x \right) \] without using l'Hôpital's rule.

Answer: -1


Question: Determine whether the set of convex combinations of product states on a tensor product of C-algebras is weak-dense in the set of all states on the tensor product. Answer 'True' or 'False'.

Answer: True


Question: Given that \(3(x+y)=x^2+y^2+xy+2\), find the maximum value of \(P=\frac{3x+2y+1}{x+y+6}\).

Answer: 1


Question: Evaluate the limit: \[ \lim \limits_{x \to 0^+} \frac{(\sin{x})^x}{x} \]

Answer: \infty


Question: Determine the maximum value of \(a\) such that \(\lim_{n\rightarrow \infty} n^{n^a} e^{-n} = 0\).

Answer: 1


Question: Determine whether the sequence of functions \( f_n(x) = \left( 1 + \frac{-e^{-x}}{n} \right)^n \) converges to \( f(x) = e^{-e^{-x}} \) for all \( x \). Justify your answer.

Answer: e^{-e^{-x}}


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

Answer: \dfrac{\pi}{2}


Question: Find the supremum of the function \( f(x) = \sin(x) + \sin(\varphi x) \), where \( \varphi \) is the golden ratio.

Answer: 2


Question:
Consider the function
\[f(x)=\frac{(1+2x)^\frac{1}{2}(1+4x)^\frac{1}{4}\cdots(1+2nx)^\frac{1}{2n}}{(1+3x)^\frac{1}{3}(1+5x)^\frac{1}{5}\cdots(1+(2m+1)x)^\frac{1}{2m+1}}\]
where \(n\) and \(m\) are positive integers satisfying \(n+m-20=f'(0)=2010\). Find the value of \(n\).

Answer: 2020


Question:
Let $W$ be a standard Brownian motion. Define the upper left Dini derivative $D^-W$ by
$$D^-W_t := \limsup_{h \to 0^-} \frac{W_{t+h} - W_t}{h}.$$ Fix $a > 0$, and define the stopping time $\tau$ by
$$\tau := \inf \{t > 0 \mid W_t \geq a\}.$$ Determine whether $D^- W_\tau = +\infty$, almost surely.

Answer: +\infty


Question:
Let $A \subseteq \mathbb{R}$ such that $A \cap A^d = \emptyset$, where $A^d$ denotes the derived set of $A$. Which of the following statements is true?

(a) $A^d$ must be empty.
(b) $A^d$ is at most a singleton set.
(c) $A^d$ is at most a finite set.
(d) $A^d$ may be infinite.

Answer: D


Question: Determine the values of \( p > 0 \) for which the integral \[ T = \sqrt{\frac{1}{2g}}\int_{0}^{b}\frac{\sqrt{1+y'(x)^2}}{\sqrt{-y(x)}} \, dx \] is convergent, given that \( y(x) = -2x^p \) and \( g \) is a constant.

Answer: (0, 2)


Question: Calculate the measure \( \mu(A) \), where \( \mu = L^1 \otimes \sum_{k = 1}^{\infty}k\delta_k \) and \( A = \{(x, y) \in \mathbb{R}^2: 3 - |x| > y > 2|x|\} \). Use Fubini's theorem to evaluate the integral \( \mu(A) = \int_{A} d\mu \).

Answer: 5


Question: Evaluate the limit: \[ \lim_{x\to 0} \left( \frac{1+x\cdot2^x}{1+x\cdot3^x} \right)^{1/x^2} \]

Answer: \dfrac{2}{3}


Question: Evaluate the integral \( \int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2} \) using the method of residues.

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral \( \oint_{|z-1|=1} \frac{2z-1}{z^2 - 1} \, dz \) using the residue theorem, where the contour is the circle of radius 1 centered at 1, traversed anticlockwise.

Answer: \pi i


Question: Let $F$ be a field of characteristic $0$, $V = F^n$ for some $n$, and $T : V \to V$ a linear transformation. If the minimal polynomial $P(x) \in F[x]$ of $T$ has an irreducible factor $P_1(x)$ of degree $r$, is it true that $V$ has a $T$-invariant subspace of dimension $r$? Justify your answer.

Answer: Yes


Question: Calculate the value of the following summation: $$\sum_{a = 1}^{p - 1}\left(\frac{a}{p}\right)$$ where \( p \) is a prime number.

Answer: 0


Question: Let $\gamma : [0,1] \to \mathbb{C}$ be the contour given by $\gamma(t) = t + it$. Suppose $f : \mathbb{C} \to \mathbb{C}$ is a complex differentiable function which satisfies $|f(z)| \le 3$ for all $z$ such that $|z| \le 100$. Determine the maximum possible value of \(\int_\gamma f(z)\,dz\).

Answer: 3\sqrt{2}


Question: Determine if the quantity \((\pi(n)!)^{1/(n-h)}\), where \(h = \pi(n/2)\) and \(n > 7\), is bounded above for all integers \(n\). If it is bounded, find the upper bound.

Answer: e


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

Answer: 1


Question: Determine the degree of the splitting field of the polynomial \(X^6 + X^3 + 1\) over \(\mathbb{Q}\).

Answer: 6


Question: If \(X\) is an integral homology circle with \(\pi_1(X)=\mathbb{Z}\), must \(X\) be homotopically equivalent to a circle?

Answer: Yes


Question: Given a prime number $p$ and a positive integer $k$, consider integer-valued polynomials $f(x)$ satisfying the property that $p \mid f(x)$ if and only if $p^k \mid x$. What is the smallest degree of such a polynomial $f(x)$?

Answer: k


Question: Calculate the integral \( I = \frac{1}{2\pi i } \oint_C \frac{f'(z)}{f(z)} dz \) where \( f(z) = \frac{1+z^2}{1-\cos{2\pi z}} \) and \( C \) is the circle around \(|z| = \pi\). Use the argument theorem to determine the value of \( I \).

Answer: -12


Question: Given complex numbers \(z_1, z_2, z_3\) such that \(|z_1| = |z_2| = |z_3| = |z_1 + z_2 + z_3| = 2\) and \(|z_1 - z_3| = |z_1 - z_2|\), with \(z_2 \neq z_3\), find the value of \(|z_1 + z_2||z_1 + z_3|\).

Answer: 8


Question: Evaluate the limit using the Squeeze Theorem: \[ \lim_{n\to \infty}\left(\dfrac{1}{\sqrt{n^2}}+\dfrac{1}{\sqrt{n^2+1}}+\cdots+\dfrac{1}{\sqrt{n^2+2n}}\right) \]

Answer: 2


Question: Determine whether the following statement is true if $f$ is a scalar function: \[ f(\nabla \cdot \mathbf{B}) = (\nabla \cdot \mathbf{B})f \] where \( \nabla \cdot \mathbf{B} = \frac{\partial B}{\partial x} + \frac{\partial B}{\partial y} + \frac{\partial B}{\partial z} \).

Answer: True


Question: Let \(A\) be a matrix in \(\mathbb{R}^{n \times k}\) with orthonormal columns. Determine the rank of the matrix \(I_n - AA^T\).

Answer: n - k


Question: Evaluate the triple integral \( \int\int\int_R \cos x \, dxdydz \), where \( R = \{(x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 \le \pi^2\} \).

Answer: 4\pi^2


Question:
Determine the interval where the equation \( f(x) = \arctan(x) - \ln(x) = 0 \) has exactly one real solution. Choose from the following options:

(a) \((1, e)\)
(b) \((e, e^2)\)
(c) \((e^2, \infty)\)
(d) \((0, 1)\)

Answer: B


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

Answer: \dfrac{1}{e}


Question:
Let $f : \mathbb{R} \to \mathbb{R}$ be a function with a continuous derivative such that $f(\sqrt{2}) = 2$ and \[f(x) = \lim_{t \to 0}\dfrac{1}{2t}\int_{x-t}^{x+t}sf'(s)\,ds \ \text{for all} \ x \in \mathbb{R}.\] Determine the value of $f(3)$. Choose from the following options:
(a) $\sqrt{3}$
(b) $3\sqrt{2}$
(c) $3\sqrt{3}$
(d) $9$.

Answer: B


Question: Given that one solution to the differential equation \( xy'' - y' + 4x^3y = 0 \) is \( y = \sin x^2 \), find a second linearly independent solution using the method of reduction of order.

Answer: \cos x^2


Question:
Consider a complex unitary matrix \( U \in \mathbb{C}^{N\times N} \). Construct a \( 2 \times 2 \) submatrix \( M \) using two diagonal and two off-diagonal elements from its \( m \)th and \( n \)th rows:
\[
M=
\begin{bmatrix}
U_{mm} & U_{mn}\\
(U_{mn})^* & U_{nn}
\end{bmatrix}
\]
Is the submatrix \( M \) always diagonalizable?

Answer: Yes


Question: Evaluate \(a+2b+3c\) given that \[ \lim_{h \to 0}\frac { \int _{ 0 }^{ h }{ f(x)dx-h(af(0)+bf(h/3)+cf(h)) }  }{ { h }^{ 4 } } \] is a finite non-zero number.

Answer: \dfrac{9}{4}


Question: Determine whether there exists an element of order 51 in the multiplicative group \( U(103) \).

Answer: Yes


Question: Determine whether the following statement is true or false: If $f \in C^\infty$ and $f^{(k)}(0) = 0$ for all $k \in \mathbb{N} \cup \{0\}$, then $f$ is identically zero.

Answer: B


Question: Given that \(\sqrt{2}+\sqrt{3}+\sqrt{6}\) is a root of the polynomial \(x^4 + ax^3 + bx^2 + cx + d = 0\), where \(a, b, c, d \in \mathbb{Z}\), find the value of \(|a+b+c+d|\).

Answer: 93


Question: Given a continuously differentiable function \( f: \mathbb{R} \to \mathbb{R} \) such that \(|f(x) - f(y)| \ge |x-y|\) for all \( x, y \in \mathbb{R} \), determine how many solutions the equation \( f'(x) = \frac{1}{2} \) has.

Answer: 0


Question: For which values of \( t \ge 0 \) does there exist a differentiable function \( f \) such that \( f(0) = 0 \), \( f'(x) > f(x) \) for all \( x > 0 \), and \( f'(0) = t \)?

Answer: [0, \infty)


Question:
Given the system of differential equations:
\[\begin{align*}
&(1) \: x' = -2x + y&\\
&(2) \: y' = -4x + 3y + 10\cos t&
\end{align*}\]

Use the elimination method to derive the characteristic equation of the homogeneous part of the general solution for \(x(t)\).

Answer: r^2 - r - 2 = 0


Question: Determine the interval of convergence for the Taylor series representation of the function \( f(x) = x^{0.5} \) centered at 6, given by \( \sum_{n=0}^\infty c_n(x-6)^n \).

Answer: [0, 12]


Question: Suppose that \( A=\int_{\alpha}^{\beta} f(B,x)\ dx, \) where \( B \) is a \( 3\times3 \) matrix. If \( B \) is diagonalized with an orthogonal matrix, is \( A \) diagonalized by the same matrix? For example, consider \( f(B,x)=\frac{1}{B+xI}. \) Provide a justification for your answer.

Answer: Yes


Question:
Determine which of the following intervals contains a solution to the equation:
\[ \prod^{45}_{k = 0} \left( 1 + \tan \frac{k \pi }{180} \right) = \left[ \log_{\frac{\sqrt{6}}{3}} \big| \sin(2x)\big| \right]^{\frac{9}{2}\sin(2x) + 20} \]

- \([0.631, 0.747]\)
- \([0.258, 0.395]\)
- \([0.412, 0.627]\)
- \([0.799, 0.814]\)
- \([0.041, 0.107]\)

Answer: B


Question: Consider the optimization problem where you want to maximize the product \( m_1^{w_1} \cdot m_2^{w_2} \cdot m_3^{w_3} \) subject to the constraint \( m_1 + m_2 + m_3 = m \), with \( w_1 + w_2 + w_3 = 1 \). If all \( m_i \) must be integers, is it always true that \( m_i \leq \lceil w_i m \rceil \) for each \( i \in \{1, 2, 3\} \)?

Answer: Yes


Question: Let $\mathcal C$ be a symmetric monoidal $\infty$-category and $L \in Pic(\mathcal C)$ a $\otimes$-invertible object. Is $L$ necessarily coherently sym-central?

Answer: Yes


Question: Determine the values of $p$ for which $\sup_n f_n(x)$ is integrable on $[0,1]$, where $f_n(x) = n^p x^n (1-x)$. Assume $\lim_{n \to \infty} f_n(x) = 0$ for $x \in [0,1]$. Provide a justification for your answer.

Answer: p < 2


Question: Let \( G = \mathbf{Z}/18\mathbf{Z} \times \mathbf{Z}/60\mathbf{Z} \) and consider the group homomorphism \( f: G \to G \) defined by \( f(x) = 4x \). Let \( f^k \) denote the \( k \)-th composite of \( f \) with itself, where \( f^1 = f \). Find the smallest integer \( k \geq 2 \) such that \( f^k = f \).

Answer: 7


Question: Calculate the integral \( \int_0^{\infty} x^{11} e^{-x^3} \, dx \).

Answer: 2


Question:
Let \(f_n:[1,2]\to[0,1]\) be defined by \(f_n(x)=(2-x)^n\) for all non-negative integers \(n\). Define \(f(x)=\lim_{n\to \infty}f_n(x)\) for \(1\le x\le 2\). Determine which of the following statements is true:

(a) \(f\) is continuous on \([1,2]\).
(b) \(\lim_{n\to \infty}\int_1^2f_n(x)dx=\int_1^2f(x)dx\).
(c) \(f_n\) converges uniformly to \(f\) on \([1,2]\) as \(n \to \infty\).
(d) For any \(a\in(1,2)\), we have \(\lim_{n \to \infty}f_n'(a)\neq f'(a)\).

Answer: B


Question: What is the minimum number of vertices in a 4-regular graph with a diameter of 2?

Answer: 6


Question: Find the angle \( \theta \) such that \( \theta = \arctan(2-\sqrt{3}) \).

Answer: \dfrac{\pi}{12}


Question: Determine if the diophantine equation \( n = \frac{3^x - 2^x}{2^y - 3^x} \), where \( x \) and \( y \) are positive integers and \( 2^y > 3^x \), has any integer solutions other than \( x=1 \) and \( y=2 \), which give \( n=1 \).

Answer: No


Question: Given the joint probability density function \( f(x,y) = \frac{3}{4}x^3y^2 \) for \( 0 \le x \le 2 \) and \( 0 \le y \le 1 \), find the probability \( P(X < Y^2) \).

Answer: \dfrac{3}{176}


Question: Determine the number of integer triples \((a, b, c)\) such that \(2^a - 5^b \cdot 7^c = 1\).

Answer: 2


Question: Evaluate the closed line integral \( \oint x \, dy \) counterclockwise around the triangle with vertices \((a,0), (0,0),\) and \((0,b)\).

Answer: \dfrac{ab}{2}


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x} \, dx \).

Answer: \dfrac{\pi}{2}


Question: Determine whether there exist two positive real numbers $a$ and $b$ such that the distance $d(a, b)$, defined as \( d(a, b) := \inf_{x \in [a], y \in [b]} |x - y| \), is greater than zero.

Answer: Yes


Question: Suppose a matrix $A$ satisfies $AGA = A$ for some matrix $G$. If $AG_1A = AG_2A = A$, is it true that $AG_1 = AG_2$?

Answer: No


Question: Suppose \( f(z) \) is a polynomial of degree at most 1 such that \( f(0) = i \) and \( |f(z)|^2 \leq 1 + |z|^3 \) for all \( z \in \mathbb{C} \). Show that \( f(z) = i \) for all \( z \in \mathbb{C} \).

Answer: i


Question: Evaluate the integral \( \int_0^\infty x^n\sin(2\pi \ln(x))e^{-\ln^2(x)} \mathrm{d}x \) and show that it equals zero.

Answer: 0


Question: Given two correlated standard normal random variables $X$ and $Y$ with correlation coefficient $\rho$, calculate the conditional expectation $\mathbb{E}[X+2Y|X-Y=a]$. Assume $X$ and $Y$ have mean $0$ and variance $1$. Express your answer in terms of $a$ and $\rho$. \(-1 \leq \rho \leq 1\).

Answer: -\dfrac{a}{2}


Question: Express the following summation in matrix form: \[ D_{βν} = \sum^{n}_{µ=1}\sum^{n}_{a=1}A_{µν}B_{αµ}C_{αβ} \]

Answer: D = C^\top B A


Question: Determine the number of distinct elements in the set \( \left\{ \left( \frac{2+i}{2-i} \right) ^n : n \in \mathbb{N} \right\} \).

Answer: \infty


Question: Given a set $X \subset [0,1]$ such that $X \cap (1-X) = \varnothing$ and $X \cup (1-X) = [0,1]$, determine if $X$ necessarily has Lebesgue measure $1/2$. Assume $1-X = \{1-x \mid x \in X\}$.

Answer: \dfrac{1}{2}


Question: Let $X = \mathrm{Spec}(R)$ be an affine variety, and let $f$ be a rational function on $X$ such that $f_{|U}$ is regular, where $U = \mathrm{Spec}(R_{g})$ for some $g \in R$. If $f_{|U} \in R_g$, is $f$ regular on $X$?

Answer: No


Question: Find the product of all solutions of the equation \(x^{\log_{10}x}=\left(100+2^{\sqrt{\log_23}}-3\sqrt{\log_32}\right)x\).

Answer: 10


Question:
Consider the following 2-variable linear regression where the error terms \(e_i\) are independently and identically distributed with mean 0 and variance 1:
\[ y_i = \alpha + \beta(x_i - \bar{x}) + e_i, \quad i = 1,2,...,n. \]
Let \(\hat{\alpha}\) and \(\hat{\beta}\) be the ordinary least squares estimates of \(\alpha\) and \(\beta\), respectively. What is the correlation coefficient between \(\hat{\alpha}\) and \(\hat{\beta}\)?

Answer: 0


Question: Let \( r \) be a rational number expressed as the infinite periodic fraction \( r = 0.12(31)_5 \). Find \( r \) in the form \( \frac{p}{q} \) where \( p, q \in \mathbb{N} \).

Answer: \dfrac{23}{75}


Question: Evaluate the contour integral \( \int_\Gamma \frac{1}{z^4 + 81} \, dz \), where \( \Gamma \) is the circle \( |z+i| = \frac{3}{4} \).

Answer: 0


Question: Calculate the expected value of the random variable $X$ over the positive integers, where $P[X=i] = \frac{C}{i^2}$ for any integer $i > 0$, and $C^{-1} = \sum_{i=1}^\infty \frac{1}{i^2}$. What is $\mathsf{E}(X)$?

Answer: \infty


Question: Show that the integral \( \int_0^x \frac{1}{1+\sqrt{t}\sin(t)^2} \, dt \) diverges as \( x \rightarrow \infty \).

Answer: \infty


Question: Evaluate the integral \( \int_{0}^{2\pi} \frac{1}{\sin^4x + \cos^4 x} \,dx \).

Answer: 2\sqrt{2}\pi


Question:
Let \((x_{n})_{n}\) be a sequence in a Banach space \(X\) such that the set \(\{x_{n}:n=1,2,\cdots\}\) is finite. Let \((f_{m})_{m}\) be a weak*-null sequence in \(X^{*}\) satisfying the following conditions:
1. The limit \(a_{m}:=\lim\limits_{n}\langle f_{m},x_{n}\rangle\) exists for each \(m\).
2. The limit \(a:=\lim\limits_{m}a_{m}\) exists.

Is \(a = 0\)?

Answer: 0


Question: Given a positive integer \( n \), let \( G \) be a simple, undirected graph with \( 2n + 1 \) vertices such that there is no cycle of length greater than 3. Find the maximum number of edges in \( G \).

Answer: 3n


Question: Calculate \( \left( \frac{3^{2020}+5^{2020}+7^{2020}}{3} \right)^{2020} \mod 8 \).

Answer: 1


Question: Determine the order of the pole or whether there is a removable singularity at \( z = -1 \) for the function \( f(z) = \frac{\sin\left(\frac{\pi}{2}(z+1)\right)}{(z^2+2z+4)(z+1)^3} \).

Answer: 2


Question: Given a concave twice continuously differentiable function $g : (0, 1) \to (0, +\infty)$, maximize the growth of the ratio \( f(x) := -\frac{g''(x)}{g(x)} \) as \( x \to 0 \). For example, if \( g(x) = x^p \) with \( p \in (0, 1) \), then \( f(x) = \frac{p(1 - p)}{x^2} \), which is maximal when \( p = 1/2 \), giving \( f(x) = 1/(4x^2) \). Another example is \( g(x) = -x\log x \), where \( f(x) = -\frac{1}{x^2\log x} \). Determine the function \( g(x) \) that maximizes \( f(x) \) as \( x \to 0 \).

Answer: \sqrt{x}


Question:
Determine the sum of the series:
\[ \frac{1}{1!} + \frac{1+2}{2!} + \frac{1+2+3}{3!} + \ldots \]

Answer: \dfrac{3e}{2}


Question: Given that $a, b, c$ are non-negative real numbers such that $a^2 + b^2 + c^2 = 1$, find the maximum value of \[ \frac{a}{1+bc} + \frac{b}{1+ac} + \frac{c}{1+ab}. \]

Answer: \sqrt{2}


Question: Consider a measurable space $X$ with a sigma-algebra generated by a family $\mathcal{G}=\bigcup_n \mathcal{G}_n$, where $(\mathcal{G}_n)$ is a sequence of sigma-algebras on $X$ of increasing fineness. Let $\sim$ be an equivalence relation on $X$, and let $X/\sim$ be the quotient measurable space. A set $Y \subseteq X/\sim$ is measurable if and only if $\bigcup_{y\in Y} y$ is measurable in $X$. Is the sigma-algebra on $X/\sim$ equal to the sigma-algebra generated by the family $$\{ Y \subseteq X/\sim\ :\ \exists G \in \mathcal{G} \quad \bigcup_{y\in Y} y = G\}?$$

Answer: Yes


Question: Compute \(|\exp{g(g(ir))|}\), where \(g(z) = \frac{1+e^z}{1-e^z}\) and \(r\) is a real number such that the function is defined.

Answer: 1


Question: Let $W(x)$ be a polynomial and $a \in \mathbb{R}$. If $W^{(i)}(a) = 0$ for all $i \leq n$, does this imply that $(x-a)^{n+1}$ divides $W(x)$? Justify your answer.

Answer: Yes


Question: Find all functions \( f: [0, +\infty) \to [0, +\infty) \) such that \( f(x) \geq \frac{3x}{4} \) and \( f\big(4f(x) - 3x\big) = x \) for all \( x \in [0, +\infty) \).

Answer: f(x) = x


Question: Evaluate the limit: $$\lim_{x\to \pi}\left(\frac{1}{x-\pi}\right)\int_{\pi}^{x}\frac{\cos t}{1-\cos t}\,dt.$$

Answer: -\dfrac{1}{2}


Question: Given the joint density function \( f(x, y) = 2 \) for \( 0 < y < x < 1 \), find the probability \( P(X - Y > z) \).

Answer: (1 - z)^2


Question: The surface given by the equation \( z = x^3 + xy^2 \) intersects the plane \( 2x - 2y = 1 \) in a curve. Determine the slope of this curve at the point where \( x = 1 \) and \( y = \frac{1}{2} \).

Answer: \dfrac{17}{4}


Question: Calculate the flux integral \( \iint_S \operatorname{curl} \vec{F} \cdot \vec{N} \,dS \) where \( S \) is the upper half of the sphere defined by \( x^2+y^2+z^2=4 \) with \( z \ge 0 \), and \( \vec{F} = (3x-y, yz, xy) \). Use Stokes' theorem to evaluate the integral, and assume \( \vec{N} \) is the outward unit normal vector to the surface.

Answer: 4\pi


Question: Let $G$ be a group and $H$ a subgroup with $[G:H]=n$. Is it true that $x^n \in H$ for all $x \in G$?

Answer: No


Question: Find the supremum of the sequence \( f(n) = (n+1)^{\frac{2}{n^2}} \) for \( n \in \mathbb{N} \).

Answer: 4


Question: Given a linear map \( \phi: V \rightarrow W \) with matrix \( A \) in the bases \( \mathbb{e} \) and \( \mathbb{f} \), determine the matrix of the dual map \( \phi^{*}: W^* \rightarrow V^* \) in the dual bases \( \mathbb{e}^* \) and \( \mathbb{f}^* \).

Answer: A^\top


Question: What is a norm on \( \mathbb{Z} \) that is similar to the norm defined for \( \mathbb{Z}[w] \) in the context of algebraic integers, where \( w \) is an element of \( \mathscr{O} \)? Is it simply \( N(n) = |n| \)?

Answer: N(n) = |n|


Question:
Given 984 people numbered from 1 to 984, select \( n \) people according to the following rules:

- Person number 12 is always picked.
- If person number \( x \) is picked and person number \( y \) is not picked, then if \( x \cdot y \le 984 \), person number \( x \cdot y \) must be picked.
- If person number \( x \) is picked and person number \( y \) is not picked, then if \( x + y \le 984 \), person number \( x + y \) must not be picked.

Determine the total number of people picked, \( n \).

Answer: 82


Question: Determine the isomorphism class of \( \operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}^m,\mathbb{Z}^n) \), where \( \operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}^m,\mathbb{Z}^n) = \{f:\mathbb{Z}^m\to\mathbb{Z}^n\mid f  \text{ is a homomorphism of } \mathbb{Z}\text{-modules}\} \).

Answer: \mathbb{Z}^{mn}


Question: Find the last two digits of the number $9^{9^9}$.

Answer: 89


Question: Evaluate the limit \( \lim_{n\to\infty}\sum_{k=0}^n\frac{(-1)^k}{k!} \).

Answer: \dfrac{1}{e}


Question:
Let $n, m,$ and $t$ be positive integers. Define the multi-family of sequences
\[ S(n,m,t)=\sum_{k_1+\cdots+k_n=m}\binom{m}{k_1,\dots,k_n}^t \]
where the sum runs over non-negative integers $k_1,\dots,k_n$. Is it always true that $n$ divides $S(n,m,t)$?

Answer: Yes


Question: Let $G/S$ be a group scheme, and assume everything is affine. Consider an $S$-point $g : S \to G$ of $G$. Define a morphism $\phi_g : G \to G$ at the level of points by $x \mapsto g$. Let $\omega \in \Gamma(G,\Omega^1_{G/S})$ be a global differential form on $G$. Is it true that $\phi_g^* \omega = 0$?

Answer: Yes


Question: A $3 \times 3$ real symmetric matrix $K$ satisfies $K = K^3$. Given that $K(1, 1, 1) = (0, 0, 0)$ and $K(1, 2, -3) = (1, 2, -3)$, it is known that $0$ and $1$ are two of the eigenvalues of $K$. Determine the third eigenvalue of $K$.

Answer: -1


Question: Let \( X \) be a Banach space and define the set \( L = \{f:[0,1]\to X\vert\, f \text{ is Borel-measurable and } \int_0^1 \Vert f \Vert < + \infty \} \), where the integral is the Lebesgue integral. Define the operator \( T:L \to X^{**} \) by \( (Tf)(x^*) = \int_0^1 x^*\circ f \) for all \( x^* \in X^* \). The operator \( T \) is well-defined, meaning \( x^*\circ f \) is integrable and \( Tf \in X^{**} \). Let \( i: X \to X^{**} \) be the canonical embedding, defined by \( i(x)(x^*) = x^*(x) \). Is there, for every \( f \in L \), an \( x \in X \) such that \( Tf = i(x) \)?

Answer: No


Question: Given the observations of the random variable \( Y \): \(0.17, 0.06, 1.76, 3.41, 11.68, 1.86, 1.27, 0.00, 0.04,\) and \(2.10\), where \( Y = X^2 \) and \( X \sim \mathrm{N}(\mu, 1) \), estimate \( \mu \) using the least squares method.

Answer: 1.11


Question: Find the symbolic expression for \( \frac{\partial (y^2(z))}{\partial (\frac{\partial y}{\partial z})} \).

Answer: 0


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

Answer: -\dfrac{9}{2}


Question: Find the maximum value of \(|\cos(z)|\) on the circle defined by \(|z| = \pi k\), where \(k\) is a positive integer.

Answer: \cosh(\pi k)


Question: Calculate the value of \((-i)^i\).

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


Question: Determine whether there exists a connected simple undirected graph $G = (V, E)$ with $|V| > \aleph_0$ such that the complement of $G$ is also connected.

Answer: Yes


Question: Determine the kernel of the homomorphism \( f: K[X,Y] \rightarrow K[T] \) defined by \( f(X) = T^2 \) and \( f(Y) = T^3 \).

Answer: (Y^2 - X^3)


Question: Find the dimension of the vector space \( \mathbb{Q}(\sqrt{3}, \sqrt{5}, \sqrt{11}) \) over \( \mathbb{Q} \).

Answer: 8


Question: Determine whether there exists an injective function \( f : S \rightarrow S \setminus \{(1,0)\} \), where \( S \) is the set of points on the unit circle centered at \((0,0)\).

Answer: Yes


Question: Determine the limit of the sequence \(x_n = \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n}\). What is \(\lim\limits_{n\to\infty} x_n\)?

Answer: \ln 2


Question: Evaluate the integral: \[ \iint_{[0,1]\times [0,1]}  \max\{x,y\}\,dx dy \]

Answer: \dfrac{2}{3}


Question: Evaluate the integral \( \int_{0}^{2\pi}\dfrac{x\sin^{100}(x)}{\cos^{100}(x)+\sin^{100}(x)}dx \).

Answer: \pi^2


Question: Evaluate the integral \( \int_0^n \cos(2\pi \lfloor x\rfloor\{x\})\,dx \), where \( n \) is a positive integer, \( \lfloor x\rfloor \) denotes the floor function, and \( \{x\} = x - \lfloor x\rfloor \) is the fractional part of \( x \).

Answer: 1


Question: Consider the heat equation \( \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \) with the initial condition \( u(x,0)=\frac{e^{2x}-1}{e^{2x}+1} \) and boundary conditions \( u(0,t)=u(\pi,t)=0 \). Find \( \lim_{t \to \infty} u(1,t) \).

Answer: 0


Question: Find the limit: \[ \lim_{x\to\infty}\frac{1}{x}\sum_{n\leq x}\frac{(-1)^n}{n} \]

Answer: 0


Question: Find the limit of \(\left[(x^m + 1)^{1/n} - (x^m - 1)^{1/n}\right]x^{(mn-m)/n}\) as \(x\to\infty\), where \(m\) and \(n\) are natural numbers.

Answer: \dfrac{2}{n}


Question: Evaluate the limit \( \lim _{(x,y)\to (0,0)} \left(\frac{\ln(5xy+1)}{5x^8y^3+5xy} \right) \) where \(x, y > 0\).

Answer: 1


Question: Evaluate the integral \( \iint_{R} \lfloor x+y \rfloor \, \mathrm{d}x\, \mathrm{d}y \) where \( R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\} \).

Answer: 30


Question: In an arithmetic progression, let the terms be denoted as \(a, b, c, d, e\). It is given that \(a + b + c + d + e\) is the cube of an integer and \(b + c + d\) is the square of an integer. If \(a, b, c, d, e\) are natural numbers in arithmetic progression, find the least number of digits of \(c\).

Answer: 3


Question: Evaluate the Riemann-Stieltjes integral \( \int_{0}^{2}{\alpha\left(\dfrac{x}{\sqrt{2}}+1\right) \mathrm{d}\alpha(x)} \), where \( \alpha(x) = 3[x] \) and \([x]\) is the greatest integer function.

Answer: 27


Question: Evaluate the integral \( \int_1^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} x \, \mathrm{d}y \, \mathrm{d}x \) using polar coordinates.

Answer: 2\sqrt{3}


Question: Given two continuous random variables \(X\) and \(Y\) with joint probability density function \(f(x,y) = 1\) for \(0 \leq x \leq 1\) and \(0 \leq y < 1\), find the expected value \(E(|X-Y|)\).

Answer: \dfrac{1}{3}


Question:
Let $X$ be a finite set, and let $G$ be a subgroup of the symmetric group $S_{X}$. Suppose that the number of orbits of $G$ acting on $n$-colorings of $X$ is given by the polynomial
\[f(n)=\frac{1}{443520}\left(n^{11}+540n^{9}+...+10n\right).\]
Determine the number of orbits of $G$ acting on $X$. \( \boxed{10} \)

Answer: 10


Question:
Let $A$ be an open subset of the real line. Define the set $S$ as follows:
\[ S = \{s \subseteq \mathbb{R} \mid \text{$s$ is open and $(A\setminus s) \cup (s \setminus A)$ has null measure}\}. \]
Determine if the union of all elements in $S$ is itself an element of $S$.

Answer: Yes


Question: Determine the value of \( \alpha \) that will suppress sinusoidal oscillations with period 6 in the process \( X_t = A\cos(\frac{\pi}{3}t)+B\sin(\frac{\pi}{3}t) \), where \( A \) and \( B \) are uncorrelated random variables with mean 0 and variance 1, using the time-invariant linear filter \( \sum_{j=-\infty}^{\infty}\psi_jX_{t-j} \) with \( \psi_0=1, \ \psi_1 = -2\alpha, \ \psi_2 = 1, \ \psi_j = 0 \) for \( j\not = 0,1,2 \).

Answer: \dfrac{1}{2}


Question: Find the integer part of the sum \( \sum^{9999}_{k=1}\frac{1}{\sqrt[4]{k}} \).

Answer: 1332


Question:
Given the character table for $S_3$:

$$\begin{array}{c|c|c|}
 & \text{1} & \text{(12)} & \text{(123)} \\ \hline
\text{$X_1$} & 1 & 1 & 1 \\ \hline
\text{$X_2$} & 1 & -1 & 1 \\ \hline
\text{$X_3$} & 2 & 0 & -1 \\ \hline
\end{array}$$

and the class function $f$ with values:

$$\begin{array}{c|c|c|}
 & \text{1} & \text{(12)} & \text{(123)} \\ \hline
\text{f} & 19 & -1 & -2 \\ \hline
\end{array}$$

Determine if $f$ is the character of a representation.

Answer: Yes


Question: Find all values of $c \in [3, \infty)$ such that there exists a measure space with the set of measure values $\{\mu(E) : E \in S\} = [0, 1] \cup [3, c]$. Provide a justification for your answer.

Answer: 4


Question: Is the fractional Sobolev space $H^s(\Omega) = W^{s,2}(\Omega)$ a Hilbert space for $s \in (0,\infty)$ and a suitable domain $\Omega$?

Answer: Yes


Question: Let $(X, \tau)$ be a Lindelöf space and $O$ an open subset of $X$. Determine whether $O$ is necessarily Lindelöf.

Answer: No


Question:
A balloon filled with hot air is modeled by the following differential equations:
\[ \dot{\theta}(t) = -0.1\cdot (\theta(t) - \theta_U) \]
\[ \dot{v}(t) = -0.5\cdot v(t) + 2 \cdot (\theta(t) - \theta_U) \]
where \(\theta(t)\) is the temperature of the air inside the balloon, \(v(t)\) is the climb rate, and \(\theta_U\) is the constant ambient temperature. The balloon starts at time \(t=0\) with an initial temperature \(\theta_0 = \theta(t=0)\). The balloon reaches the same maximum height when \(\theta_U = 10\) and \(\theta_U = 0\). What is the difference between the starting temperatures \(\theta_0\) in each case?

Answer: 10


Question: Determine the number of digits in the sum \( \sum_{n=1}^{256}\frac{256!}{(256 - n)!} \).

Answer: 508


Question: Given the function $f: \mathbb{N}^+ \rightarrow \mathbb{N}$ defined by the rules $f(2^n) = n^2$ and $f(3n) = f(2n) + 5$, find the value of $f(3^{100})$. Provide your answer as a single integer.

Answer: 10500


Question: Determine whether there exists a prime number of the form $C(n) + P(n) + S(n)$, where $C(n)$ is the concatenation of the first $n$ primes, $P(n)$ is the product of the first $n$ primes, and $S(n)$ is the sum of the first $n$ primes.

Answer: No


Question: Evaluate the line integral \( I = \oint_R (12x^2 - 32y^2) \, dx + (16y - 24xy) \, dy \) using Green's Theorem, where \( R \) is the region bounded by the lines \( y = 0 \), \( x = 0 \), and \( x + y = 4 \).

Answer: \dfrac{1280}{3}


Question: Let $A=(a_{nk})_{n,k=1}^{\infty}$ be an infinite matrix that defines a bounded operator on $\ell^2(\mathbb{N})$ by matrix multiplication. Suppose that $B=(b_{nk})_{n,k=1}^{\infty}$ satisfies $|b_{nk}|\leq |a_{nk}|$ for all $n,k\in \mathbb{N}$. Does $B$ act boundedly on $\ell^2(\mathbb{N})$ by matrix multiplication?

Answer: Yes


Question: Using the integral test, find an upper bound for the error in approximating the series \( S = \sum_{n=3}^\infty \frac{1}{n(\ln n)^4} \) by the partial sum \( S_{30} = \sum_{n=3}^{30} \frac{1}{n(\ln n)^4} \). Provide your answer accurate to three significant digits.

Answer: 0.00847


Question: A merchant bought a quantity of cotton and exchanged it for oil, which he then sold. The number of kilograms of cotton, the number of liters of oil obtained per kilogram, and the number of rupees for which he sold the oil form a decreasing geometric progression. If he had obtained 1 kilogram more of cotton, 1 liter more of oil per kilogram, and one rupee more per liter, he would have received Rs. 10169 more. Conversely, if he had obtained 1 kilogram less of cotton, 1 liter less of oil per kilogram, and one rupee less per liter, he would have received Rs. 9673 less. How much did he actually receive in rupees?

Answer: 64000


Question: Calculate the line integral \( I = \int_{C} \mathbf{v} \cdot d\mathbf{r} \) where \( \mathbf{v}(x,y) = y\mathbf{i} - x\mathbf{j} \) and \( C \) is the semicircle of radius 2 centered at the origin, from \((0,2)\) to \((0,-2)\) along the negative x-axis (left half-plane). Use the parametrization \( \mathbf{r}(t) = (2\cos t, 2\sin t) \) for \( t \in [0, \pi] \).

Answer: -4\pi


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

Answer: \dfrac{13}{4}


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \( \lim\limits_{x \to \infty} \frac{f(x)}{x} = c \), where \( c \neq 0 \) or \( \infty \). If \( \lim\limits_{x \to \infty}(f(x+1)-f(x)) \) exists, is it necessarily equal to \( c \)? Provide a proof or a counterexample.

Answer: c


Question: Evaluate the expression \( \sum_{n=1}^{1009} \gcd(2n, 2020) - \sum_{n=0}^{1008} \gcd(2n+1, 2019) \).

Answer: 6481


Question: Evaluate the sum of the series: \[ \frac{2+6}{4^{100}} + \frac{2+2\cdot6}{4^{99}} + \frac{2+3\cdot6}{4^{98}} + \cdots + \frac{2+99\cdot6}{4^2} + \frac{2+100\cdot6}{4} \]

Answer: 200


Question:
Determine the number of differentiable functions \( y: (-\infty, +\infty) \to [0, \infty) \) that satisfy the differential equation \( y' = 2\sqrt{y} \) with the initial condition \( y(0) = 0 \). Choose the correct option:

(A) 1
(B) 2
(C) Finite but more than 2
(D) Infinite

Answer: D


Question: Let \( A \) and \( B \) be \( n \times n \) matrices. If \( (A^2)(B^2) = (B^2)(A^2) \), does it follow that \( AB = BA \)? Provide a justification for your answer.

Answer: No


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}\frac{1}{n\sqrt{n}}\sum^n_{k=1}\sqrt{k} \]

Answer: \dfrac{2}{3}


Question: Let $f(z)$ be an analytic function on an open disk $D$ centered at zero, satisfying the functional equation $f(z^3) = f(-z^3)$ for all $z \in D$. Can you conclude that $f(z) = f(-z)$ for all $z$ in some open neighborhood of the origin? Justify your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^{2\pi} \delta(\cos x) \, dx \).

Answer: 2


Question: Evaluate the integral \( \int_0^1 f(x) \, dx \), where \( f(x) \) is defined as follows: for each \( x \in [0,1) \), let \( x = 0.x_1x_2x_3\ldots \) be the decimal expansion of \( x \) (not eventually all 9's). Define \( f(x) = x_1 \), the first digit in the expansion.

Answer: \dfrac{9}{2}


Question: Evaluate the limit: \[ \lim_{x\to +\infty} \left((x^2 +1)^{1/2} -2(x^3 + x)^{1/3} +(x^4 + x^2)^{1/4}\right) \]

Answer: 0


Question: Calculate the sum of the determinants of all possible 3x3 matrices that can be formed using the numbers 1 to 9, with each number used exactly once in each matrix.

Answer: 0


Question: Calculate the fundamental group \( \pi_1 \) of the space \( S^2 - \{N, S\} \), where \( N \) and \( S \) are the north and south poles of the 2-sphere \( S^2 \).

Answer: \mathbb{Z}


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

Answer: 2


Question: Let $G$ be a finite group and $H$ a normal subgroup of $G$ with $|H| = p$, where $p$ is the smallest prime dividing $|G|$. Show that for any non-identity element $a \in H$, $G = HC_G(a)$, where $C_G(a)$ is the centralizer of $a$ in $G$. Prove this statement.

Answer: G = HC_G(a)


Question: Find the units digit of the coefficient 'a' in the remainder when \(x^{50}\) is divided by \((x-3)(x+2)\).

Answer: 5


Question: A group of order 15 acts on a set of order 22 with no fixed points. Determine the number of orbits in this group action.

Answer: 6


Question:
Evaluate the following expression:
\[ \frac{\sum_{n=0}^{\infty} \left( \frac{1}{\sqrt{3n+1}} - \frac{1}{\sqrt{3n+2}} \right)}{\sum_{n=0}^{\infty} \left( \frac{1}{\sqrt{6n+1}} - \frac{1}{\sqrt{6n+5}} \right)} \]
and verify that it equals \( 2 - \sqrt{2} \).

Answer: 2 - \sqrt{2}


Question: Given that \(a+b+c+d=0\) and \(a^2+b^2+c^2+d^2=1\), find the maximum value of \(ab+bc+cd+da\).

Answer: 0


Question: Determine the order of the quotient ring \( \mathbb{Z}[\sqrt{-5}] /(2) \).

Answer: 4


Question: Let $(X, \mathcal{A}, \mu)$ be a measurable space and $f$ a non-negative measurable function. Define the set $E=\{x \in X: f(x) < 1\}$. Calculate the limit: $$\lim_{n \to \infty} \int_E e^{-f^n} \, d\mu.$$

Answer: \mu(E)


Question: Calculate the limit superior of the sequence: \[ \limsup_{n \to \infty} \left(1+\frac{(-1)^n-3}{n} \right)^{n} \] Is the step \[ \limsup_{n \to \infty} \left(1+\frac{(-1)^n-3}{n} \right)^{n} =\lim_{n \to \infty} \left(1+\frac{1-3}{n} \right)^{n} \] correct?

Answer: e^{-2}


Question: If $k$ is a Noetherian ring, do the Kahler forms of a smooth affine $k$-algebra of dimension $d$ vanish above $d$? Specifically, is $\Omega^{d+1}_{A|k} \cong 0$?

Answer: \Omega^{d+1}_{A|k} \cong 0


Question:
Given $n+2$ points $P_0, P_1, \ldots, P_{n+1}$ in projective space such that no $n+1$ points lie on a hyperplane, determine if there exists a linear automorphism $f$ such that:

- $f(P_0) = (1:0:\dots:0)$
- $f(P_1) = (0:1:0:\dots:0)$
- $f(P_{n+1}) = (1:1:\dots:1)$.

Answer: Yes


Question:
Find all functions \( f: (0, \infty) \to (0, \infty) \) such that for all positive real numbers \( x, y, z \) with \( xyz = 1 \), the following equation holds:
\[
f\left(x + \frac{1}{y}\right) + f\left(y + \frac{1}{z}\right) + f\left(z + \frac{1}{x}\right) = 1.
\]

Answer: f(x) = \dfrac{1}{3}


Question: Compute \( \lim\limits_{x\to0} \left(\sin x + \cos x\right)^{1/x} \).

Answer: e


Question: Given a line segment $AB$ of arbitrary length and a unit length line $CD$, construct a line segment whose length is equal to the square of the length of $AB$ using only a compass and straightedge.

Answer: AB^2


Question: Evaluate the integral: \[ I = \int_0^{\pi/2} \frac{1}{1+\tan^n x} \, dx \] and show that \( I = \frac{\pi}{4} \).

Answer: \dfrac{\pi}{4}


Question: Determine the cardinality of the set of increasing functions from \( \mathbb{Z} \) to \( \mathbb{Z} \).

Answer: 2^{\aleph_0}


Question: If \( f(x) = x^{x^x} \), find \( f''(1) \).

Answer: 2


Question:
Let \(f(x)\) and \(g(x)\) be two cubic polynomials such that \(g(-x+1)=f(x)\). If the sum of the distinct roots of \(f(x)\) is 10, which of the following cannot be the sum of the distinct roots of \(g(x)\)?

(a) \(-10\), (b) \(-9\), (c) \(-8\), (d) \(-7\).

Answer: A


Question:
Let $f(x)$ be a quartic polynomial with extreme values at $x=1$ and $x=2$. Given that \( \lim_{x\to0}\left[1+\frac{f(x)}{x^2}\right]=3 \), find the value of $f(2)$. Choose from the following options:

(1) $-8$
(2) $-4$
(3) $0$
(4) $4$

Answer: C


Question: Determine all real numbers \(a\) such that the equation \(x^2a - 2x + 1 = 3|x|\) has exactly three distinct real solutions for \(x\).

Answer: \dfrac{1}{4}


Question: Determine whether the following statement about product topology is correct: Let $I$ be a set of indices. Also, let $\Omega \subset \mathbb{R}^I$ be compact in the sense of product topology. Assume $x \in \mathbb{R}^I$ and $x \notin \Omega$. Then there exists a finite subset of indices $J \subset I$, such that the corresponding projection operator $P_J$ satisfies $P_J(x) \notin P_J(\Omega)$. Is this statement true or false?

Answer: A


Question:
Determine whether the equilibrium states of the following system of differential equations are only possible if $S=0$:

$$ \begin{aligned} I' &= -bkIS \\ S' &= bkIS - akS(S+R) \\ R' &= akS(S+R) \end{aligned} $$

where $I'=S'=R'=0$. Provide a justification for your answer.

Answer: Yes


Question: Assume that $F$ is a field such that $GL_n(F)$ is a divisible group. Is it true that the multiplicative group $F^*$ is also divisible?

Answer: Yes


Question: Given a partially separable 3-qubit state \( \phi = \left(a_0\left|0\right\rangle + a_1\left|1\right \rangle\right) \otimes \left(b_{00}\left|00\right \rangle + b_{01}\left|01\right \rangle + b_{10}\left|10\right \rangle + b_{11}\left|11\right \rangle\right) \), where the second and third qubits are entangled, and the first qubit is separable, the unseparated state is given by \( \phi = c_{000}\left|000\right\rangle + c_{001}\left|001\right\rangle + c_{010}\left|010\right\rangle + c_{100}\left|100\right\rangle+ c_{011}\left|011\right\rangle  + c_{101}\left|101\right\rangle + c_{110}\left|110\right\rangle + c_{111}\left|111\right\rangle \) with \( c_{ijk} = a_i b_{jk} \). If a unitary transformation is applied to the first two qubits using a 4x4 unitary matrix \( U = (u_{nm}) \), determine the form of the overall 8x8 matrix that represents the effect on the entire 3-qubit system, including the third qubit.

Answer: U \otimes I


Question: Evaluate the expression \( \frac{1}{\cos0^\circ \cos1^\circ} + \frac{1}{\cos1^\circ\cos2^\circ} + \ldots + \frac{1}{\cos44^\circ\cos45^\circ} \) in terms of \( x = \sin 1^\circ \).

Answer: \dfrac{1}{x}


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{a_{12}(n)}{a_{22}(n)} \), where \( a_{ij}(n) = \begin{pmatrix}2 & 1\\ 1 & 0\end{pmatrix}^n \).

Answer: 1 + \sqrt{2}


Question: Calculate the number of elements in the symmetric group \( S_{10} \) that commute with the permutation \( \sigma = (1\ 3\ 5\ 7\ 9) \).

Answer: 600


Question: Given a bi-regular bipartite graph $G=(V,E)$, where $V=V_1 \cup V_2$ and $V_1 \cap V_2=\emptyset$, with $\text{deg}(u)=m$ for all $u\in V_1$ and $\text{deg}(v)=n$ for all $v\in V_2$, such that $m|V_1|=n|V_2|$. Suppose there exists an integer $k < m$ such that $k|V_1|=n L$ where $L < |V_2|$ is an integer. Can an induced bi-regular subgraph be found by selecting $L$ vertices from $V_2$ and all vertices from $V_1$?

Answer: Yes


Question: Is it correct to assume that the algebraic multiplicities of the eigenvalues of an $n \times n$ projection matrix with rank $m$ are equivalent to their geometric multiplicities?

Answer: Yes


Question: Evaluate the surface integral \( \int_S { \vec{D} \cdot} d\vec{s} \) over the surface of the cube defined by the vector field \( \vec{D} = 2y\vec{a_x} +  xz\vec{a_y} +z\vec{a_z} \), where the cube's coordinates are from \(x, y, z \in (0, 2)\).

Answer: 8


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

Answer: 1


Question: Determine the crossing number of the complete graph $K_5$ on the tubular neighborhood of a trefoil knot.

Answer: 0


Question: Find the remainder when the number formed by concatenating all integers from 1 to 2013 is divided by 75.

Answer: 63


Question: Evaluate the limit \( \lim_{x \to 0} \frac{\sin(x^n) - \sin^n(x)}{x^{n+2}} \) for \( n \geq 2 \).

Answer: \dfrac{n}{6}


Question: Given the sequence defined by $U_{n+1} = 1 - \sqrt{1 - U_n}$ with $0 < U_1 < 1$, find the value of $\lim_{n \to \infty} \frac{U_{n+1}}{U_n}$. Provide your answer as a single number.

Answer: \dfrac{1}{2}


Question: Find the residue of the function \( f(z) = \frac{z}{8-z^3} \) at \( z = 2 \).

Answer: -\dfrac{1}{6}


Question: Find the number of elements of order 4 in the group \((\mathbb{Z}/525\mathbb{Z})^\times\).

Answer: 8


Question: Let \( f \in \mathbb{R}[x_1, \ldots, x_n] \) be a homogeneous real polynomial. Suppose there exists a polynomial \( h \in \mathbb{R}[x_1, \ldots, x_n] \) such that \( f = h \cdot (x_1 + \cdots + x_n - 1) \). Is it necessarily true that \( f = 0 \)? Justify your answer.

Answer: 0


Question: Evaluate the sum \( \sum_{r=0}^8 (-1)^r \binom{20}{r} \binom{20}{8-r} \).

Answer: 4845


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

Answer: \dfrac{\pi}{e}


Question: Evaluate the limit: \[ \lim_{n \to \infty} 2n \int_0^1 \frac{x^{n-1}}{1+x} \, dx. \]

Answer: 1


Question: Given the function \( f(x) = \frac{4x^3}{x^2+1} \), find the value of \( \frac{d}{dx} (f^{-1}(2)) \).

Answer: \dfrac{1}{4}


Question: Let $R$ be a discrete valuation ring (dvr), and $X$ a flat, projective, integral, normal $R$-scheme such that every closed fiber is integral and normal. Suppose $F$ is a torsion-free coherent sheaf on $X$, flat over $R$. Is the restriction of $F$ to the special fiber torsion-free?

Answer: Yes


Question: Find \( \lim_{x \to 0}{\dfrac{\cos{x}-e^{-\left(x^2\right)/2}}{20x^4}} \).

Answer: -\dfrac{1}{240}


Question: Let \( C[0, 1] \) be the space of all continuous functions on the interval \([0, 1]\) equipped with the norm \( \|f\| = \max_{0 \leq x \leq 1} |f(x)| \). Define the operator \( A \) by \( Af(x) = \int_0^1 x f(y) \, dy \). Find the operator norm \( \|A\| \).

Answer: 1


Question: Prove that \(2^{1194} + 1\) is divisible by 65.

Answer: 65


Question: Find the minimum distance between the curves \( f(x) = e^x \) and \( g(x) = \ln x \).

Answer: \sqrt{2}


Question: Find the sum of the radicals: \( \sqrt{3-\sqrt{5}} + \sqrt{3+\sqrt{5}} \).

Answer: \sqrt{10}


Question: Determine the number of ways to distribute 7 different balls into 4 identical boxes such that no box is empty.

Answer: 350


Question: Evaluate the limit of the sequence: \[ \lim_{n \rightarrow \infty} n \sin(2 \pi e n!) \]

Answer: 2\pi


Question: Calculate the residue of the function \( f(z) = \frac{1-e^{2z}}{z^4} \) at the pole \( z = 0 \).

Answer: -\dfrac{4}{3}


Question: Verify the calculation of the Chern number for the product of complex projective lines \( \mathbb{C}P^1 \times \mathbb{C}P^1 \). Given the Chern class \((1 + 2a^2)(1 + 2a^2) = 1 + 4a^2 + 4a^4\), determine if the Chern number \(c_1c_1[\mathbb{C}P^1\times \mathbb{C}P^1]\) is 8 or 16.

Answer: 8


Question: Evaluate the double integral \( \int_{-5}^5 \int_{-5}^5 -\frac{3}{2}|x+y| - \frac{3}{2}|x-y| + 15 \,\mathrm{d}x\,\mathrm{d}y \).

Answer: 500


Question:
Given the four-potential \( A = (x^1, 2x^1, x^3 - 2x^2, 2x^3) \), determine which electric and magnetic fields are generated. Choose the correct option:

1. \( E = (-c,0,2c), B = (0,1,-2) \)
2. \( E = (c,0,2c), B = (1,0,-2) \)
3. \( E = (-c,-2c,2c), B = (1,0,1) \)
4. \( E = (-c,0,0), B = (1,0,0) \)
5. \( E = (c,0,0), B = (0,-2,-2) \)

Answer: 4


Question: Determine the degree of the map \( f: T^2 \rightarrow T^2 \) given by \( f(z,w) = (z^2, w^3) \).

Answer: 6


Question: Evaluate the contour integral \( \oint_{|z-i|=1} \frac{z^3}{e^z+i} \, dz \) by finding the singularities of the integrand inside the contour.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty}\frac{n^k}{p^{\operatorname{val}_p(n!)}}. \]

Answer: 0


Question: In a finite-dimensional inner product space over \(\mathbb{R}\), does every self-adjoint operator have an eigenvalue? Provide a justification for your answer.

Answer: Yes


Question:
Consider the matrix polynomial \( P_\Delta(\lambda) = (A_m + \Delta_m)\lambda^m + \cdots + (A_1 + \Delta_1)\lambda + (A_0 + \Delta_0) \), where \( A_j, \Delta_j \in \mathbb{C}^{n \times n} \) for \( j = 0, 1, 2, \ldots, m \), and \( \lambda \) is a complex variable. Determine if the following limit holds:
\[
\lim_{\lambda \to \infty} \frac{\left| \lambda \right|^{mn}}{\left| \det(P_\Delta(\lambda)) - \det(A_m + \Delta_m)\lambda^{mn} \right|} = \infty.
\]

Answer: \infty


Question: Determine the number of solutions to the equation \(x^4 - 1 = e^x\).

Answer: 3


Question: Determine the number of symmetric relations on the set \( A = \{1, 2, 3, 4, 5, 6, 7\} \) that contain exactly four ordered pairs.

Answer: 686


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

Answer: \ln 2


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} (\sin n) x^n \).

Answer: 1


Question: Evaluate the integrals \( J = \int_0^9 e^x \left(1+\{x\}+\{x\}^2\right)dx \) and \( I = \int_0^1 e^x \left(1+\{x\}+\{x\}^2\right)dx \), where \( \{x\} \) denotes the fractional part of \( x \). If \( \frac{J}{I} = \sum_{n=0}^k e^n \), find the value of \( k \).

Answer: 8


Question:
If \(x^2 + px + 1\) is a factor of \(ax^3 + bx + c\), where \(a, b, c \in \mathbb{R}\), determine the correct relationship among the coefficients from the following options:

A) \(a^2 + c^2 + ab = 0\)

B) \(a^2 - c^2 + ab = 0\)

C) \(a^2 - c^2 - ab = 0\)

D) \(ap^2 + bp + c = 0\)

Answer: C


Question: If $f^n$ is measurable for all $n > 4$, is the function $f$ necessarily measurable?

Answer: Yes


Question: Calculate the supremum \( \sup_{ x\in X-\{0\}} \frac{d(x,E)}{\|x\|} \), where \(E \subseteq X\) is a closed, proper subspace of a normed \(\mathbb{K}\)-linear space \(X\).

Answer: 1


Question: In a town, earthquakes, floods, and meteor strikes occur as Poisson processes with averages of 3, 2, and 1 per decade, respectively. Let $E$, $F$, and $M$ represent the time in decades until the first earthquake, flood, and meteor strike. Calculate the covariance $\text{Cov}(\min\{E, F, M\}, M)$. Provide your answer as a single numerical value.

Answer: \dfrac{1}{36}


Question: Find the value of \( \sum_{k=1}^{\infty}\frac{1}{z_k^2} \), where \( z_1, z_2, \dots, z_k, \dots \) are all the roots of the equation \( e^z = z \). Assume that \( \lim_{N\to \infty} \int_{C_N}\frac{e^z-1}{z^2(e^z-z)}dz=0 \), where \( C_N \) is the square in the complex plane centered at the origin with sides parallel to the axes and each of length \( 2\pi N \).

Answer: -1


Question:
Determine whether the following statement is true:
\[ \sum_{i=1}^n \frac{s_i}{i + \sqrt{s_i}} = O( \sqrt{\sum_{i=1}^n s_i} \log n ) \]
where \(s_i \geq 1\) for all \(i\).

Answer: False


Question: Given a simple, undirected graph $G=(V,E)$, determine if there exists a regular graph $G_R$ such that $G$ is isomorphic to an induced subgraph of $G_R$ and the chromatic number of $G$ is equal to the chromatic number of $G_R$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \frac{i}{4-\pi}\int_{|z|=4}\frac{1}{z\cos{z}}dz \).

Answer: 2


Question: In a box containing 91 balls, $m$ of them are red and the rest are blue. A sample of 19 balls is drawn, resulting in 7 red balls and 12 blue balls. Estimate the number of red balls, $m$, using the method of maximum likelihood.

Answer: 33


Question: A clerk drops $n$ matching pairs of letters and envelopes. He then places the letters into the envelopes in a random order. Let $X$ be the number of correctly matched pairs. Find the variance of $X$. \( \text{Var}(X) = ? \)

Answer: 1


Question: Let $V$ be a vector space with a constant metric, and let $\nabla$ be the Levi-Civita connection with respect to this metric. Given a constant vector field $w$ on $V$ and a curve $\gamma$ on $V$, determine if $\nabla_{\dot{\gamma}_t}w = 0$ holds.

Answer: 0


Question: Let \( A = \begin{bmatrix} 0 & 1 \\ -1 & 1 \end{bmatrix} \). Find the smallest positive integer \( n \) such that \( A^n = I \), where \( I \) is the identity matrix. Choose from the following options: (a) 1, (b) 2, (c) 4, (d) 6.

Answer: D


Question: Given a positive integer $N$ and two integers $r$ and $s$ in $\mathbb{Z}_N$ such that $\gcd{(N, r)}=1$ and $\gcd{(N, s)}=1$, define $g=\gcd{(N, r-s)}$, $u=N/g$, and $v=(r-s)/g$. Let $r^{-1}$ and $s^{-1}$ be the multiplicative inverses of $r$ and $s$ in $\mathbb{Z}_N$, respectively. Since $r^{-1} - s^{-1} \equiv 0 \pmod{g}$, we can write $r^{-1} - s^{-1} = v'g$. Are $u$ and $v'$ coprime?

Answer: Yes


Question: In a bank, people arrive according to a Poisson process with a rate of \( \lambda = 1 \) client per minute. Each client makes a deposit of \( Y \sim \mathrm{Unif}\{1,2\} \) thousand dollars. Calculate the probability that at time \( t=5 \) minutes, the total deposit is exactly 6 thousand dollars.

Answer: \dfrac{154625}{9216} e^{-5}


Question: Determine whether the function \( f(x) = \begin{cases} 1, & x = \frac{1}{n}, \; n \in \mathbb{N} \\ 0, & \text{otherwise} \end{cases} \) is Riemann integrable over the interval \([0, 1]\).

Answer: Yes


Question:
Find the eigenvalues of the linear transformation \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) that maps the circle \(C=\{(x,y):x^2+y^2=1\}\) to the ellipse \(E=\{(x,y): \frac{x^2}{9} + \frac{y^2}{4}=1\}\). Choose the correct option:
(1) \(1,-1\)
(2) \(3,2\)
(3) \(-3,-2\)
(4) None of the above.

Answer: D


Question: Determine the closure of the set \( S = \left\{ \frac{a}{p^n} \in \mathbb{R} \mid a \in \mathbb{Z}, n \in \mathbb{N} \right\} \) in \( \mathbb{R} \) with respect to the usual topology, where \( p \) is a fixed prime number.

Answer: \mathbb{R}


Question:
Consider the function field \( \mathbb{F}_2(t) \) over \( \mathbb{F}_2 \). Let \( \mathbb{F}_2(t)(\alpha) \) be an extension of degree 4 of \( \mathbb{F}_2(t) \), where the minimal polynomial of \( \alpha \) over \( \mathbb{F}_2(t) \) is given by:
\[
\alpha^4 + a\alpha^2 + b = 0 \quad (a, b \in \mathbb{F}_2(t)).
\]
It is known that \( \mathbb{F}_2(t)(\alpha) \) is an inseparable extension of \( \mathbb{F}_2(t) \) and that \( \mathbb{F}_2(t)(\alpha^2) \) is a separable closure of \( \mathbb{F}_2(t) \) of degree 2 inside \( \mathbb{F}_2(t)(\alpha) \).

Does \( \mathbb{F}_2(t)(\alpha) \) always contain an intermediate field \( L \) such that \([L: \mathbb{F}_2(t)] = 2\) and \( L \) is inseparable over \( \mathbb{F}_2(t) \)?

Answer: No


Question: Find the largest positive integer \( n \) such that \( \frac{(n!)^2}{(2n)!} \leq \frac{1}{500} \).

Answer: 6


Question: Let \((X, Y)\) have a uniform distribution on the set \((x, y) : 0 < x < 2\) and \(0 < y < 8\) and \(x < y\). Find the probability \(P(Y < X^3)\).

Answer: \dfrac{9}{56}


Question: Determine whether there exists a character \( \chi: G_m^n \to G_m \), which is both a morphism and a group homomorphism, such that a given proper subvariety \( V \subseteq G_m^n \) (where \( V \) is also a subgroup) satisfies \( V \subseteq \ker \chi \).

Answer: Yes


Question: Given a pair (N, S) in the Tonalpohualli calendar, where N is the number and S is the symbol, determine the original day number from 1 to 260 that corresponds to this pair. For example, find the day number that produces the pair (3, 12).

Answer: 172


Question: Evaluate the improper integral: $$\int_0^1\frac{dx}{\sqrt{-\ln x}}.$$

Answer: \sqrt{\pi}


Question: Evaluate the limit \( \lim _{n\to \infty }\int_1^2\:\frac{x^n}{x^n+1}dx \).

Answer: 1


Question: Consider a matrix $A \in \mathbb{R}^{m \times n}$ with $m \geq n$ and $\mathrm{rank}(A) = n$. There exists at least one non-vanishing minor of order $n$. Now, append $A$ with a matrix $B \in \mathbb{R}^{p \times n}$ to form $\left[ \begin{array}{c} A\\ B \end{array}\right] \in \mathbb{R}^{(m+p) \times n}$, where $\mathrm{rank}(B) < n \leq p$. Can the number of non-vanishing minors of order $n$ increase?

Answer: Yes


Question: Evaluate the expression \( \int_1^2 f(x)\,dx + 2\int_1^5 f^{-1}(2x)\,dx \) where \( f(x) = x^3 + x \).

Answer: 18


Question: Draw cards one at a time, with replacement, from a standard deck of playing cards. What is the expected number of draws until four successive cards of the same suit appear?

Answer: 85


Question: Calculate the flux of the vector field \( \mathbf{A}(x,y,z) = \left(\frac{-6x}{x^2+y^2},\frac{-6y}{x^2+y^2},z+1\right) \) through the surface \( S: x^2+4y^2=4, 0 \leq z \leq 1 \). The flux is directed away from the z-axis. Use the integral \( I = -12 \cdot \int_0^1 \int_0^{2\pi} \frac{1}{1+3\cos^2(\theta)} \,d\theta \,dz \) to find the flux. Verify that the solution is \(-12\pi\).

Answer: -12\pi


Question: Find the surface area of the part of the sphere defined by the equation \(x^2 + y^2 + z^2 = 4\) that lies above the plane \(z = 1\).

Answer: 4\pi


Question: Calculate the volume of the solid paraboloid of revolution defined by the equation \( y = 10 - x^2 - z^2 \) with the condition \( y \geq 0 \).

Answer: 50\pi


Question: Find the volume of the solid region lying below the surface \(z = (7x + 2y)^2 \sqrt{2y-x}\) and above the plane region \(R\), where \(R\) is the region bounded by the parallelogram with vertices \((0, 0), (-1,7/2), (2,5), (3,3/2)\). Use a change of variables to solve the problem.

Answer: 3072\sqrt{2}


Question: In the quadrilateral $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively. The centroid of triangle $BNP$ is $F$, and the centroid of triangle $NPD$ is $G$. The line $MG$ intersects $FQ$ at point $K$. Given that $FK = 6$, prove that $KQ = 9$.

Answer: 9


Question:
Given a linear model \(Y = X\beta + \epsilon\) with three treatments and six subjects, where \(X\) is the design matrix, consider the following design matrices:

\[ X = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & -1 & -1 \\ 1 & -1 & -1 \end{bmatrix} \]

and

\[ X' = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \end{bmatrix} \]

with the response vector \(Y = [Y_{11}, Y_{12}, Y_{21}, Y_{22}, Y_{31}, Y_{32}]^T\) and the constraint \(\sum_{i=1}^{3}\beta_i = 0\). Do these two matrices represent the same model? Justify your answer.

Answer: Yes


Question: Evaluate the contour integral \( \int_{C(0,7)} \frac{1}{(z-1)(z-3)} \, dz \) along the circular path centered at the origin with radius 7.

Answer: 0


Question: If \(A = \sin\left(\frac{2\pi}{7}\right) + \sin\left(\frac{4\pi}{7}\right) + \sin\left(\frac{8\pi}{7}\right)\) and \(B = \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{8\pi}{7}\right)\), find the value of \(A^2 + B^2\).

Answer: 2


Question: Find the order of the element \( \alpha \) in the field \( \mathbb{F}_2[\alpha] \), where \( \alpha^6 = \alpha^4 + \alpha^3 + \alpha + 1 \). Verify if the calculated order is consistent with the possible divisors of the multiplicative group order.

Answer: 63


Question: Suppose you have a straight line of length \( L \). A point is chosen at random along the line, and a second point is chosen at random to the left of the first point. What is the expected distance between these two points?

Answer: \dfrac{L}{4}


Question:
For what values of \( t \in \mathbb{R} \) is the function \( f(x) \) differentiable at \( x = 0 \), where
\[
f(x) =
\begin{cases}
x^t \sin \frac{1}{x^2} & \text{if } x \neq 0 \\
0 & \text{if } x = 0
\end{cases}?
\]

Answer: t > 1


Question: The circumferences of Lebanese cedar cones are modeled as Gamma-distributed random variables with a mean of 5.40 cm and a variance of 3.24 cm². A botanist is interested in collecting cones with a circumference greater than 6 cm. Calculate the expected number of cones she needs to inspect to find her third cone with a circumference greater than 6 cm.

Answer: 9.01


Question: Determine whether the expression \( d(n) = \sum_{q=1}^{\lceil n/2\rceil} R_q(n) \) equals the number of divisors of \( n \), where \( R_q(n) = \begin{cases} r\left(\frac{n}{2q-1}\right) & \text{if } (2q-1) \mid n \\ 0 & \text{otherwise} \end{cases} \) and \( r(n) \) is the 2-adic valuation of \( 2n \).

Answer: Yes


Question: Let \( g(n) \) be a function defined for all integers \( n \geq 0 \) as follows: \[ g(n) = \begin{cases} 0, & \text{if } n=0 \\ 1+g(n/2), & \text{if } n > 0 \text{ and } n \text{ is even} \\ 1+g(n-1), & \text{if } n > 0 \text{ and } n \text{ is odd} \end{cases} \] Determine the value of \( g(2^l + 2^k) \) where \( l > k \geq 0 \) are integers.

Answer: l + 2


Question: Evaluate the limit: $$\lim_{x \to 0^+} \frac{-1+\sqrt{(\tan x-\sin x) +\sqrt{(\tan x-\sin x) +\sqrt{(\tan x-\sin x})\ldots\infty } }}{-1+\sqrt{x^3+\sqrt{x^3}+ \sqrt{x^3}...\infty}}.$$

Answer: \dfrac{1}{2}


Question: Is it true that every oriented vector bundle with odd-dimensional fibers has a global section that vanishes nowhere?

Answer: No


Question: Let $A$ be a random $n \times n$ matrix, where each entry $X_{ij}$ is independent and $P(X_{ij}=1)=P(X_{ij}=-1)=1/2$. Compute $\text{Var}(\text{det}(A))$. Provide your answer as a function of $n$.

Answer: n!


Question: Evaluate the limit: \[ \lim_{R \to \infty} \int_0^{\frac{\pi}{2}} e^{-R \sin{x}} \, dx \]

Answer: 0


Question: Let $A$ and $B$ be Riemann surfaces, and let $f: A \to B$ be a bijective morphism of Riemann surfaces, i.e., a bijective holomorphic function. Is the inverse of $f$ also holomorphic?

Answer: Yes


Question: How many distinct words can be formed using all the letters of the word "MATHEMATIK" such that no two identical letters are adjacent?

Answer: 236880


Question:
Determine the general solution of the differential equation \(x\frac{d^2y}{dx^2}-\frac{dy}{dx}+xy=0\) in terms of Bessel's functions of order 0 or 1. Choose the correct option:

1. \(axJ_1(x)+bxY_1(x)\)
2. \(aJ_1(x)+bY_1(x)\)
3. \(aJ_0(x)+bY_0(x)\)
4. \(axJ_0(x)+bxY_0(x)\)

Answer: 1


Question: Find all natural numbers \(x, y, z\) such that \(7^x + 1 = 3^y + 5^z\).

Answer: (1, 1, 1)


Question: Evaluate the principal value integral \( PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2} \, dx \) for \( a > 0 \) using the method of residues.

Answer: \pi a


Question: Given a function \( f \in C^0(I \times \mathbb{R}) \) where \( I = [\xi, \xi + a] \) and \( a > 0 \), suppose \( \max_{x \in I} |f(x, \eta)| = M < \infty \) for some \( \eta \in \mathbb{R} \). Assume \( f \) is differentiable with respect to \( y \) and \( |f_y(x, y)| \leq K|f(x, y)| \) uniformly for \( x \in I \). For any \( x \in I \), \( f(x, \cdot) \in C^1(\mathbb{R}) \) is locally Lipschitz, meaning there exists \( \delta_x > 0 \) and \( L_x \geq 0 \) such that for all \( y_1, y_2 \in [\eta - \epsilon_x, \eta + \epsilon_x] \), \( |f(x, y_1) - f(x, y_2)| \leq L_x |y_1 - y_2| \). Is the Lipschitz constant \( L_x \) continuous with respect to \( x \)? Assume \( L_x \) is taken as the infimum of all possible values.

Answer: Yes


Question: Does there exist an infinite-dimensional Hilbert space $X$ such that every convergent sequence in $X$ is relatively compact?

Answer: Yes


Question: Evaluate the triple integral: $$\int_{0}^{2}\int_{6}^{1}\int_{0}^{1} xyz\sqrt{2-y^2-x^2}\,dx\,dy\,dz$$

Answer: \dfrac{8\sqrt{2} - 4}{15}


Question: Let $X$ be a non-empty set and $A, B, C, A', C' \in \mathcal{P}(X)$ such that $A \subseteq B \subseteq C$, $A' \subseteq C'$, $|A| = |A'|$, and $|C| = |C'|$. Is it true that there exists a set $B' \in \mathcal{P}(X)$ such that $A' \subseteq B' \subseteq C'$ and $|B| = |B'|$? (Here, $|A| = |B|$ means there is a bijection between $A$ and $B$.)

Answer: Yes


Question: You are fighting a dragon with three heads. Each time you swing at the dragon, you have a 20% chance of hitting off two heads, a 60% chance of hitting off one head, and a 20% chance of missing altogether. If you knock off one head, the head grows back immediately before the next iteration. If you miss, an additional head grows immediately before the next iteration. If you knock off two heads, the heads stay knocked off and you move to the next iteration. You win if you knock off all of the dragon's heads, and the dragon wins if at any time it has five heads. What is the probability that you win the game?

Answer: \dfrac{2}{3}


Question: Determine the region of convergence for the complex series \( \sum \frac{z^n}{1-z^n} \).

Answer: |z| < 1


Question: Determine the number of real roots of the equation \(11^x + 13^x + 17^x - 19^x = 0\).

Answer: 1


Question: Determine the number of zeroes of the function \(6z^3 + e^z + 1\) within the unit disc \(|z|<1\).

Answer: 3


Question: Find the variance of a binomial distribution $B(n,p)$ where $n$ is a random variable following another binomial distribution $B(m,q)$. In other words, calculate $Var(B(B(m,q),p))$. Provide your answer in terms of $m$, $q$, and $p$.

Answer: m p q (1 - p q)


Question: Suppose $p$ is a projection in a $C^*$ algebra and $vv^*$ is another projection such that $vv^* \leq p \leq Cvv^*$, where $C$ is a constant. Prove that $p = vv^*$.

Answer: p = vv^*


Question: Does there exist a family of compact complex manifolds over a unit disk such that the Hodge numbers are not constant in the family?

Answer: Yes


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

Answer: -2e^2


Question: Given the equation \(u^2 + (u-2v-1)^2 = -4v(u+v)\), find the value of \(u + 3v\).

Answer: -\dfrac{1}{4}


Question: Suppose you have a set of coins, N, and in each round, you flip all the coins. Given infinite rounds, what is the probability that there will be at least one round where all N coins show either heads or tails?

Answer: 1


Question: Is the non-falling rank of smooth maps an open condition? Specifically, if $f \colon M \to N$ is a smooth map of smooth manifolds, for any point $p \in M$, is there an open neighborhood $V$ of $p$ such that for all $q \in V$, $\mathrm{rnk}_q (f) \geq \mathrm{rnk}_p (f)$?

Answer: Yes


Question: Given a normal distribution $X \sim N(1,2)$, find the expected value $E[X^3]$. Use the properties of the normal distribution to determine this value.

Answer: 7


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

Answer: 1


Question:
Let $U$ and $V$ be random variables jointly uniformly distributed over the square with corners $(0,0), (1,0), (1,1)$, and $(0,1)$. Define $X = UV$. Find the cumulative distribution function (CDF) and probability density function (PDF) of $X$. Specifically, verify if the following CDF is correct:

$$\mathbb{P}(UV \leq x) = \begin{cases}0 & \text{if } x > 1 \text{ or } x < 0,\\ x & \text{if } 0 \leq x \leq 1.\end{cases}$$

Additionally, determine the PDF of $X$. Assume the PDF is obtained by differentiating the CDF with respect to $x$. Provide your answer for the PDF.

Answer: -\ln x


Question:
Calculate the length of the curve that is the intersection of the surfaces given by the equations:
\[ 4x = (y+z)^2 \]
\[ 4x^2 + 3y^2 = 3z^2 \]
Find the length of the curve from the point \( O(0,0,0) \) to the point \( M(x,y,z) \). Express your answer in terms of \( z \).

Answer: \sqrt{2} z


Question: If a subset of \( \mathbb{R} \) has Lebesgue outer measure zero, does its Cartesian product with an arbitrary subset of \( \mathbb{R} \) also have Lebesgue outer measure zero in \( \mathbb{R}^2 \)?

Answer: Yes


Question: Find a cluster point of the sequence \( x_{n} = (\pi \cdot n) \mod 2 \).

Answer: 1


Question:
If $a, b, c$ are the roots of the polynomial $x^3 - px^2 + qx - r = 0$, find the value of $(a+b-c)(b+c-a)(c+a-b)$. Choose the correct answer from the following options:

A) $p^3 - 8r$

B) $4pq - p^3$

C) $4pq - p^3 - 8r$

D) $4pq - 8r$

Answer: C


Question: If two domains $A$ and $B$ in the complex plane are conformally equivalent and $A$ is simply connected, is $B$ also simply connected?

Answer: Yes


Question: For a given integer $n$, does there always exist a prime number $p$ such that $p \equiv 1 \pmod{n}$?

Answer: Yes


Question: Determine if there exists a large finite integer \( n \) such that \( \sum_{k=n}^{\infty}\frac{1}{(2k-1)^2} < \epsilon \) for any positive real number \( \epsilon \).

Answer: Yes


Question: Determine the spectrum of the operator defined by multiplication by $x$ on $L^2([0,1])$. Is the spectrum $\sigma(T) = [0,1]$?

Answer: [0,1]


Question: Find the last digit of \((K+6)^{(K+6)!}\), where \(K=\sum_{n=1}^{\infty}\frac{6^n}{(3^n-2^n)(3^{n+1}-2^{n+1})}\).

Answer: 6


Question: Let \( f \) be a nonnegative measurable function on \( \mathbb{R}^d \) such that \( m(\{x \mid f(x) > \lambda\}) = \frac{1}{1+\lambda^2} \). For which values of \( p \) is \( f \in L^p(\mathbb{R}^d) \)?

Answer: (0, 2)


Question: Find the polynomial \( p(x) \) such that \( p(x) = p(x+3) \) for all \( x \).

Answer: c


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{((2n)!)^2}{2^{4n}(n!)^4} \]

Answer: 0


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

Answer: \dfrac{2}{3}


Question: Determine the number of ones $n$ in the number $N = 11\dots1$ such that $N$ is a prime number. Exclude the cases where $n = 1$ or $n = 3k$ for some integer $k$. For which values of $n$ is $N$ prime?

Answer: 2


Question:
Find the largest natural number \( n \) such that there exists a real number \( C \) for which the function \( g(x) = \begin{cases} \frac{-\frac{1}{12}x^4 + o(x^5)}{x^n} & \text{if } x \neq 0 \\
C & \text{if } x = 0 \end{cases} \) is differentiable at \( x = 0 \).

Answer: 4


Question: Let \( k \) be a positive integer and \( w \) be an integer such that two of the roots of the cubic equation \( x^3 - 17x^2 + kx + w = 0 \) are consecutive positive integers. Find the value of \( k \) that maximizes the absolute value of \( k - w \).

Answer: 96


Question: Evaluate the limit: \[ \lim_{\eta\rightarrow\infty}[(2\eta + 5)^x-(2\eta)^x + 1] \] where \(x\in (0,1)\).

Answer: 1


Question: Calculate the average value of the function $F(x,y,z) = z$ over the region $E$, which is the top half of a sphere with radius 2. The region is defined by $z \geq 0$ and $x^2 + y^2 + z^2 \leq 4$. Use the formula for the average value of a function over a volume: \( \overline{z} = \frac{1}{V} \int \int \int_{E} z \, dV \), where \( V \) is the volume of the region.

Answer: \dfrac{3}{4}


Question:
Evaluate the limit of the integral as \( n \to \infty \):
\[ \lim_{n \to \infty} \int_0^1 \frac{1}{1+x+x^2+\cdots+x^n} \, dx \]

Answer: \dfrac{1}{2}


Question: Let $F: \mathbb{R}^n \to \mathbb{R}^m$ ($m < n$) be a Lipschitz function whose Jacobian determinant $J F$ does not vanish on a compact set $A \subseteq \mathbb{R}^n$. Assume $J F$ exists everywhere. Is $J F(x)$ bounded away from zero for all $x \in A$?

Answer: No


Question: Evaluate the integral \( \int_{\{ p \}} f(x) \, \text{dvol}_{\{ p \}}(x) \), where \( f: S^2 \rightarrow \mathbb{R} \) is a continuous function on the standard sphere, \( p \in S^2 \) is a fixed point, and \( \text{dvol}_{\{ p \}}(x) \) is the Riemannian volume element of the manifold \( \{ p \} \).

Answer: f(p)


Question: Find the length of the curve defined by the parametric equations \(x(t) = 1 + 2\cos(t) + \cos(2t)\) and \(y(t) = 2\sin(t) + \sin(2t)\) for \(0 < t < 2\pi\).

Answer: 16


Question: Evaluate the limit: \[ \lim_{n\to\infty}(n^2 + \sqrt[3]{n^4-n^6}). \]

Answer: \dfrac{1}{3}


Question:
Let \( f(x) \) be a polynomial of degree four such that:
\[ f(1) = 1, \quad f(2) = 4, \quad f(3) = 9, \quad f(4) = 16, \quad f(7) = 409 \]
Find \( f(5) \).

Answer: 49


Question:
Calculate the greatest integer less than or equal to the quotient of the following improper integrals:
\[
 \left\lfloor{\ \frac{\int_0^{\infty} e^{-x^{2}}\,dx}{\int_0^{\infty} e^{-x^{2}}\cos 2x\, dx}}\ \right\rfloor
\]
Given that \( \int_0^{\infty} e^{-x^{2}}\,dx = \sqrt{\pi}/2 \).

Answer: 2


Question: Determine a formula in terms of \(n\) for the maximum number of distinct \(n\)-letter words that can be read from outside a convex \(n\)-gon, where each corner is labeled with a distinct letter. Do not count words in which some letters are obscured from view.

Answer: 2n