Question: Find the outward flux of the vector field \( V = x(x^2+y^2+z^2)^{-3/2}\mathbf{i} + y(x^2+y^2+z^2)^{-3/2}\mathbf{j} + z(x^2+y^2+z^2)^{-3/2}\mathbf{k} \) through the boundary of the rectangular box \( R \), where \( R \) consists of all points \((-1 \leq x \leq 1, -1 \leq y \leq 1, -1 \leq z \leq 1)\). Note that the vector field is not defined at the origin, so the Divergence Theorem cannot be directly applied to the box and its interior.

Answer: 4\pi


Question: Find the positive constant $c$ such that the matrix $B = cA$, where $A = \begin{pmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{pmatrix}$, is an orthogonal projection matrix.

Answer: \dfrac{1}{3}


Question: Let $G$ be a group and $m_1, m_2: G \to \mathbb{C}$ be unbounded functions satisfying the exponential functional equation \( m_j(x+y) = m_j(x)m_j(y) \) for all \( x, y \in G \) and \( j = 1, 2 \). If \( |m_1(x) - m_2(x)| \) is a bounded function, determine whether \( m_1 = m_2 \).

Answer: m_1 = m_2


Question: Find the remainder when $870^{479}$ is divided by 65.

Answer: 25


Question: Determine the Galois group \(\mathrm{Gal}(\mathbb{Q}(\sqrt[5]{3})/\mathbb{Q})\).

Answer: \{e\}


Question: Determine whether the following statement is true or false: Let $X$ be a topological space such that any two nonempty open sets in $X$ intersect. Then $X$ is compact. Provide a proof or counterexample.

Answer: False


Question: Given that $\beta,\gamma\in(0,\pi)$ satisfy the equations $\cos\alpha+\cos(\alpha+\beta)+\cos(\alpha+\beta+\gamma)=0$ and $\sin\alpha+\sin(\alpha+\beta)+\sin(\alpha+\beta+\gamma)=0$, and the functions $f(x)=\frac{\sin2x}{1+\cos2x}$ and $g(x)=\frac{1+\sin x-\cos x}{1+\sin x+\cos x}$, find the value of $f'(\beta)+\lim_{x\to\gamma}g(x)$. Assume $f(x)=\tan x$ and $g(x)=\tan\frac{x}{2}$.

Answer: 4 + \sqrt{3}


Question: Determine the values of \(x\) for which the series \(\sum_{n=1}^\infty \frac{n! \> x^n}{(x+1)(2x+1) \cdots (nx+1)}\) converges.

Answer: [0, 1)


Question:
Given $a+b+c=0$, simplify the expression:
\[ \frac{(a^3-abc)^3+(c^3-abc)^3+(b^3-abc)^3}{(c^2-ab)(b^2-ac)(a^2-bc)} \]

Answer: 3abc


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

Answer: \sqrt{e}


Question: Find the limit: \[ \lim_{n \rightarrow \infty} \left(\sum_{i=(n+1)/2}^n {n \choose i} \times 0.51^i \times 0.49^{n-i}\right) \]

Answer: 1


Question: Find all functions \( f: \mathbb{R}^+ \to \mathbb{R}^+ \) such that \( f(1 + x f(y)) = y f(x + y) \) for all \( x, y \in \mathbb{R}^+ \).

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


Question: Evaluate the limit of the sequence defined by \( a(n) = \int_n^{n+1} \ln\left(1+\frac{|\sin{x}|}{x}\right) dx \) as \( n \to \infty \).

Answer: 0


Question: Determine whether there exists a linear hypergraph $H = (\omega, E)$ with $E \subseteq [\omega]^\alpha$ such that $\chi(H) = \aleph_0$, where $\alpha \in (\omega \cup \{\omega\}) \setminus \{0,1,2\}$. Provide a justification for your answer.

Answer: Yes


Question: Let $K$ be a field and $K \subseteq M \subseteq \overline{K}$ be an intermediate field. Suppose the Galois group $\text{Gal}(\overline{K}/K)$ acts naturally on $M$. Is the extension $M/K$ a normal extension?

Answer: Yes


Question: Determine whether there exist 2000 distinct unit fractions whose sum is equal to 1.

Answer: Yes


Question: Solve the equation $f(x) = xe^x - 1 = 0$ using the successive approximation method by considering the fixed point problem $x = e^{-x}$. Apply Banach's fixed-point theorem to find the solution.

Answer: 0.567143


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

Answer: \dfrac{2}{3}


Question: Calculate the flow of the vector field \( \vec{F}= (x-z) \vec{i}+(y+z)\vec{j}+(x-y)\vec{k} \) through the outer side of the cylindrical surface \( x^2+y^2=a^2 \), bounded by the planes \( z=x \) and \( z=0 \) with \( z \geq 0 \).

Answer: \dfrac{4}{3} a^3


Question: In \(\triangle ABC\), point \(D\) is the midpoint of \(AB\), and point \(E\) trisects \(BC\) such that \(BE > CE\). Given that \(\angle ADC = \angle BAE\), find \(\angle BAC\).

Answer: 90^\circ


Question:
Consider the sets $A=\{x^2\mid0<x<1\}$ and $B=\{x^3\mid1<x<2\}$. Determine which of the following statements is true:
1. There is a one-to-one, onto function from $A$ to $B$.
2. There is no one-to-one, onto function from $A$ to $B$ taking rationals to rationals.
3. There is no one-to-one function from $A$ to $B$ which is onto.
4. There is no onto function from $A$ to $B$ which is one-to-one.

Which statement is correct?

Answer: 1


Question:
Let $X$ and $Y$ be independent and identically distributed uniform random variables on $[0,1]$. Define the random variable $Z$ as follows:
\[Z = \begin{cases} X+Y & \text{if } X>\frac{1}{2} \\ \frac{1}{2} + Y & \text{if } X\leq\frac{1}{2} \end{cases}\]
Calculate the conditional expectation $E\{Z|Z\leq 1\}$. Provide your answer as a single value.

Answer: \dfrac{7}{9}


Question: Find the minimum value of the square of the distance between the points \(a=(u,\sqrt{2-u^2})\) and \(b=(v,\frac{9}{v})\).

Answer: 8


Question: Compute the integral \( \int_M \omega \) where \( M = \{(x,y,z) : z = x^2 + y^2, z < 1\} \) is a smooth 2-manifold in \( \mathbb{R}^3 \) and \( \omega = x \, dy \wedge dz + y \, dz \wedge dx + z \, dx \wedge dy \in \Omega^2(\mathbb{R}^3) \).

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


Question: Let \( f \in C^{\infty}(\mathbb{R}^n) \) be a real function such that \( \underline{0} \) is a local minimum point for \( f \) on every algebraic curve. That is, for every algebraic curve \( C \), there exists an \( \epsilon > 0 \) such that if \( x \in B_{\epsilon}(\underline{0}) \cap C \), then \( f(0) \leq f(x) \). Here, \( B_{\epsilon}(0) := \{ x \in \mathbb{R}^n \; : \; ||x|| < \epsilon \} \) and \( C \) is the set of points that belong to the algebraic curve. Is it true that \( \underline{0} \) is necessarily a local minimum point for \( f \)?

Answer: Yes


Question: Determine the function \( y(x) \) that makes the integral \( \int_{0}^{1} (x^{2} + y'^2) \, dx \) stationary, subject to the constraints \( \int_{0}^{1} y^2 \, dx = 2 \), \( y(0) = 0 \), and \( y(1) = 0 \).

Answer: 2 \sin(\pi x)


Question: Let \( \alpha \) be a real root of the equation \( x^5 - x^3 + x - 2 = 0 \). Determine the value of \( \lfloor \alpha^6 \rfloor \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

Answer: 3


Question: Evaluate the expression \( \int\limits_1^2 {f\left( x \right)dx} + 2\int\limits_1^5 {{f^{ - 1}}\left( {2x} \right)dx} \) where \( f(x) = x^3 + x \).

Answer: 18


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

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{x\to 0} \left(\csc(x^2)\cos(x)-\csc(x^2)\cos(3x) \right)$$

Answer: 4


Question: Evaluate the contour integral \( \int _\gamma \frac{1}{z+\frac{1}{2}} \, dz \), where \( \gamma(t) = e^{it} \) for \( 0 \leq t \leq 2\pi \).

Answer: 2\pi i


Question: Compute the limit: \[ \lim_{x \to +\infty} x^2 \log\left(\frac{x^2+1}{x^2+3}\right) \]

Answer: -2


Question:
Determine the number of positive real solutions for the system of equations given by the curves:
\[ x = \frac{-(ab+c)y^2+(a-m)y}{cy-m} \]
\[ y = \frac{-(ef+d)x^2+(e-m)x}{dx-m} \]
where \(a, b, c, d, e, f, m\) are positive nonzero constants, and \(cy-m \neq 0\), \(dx-m \neq 0\). Additionally, identify the necessary conditions on the coefficients to ensure that number of solutions.

Answer: 1


Question: Evaluate the line integral \( \int_C \sqrt{1+4x^2 z^2} \, ds \), where \( C \) is the curve of intersection of the surfaces \( x^2 + z^2 = 1 \) and \( y = x^2 \).

Answer: 3\pi


Question: True or false: A function $f$ defined on an interval $I$ is continuous if the image of each interval $I_0 \subset I$ is an interval.

Answer: B


Question: Given the recurrence relation $a_1 = 2$ and $a_{n+1} = 2^n + a_n$, use generating functions to find a closed form expression for $a_n$. Verify your result by deriving the generating function $A(x) = \sum_{n \geq 1} a_n x^n$ and simplifying it to obtain the closed form.

Answer: 2^n


Question: Given that $a^2 = 2b^3 = 3c^5$, where $a$, $b$, and $c$ are natural numbers, find the smallest possible value of the product $abc$.

Answer: 746496


Question:
Consider the following subsets of \( \{ a, b, \$ \} ^* \): \( A = \{ xy \mid x,y \in \{ a, b, \} ^*, \#a(x) = \#b(y) \} \) and \( B = \{ x \$ y \mid x,y \in \{ a, b, \} ^*, \#a(x) = \#b(y) \} \). Determine which of the following statements is true:

1. \( A \) and \( B \) both are regular.
2. \( A \) is regular but \( B \) is not.
3. \( A \) is not regular but \( B \) is.
4. Both are non-regular.

Answer: 2


Question: Find the maximum value of \( \sum_{i=1}^n x_i \) subject to the constraint \( \sum_{i=1}^n x_i^2 \leq c \), where \( c \) is a given constant.

Answer: \sqrt{nc}


Question: Evaluate the series: \( \tan\theta + 2\tan(2\theta) + 2^2\tan(2^2\theta) + \dots + 2^{14}\tan(2^{14}\theta) + 2^{15}\cot(2^{15}\theta) \). Find the value of this expression.

Answer: \cot \theta


Question: Determine the values of \( p \) for which the series \( \sum\limits_{n=1}^\infty n(1+n^2)^p \) converges.

Answer: (-\infty, -1)


Question: Determine whether the following statement is true or false: If \( f : (0,1) \to \mathbb{R} \) is a strictly monotonically increasing function and \( f \) is differentiable at some \( x \in (0,1) \), then \( f^{-1}(y) \) is differentiable at \( y = f(x) \).

Answer: B


Question: A fair coin is flipped until a head appears. The payoff of the game is $2^N$, where $N$ is the number of flips needed to get the first head. Calculate the expected value of the game.

Answer: \infty


Question: Calculate the limit \( \lim_{n\rightarrow\infty} \sum_{k=1}^{n}\cos\left( \frac{2\pi k}{2n+1} \right) \).

Answer: -\dfrac{1}{2}


Question: Given a rectangular box with a diagonal of length 10 units, determine the maximum possible perimeter of the box.

Answer: 40\sqrt{3}


Question: Consider a vector space $V$ over a field $F$ with $\dim V = \infty$. Let $\varphi_1, \ldots, \varphi_k \in V^*$ be linearly independent. Is it true that there exist $x_j \in V$ such that $\varphi_i(x_j) = \delta_{ij}$ for all $i, j$? Justify your answer.

Answer: Yes


Question: Determine the completion of the space of polynomials \( \mathcal{P} \) with respect to the norm \( \| a_0 + a_1 x + \dots + a_nx^n\| = |a_0| + |a_1| + \dots + |a_n| \).

Answer: \ell^1


Question: Determine all possible values of the positive constant \( k \) such that the series \( \sum_{n=1}^{\infty}\ln\left(1+\frac{1}{n^k}\right) \) is convergent.

Answer: k > 1


Question: Compute the limit: $$\lim_{x\rightarrow 3}\frac{ \tan\frac{x-3}{x+3}\sin(9\sin(x-3)) }{ \sin(x-3)\sin(x^3-27))}$$

Answer: \dfrac{1}{18}


Question: Determine the radius of convergence \( R \) for the power series \( \sum_{n=0}^{\infty} \frac{(-1)^n 2^{-n}}{n!} z^{2n+1} \), where \( z \in \mathbb{C} \).

Answer: \infty


Question: Let $f(x)$ be a polynomial satisfying the equation $f(x)f(y) = f(x) + f(y) + f(xy) - 2$ for all $x, y$. Given that $f(2) = 1025$, find the value of $\lim_{x \to 2} f'(x)$. \( \text{(Assume that } f(1) \neq 1 \text{ if necessary.)} \)

Answer: 5120


Question: Determine whether the group presentation \( \langle a, b \mid ab = ba = 1 \rangle \) is equivalent to the group \( \langle a \mid \rangle = \mathbb{Z} \). Provide a justification for your answer.

Answer: Yes


Question: Suppose that $(X,Y)$ is a bivariate normal distribution where both $X$ and $Y$ have mean $0$ and variance $1$. Given that $\operatorname{corr}(X,Y) = \rho$, find $\operatorname{corr}(X^2,Y^2)$. Use conditional expectations in your solution.

Answer: \rho^2


Question: Find the number of subgroups of index 3 in the group \( P = \Bbb Z_4 \times \Bbb Z_6 \times \Bbb Z_9 \). Here, \( \Bbb Z_m \) denotes the cyclic group of order \( m \).

Answer: 4


Question: What is the cardinality of the set of countably infinite subsets of a countably infinite set?

Answer: 2^{\aleph_0}


Question: Suppose $f \in L_1(-\pi, \pi)$ satisfies $\hat{f}(n) = 0$ for all $n \in \mathbb{Z}$, where the Fourier coefficients are defined by $\hat{f}(n) = \int_{-\pi}^{\pi} f(\theta) e^{-in\theta} d\theta$. Does it follow that $f = 0$ almost everywhere on $(-\pi, \pi)$?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to\infty}\sqrt{n}\int_{-\infty}^{+\infty}\frac{\cos t}{\left(1+t^2\right)^n}dt \).

Answer: \sqrt{\pi}


Question: Determine whether there exists a function \( \mathcal{f} : \mathbb{C} \to \mathbb{C} \), where \( \mathcal{f} = u + iv \), such that \( u_x, u_y, v_x, \) and \( v_y \) exist and satisfy the Cauchy-Riemann equations at \((0,0)\), but \( \mathcal{f} \) is not differentiable at \((0,0)\).

Answer: Yes


Question: Do there exist plane curves \(f, g \in \mathbb{C}[x,y,z]_d\), with \(d \ge 2\), that intersect in exactly one point in \(\mathbb{P}^2\)?

Answer: Yes


Question: Does there exist a continuous density function $f$ on the real line such that $f(x) > 0$ for all $x \in \mathbb{R}$ and the expected value $E[X] = \infty$ where $X \sim f$?

Answer: Yes


Question: Find the number of sequences of length 10 that can be formed using the digits 0, 1, 2, 3, and 4, where each digit appears exactly twice and no two adjacent elements are the same.

Answer: 39480


Question:
Consider a path-connected metric space $(X,d)$ with at least two elements, and define the set $S = \{d(x,y) : x, y \in X\}$. Determine which of the following statements is not necessarily true:

A) $S$ is infinite.
B) $S$ contains a non-zero rational number.
C) $S$ is connected.
D) $S$ is a closed subset of $\mathbb{R}$.

Select the statement that is false.

Answer: D


Question: Let $\alpha,\beta:[0,1]\to\mathbb{R}^2$ be two smooth curves satisfying $|\alpha(t)| = |\beta(t)|$ and $|\dot{\alpha}(t)| = |\dot{\beta}(t)|$ for all $t\in[0,1]$. Does there exist a rotation matrix $Q$ (with possibly negative determinant) such that $\alpha(t) = Q\beta(t)$ for all $t$?

Answer: Yes


Question: Is the order of the torsion subgroup of the quotient module $N / \text{image}(f)$, where $M$ and $N$ are finitely generated free $\mathbb{Z}$-modules and $f: M \to N$ is a $\mathbb{Z}$-linear map corresponding to a matrix $F$, bounded by the product of the lengths of the nonzero columns of $F$?

Answer: Yes


Question: Consider a continuous map \( f: B \to \mathbb{R}^n \) where \( B \) is the \( n \)-ball and \( B \subset f(B) \). Does such a map \( f \) necessarily have a fixed point? Justify your answer.

Answer: Yes


Question: Find the value of the 50th derivative of the function \( f(x) = \frac{3x-1}{x^2-1} \) evaluated at \( x = 0 \).

Answer: 50!


Question:
Consider the sequence of functions \((f_{m})_{m \geq 1}\) defined on the set \(E = \{ \frac{1}{n} \mid n \in \mathbb{N} \}\) by:
\[
f_{m}(x) =
\begin{cases}
\cos{(m x)} & \text{if }\,x \geq \frac{1}{m}\\
0 & \text{if }\,\frac{1}{m+10}<x<\frac{1}{m}\\
x&\text{if } x \le \frac{1}{m+10}\\
\end{cases}
\]
Determine which of the following statements is true:
1. No subsequence of \((f_{m})_{m \geq 1}\) converges at every point of \(E.\)
2. Every subsequence of \((f_{m})_{m \geq 1}\) converges at every point of \(E.\)
3. There exist infinitely many subsequences of \((f_{m})_{m \geq 1}\) which converge at every point of \(E.\)
4. There exists a subsequence of \((f_{m})_{m \geq 1}\) which converges to 0 at every point of \(E.\)

Answer: 3


Question: If a sequence of functions \( f_n \) converges to \( f \) in \( L^1_{\text{loc}}(\mathbb{R}^n) \), does this imply that the corresponding sequence of distributions \( \Lambda_{f_n} \) converges to \( \Lambda_f \) in \( \mathcal{D}'(\mathbb{R}^n) \)?

Answer: Yes


Question: Consider the set \( \mathcal{K}_B \) of convex bodies (convex and compact sets) that are contained within the unit closed ball in \( \mathbb{R}^d \). Endow this set with the Hausdorff distance. Is \( \mathcal{K}_B \) compact?

Answer: Yes


Question: Let $a, b \in \mathbb{R}$ with $a < b$, and let $f \in C^{1}([a, b])$. Show that \[ \lim_{n \rightarrow \infty} \int_{a}^{b} f(x) \sin (n x) \, dx = 0. \]

Answer: 0


Question: Let \( M \) and \( N \) be any two \( 4 \times 4 \) matrices with integer entries satisfying \( MN = 2 \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \). Determine the maximum value of \( \det M + \det N \).

Answer: 17


Question: For every infinite cardinal \( \kappa \), does there exist a \( T_2 \)-space of cardinality \( \kappa \) where every non-empty open set is homeomorphic to the space itself?

Answer: Yes


Question: Given the matrix \( A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix} \), find the value of \( E_3(A) \) such that the characteristic polynomial \( P_A(t) = t^4 - 4t^3 + 3t^2 - E_3(A)t - 1 \) is satisfied.

Answer: -2


Question: Evaluate the limit \( \lim_{n\to \infty} n \int_{0}^{1} x^n f(x) \, dx \) where \( f: [0,1] \to \mathbb{R} \) is a continuous function.

Answer: f(1)


Question: Let $\Gamma$ be a discrete group and $\text{Prob}(\Gamma)$ be the space of all probability measures on $\Gamma$ defined as $\text{Prob}(\Gamma)=\{\mu\in l^{1}(\Gamma): \mu\geq0,~\sum\limits_{t\in\Gamma}\mu(t)=1\}$. Define $F(\mu, r)=\{t\in \Gamma:~\mu(t)>r\}$. Determine whether the following integral holds: \[ \int_0 ^1 |F(\mu,~r)|dr=1 \] where $|\cdot|$ denotes the cardinality of a set.

Answer: 1


Question: Find the value of \( n \) such that the expression \( 2^{200} - 2^{192} \cdot 31 + 2^n \) is a perfect square.

Answer: 198


Question: In triangle $\triangle ABC$, sides $AC$ and $BC$ are equal, and $\angle ACB = 96^\circ$. Point $D$ is inside $\triangle ABC$ such that $\angle DAB = 18^\circ$ and $\angle DBA = 30^\circ$. Determine the measure of $\angle ACD$ in degrees.

Answer: 78


Question: Determine the values of \( b \) for which resonance occurs in the differential equation \( y'' + 100y = \frac{1}{3} \cos(bx) \) with initial conditions \( y(0) = 0 \) and \( y'(0) = 0.1 \).

Answer: 10


Question: Compute the area between the u-axis and the curve \(\alpha(t)=(r\cos t, r\sin t)\) for \(0 < t < \pi\) in the Poincare half-plane \(\{(u,v)|v>0\}\) using the area form \(du\wedge dv/v^2\). Is this area finite or infinite?

Answer: \infty


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\tan 2x + \tan 4x - \tan 6x}{x^3} \]

Answer: -48


Question: Find the remainder when \(2^{2^{517}}\) is divided by 23.

Answer: 13


Question: Assume there is a totally ordered local basis at a point \(x\) in a topological space \(X\). Does this imply that there is a countable local basis at \(x\)?

Answer: No


Question: What is the probability that the determinant of a 2x2 matrix, whose elements are independent and uniformly distributed random variables in the interval (0,1), is positive?

Answer: \dfrac{1}{2}


Question: Determine the number of zeros of the function \( f(z) = z^2 + \cosh(iz) \) within the disk \( |z| < \frac{1}{2} \).

Answer: 0


Question: Given a random variable $X \sim \mathcal{N}(0,1)$, determine the transformation needed to obtain a zero-censored normal distribution from $X$.

Answer: Y = \max(X, 0)


Question: Consider a Riemannian hypersurface $M^n \subset \mathbb{R}^{n+1}$ with an isometrically immersed shape operator $s$. The characteristic polynomial of $s$ is given by $p(\lambda) = \lambda^n + a_{n-1}\lambda^{n-1} + \ldots + a_0$. For $n > 2$, determine which coefficients $a_i$ are local isometry invariants of $M$. Provide a justification for your answer.

Answer: a_{n-2}


Question: Let \( \{X_n\} \) be a sequence of independent random variables taking only positive integer values. If for every integer \( k \ge 1 \), the limit \( \lim_{n \to \infty} P(X_n =k) \) exists, determine whether the series \( \sum_{k=1}^\infty ( \lim_{n \to \infty} P(X_n =k)) \) is convergent.

Answer: Yes


Question: Determine whether any integer can be the smallest number in more than one primitive Pythagorean triple. Provide a justification for your answer.

Answer: Yes


Question: Find the 2021st derivative of the function \( f(x) = \frac{1+x}{1+x+x^2} \) evaluated at \( x = 0 \).

Answer: -2021!


Question: In a metric space (X, d) where a finite subset A is dense, determine all open sets in (X, d).

Answer: \mathcal{P}(X)


Question: Determine whether the matrix expression \( B^T (M M^T) B \) is positive definite or semi-positive definite, given that both \( B \) and \( M \) are real invertible matrices.

Answer: positive definite


Question: Determine the value of \( \beta \) at which the number of fixed points of the equation \( \tau\frac{dx}{dt}=-x+\frac{e^{\beta(x-\frac{1}{2})}}{1+e^{\beta(x-\frac{1}{2})}} \) changes, given that for \( 4<\beta\leq5 \) there are 2 additional fixed points.

Answer: 4


Question: Find the limit of the sum: $$\lim_{n\to\infty}\sum_{k=1}^n \ln\Big(1+\frac{k}{n^2}\Big)$$

Answer: \dfrac{1}{2}


Question: Let \( f(x):\mathbb{R}\rightarrow\mathbb{C} \) be a \( C^\infty \)-smooth function such that \( f(x) \neq 0 \) for all \( x \in \mathbb{R} \). Is \( \frac{1}{f(x)} \) also \( C^\infty \)-smooth?

Answer: Yes


Question: Given that \( \left|\frac{z_1-3z_2}{3-z_1\overline{z}_2}\right|=1 \) and \( |z_2| \neq 1 \), find the modulus \( |z_1| \) of the complex number \( z_1 \).

Answer: 3


Question: Consider the set of continuous functions \( S = \{ f : [0,1] \to \mathbb{R} \mid \int_0^1 f(x) \, \text{d}x = 5, \int_0^1 x f(x) \, \text{d}x = 3 \} \). Determine the infimum \( I = \inf_{f \in S} \int_0^1 f(x)^2 \, \text{d}x \). Find the value of \( I \).

Answer: 28


Question: Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(x) = x$ for $x \notin [-T, T]$ for some $T>0$, and $0$ is a regular value of $f$. Compute the sum \( \sum\limits_{x\in f^{-1}(0)}\frac{f'(x)}{|f'(x)|} \).

Answer: 1


Question: Let $\mathbb{R}^A$ be the vector space of functions from a set $A$ to $\mathbb{R}$. Suppose $x_1, \ldots, x_n \in \mathbb{R}^A$ are linearly independent. Determine whether there always exist $\alpha_1, \ldots, \alpha_n \in A$ such that the vectors $$(x_1(\alpha_1), \ldots, x_1(\alpha_n) ),$$ $$\ldots,$$ $$(x_n(\alpha_1), \ldots, x_n(\alpha_n) )$$ are linearly independent in $\mathbb{R}^n$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \oint_{S^1} dx \), where \( S^1 \) is the unit circle in the plane. Use an appropriate parametrization to compute the integral and determine its value.

Answer: 0


Question: Evaluate the line integral \( \int_\gamma \frac{x\,dy-y\,dx}{x^2+y^2} \), where \( \gamma \) is the oriented piecewise \( \mathcal{C}^1 \)-arc consisting of the line segment from \((1,0)\) to \((0,1)\), followed by the line segment from \((0,1)\) to \((-1,0)\).

Answer: \pi


Question: Maximize the function \( A(x,y) = \frac{1}{2}(x(12-x) + y(13-y)) \) subject to the constraint \( x^2 + (12-x)^2 - y^2 - (13-y)^2 = 0 \).

Answer: 36


Question: Compute \( \lim_{n\to \infty} \int_{1}^{\infty} \exp(-x)(\cos(x))^n \, dx \).

Answer: 0


Question: Find all pairs of positive integers \((m, n)\) such that \(10^m - 8^n = 2m^2\).

Answer: (1, 1)


Question: Determine the necessary and sufficient condition for which \( \left(2+\frac{x}{y}+\frac{y}{x}\right)^x > (1+x)^{1+x} \) given that \( 0 < y < x < 1 \).

Answer: 0 < y < x < 1


Question: Does there exist an open subset of \(\mathbb{R}^n\) that has a dimension less than \(n\) in the standard topology?

Answer: No


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \left(\frac{\sin x}{x}\right)^{2} dx \) using the Plancherel theorem.

Answer: \pi


Question: Let \( f(x) \) be a polynomial with a leading coefficient of 1 and the least degree such that \( f(1) = 5 \), \( f(2) = 4 \), \( f(3) = 3 \), \( f(4) = 2 \), and \( f(5) = 1 \). Find \( f(0) \).

Answer: -114


Question: Calculate the Fourier transform \( \mathscr{F}(f_a) \) of the function \( f_a(x) = \frac{a}{\pi(x^2+a^2)} \) for \( a > 0 \), given that \( \mathscr{F}(\mathscr{F}(f))(x) = 2\pi f(-x) \) and \( \mathscr{F}(e^{-a|x|}) = \frac{2a}{a+\xi^2} \).

Answer: e^{-a|\xi|}


Question: Compute the flux of the vector field \( F(x,y,z) = \langle z^2 - y^2 e^z, z \ln(1-2x^2), 3 \rangle \) across the surface \( S \), where \( S \) is the portion of the graph \( z = 5 - x^2 - y^2 \) that sits above the plane \( z = 0 \), with an upward orientation. Calculate \( \iint_S F \cdot dS \).

Answer: 15\pi


Question: Let $f$ be a holomorphic function on the open unit disk. Determine if $f$ has no poles inside this disk and if the Taylor series expansion of $f$ at $0$ has a convergence radius of at least 1.

Answer: Yes


Question: For each integer \( n \geq 1 \), determine whether there exists a prime \( p \) and an integer \( k \) such that \( p^k - 1 \) has exactly \( n \) distinct prime divisors.

Answer: Yes


Question: Determine all the points in the complex plane where the function \( f(z) = \tan(z) \) is differentiable and calculate the derivative at those points.

Answer: \sec^2(z)


Question: Find the radius of convergence of the power series \( \sum_{n=0}^{\infty} c_n z^n \), where \( c_n = \frac{1}{n!} f^{(n)}(0) \) and \( f(z) = (z-1-i)^{-2} \cdot \sin(z) \).

Answer: \sqrt{2}


Question: Let $GF(q)$ be the finite field of order $q$, where $q$ is an odd prime power. Determine the number of points $(x, y, z)$ in $GF(q)^3$ that satisfy the equation $x^2 + y^2 + z^2 = d$, where $-d \neq 0$ is a non-square in $GF(q)$. Show that this number is equal to $q^2 - q$.

Answer: q^2 - q


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0}\left(\frac{16^x+9^x}{2}\right)^{\frac{1}{x}} \]

Answer: 12


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sqrt[n^n]{(3n)! + n^n} \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{(x,y)\rightarrow (0,0)}\frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4} \]

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \frac{\sin(\arctan(\sin x))}{\sqrt{x} \sin 3x +x^2+ \arctan 5x} \] without using L'Hospital's rule.

Answer: \dfrac{1}{5}


Question: Let $X$ be an exponential random variable with parameter $\lambda=4$ and $Y$ be an exponential random variable with parameter $\lambda=5$. $X$ and $Y$ are independent. Find the probability that $3X < 2Y$. Express your answer as a single probability value.

Answer: \dfrac{8}{23}


Question: Does there exist a set $A \subset \mathbb{R}^n$ such that $A$ is compact, contains the origin, and has measure zero?

Answer: Yes


Question: Find the value of $r$ for which a superstable period-2 orbit exists in the logistic map defined by $G(x) = rx(1-x)$. Provide your answer as a numerical value.

Answer: 1 + \sqrt{5}


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} \sqrt[n]{\frac{1^n+2^n+3^n+\cdots +n^n}{1+2!+3!+\cdots +n!}} \]

Answer: e


Question: Evaluate the limit: \[ \lim_{x\to a^+} \frac{\cos(x)\ln(x-a)}{\ln(e^x-e^a)} \]

Answer: \cos a


Question: Let $X$ be a Banach space and $P : \mathcal{D}(P) \subset X \to X$ be a linear operator such that $P\mathcal{D}(P)\subset \mathcal{D}(P)$ and $P^2x = Px$ for all $x \in \mathcal{D}(P)$. Does this imply that $P$ is bounded? Provide a justification for your answer.

Answer: No


Question: Evaluate the definite integral: \[ I = \int_0^{\frac{\pi}{2}} \frac{\cos^{2008} (x)}{\sin^{2008} (x)+\cos^{2008} (x)}~dx \]

Answer: \dfrac{\pi}{4}


Question: Find the global minimum of the function \( f(x, y) = x^4 + y^4 + 4x^2 - xy + y^2 + 6 \).

Answer: 6


Question: Determine whether the sequence \( u_n = \frac{1}{n!}\sum_{k=1}^n k! \) converges.

Answer: 1


Question: Let \( G \) be a free abelian group of rank \( k \). If \( S \) is a subset of \( G \) such that \( |S| = k \) and \( S \) generates \( G \), are the elements of \( S \) linearly independent over \( \mathbb{Z} \)? Justify your answer.

Answer: Yes


Question: Evaluate the triple integral \( \int \int \int _{E} \frac{1}{x^2+y^2+z^2 }dV \), where the region \( E \) is defined as \( E=\{(x,y,z) \in \mathbb{R}^3 : z\geq \sqrt{\frac{x^2+y^2}{3}}, 2z\leq x^2+y^2+z^2 \leq 9\} \).

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


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sqrt[n]{\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}} \]

Answer: 1


Question: Find the coefficient of $z$ in the Laurent series expansion of $\frac{e^z}{z-1}$ in the region $|z|>1$.

Answer: e - 2


Question: Evaluate the integral \( \int_1^{\sqrt{3}} \left( x^{2x^2+1} + \ln x^{\left(2x^{2x^2+1}\right)} \right) \, dx \).

Answer: 13


Question: Find the value of the series \( \sum_{n=1}^{\infty} \left( \frac{2}{2n+1} - \frac{1}{n+1} \right) \).

Answer: 2 \ln 2 - 1


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

Answer: \dfrac{\pi}{2}


Question: Let \( A \) and \( B \) be \( n \times n \) matrices. Prove that if \( ABA = 0 \), then \( \text{tr}(AB) = 0 \).

Answer: 0


Question: A particle attached to a string of length 2 meters is given an initial velocity of 6 m/s. The string is attached to a peg, and as the particle rotates about the peg, the string winds around the peg. Determine the length of the string that has wound around the peg when the velocity of the particle is 20 m/s.

Answer: 1.4


Question: Consider a $100 \times 100$ matrix $A$ where the element in the $i^{th}$ row and $j^{th}$ column is given by $i^2 + j^2$. Determine the rank of matrix $A$. \( \text{Rank}(A) = \boxed{?} \)

Answer: 2


Question: Calculate the limit: \[ \lim_{x \to 0^+} \frac{\frac{4}{\pi}\arctan\left(\frac{\arctan x}{x}\right) - 1}{x} \] without using Taylor's theorem or L'Hôpital's rule.

Answer: 0


Question: Determine the number of zeros of the function \( f(z) = z - \phi(z) \) that are located within the unit disk \( |z| < 1 \), given that \( \phi(z) \) is analytic and \( |\phi(z)| < 1 \) whenever \( |z| < 1 \). Use Rouche's Theorem to find the answer.

Answer: 1


Question: For subsets $U, W, W'$ of a vector space $V$, if $U \oplus W = U \oplus W'$, does it necessarily follow that $W = W'$? Determine whether this statement is true or false.

Answer: False


Question: Determine whether the 3-dimensional cone in 4-dimensional space, known as the hypercone, is contractible.

Answer: Yes


Question:
The roots of the polynomial \(x^3 - 4x^2 + x + 6\) are \(\alpha\), \(\beta\), and \(\omega\). Evaluate the expression:
\[
\frac{\alpha+\beta}{\omega} + \frac{\alpha+\omega}{\beta} + \frac{\beta+\omega}{\alpha}
\]

Answer: -\dfrac{11}{3}


Question: Find a vector field \( Z = a\partial_x + b\partial_y + c\partial_z \) in \( \mathbb{R}^3 \) such that \( \alpha(Z) = 1 \) and \( d\alpha(Z,.) = 0 \), where \( \alpha = dz - ydx \).

Answer: \partial_z


Question: Find the set \(S = \{(x, y) \in \mathbb{C} \times \mathbb{C} : \begin{pmatrix} x & i \\ iy & 0 \end{pmatrix}^T A^T = \begin{pmatrix} 0 & -5i \\ i & 2i \end{pmatrix} \}\), where \(A = \begin{pmatrix} 1+i & -1 \\ 1 & -i \end{pmatrix}\).

Answer: \emptyset


Question: Find the maximum value of the function \( f(x, y, z, t) = (-2(x+5)^2 + 200x) + (-2(y+10)^2 + 200y) + (-2(z+15)^2 + 200z) + (-2t^2 + 200t) \) subject to the constraint \( x + y + z + t = 150 \), where \( x, y, z, \) and \( t \) are positive parameters.

Answer: 13800


Question: Evaluate the sum \( \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\binom{n}{k}^{-1} \).

Answer: 2


Question: Evaluate the limit: \[ \lim\limits_{x\to \infty} x\big(\log(x+1) - \log(x-1)\big) \]

Answer: 2


Question: Let $G$ be a finite-by-cyclic group, meaning it contains a finite normal subgroup $N$ such that $G/N$ is cyclic. Determine whether $G$ contains only finitely many finite subgroups.

Answer: Yes


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

Answer: 0


Question:
Consider the equation \(x^{2007} - 1 + x^{-2007} = 0\). Let \(m\) be the number of distinct complex non-real roots and \(n\) be the number of distinct real roots of the equation. What is the value of \(m - n\)?

1. \(0\)
2. \(2006\)
3. \(2007\)
4. \(4014\)

Answer: D


Question: Let $f, g: \mathbb{C} \to \mathbb{C}$ be functions such that their $L^p$ norms are equal, i.e., \( \int_{\mathbb{C}} |f(z)|^p \, dz = \int_{\mathbb{C}} |g(z)|^p \, dz. \) Define the weight function \( w(z) = \left( 1 + \left| \frac{z - \bar{z}}{2} \right|^2 \right)^{s/2} \) for \( s > 0. \) Can we conclude that \( \int_{\mathbb{C}} |f(z) w(z)|^p \, dz = \int_{\mathbb{C}} |g(z) w(z)|^p \, dz? \)

Answer: No


Question: Calculate the probability that a randomly chosen 4-digit number is "non-adjacent," meaning that for any pair of digits, the absolute difference between them is greater than 1. Express your answer as a fraction.

Answer: \dfrac{2}{25}


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{\sum^n_{r=0}\binom{2n}{2r}\cdot2^r}{\sum^{n-1}_{r=0}\binom{2n}{2r+1}\cdot2^r} \). Provide a proof for the value of this limit.

Answer: \sqrt{2}


Question: Let \( \omega \) be a primitive 10th root of unity. Find the irreducible polynomial of \( \omega + \omega^{-1} \).

Answer: x^2 - x - 1


Question: A princess is equally likely to sleep on any number of mattresses from six to twelve each night. On half of the nights, a pea is placed under the lowest mattress. She never sleeps well if a pea is under six mattresses. If the pea is under seven mattresses, she sleeps well 10% of the time; under eight mattresses, 20% of the time; and so on, up to 60% of the time under twelve mattresses. One morning, she reports having slept incredibly well. What is the expected number of mattresses she slept on that night?

Answer: \dfrac{121}{13}


Question:
Let \( \alpha_1, \alpha_2, \alpha_3, \alpha_4 \in \mathbb{R} \) be the roots of the polynomial \( p(x) = x^4 + 4x^3 + x^2 - 6x - 1 = 0 \). Calculate the value of the following sum:
\[ S = \sum_{i=1}^{4} \frac{1}{\alpha_i^4 - 15\alpha_i^2 - 10\alpha_i + 24} = \frac{m}{n} \]
Express the result in the form \( \frac{m}{n} \) where \( \text{G.C.D}(m,n) = 1 \) and \( m, n \in \mathbb{N} \). Provide \( m+n \) as your answer.

Answer: 715


Question: Given the probability density function (PDF) \( f_X(x) = \frac{1}{4}(x+1) \) for \( 0 < x < 2 \), find the PDF of the transformed variable \( Z = \frac{9}{(X+1)^2} \). The correct PDF is known to be \( f_Z(z) = \frac{9}{8z^2} \) for \( 1 < z < 9 \). Verify this result and identify any errors in the calculation process.

Answer: f_Z(z) = \dfrac{9}{8z^2}


Question: Does every open cover of \(\mathbb{R}\) have a countable subcover?

Answer: Yes


Question: Calculate the integral \( \int_{\mathbb{R}} x^2 \, d\mu \) where the measure \( \mu \) is defined by \( \mu(E) = \int_E \chi_{[0,\infty)}(x) e^{-ax} \, dx \) for a constant \( a > 0 \).

Answer: \dfrac{2}{a^3}


Question: Compute the residue of \( \frac{e^{2iz}-1}{z^2} \) at \( z=0 \).

Answer: 2i


Question: Evaluate the integral \( I = \int_{-\infty}^{\infty} f(x) \, dx \) where \( f(x) = \delta(x) \) for \( x \in (-\infty, 0] \) and \( f(x) = 0 \) for \( x \in (0, \infty) \). Here, \( \delta(x) \) is the Dirac delta function.

Answer: 1


Question: Determine the number of real roots of the equation \(3^{x^{22}} + 28^{(1-x)^2} = 2020\) without using derivatives.

Answer: 2


Question: Calculate the area of the region that lies inside the polar curve \( r = 1 + \cos\theta \) and outside the polar curve \( r = 2\cos\theta \).

Answer: \dfrac{\pi}{2}


Question: Determine the number of solutions to the congruence \(x^2 \equiv 1 \pmod{2^k}\) for \(k \ge 3\).

Answer: 4


Question:
Consider the region \( D = \{(x,y) \in \mathbb{R}^2 \mid 1 \leq x^2 + y^2 \leq 3 \} \). Evaluate the integral \( I = \iint_D (|x+y| - x - y) \, dA \). Determine which of the following statements is true:

\[ I = 0 \]
\[ I > 0 \]
\[ I < 0 \]

Answer: I > 0


Question: Evaluate the limit of the integral: \[ \lim_{y \to 0} \int_{0}^{2} \frac{e^{(x+y)^2} - e^{x^2}}{y} \, dx \]

Answer: e^4 - 1


Question: Let $A$ be a reducible non-negative matrix. Is it true that $A$ has a positive eigenvector?

Answer: No


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

Answer: 1


Question: Determine the number of semi-direct products \(H \rtimes Q\) that can be constructed for \(H \cong C_{42}\) and \(Q \cong C_{3}\).

Answer: 2


Question: Is the Knaster-Kuratowski fan a Borel subset of the Euclidean plane?

Answer: Yes


Question: Calculate the residue of the function \( f(z) = \frac{(z^2 - 1)^4}{z^5} \) at \( z = 0 \).

Answer: 6


Question: How many distinct 3x3 matrices with entries in \(\{0,1\}\) are there such that each row is non-zero, distinct, and each matrix is of real rank 2 or 3, considering that if a matrix \(M\) is included in the count, \(PM\) where \(P\) is a permutation matrix, is not included?

Answer: 35


Question: Let $x = -\frac{(2\;W({\pi\over2}))}{\pi}$, where $W$ denotes the Lambert W-function. Given that \( {\log(i^2)\over i} = \pi \) and \( {\log(x^2)\over x}=\pi \), determine if $x = i$. Provide a justification for your answer.

Answer: x \neq i


Question: Find \( \frac{dy}{dx} \) at the point \((0,1)\) for the equation \( y^5 = (x+2)^4 + e^x \ln y - 15 \).

Answer: 8


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \). Suppose that for every sequence \( x_n \rightarrow 0 \), the sequence \( f(x_n) \) converges. Is it true that for any sequences \( x_n \rightarrow 0 \) and \( y_n \rightarrow 0 \), we have \( \lim\limits_{n\to\infty}{f(x_{n})} = \lim\limits_{n\to\infty}{f(y_{n})} \)?

Answer: Yes


Question: Does the principal branch of the complex logarithm, \( \log: \mathbb{C} \setminus (-\infty,0] \rightarrow \mathbb{C} \), have a primitive on the set \( \{ z \in \mathbb{C} : \operatorname{Re} z > 0 \} \)?

Answer: Yes


Question: Evaluate the integral by reversing the order of integration: \[ \int_0^1\int_{\arcsin(y)}^{\frac{\pi}{2}} \sec ^2{(\cos x)} \, dx \, dy \]

Answer: \tan(1)


Question: Determine the order of the pole of the function \( f(z) = \frac{1}{2-e^z} \) at the points \( z_k = \log(2) + 2\pi ik \), where \( k \in \mathbb{Z} \).

Answer: 1


Question: Determine the quotient of \( \mathbb{C}^* \) by the inverse action, where the map \( f: \mathbb{C}^* \longrightarrow \mathbb{C}^* \) is defined as \( z \mapsto z^{-1} \). What is the resulting quotient space?

Answer: \mathbb{C}


Question: Let $X$ and $Y$ be Banach spaces, and let $\alpha$ be a cross-norm on $X \otimes Y$ such that $\|x \otimes y \| = \|x\|\|y\|$ for $x \in X, y \in Y$ and $\|f \otimes g\|=\|f\|\|g\|$ for $f \in X^*$ and $g \in Y^*$. Consider the operator $I_X \otimes P: X \otimes_\alpha Y \to X \otimes_\alpha Y$, where $P$ is a rank-one projection onto a unit vector $y_0 \in Y$ given by $Py = g(y)y_0$ with $g \in Y^*$ such that $g(y_0)=1$. Is the operator $I_X \otimes P$ bounded?

Answer: Yes


Question: Find the maximum value of \(|f(z)|\) on the boundary of the disk \(|z| = 1\), where \(f(z) = \frac{1}{z^2 - 2z + 2}\).

Answer: \sqrt{2}


Question: Evaluate the integral \( \int_{\vert z \vert = 1} \frac{1}{8z^3-1} \, dz \).

Answer: 0


Question: If $H \trianglelefteq K$ and $K \trianglelefteq D_4$, does it follow that $H \trianglelefteq D_4$? ($D_4$ is the dihedral group of order 8.)

Answer: No


Question: Evaluate the limit: \[ \lim_{x \to \infty}\frac{e^{\frac{1}{x^2}}-1}{2\arctan(x^2)-\pi} \]

Answer: -\dfrac{1}{2}


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

Answer: \infty


Question: Let $A$ be a bounded self-adjoint operator on a Hilbert space $\mathcal{H}$ such that $A^2 \leq A$. Prove that $0 \leq A \leq 1$, meaning that $0 \leq \langle Av, v \rangle \leq \langle v, v \rangle$ for all $v \in \mathcal{H}$. Provide a justification using either partial fraction decomposition or the Spectral Theorem and functional calculus.

Answer: 0 \leq A \leq 1


Question: Find the minimum value of the perimeter of a triangle with integer sides $a > b > c$ such that $3^a \equiv 3^b \equiv 3^c \pmod{10000}$. What is the minimum perimeter of such a triangle?

Answer: 3003


Question: Find all continuous functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( \int_{0}^{x} f(t) \, dt = \frac{f(x)}{2} + 1 \).

Answer: -2e^{2x}


Question: Calculate the volume of the region bounded by the two cylinders: \(x^2 + y^2 = 1\) and \(x^2 + z^2 = 1\).

Answer: \dfrac{16}{3}


Question: Find the radius of circle $S_3$ which touches circles $S_1$ and $S_2$ externally and circle $S$ internally, given that $S, S_1, S_2$ are circles with radii 5, 3, and 2, respectively, and $S_1$ and $S_2$ touch each other externally and both touch $S$ internally.

Answer: \dfrac{30}{19}


Question:
Given a Markov chain \(\{X_n, n \geq 0\}\) with states \(0, 1, 2\) and the transition probability matrix:
\[
\begin{bmatrix}
\frac{1}{2} & \frac{1}{3} & \frac{1}{6} \\
0 & \frac{1}{3} & \frac{2}{3} \\
\frac{1}{2} & 0 & \frac{1}{2}
\end{bmatrix}
\]
If \(P(X_0=0) = P(X_0=1) = \frac{1}{4}\) and \(P(X_0=2) = \frac{1}{2}\), find \(E[X_3]\).

Answer: \dfrac{53}{54}


Question: Does a $C^1$ strictly increasing function from $\mathbb{R}$ to $\mathbb{R}$ have an inverse that is also $C^1$?

Answer: No


Question: Let $f$ be a holomorphic function defined in a neighborhood of $\overline{D(0,R)}$ with no zeros on $\partial D(0,R)$. Let $N$ be the number of zeros of $f$ in $D(0,R)$. Show that \[ \max_{|z|=R} \operatorname{Re}\left(z\frac{f'(z)}{f(z)}\right) \geq N. \]

Answer: N


Question: Determine if the set of matrices \( \{A \in M(n,\mathbb{R}) : (Ax)^TQ(Ay) = x^TQy, \forall x,y\} \) is isomorphic to the set \( \{A \in M(n,\mathbb{R}) : (Ax)^TE(Ay) = x^TEy, \forall x,y\} \), where \( Q = P^TEP \) for some invertible matrix \( P \) and \( E \) is a diagonal matrix with \( p \) 1's, \( q \) -1's, and some zeros. This would imply that the definition of \( SO(p,q) \) only depends on the signature \((p,q)\).

Answer: Yes


Question:
Given the partial differential equation \( \frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0 \), determine the general solution of the equation from the following options:

(a) \( u(x,y) = 2f(x+2y) + g(x-2y) \)

(b) \( u(x,y) = (x+y)f(x-y) + g(x-y) \)

(c) \( u(x,y) = (x+2y)f(x+y) + g(x-y) \)

Answer: b


Question: Calculate the limit: \[ \lim_{x \to 0} \left(\frac{1- (\cos x)^{\sin x}}{x^3}\right). \]

Answer: \dfrac{1}{2}


Question: Determine whether every element in the alternating group $A_n$ can be expressed as the square of an element in the symmetric group $S_n$. Provide a justification for your answer.

Answer: Yes


Question:
Calculate the area of the surface $S$ above the region $\Omega$ in the $(xy)$-plane defined by $x^2+y^2\leq a^2$, given that $\left|\frac{\partial z}{\partial x}\right|\leq 2$ and $\left|\frac{\partial z}{\partial y}\right|\leq 2$. Express your answer in terms of $a$. Use the formula:
$$
\iint_\Omega \sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2} \,dx\,dy
$$

Answer: 3\pi a^2


Question: Find the limit of the series \( \sum_{n=1}^{\infty} \frac{-2}{n^x} \) as \( x \to 0 \).

Answer: 1


Question: For the function \(f: \mathbb{R}^{<0} \to \mathbb{R}\) defined by \(f(x) = 2x^2 - 3\), find the values of \(a\) such that \(f(a) = f^{-1}(a)\).

Answer: -1


Question:
Let \( A \) be a \( 3 \times 3 \) real matrix satisfying \( A^8 = I \). Determine which of the following statements is true:

1. The minimal polynomial of \( A \) is of degree 3.
2. The minimal polynomial of \( A \) is of degree 2.
3. \( A = I \) or \( A = -I \).
4. There are uncountably many such \( A \).

Answer: 4


Question: Evaluate the integral \( \int_0^{\infty}\frac{x}{x^4+1}dx \).

Answer: \dfrac{\pi}{4}


Question: How many $4 \times 4$ matrices can be formed using only $1$ and $-1$ as elements such that the sum of all elements in any row and column is $0$?

Answer: 90


Question: Compute the limit \( \lim_{n \to \infty} \frac{x_n}{\sqrt{n}} \) for the sequence defined by \( x_{n+1} = x_n + \frac{2}{x_n} \) with initial condition \( x_0 = 1 \).

Answer: 2


Question: Find all rational solutions to the Diophantine equation: \( y^2 = 25x^6 - 20x \).

Answer: (0, 0)


Question: Given two independent random variables $X$ and $Y$ uniformly distributed over $[0, 1]$, find the probability density function (PDF) of the product $Z = XY$. Provide the expression for the PDF of $Z$.

Answer: -\ln z


Question: Convert the iterated integral \( \int_0^1 \int_0^{2-x} dydx \) to polar coordinates and evaluate the integral.

Answer: \dfrac{3}{2}


Question: Find the radius of convergence of the Taylor series of \(\frac{1}{2+x^2}\) about \(x=2\).

Answer: \sqrt{6}


Question: Evaluate the integral \( \int x^x \ln(xe) \,\mathrm{d}x \).

Answer: x^x + C


Question:
Find the 100th derivative of the function \( y = \ln(2x-x^2) \) evaluated at \( x = 1 \). Choose the correct answer from the following options:

\( a)\ 2\times99! \)

\( b)\ -2\times99! \)

\( c)\ 2\times101! \)

\( d)\ -2\times101! \)

Answer: B


Question: Let $X_1, X_2, \ldots$ be independent random variables where $X_n \sim B(p_n)$ and $p_n = \frac{1}{n}$. Calculate $P\left(\limsup \left(X_n=0, X_{n+1}=1, X_{n+2}=0 \right)\right)$. Use the Borel-Cantelli lemma if necessary.

Answer: 1


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{\sin(x^2)-\sin^2(x)}{x^2\ln(\cos x)} \] without using L'Hospital's rule.

Answer: -\dfrac{2}{3}


Question: Determine the cardinality of the set of all partially ordered sets of natural numbers that have one least element and an infinite number of maximal elements.

Answer: 2^{\aleph_0}


Question: Is the infinite product space \(\Pi_{i = 1}^{\infty}\mathbb{N}_n\) compact, where each \(\mathbb{N}_n\) is equipped with the discrete topology and is compact for finite \(n\)?

Answer: Yes


Question: Consider a full column-rank square matrix with entries of either $0$ or $1$. Is it always possible to rearrange the columns of this matrix such that the diagonal of the resulting matrix consists entirely of $1$'s?

Answer: Yes


Question: Find the Hermitian interpolation polynomial \( p(x) \) that satisfies the following conditions: \( p(x_0) = -1 \), \( p'(x_0) = 1 \), and \( p(x_1) = 1 \), where \( x_0 = 0 \) and \( x_1 = 1 \), using the generalized Lagrangian polynomials.

Answer: x^2 + x - 1


Question: Find the coefficient of \(x^6\) in the expansion of \((1+x+\frac{x^2}{2})^{10}\) using combinatorial methods.

Answer: 1170


Question: Determine the value of \( n \) for which the improper integral \( \int_1^{+\infty} \left( \frac{n}{x+1} - \frac{3x}{2x^2 + n} \right) \, dx \) is convergent.

Answer: \dfrac{3}{2}


Question: Find all values of the constant \(a \in \mathbb{C}\) such that the function \[ f(z) = \frac{z^2 - z +1}{z(z-1)^2} + \frac{a}{\sin z} \] has an antiderivative in the domain \( U = \{ z : |z| < 2 \} \setminus \{0, 1\} \).

Answer: -1


Question: Determine the maximum number of algebraically independent homogeneous polynomials of degree $d \geq 2$ in the polynomial ring $\mathbb{Z}[x_1, \dots, x_n]$. Can this number be as large as $n^2$ or $n^3$, or is it limited to $n$?

Answer: n


Question: Let \(a\) be a complex number such that \(a^2 + a + \frac{1}{a} + \frac{1}{a^2} + 1 = 0\). Let \(m\) be a positive integer. Find the value of \(a^{2m} + a^m + \frac{1}{a^m} + \frac{1}{a^{2m}}\).

Answer: -1


Question: Find the sum of the series \( \sum_{n=0}^{\infty } \frac{1}{n(n+1)(n+2)} \) if it converges.

Answer: \dfrac{1}{4}


Question: Let \( X \) and \( Y \) be independent random variables with densities \( f_X(x) = 4x^2e^{-2x} \) for \( x > 0 \) and \( f_Y(y) = \frac{8}{3}y^3e^{-2y} \) for \( y > 0 \). Define \( V = \frac{X}{X+Y} \). Find the variance of \( V \), \( \text{Var}(V) \).

Answer: \dfrac{3}{98}


Question: Find the $n$-dimensional volume of the region defined by $\{ x \in \mathbb{R}^n \mid x_i \geq 0$ for all $i = 1,...,n$ and $x_1 + ... + x_n \leq 1 \}$.

Answer: \dfrac{1}{n!}


Question: Compute the integral \( \int_{|z|=1} \frac{z^7}{10z^8-5z^4+2z^2+iz+1} \, dz \).

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


Question: Consider a random walk on the complex plane starting at the origin, where each step is taken in the direction of a randomly chosen nth root of unity, and each step is of unit length. Let $X_N$ denote the distance from the origin after $N$ steps. Determine an upper bound for the expected value $E(X_N)$. Provide your answer in terms of $N$ and $n$.

Answer: \sqrt{N}


Question:
Suppose \(z\) is a complex number satisfying \(z^5+z^3+z+3=0\). Determine which of the following is true for \(|z|\):

A. \(|z| < 1\)

B. \(|z| \geq 1\)

C. \(|z| \leq \frac{1}{2}\)

D. \(|z| \leq \frac{1}{3}\)

Answer: B


Question: Consider the ideal \( I = \langle X-2 \rangle \) in the ring \( \mathbb{Z}_{11}[X] \). Is \( I \) a maximal ideal in \( \mathbb{Z}_{11}[X] \)?

Answer: Yes


Question: Determine whether the following statement is true: Given any field $F$, there exists a proper field extension $K$ of $F$. Provide a justification for your answer.

Answer: True


Question: Given the equation \( xy = e - e^y \), find the value of \( \ln\left(\frac{d^2y}{dx^2}\right)_{x=0} \).

Answer: -2


Question: Consider two Riemannian manifolds $(M_1, g_1)$ and $(M_2, g_2)$ of the same dimension, and a smooth map $\phi: M_1 \to M_2$. The map $\phi$ is a local isometry if for all $m \in M_1$ and $X, Y \in T_m M_1$, the condition $g_2 (\phi_* X, \phi_* Y ) = g_1 (X, Y )$ holds, where $\phi_* : T_m M_1 \to T_{\phi(m)} M_2$ is the derivative of $\phi$ at $m$. Is the relation of being locally isometric for Riemannian manifolds transitive?

Answer: Yes


Question: Evaluate the integral: \[ \int_{0}^b x\frac{2b(b^2-x^2)}{(b^2+x^2)^2} \, dx \]

Answer: b(1 - \ln 2)


Question:
Find the determinant of the matrix:
\[
\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 2 \\
0 & 1 & 0 & 0 & 2 & 0 \\
0 & 0 & 1 & 2 & 0 & 0 \\
0 & 0 & 2 & 1 & 0 & 0 \\
0 & 2 & 0 & 0 & 1 & 0 \\
2 & 0 & 0 & 0 & 0 & 1
\end{bmatrix}
\]

Answer: -27


Question: Evaluate the line integral \( \oint _C \textbf{F} \cdot d\textbf{r} \) using Stoke's Theorem, where \( \textbf{F} = \langle 1, -2x, 3 \rangle \) and \( C \) is the circle defined by \( x^2 + y^2 = 1 \). Choose \( S \) to be the hemisphere \( x^2 + y^2 + z^2 = 1 \) with \( z \geq 0 \).

Answer: -2\pi


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

Answer: 1


Question: What is the name of a lattice where for every subset \( S \) of the lattice, both the infimum and supremum of \( S \) are elements of \( S \)?

Answer: finite chain


Question: Determine the number of solutions to the equation \(1+e^{\cot^2x} = \sqrt{2| \sin x | -1} + \frac{1-\cos2x}{1 + \sin^4x}\) for \(x\in(0,5\pi)\).

Answer: 5


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

Answer: -\dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{1+2^{1/2}+3^{1/3}+\cdots+n^{1/n}}{n} \]

Answer: 1


Question: If two different functions have the same derivative and are equal at at least one point, will the two functions be equal at all points in their mutual domain? Justify your answer.

Answer: No


Question: Determine the number and nature of the roots of the equation \(x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}\).

Answer: 3


Question: Find the sum of the infinite series: \[ \sum_{k=1}^{\infty} \frac{k^2}{2^k} \]

Answer: 6


Question:
Given that \( p = 10007 \) and \( 5 \) is a primitive root, with known logarithms \( L_5(2) = 6578 \), \( L_5(3) = 6190 \), and \( L_5(7) = 1301 \), find \( L_5(100) \) using the Index-Calculus method. The equation to solve is:
\[ 100 \times 5^r \equiv 42 \pmod{10007} \]
Determine the correct value of \( r \) such that this congruence holds.

Answer: 911


Question: Determine whether the set of computable real numbers has a positive Hausdorff dimension.

Answer: 0


Question: Find all real values of \(a\) such that the quadratic equation \(x^2 + (a+i)x - 5i = 0\) has at least one real solution.

Answer: -5


Question: Evaluate the limit: \[ \lim_{t \rightarrow \infty}\dfrac{2\log(tx) + \sin(\log(tx))}{2\log(t) + \sin(\log(t))} \] where \( x > e \).

Answer: 1


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

Answer: \dfrac{1}{2}


Question: Let $I\subseteq\mathbb{R}$ be a non-degenerate interval and let $f: I\longrightarrow\mathbb{R}$ be a function with the property that for every pair $(x_0, y_0)\in I\times\mathbb{R}$, there exists a unique function $F: I\longrightarrow\mathbb{R}$ such that $F'=f$ and $F(x_0)=y_0$. Must $f$ be continuous? Answer yes or no.

Answer: No


Question: Let $1 \leq p < \infty$ and consider the space $L^p(\mathbb{R}^d)$ equipped with the Lebesgue measure. Prove that if $f \in L^p(\mathbb{R})$, then \(\lim_{|h| \to 0} ||f(x+h)-f(x)||_{L^p} = 0\).

Answer: 0


Question: Solve the partial differential equation \( \frac{\partial ^2u}{\partial x^2}=\frac{\partial u}{\partial t} \) for \( 0<x<1, 0<t \), with the boundary conditions \( u(0,t)=-1 \) and \( -\frac{\partial u(1,t)}{ \partial x}=(u(1,t)-1) \), and the initial condition \( u(x,0)=x-1 \), using the method of separation of variables.

Answer: u(x, t) = x - 1


Question: Evaluate the integral \( \int_{\Gamma:|z|=1}\frac{e^z-1-z}{z^2}dz \), where \( \Gamma \) is the unit circle in the complex plane.

Answer: 0


Question: Calculate the limit \( \lim_{k\rightarrow\infty} x_k \) where \( x_k = \sum_{n=1}^{\infty} \frac{1}{n^2} \left(1-\frac{1}{2n}+\frac{1}{4n^2}\right)^{2k} \) for every \( k \in \mathbb{N} \).

Answer: 0


Question: Determine whether the chromatic number of the graph \( G = ([\omega]^\omega, E) \), where \( E = \{\{a,b\}: a,b\in [\omega]^\omega\text{ and } |a\cap b| = 1\} \), is countably infinite, i.e., \( \chi(G) = \aleph_0 \).

Answer: \aleph_0


Question: Calculate the probability that at least 40 people share the same birthday in a group of 350 people.

Answer: 0


Question: Show that the semidirect product \( \mathbb{Z}_3 \rtimes_{\varphi} (\mathbb{Z}_2 \oplus \mathbb{Z}_2) \) is isomorphic to the dihedral group \( D_6 \), where the homomorphism \( \varphi: \mathbb{Z}_2 \oplus \mathbb{Z}_2 \to \text{Aut}(\mathbb{Z}_3) \) is defined by \((1,0) \mapsto -1\) and \((0,1) \mapsto -1\).

Answer: D_6


Question: Consider the power set of \( \{0, 1, 2, 3\} \), and let \( C = \{ \{0\}, \{1, 2\}, \{1, 3\}, \{2, 3\} \} \). Calculate the intersection number of \( C \).

Answer: 3


Question: Calculate the limit: \[ \lim_{n\rightarrow \infty} \sqrt{n}\int_{0}^{\infty} \frac{1-\cos(x)}{1+nx^6}dx \]

Answer: \dfrac{\pi}{12}


Question: Find the polynomial function $f(x)$ that satisfies the equations $f(x^2) = f^2(x)$ and $f(0) = 1$. Provide the explicit form of $f(x)$.

Answer: 1


Question:
Given that \( g(x) = f(x) + f(1-x) \) and \( f'(x) < 0 \) for all \( x \in (0,1) \), determine the interval in which \( g \) is monotone increasing. Choose the correct option:

(A) \((1/2,1)\).
(B) \((0,1/2)\).
(C) \((0,1/2) \cup (1/2,1)\).
(D) none.

Answer: D


Question: Given a circle \( \circ O \) centered at point \( O \) with a point \( A \) on it, inscribe a square \( \square ABCD \) within the circle using exactly seven compass-and-straightedge steps. The first two steps should use the compass, the third step should use the straightedge, and the last four steps should use the straightedge to draw the sides of the square. Note that the problem is not considered solved until each line containing one of the four sides of the square is drawn. Naming and creating points are allowed and do not count toward the seven steps.

Answer: ABCD


Question:
Given a differentiable function \( f(x,y) \), the maximum value of the directional derivative \( D_uf(0,2) \) is 2, occurring when \( u = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \). If \( x(s,t) = s^2t \), \( y(s,t) = 2se^t \), and \( F(s,t) = f(x(s,t), y(s,t)) \), find \( \frac{\partial F}{\partial t}(1,0) \). Choose the correct answer from the following options:

(a) \( 2\sqrt{2} \)
(b) \( 3\sqrt{2} \)
(c) \( 0 \)
(d) \( \sqrt{2} \)
(e) \( 3\frac{\sqrt{2}}{2} \)

Answer: B


Question: Find the maximum area of a hexagon that can be inscribed in the ellipse given by \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \).

Answer: 18\sqrt{3}


Question: A stick lies on the interval $[0,1]$ and is broken at a point $X$ where $X$ is uniformly distributed over $(0,1)$. The left part of the stick is then broken again at a point $Y$, where $Y$ is uniformly distributed over $(0,X)$. Find the unconditional distribution of $Y$.

Answer: -\ln(y)


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

Answer: 2\pi


Question: Find the expected value of a random variable $X$ with the density function $f(x) = \frac{5}{x^2}$ for $x > 5$ and $f(x) = 0$ elsewhere.

Answer: \infty


Question: Evaluate the integral \( \int\int\int_E z \, dV \), where \( E \) is the region above the paraboloid \( z = x^2 + y^2 \) and below the plane \( z = 6y \). Express the integral in cylindrical coordinates and compute the volume, rounding your answer to the nearest hundredth.

Answer: 1908.52


Question: Let \( p, q \in \mathbb{R}[x] \) be polynomials with non-negative coefficients. Assume that \( p(r) \leq q(r) \) for every \( r \geq 0 \). Determine whether the polynomial \( q - p \) has non-negative coefficients.

Answer: No


Question: Let $\Gamma$ be a convex body contained within a unit sphere $B(0,1)$ in $\mathbb{R}^n$. Is the surface area of $\Gamma$ less than or equal to the surface area of the sphere? Justify your answer.

Answer: Yes


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

Answer: 1


Question:
Given the vector field \( F = (x + 2y)i + (3y + 2z)j + (x^2 − 4z)k \), determine which of the following is a vector potential for \( F \):

A) \((yz + 3z^2 − xy)i + (4xyz)j\)
B) \((2xy − z/2)i + (x^3 − 2yz)j\)
C) \((z^3/3 + 2(x^2)y)i + (xy − 3yz)j\)
D) \((3yz+z^2 − x^2y)i − (xz + 2yz)j\)
E) \(z(x+2y)i + x(3y + 2z)j + y(x^2 − 4z)k\)

Select the correct option.

Answer: D


Question: Calculate the winding number of the curve \(|z|=3\) around the point \(\alpha=i\).

Answer: 1


Question: Calculate \( p^3(-1) \) for the polynomial \( p(z) = z^2 + 6z + 1 \) using the transformation \( p = \varphi^{-1} \circ f \circ \varphi \), where \( \varphi(z) = z + 3 \).

Answer: 8


Question: Given a convex quadrilateral $ABCD$ circumscribed about a circle of diameter $1$, and a point $M$ inside $ABCD$ such that $MA^2 + MB^2 + MC^2 + MD^2 = 2$, find the area of the quadrilateral.

Answer: 1


Question: Evaluate the surface integral \( \int_S \mathbf{F} \cdot \mathbf{\hat{n}} \, dS \), where \( \mathbf{F} = 2y\mathbf{i} - z\mathbf{j} + x^2\mathbf{k} \) and \( S \) is the surface of the parabolic cylinder \( y^2 = 8x \) in the first octant, bounded by the planes \( y = 4 \) and \( z = 6 \). The unit normal vector \( \mathbf{\hat{n}} \) points in the direction of increasing \( x \). Evaluate the integral by projecting onto the plane \( x = 0 \).

Answer: 132


Question: Given a strictly increasing function $f: [0, 1] \to \mathbb{R}$ that is continuous on $[0,1]$ and differentiable on $(0,1)$, with $f(0)=0$, and satisfying the condition $f'(x) \geq xf(x) + x$ for $x \in (0,1)$, find the minimum value of $f(1)$. Assume $f'(x) \geq 2xf(x)$ for $x \in (0,1)$ as well.

Answer: \sqrt{e} - 1


Question: Let \( f \) be a function that is twice differentiable on \((0, \infty)\) with \( f'' \) bounded on \((0, \infty)\). Given that \( f(x) \to 0 \) as \( x \to \infty \), prove that \( f'(x) \to 0 \) as \( x \to \infty \).

Answer: 0


Question: Find the length of the curve formed by the intersection of the unit sphere \(x^2 + y^2 + z^2 = 1\) and the plane \(x + y = 1\).

Answer: \pi \sqrt{2}


Question: Calculate the limit: \[ \lim _{x\to 3}\left(3x-8\right)^{\frac{x}{\sin(x-3)}} \] without using L'Hôpital's rule.

Answer: e^9


Question: Calculate the volume of the torus formed by rotating the circle \((x-a)^2+y^2=b^2\), where \(a > b\), around the y-axis.

Answer: 2\pi^2 a b^2


Question: Evaluate the integral \( \int_{0}^{e^2} \left( \frac{1}{\log{x}} - \left(\frac{1}{\log x}\right)^2 \right) \, \mathrm{d}x \).

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


Question: Find the derivative of the expression $a^{T}Xb$ with respect to the vector $b$, where $a$ and $b$ are $d$-dimensional vectors and $X$ is a $d \times d$ matrix.

Answer: X^\top a


Question: Determine whether the following statement is true: For the connected sum of real projective spaces, \( \beta_{2n-1}(\sharp_{h}\mathbb{RP}^{2n}) = h-1 \).

Answer: True


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{r=1}^n \frac{r^4}{4r^2-1} \]

Answer: \infty


Question: Determine the value of \( k \) such that the polynomial equation \( 3x^4 - 16x^3 + 18x^2 + k = 0 \) has four real solutions.

Answer: (-5, 0)


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

Answer: \sqrt{e}


Question: Evaluate the integral \( \oint_{C}\frac{z + 2}{z^2(z^2-1)}dz \), where \( C = \{z : |z + 1| = \frac{3}{2}\} \).

Answer: -3\pi i


Question: Given a finite subset $S \subset \mathbb{Z}^d$, let $\varphi(S) = \sum_{z \in S}|z|^2$. For a given $m \in \mathbb{N}$, determine a rough upper bound for the size of $\varphi^{-1}(m)$, the set of all finite subsets of $\mathbb{Z}^d$ with sum of squares equal to $m$.

Answer: 2^{O(m^{d/2})}


Question: Evaluate the limit: \[ L = \lim_{n\to \infty} \int_{0}^{\pi} \sqrt[n]{x}\sin x\, dx \]

Answer: 2


Question: Determine whether the Laplace transform of a piecewise continuous function is injective. Specifically, if \((\mathcal{L}f)(s) = 0\) for all \(s \in \mathbb{R^+}\), does this imply that \(f(t) = 0\) for all \(t \geq 0\)?

Answer: No


Question: Let $a$ be a zero of the polynomial $x^3 + x^2 + 1$ in some extension of $\mathbb{Z}_2$. Find the multiplicative inverse of $a+1$ in this extension.

Answer: a^2


Question: Find the value of \(s\) that satisfies the equation: \[2^s = 2s + 2\]

Answer: 3


Question: Evaluate the limit: $$\lim_{m \to +\infty} \left[ \lim_{x \to 0^+} \int_{0}^{x} \left( \lim_{n \to \infty} \left( 1 - \frac{1}{n} \right)^n \right)^{t^m} t \, dt \right]$$

Answer: 0


Question: Given a positive integer \( n \), determine if there exists a positive integer \( k \) such that for every finite, simple, undirected graph \( G \) with maximum degree \( \Delta(G) = n \), the chromatic number \( \chi(G) \leq k \).

Answer: n + 1


Question: Determine whether the assertion about the geometrical interpretation of the line integral of a function $f$ along a curve $C$ with respect to $x$ is true or false.

Answer: B


Question: Find the value of \( a \) such that \( \lim_{t \to 0} \left(\frac{a}{t^2} - \frac{\sin 6t}{t^3 \cos^2 3t}\right) = -18 \).

Answer: 6


Question: Evaluate the integral \( \int_0^\pi \left[\frac{\sin(nx)-\sin((n-1)x)}{\sin(x/2)}\right]^2 \, dx \) analytically, where \( n \in \mathbb{N} \).

Answer: 2\pi


Question: Let $Y$ be a proper subspace of a real normed linear space $X$. Assume there exists an $x \in X$ such that $\|x\| = 1$ and $\text{dist}(x, Y) = 1$. Is $Y$ necessarily finite-dimensional?

Answer: No


Question: 100 people are seated at a round table, and initially, no one is an enemy with their neighbor. Each night, every person becomes an enemy with one of their neighbors. Determine the number of nights that must pass until there is no possible seating arrangement where no one sits beside their enemy.

Answer: 50


Question: Consider the expansion of \((1+x+x^2)^n\) as \(\sum_{r=0}^{2n} a_r x^r\). Find the sum of the coefficients \(a_0 + a_3 + a_6 + a_9 + \ldots\).

Answer: 3^{n-1}


Question: Find the number of pairs \((X, Y)\) such that \(X, Y \subseteq \{1, 2, \ldots, n\}\) and \(|X \cap Y| = 1\).

Answer: n \cdot 3^{n-1}


Question: Given a transition function \( \delta : (Q \setminus F) \times \Gamma \to Q \times \Gamma \times \{L, R, S\} \), where \( \Sigma = \{0, 1\} \) and \( \Gamma = \{0, 1, \flat\} \), and \( |Q| = t \), determine why the number of possible functions \( \delta \) is \((9t)^{3t}\). Explain the combinatorial reasoning behind this expression.

Answer: (9t)^{3t}


Question: Evaluate the triple integral \( \iiint_T xy\,dx\,dy\,dz \) where \( T \) is the region in the first octant bounded by the coordinate planes and the upper half of the sphere \( x^2 + y^2 + z^2 = 4 \).

Answer: \dfrac{32}{15}


Question: Compute the limit \( \lim\limits_{x\to 0} (-1)^{n+1} \frac{\int_0^x \frac{t^{2n+2}}{1+t^2}dt}{x^{2n+1}} \).

Answer: 0


Question: Suppose $M$ is a complex manifold and $\Omega$ is a bounded pseudoconvex domain in $M$. Let $u:M\setminus\Omega\to\mathbb{R}$ be a pluriharmonic function. Determine whether $u$ can be extended as a pluriharmonic function to the entire manifold $M$. Assume $\dim_{\mathbb{C}}(M)\geq 2$. Provide a justification for your answer.

Answer: Yes


Question: How many Riemann surfaces are homeomorphic to the sphere?

Answer: 1


Question:
Determine which of the following formulas defines a metric:

1. \(d(f,g)=\sup_{x\in[-1,1]} |x|\cdot|f(x)-g(x)|\) on \(B([-1,1],\mathbb{R})\)

2. \(d(f,g)=\int_{-1}^1 |x|\cdot|f(x)-g(x)|\ dx\) on \(C^0([-1,1],\mathbb{R})\)

where \(B(S,\mathbb{R}):=\{f:S\to\mathbb{R}|f\) is bounded\}. A function \(d:S\times S\to\mathbb{R}_0^+\) is a metric on a set \(S\) if for all \(x,y,z\in S\):

- \(d(x,y)\geq 0\) and \(d(x,y)=0\Leftrightarrow x=y\),
- \(d(x,y)=d(y,x)\),
- \(d(x,y)\leq d(x,z)+d(z,y)\).

Answer: 2


Question: Find the last two digits of \((1! + 2! + 3! + 4! + \ldots + 2016!)^{42}\).

Answer: 69


Question: Is the expectation value map Lipschitz with respect to the 1-Wasserstein metric on the space of Borel measures on the interval $[0,1]$? Specifically, can a constant $C$ be found such that for every pair of measures $p$ and $q$, the inequality $|E_p - E_q| \le C \cdot d_W(p, q)$ holds, where $E_p = \int_{[0,1]} x \, dp(x)$?

Answer: 1


Question: Suppose $\hat{f}(n)$ is the Fourier coefficient of a function $f$ on the circle. Is it true that if $\sum |\hat{f}(n)| < \infty$, then $f$ is in $L^1$?

Answer: Yes


Question: Let \( \{a,b,c,d\} \) be an algebraically independent set over \( \mathbb{Q} \). Prove or disprove the statement: \( \mathbb{Q}(a,b) \cap \mathbb{Q}(c,d) = \mathbb{Q} \), where \( \mathbb{Q}(a,b) \) is the smallest subfield containing \( \mathbb{Q} \cup \{a,b\} \).

Answer: \mathbb{Q}


Question:
Find the smallest positive integer \( n \) such that \( 1 + 5u_nu_{n+1} = k^2 \), where \( k \) is a natural number, given the sequence \( \{u_n\} \) defined by:
\[\begin{cases}u_1=20;\\u_2=30;\\ u_{n+2}=3u_{n+1}-u_{n},\; n \in \mathbb{N}^*.\end{cases}\]

Answer: 3


Question: Find \( f'(1) \) if \( f(x) = e^{g(x)} \) where \( g(x) = \int_{x^2}^{x^3} \frac{t^2}{1+t^2} \, dt \). Use the Leibniz Formula to solve the problem.

Answer: \dfrac{1}{2}


Question: Given the joint probability density function \( f(x,y) = \frac{1}{2\pi\sqrt{1-p^2}}e^{-\frac{x^2+y^2-2pxy}{2(1-p^2)}} \) for \(-\infty < x, y < \infty\), find the conditional expectation \( E(Y|X=1) \).

Answer: p


Question:
Given that $a, b, c$ are in geometric progression, determine the type of progression that $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ must be in for the quadratic equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ to have a common root. Choose from the following options:

- Arithmetic Progression
- Geometric Progression
- Harmonic Progression

Answer: Arithmetic Progression


Question: In an L-space (Limit-space), is a single point $x \in X$ closed? Provide a justification for your answer.

Answer: Yes


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

Answer: 6e^2


Question: How many directed graphs of size \( n \) are there where each vertex is the tail of exactly one edge?

Answer: n^n


Question: Evaluate the integral \( \int_\gamma \overline{z} \, dz \), where \( \gamma \) is the circular path defined by \( |z| = 2 \).

Answer: 8\pi i


Question: How many orthogonal $4 \times 4$ matrices $A$ are there such that $0 < |(A)_{ij}| \leq 1/2$ for all entries $(A)_{ij}$?

Answer: 768


Question: For every integer $n \geq 3$, is it possible to construct an undirected graph with $n$ vertices, no loops, and no double edges, such that $n-1$ vertices have distinct degrees?

Answer: Yes


Question: Determine whether the sequence \( a_n = \frac{n^2}{((2n)!)^{\frac{1}{n}}} \) converges, and if so, find its limit.

Answer: \dfrac{e^2}{4}


Question: Let $A$ be a set with positive measure, and let $(A_n)_n$ be a pairwise disjoint sequence of subsets of $A$, each with positive measure. Consider two subsets $S, T \subset \mathbb{N}$ with $S \neq T$. Let $\chi_S$ and $\chi_T$ be the characteristic functions of $\cup_{n \in S} A_n$ and $\cup_{n \in T} A_n$, respectively. Compute the $L^\infty$-norm of the difference $||\chi_S - \chi_T||_\infty$.

Answer: 1


Question: Find all functions \( f: \mathbb{N} \to \mathbb{N} \) such that for all natural numbers \( n \) and \( m \), the expression \( f(m)^2 + f(n) \) divides \( (m^2+n)^2 \).

Answer: f(n) = n


Question:
Let $A$ be a $10 \times 10$ matrix with complex entries such that all its eigenvalues are non-negative real numbers, and at least one eigenvalue is positive. Determine which of the following statements is always false:

A. There exists a matrix $B$ such that $AB - BA = B$.
B. There exists a matrix $B$ such that $AB - BA = A$.
C. There exists a matrix $B$ such that $AB + BA = A$.
D. There exists a matrix $B$ such that $AB + BA = B$.

Choose the correct option.

Answer: B


Question: Find all positive integer pairs \((a, b)\) such that the equation \(5^b + 2 = 7^c\) is satisfied.

Answer: (1, 1)


Question: Determine the disk of convergence for the power series \( \sum_{n=1}^\infty n!\,(z-i)^{n!} \).

Answer: |z - i| < 1


Question: Evaluate the integral \( \int_{-\infty}^{\infty} x^3 e^{-ax^2} \, dx \).

Answer: 0


Question: Evaluate the expression $(-1)^i$, where $i^2 = -1$, and determine its value.

Answer: e^{-\pi}


Question: Let $X$ be a complex projective variety, and let $L$ be a line bundle on $X$ of order $r$, meaning $L^{\otimes r} \simeq \mathcal{O}_X$. Determine whether $L$ can be viewed as a $\mu_r$-bundle, where $\mu_r = \{\xi \in \mathbb{C}^* \mid \xi^r = 1\}$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the convolution of the function \( \frac{\sin(t)}{t} \) with the constant function 1. Express your answer as a function or an expression.

Answer: \pi


Question:
Determine the total number of constraints provided by a 2D rigid body transformation matrix in homogeneous representation, given by:
\[
\begin{bmatrix}
\textbf{R} & \textbf{t} \\
0^T & 1
\end{bmatrix}
\]
where \(\textbf{R}\) is a rotation matrix with the orthonormal constraint \(R^TR=1\). Consider any additional constraints from the homogeneous representation and the translation vector \(\textbf{t}\). What is the total number of constraints in the 2D case?

Answer: 6


Question: Evaluate the expression: $$\frac{{\displaystyle \sum_{n=1}^{99}} \sqrt{10+\sqrt{n}}}{{\displaystyle \sum_{n=1}^{99}} \sqrt{10-\sqrt{n}}}$$

Answer: 1 + \sqrt{2}


Question: Does there exist a nonabelian finite simple group $S$ such that $H^2(S, \mathbb{Z}/p\mathbb{Z})$ is nontrivial, where $p$ is an odd prime?

Answer: Yes


Question: Let $A$ be a reduced commutative ring, and let $P_1$ and $P_2$ be two minimal prime ideals of $A$. Is the intersection $P_1 \cap P_2$ equal to zero? Justify your answer.

Answer: No


Question: Let $f: \mathbb{C} \to \mathbb{C}$ be an entire function such that $f(L) \subset L$, where $L = \{ x + iy : x = y \}$. Given that $f(1) = 0$, find $f(i)$.

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{1}{x^4}\int_{-x}^{0}\sin(t^3)\,dt. \]

Answer: -\dfrac{1}{4}


Question: Determine all positive integers \( n \) such that if \( 1 = d_1 < d_2 < \ldots < d_k = n \) are the divisors of \( n \), then \( n = d_6^2 + d_7^2 - 1 \).

Answer: 144


Question: Evaluate the complex contour integral \( g(w) = \int_C \frac{z e^{z^3}}{(z-w)^3} \, dz \) where \( C \) is the positively oriented ellipse given by \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \). Find \( g(2+3i) \) in the form \( a + bi \).

Answer: 0


Question: Determine the space of sections of the vector bundle \( \pi: \mathfrak{g} \longrightarrow \{e\} \), where \( \mathfrak{g} \) is a Lie algebra and \( \{e\} \) is a single point. What is the space of sections of this bundle?

Answer: \mathfrak{g}


Question: Find the flux of the vector field \( \mathbf{F} = (2x-xy, -y, yz) \) out of the region \( W \), where \( W \) is the 3D region under the graph of \( f(x,y) = \exp(x^2+y^2) \) over the region in the plane defined by \( 1 \leq x^2 + y^2 \leq 2 \). Verify your answer using the divergence theorem.

Answer: \pi (e^2 - e)


Question: Evaluate the integral \( \int_3^4 \sqrt{(x - 3)(4 - x)} \, dx \). Choose the correct answer from the following options: \((a) \frac{\pi}{4} \quad (b) \frac{\pi}{8} \quad (c) \frac{\pi}{2} \quad (d) \text{none}\).

Answer: B


Question: A small elevator has a maximum capacity $C$, which is normally distributed with a mean of $400$ kg and a standard deviation of $4$ kg. The weight of the boxes being loaded into the elevator is a random variable with a mean of $30$ kg and a standard deviation of $0.3$ kg. Assume that the weights of the boxes and the maximum capacity are independent random variables. Determine the maximum number of boxes that can be loaded into the elevator before the probability of exceeding the elevator's capacity exceeds $20\%$. Use the Central Limit Theorem to find your answer.

Answer: 13


Question: Find the minimum value of \(a^2 + b^2\) given that all roots of the polynomial \(x^4 + ax^3 + bx^2 + ax + 1 = 0\) are real.

Answer: 4


Question: In a queuing system denoted as $(M | M | 2 | 5)$, the input flow rate is $240$ requests per hour, and the average service time for one request is $30$ seconds. Calculate the average waiting time for a request in the queue.

Answer: 20


Question: Determine the points in \(\mathbb{R}\) where the identity map \(i: (\mathbb{R}, \zeta) \to (\mathbb{R}, \zeta')\) is continuous, where \(\zeta\) is the usual topology and \(\zeta'\) is the discrete topology on \(\mathbb{R}\).

Answer: \emptyset


Question: Find the stationary points \((x, y, z)\) of the function \(f(x, y, z) = 5x^2 - 8xy - 4xz + 5y^2 - 4yz + 8z^2 + 1\) subject to the constraint \(x + y - 4z = 8\) using the method of Lagrange multipliers.

Answer: (2z + 4, 2z + 4, z)


Question: Let \( \mathbb{K} \) be a subfield of \( \mathbb{C} \) and consider \( \mathbb{K}^n \) with the Euclidean topology. If a polynomial \( p \in \mathbb{K}[x_{1},...,x_{n}] \) vanishes on a nonempty open subset of \( \mathbb{K}^n \), is \( p \) necessarily the zero polynomial?

Answer: Yes


Question: Find the positive value of \(\alpha\) such that the function \(f(x,y) = \frac{x^3}{3} - \frac{yx^2}{2} - \frac{\alpha x^2}{2} + \alpha yx + \frac{2y^3}{3} + y^2\) has three distinct critical points.

Answer: 1


Question: Evaluate the limit: \[ \lim \limits_{x \to \infty} x\int_{0}^{x}e^{t^2-x^2}dt \]

Answer: \dfrac{1}{2}


Question: Find the third derivative \( y'''(x) \) of the function implicitly defined by the equation \( x^2 + xy + y^3 = 1 \) at the point where \( x = 1 \).

Answer: 42


Question: Find the sum of the infinite series: \[ \frac{1}{4} + \frac{1 \cdot 3}{4 \cdot 6} + \frac{1 \cdot 3 \cdot 5}{4 \cdot 6 \cdot 8} + \cdots \]

Answer: 1


Question: Use the divergence theorem to evaluate the surface integral \(\iint_S \vec{A} \cdot \vec{n} \, dS\), where \(\nabla \cdot \vec{A} = 4r \cos^2(\theta)\) and \(S\) is the spherical surface of unit radius centered at the origin.

Answer: \dfrac{4\pi}{3}


Question: Find the sum of the infinite series: \[ \sum\limits_{k=1}^{\infty}\frac{1}{(2k-1)(2k+1)(2k+3)} \]

Answer: \dfrac{1}{12}


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

Answer: 2x + 2


Question:
Determine whether the following sum diverges to infinity:
\[
\sum_{1\leq n \leq (\log x)^{100} } \mu^2(2n) \sum_{(\log x )^{100} < m \leq x} \left(\frac{m}{n}\right) \frac{\mu^2(2  m )}{m},
\]
where \(\mu\) is the M"obius function and \(\left(\frac{m}{n}\right)\) is the Jacobi quadratic symbol for positive squarefree odd integers \(n,m\).

Answer: \infty


Question: Can there exist two nonisomorphic number fields with the same degree, the same discriminant, and the same simple Galois group for their splitting field?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to +\infty} \left(\sinh x\right)^{\arctan\frac{1}{x}} \] without using L'Hôpital's rule.

Answer: e


Question:
A driver and a passenger are in a car accident. Each of them independently has a probability of 0.3 of being hospitalized. When a hospitalization occurs, the loss is uniformly distributed on [0,1]. When two hospitalizations occur, the losses are independent. Calculate the expected number of people in the car who are hospitalized, given that the total loss due to hospitalizations from the accident is less than 1. Choose the correct answer:

(A) 0.510
(B) 0.534
(C) 0.600
(D) 0.628
(E) 0.800

Answer: B


Question: Let \( \alpha \) be an element in an extension field \( K \) of \( \mathbb{Q} \) generated by the cubic root of a non-cubic \( u \in \mathbb{N} \), where \( K = \mathbb{Q}[\sqrt[3]{u}] \) and \( \alpha \notin \mathbb{Q} \). Determine the degree of the minimum polynomial of \( \alpha \) over \( \mathbb{Q} \).

Answer: 3


Question: Can a continuous bounded function \( u:[0,\infty)\to \mathbb{R} \) be approximated by linear combinations of exponential functions?

Answer: No


Question: Calculate the rate of heat inflow through the surface defined by the equation \( y^2 + z^2 = 6 \) for \( x \in [0, 4] \), given that the temperature distribution is \( u(x, y, z) = 2y^2 + 2z^2 \) and the thermal conductivity is \( K = 6.5 \). Use the formula for heat flow \( \int \int_S -K \nabla u \cdot d\vec{S} \) and find the inward heat flow through the surface.

Answer: 1248\pi


Question: Suppose $(X_n)_{n \in \mathbb{N}}$ is a sequence of real random variables that are uniformly bounded $P$-a.s. If $\lim_{n \to +\infty} E[ \vert X_n \vert] = 0$, does it necessarily follow that $\lim_{n \to +\infty} E[ (X_n)^2] = 0$? Justify your answer.

Answer: Yes


Question: Find the minimum value of the expression \(x_1^2 + y_1^2 + x_2^2 + y_2^2 - 2x_1x_2 - 2y_1y_2\) subject to the conditions that \((x_1, y_1)\) and \((x_2, y_2)\) lie on the curve \(xy = 1\), with \(x_1 > 0\) and \(x_2 < 0\).

Answer: 8


Question: Consider a flight of $N$ stairs. You start at the bottom and want to reach the top. Each time you attempt to take a step, you flip a fair coin. If it lands on heads, you take a step up; if it lands on tails, you take a step down. However, if you are at the bottom of the stairs, a tails result means you stay at the bottom and flip again. What is the expected number of coin flips needed to reach the top of the stairs?

Answer: N(N+1)


Question: How many complex numbers $z$ are there such that $|z|=1$ and $z^{5040} - z^{720}$ is a real number?

Answer: 10080


Question:
Find all polynomials \( P(a) \in \mathbb{R}[a] \) such that for all \( x, y, z \in \mathbb{R} \setminus \{0\} \), the following equation holds:
\[ P(x + y + z) = \frac{P(y)P(z)}{x^2} + \frac{P(z)P(x)}{y^2} + \frac{P(x)P(y)}{z^2}. \]

Answer: 0


Question: Let $X$ be an $n$-dimensional Alexandrov space with curvature bounded below. Let $f_1,\dots, f_n\colon X\to \mathbb{R}$ be $\lambda$-concave functions. Assume that at a fixed point $p$ there exist directions $\xi_1^\pm,\dots,\xi_n^\pm\in \Sigma_p$ such that $f'_i(\xi_i^+)>1,\, f'_i(\xi^-_i)<-1$ and $|f'_i(\xi_j^\pm)|<\frac{1}{100 n}$ for $i\ne j$. Is the map $$(f_1,\dots,f_n)\colon X\to \mathbb{R}^n$$ injective in a small neighborhood of $p$?

Answer: Yes


Question: Find the closed form expression for the number of neighboring integer points in an n-dimensional space, given the recursive formula: \( n_i = 3n_{i-1} + 2 \).

Answer: 3^n - 1


Question: Find the number of ordered pairs \((a, b)\) of positive integers such that \(1 \leq a, b \leq 50\) and \(\gcd(a, b) = 1\).

Answer: 1547


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a non-constant, three times differentiable function such that \( f(1 + \frac{1}{n}) = 1 \) for all integers \( n \). Find \( f''(1) \).

Answer: 0


Question: Determine whether the cardinality of the Borel \(\sigma\)-algebra is equal to the cardinality of the continuum.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{ x\to 0} \frac{ \left(\arctan(2x)-2\arctan( x)\right)^x}{\mathrm e^{x^2}-1-\sin^2(x)}. \]

Answer: \infty


Question: Let $L$ be a normal extension over $k$, and let $a \in \overline{k} \cap L^c$ such that $L(a)$ is Galois over $L$. Prove that there exists a finite extension $L'$ over $k$ such that $L'$ is contained in $L$, $L'(a)|L'$ is Galois, and $\text{Gal}(L'(a)|L)$ is isomorphic to $\text{Gal}(L(a)|L)$. Additionally, if $L$ is Galois over $k$, show that $L'$ can be chosen to be Galois over $k$. Provide a detailed proof of your result.

Answer: L'


Question: Consider a set where none of its proper subsets' sums is divisible by the cardinality of the set. Does this imply that all elements of the original set have the same remainder when divided by the cardinality of the set? Answer yes or no.

Answer: Yes


Question: Find all points at which the surfaces $x^2+y^2+z^2=3$ and $x^3+y^3+z^3=3$ share the same tangent line.

Answer: (1, 1, 1)


Question: Assume that \(\lim_{x \to 1} \frac{16^x-16x^4}{a^x-2x^b} =32.\) Find the value of \(a+b\) without using L'Hôpital's rule.

Answer: 3


Question: Evaluate the contour integral \( \oint \limits_C \frac{z^2+1}{(2z-i)^2}dz \) using the residue theorem, where \( C:|z|=1 \) is a circle with positive orientation.

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


Question: Given two rotations $R_1$ and $R_2$ in $SO(3)$, where $R_1$ is a rotation by an irrational multiple of $\pi$ about the z-axis, and $R_2$ is a rotation by a rational multiple of $\pi$ about an axis at an angle $\phi$ from the z-axis with $0<\phi<\frac{\pi}{2}$, determine if $R_1$ and $R_2$ densely generate $SO(3)$. Provide a justification for your answer.

Answer: Yes


Question: Consider an infinite-rank operator that generates a complete set of eigenfunctions and eigenvalues with finite multiplicity, where the eigenvalues accumulate around zero and can be ordered such that \( \lambda_i \to 0 \). Determine whether such an operator is compact.

Answer: Yes


Question: Determine all real numbers \(x\) such that the series \[ S = \sum_{n=1}^{\infty} \frac{1}{n^2+n} \cdot \tan^2\left(\frac{1}{\sqrt{n}}\right) \cdot n^x \] converges.

Answer: (-\infty, 2)


Question: Find all pairs of prime numbers \((p, q)\) such that the expression \(p^{q+1} + q^{p+1}\) is a perfect square.

Answer: (2, 2)


Question: Let $S$ be a finite set of square matrices $A_i$ of order $n$ with entries in $\mathbb{C}$, such that the null vector is the only element in the intersection of their kernels: $$\cap_i \ker(A_i) = \{0\}.$$ Does the left ideal generated by $S$ coincide with the unit ideal of $M_n(\mathbb{C})$?

Answer: Yes


Question: Let $L$ be the splitting field of the polynomial $x^3-10$, so $L=\mathbb{Q}(\sqrt[3]{10},\zeta)$ where $\zeta$ is a complex cubic root of unity. Let $K=\mathbb{Q}(\sqrt{5},\sqrt{-7})$. Compute the degree of the intersection $K\cap L$ over $\mathbb{Q}$, denoted as $|K\cap L:\mathbb{Q}|$. Determine whether this degree is 1 or 2.

Answer: 1


Question: Determine the values of \( n \ge 4 \) for which the local ring \( R_n = k[[x,y,z,w]]/(x^2y+y^{n-1}+z^2+w^2) \) is not a unique factorization domain (UFD), where \( k \) is an algebraically closed field of characteristic 0.

Answer: n \geq 4


Question: Find the Fourier transform of the product \( \delta(x)f(x) \). Is the result \( f(0) \)?

Answer: f(0)


Question: If $f_n \xrightarrow{wk\star} f$, does it follow that $\int_{\mathbb{R}^d} f_n g \to \int_{\mathbb{R}^d} f g$ for all $g \in L^q(\mathbb{R}^d)$, where $1/p + 1/q = 1$?

Answer: Yes


Question: Determine the smallest positive integer \(m\) such that the dihedral group \(D_4\) can act faithfully on a set of size \(m\).

Answer: 4


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

Answer: 0


Question: Determine whether there exists a commutative integral domain $R$ such that every nonzero prime ideal $Q$ is maximal, and there are maximal ideals $Q$ with residue fields of sizes $3$, $11$, and $27$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the complete graph $K_5$ can be embedded in a Möbius band.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{1}{1-\cos(x^2)}{\sum_{n=4}^\infty {n^5x^n} } \]

Answer: 2048


Question: In NBG set theory, consider a category-theoretic product function \( f: V \times V \to V \) that satisfies the universal property for products. Determine whether it is true that for any such product function \( f \), there exists an ordered pair function \( OP: V \times V \to V \) such that for all sets \( S \) and \( T \), \( f(S,T) \) is the set of all ordered pairs whose first component is in \( S \) and whose second component is in \( T \).

Answer: True


Question: Find the multiplicative inverse of the element $x^2 + 1$ in the ring $\Bbb{F}_3[x]/(x^3 + 2x + 2)$. Use the Euclidean Algorithm to assist in your solution.

Answer: 2x^2 + 2x + 2


Question: Find the maximum value of the function \( f(x, y) = 2xy^2 - 2x^2 \) subject to the constraint \( x^2 + y^2 \leq 1 \) and \( x \geq 0 \).

Answer: \dfrac{10}{27}


Question: Consider a loop of string with length N units. Mark N points on it at independent and uniformly random positions along its length. Cut the string at each of these N points, resulting in N pieces. As N approaches infinity, determine the probability density function of the lengths of these pieces. What is this function, and does it have a name?

Answer: e^{-x}


Question: Given the system of equations: \(x^2 + xy + y^2 = 25\), \(x^2 + xz + z^2 = 49\), and \(y^2 + yz + z^2 = 64\), find the value of \(x + y + z\).

Answer: \sqrt{129}


Question: Minimize the function \( x^2 + y^2 \) subject to the constraints \((x-2)^2 + (y-3)^2 \le 4\) and \(x^2 = 4y\) using the Lagrange multiplier method.

Answer: 5


Question: Let \(A_n=\frac{(n+1)+(n+2)+(n+3)+\ldots+(n+n)}{n}\) and \(B_n=[(n+1)(n+2)(n+3)\ldots(n+n)]^{1/n}\). If \(\lim_{n\to\infty}\frac{A_n}{B_n}=\frac{ae}{b}\) where \(a,b\in \mathbb{N}\) and are relatively prime, find the value of \(a+b\).

Answer: 11


Question: Determine whether the statement "There are $n$ homomorphisms from the group $\mathbb{Z}/n\mathbb{Z}$ to the additive group of rationals $\mathbb{Q}$" is true or false.

Answer: False


Question: Determine the number of positive integer triples \((x, y, z)\) that satisfy the equation \(x^2 + 2021y^2 = z^2\).

Answer: \infty


Question: Does there exist a sequence of real-valued functions \( \{f_n\} \) with domain \( \mathbb{R} \) that is uniformly convergent on some subset of \( \mathbb{R} \) to a continuous function, where each \( f_n \) is continuous at points forming a proper dense subset of \( \mathbb{R} \)?

Answer: Yes


Question: Let $X$ be a projective surface over $\mathbb{C}$, and let $E$ be a rank $2$ vector bundle on $X$ such that a general global section of $E$ vanishes on a finite set of points (i.e., $\text{Codim} \, Z(s) = 2$ for a general $s \in H^0(X, E)$). Additionally, assume $\dim_{\mathbb{C}} H^0(X,E) \geq 2$. Is the vector bundle $E$ generically globally generated? (A vector bundle $E$ is generically globally generated if the evaluation map $\text{ev}: H^0(E) \otimes \mathcal{O}_X \rightarrow E$ is surjective on an open set $U \subseteq X$.)

Answer: Yes


Question: Find the minimum positive integer $k$ such that $a + nk$ is a prime number for all $n \in \{0, 1, 2, 3, 4, 5\}$.

Answer: 30


Question: Let $N$ be the product of $m$ distinct primes $P_1, P_2, \ldots, P_m$. Define the set $S$ as the set of all integers $K < N$ such that $N$ divides $K(K+1)$. Determine the size of the set $S$.

Answer: 2^m


Question: Given that $f'(0) = f''(0) = 1$, $f^{(12)}$ exists, and $g(x) = f(x^{10})$, find $g^{(11)}(0)$.

Answer: 0


Question: Find the norm of the linear functional \( l(x) = x(0) + \int_0^1 (2t-1)x(t) \, dt \) on the space of continuous functions \( C[0,1] \).

Answer: \dfrac{3}{2}


Question: Calculate the residue of the function \( f(z) = \sin\left(\frac{z^2+1}{z}\right) \) at the point \( z = 0 \).

Answer: J_1(2)


Question: Find the residue of the function \( f(z) = z^4 \sin(1/z) \) at \( z = 0 \).

Answer: \dfrac{1}{120}


Question: In a cyclic quadrilateral $ABCD$, it is given that $BC = BD = 1$ and the ratio of the sides $AD:AC:AB$ is $1:7:5$. Determine the value of $AD^2$.

Answer: \dfrac{1}{32}


Question: Evaluate the integral \( \int_1^\infty \frac{1}{x+x^3} \, dx \) and show that it equals \( \frac{\ln 2}{2} \).

Answer: \dfrac{\ln 2}{2}


Question: Determine the measure of the set \(S = \{w \in \{0,1\}^\omega \mid \text{for every } N \in \mathbb{N}, w \text{ has a prefix of the form } pp \text{ with } |p| \ge N \}\).

Answer: 0


Question: Evaluate the limit: \( \lim_{x \to \infty} x \cdot o\left(\frac{1}{x^2}\right) \)

Answer: 0


Question: Evaluate the infinite product: \[ \lim_{n \to \infty} \prod_{r=1}^n \left(1+ \frac{1}{a_r}\right), \] where \( a_1=1 \) and \( a_r=r(1+a_{r-1}) \) for \( r \geq 2 \).

Answer: e


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$ Choose the correct option: (a) \(\frac{1}{3}\) (b) \(\frac{1}{9}\) (c) \(\frac{1}{81}\) (d) 0.

Answer: B


Question: Compute the double integral \( I = \iint_R \left|\cos(2x)-\cos(y)\right|\mathrm{d}x\mathrm{d}y \) over the region \( R = [0,\pi]\times [0,\pi] \).

Answer: 8


Question: Given the joint probability density function \( f_{X,Y}(x,y) = \begin{cases} 2e^{-x-y} & 0<x<y \\ 0 & \text{otherwise} \end{cases} \), find the probability \( P(Y < 3X) \).

Answer: \dfrac{1}{2}


Question: Consider the sequence \((u_{n})_{n\geqslant 0}\) defined by \[ u_{n}=\dfrac{2}{\log^{2}\dfrac{p_{n+2}^{2}+3}{2}}\prod_{i=2}^{n+2}\dfrac{p_{i}}{p_{i}-2} \] where \(p_{k}\) is the \(k\)-th prime number. Determine if \(u_{0}\) is the greatest term of this sequence.

Answer: u_0


Question: Determine all real numbers \(x\) such that the series \( \sum_{k=1}^{\infty}(k-\sqrt{k^2+1})(x+2)^k \) is convergent.

Answer: [-3, -1)


Question: Evaluate the limit: $$ \lim _{x\to 0}\left(\frac{\ln \left(x\right)}{x}\cdot \:e^{\ln ^2\left(x\right)}\right). $$

Answer: -\infty


Question: If there is an equivalence of categories between categories \( C \) and \( D \), does it imply that there is also an equivalence of categories between \( D \) and \( C \)?

Answer: Yes


Question: Find a positive integer \( x \) such that the last three digits of \( 7^{7^x} \) are 007. Do not use trial and error.

Answer: 4


Question: Determine the value of \( m \) such that the equation \( y^2 = x^3 + m \) has no integral solutions.

Answer: 7


Question: Let $V$ be a 4-dimensional vector space, and let $T: V \to V$ be a linear map with characteristic polynomial $\chi_T = x^4 - x^3$. Prove that $V$ has $T$-invariant subspaces of dimensions 1, 2, and 3.

Answer: V


Question: Evaluate the infinite series: \[ \sum^{\infty}_{n=1}\int^{2(n+1)\pi}_{2n\pi}\frac{x\sin x+\cos x}{x^2} \, dx \]

Answer: \dfrac{1}{2\pi}


Question: Consider the set \( A = \{1, 2, 3, 4, 5\} \). A function \( f: A \to A \) is chosen at random from the set of all nonempty relations on \( A \). What is the probability that \( f \) is a bijective function such that \( f(x) \neq x \) for all \( x \in A \) and \( f(1) \neq 2 \)?

Answer: \dfrac{11}{40}


Question: Consider the polynomial \( P(x) = 5x^6 - ax^4 - bx^3 - cx^2 - dx - 9 \), where \( a, b, c, d \) are real numbers. If the roots of \( P(x) \) are in an arithmetic progression, find the value of \( a \).

Answer: 35


Question: Determine the number of values of \( b \) modulo 55 for which the congruence \( x^2 + x + b \equiv 0 \pmod{55} \) has exactly 2 solutions.

Answer: 7


Question: Find all polynomials \( p(x) \) such that \( p(q(x)) = q(p(x)) \) for every polynomial \( q(x) \).

Answer: x


Question: Mr. Noeth's BC Calculus class has 17 students who live in consecutive houses on the same street. He needs to select at least three out of every four consecutive students to take a makeup test. How many ways can Mr. Noeth select the students to take the makeup tests?

Answer: 345


Question: Given that $\nabla f(1, -4) = (3, -2)$, find $\frac{\partial h}{\partial y}(2, -1)$ for the function $h(x,y) = f(3y^2 - x, 2xy)$. Assume $f$ is $C^1$. Use the chain rule to express $\frac{\partial h}{\partial y}$ in terms of the partial derivatives of $f$ and evaluate it at the given point.

Answer: -26


Question: Evaluate the limit: $$\lim_{x\to \infty} \frac{5x-2x\sqrt{x^2+1}+\sqrt{4x^4+3}}{2-\sqrt{x^2+1}}$$ without using L'Hospital's Rule.

Answer: -5


Question: Consider the set of numbers \( \{1, 3, 7, 9, 11, 13, 17, 19\} \). Determine the number of subgroups of this set under multiplication modulo 20, and identify the generators of these subgroups.

Answer: 8


Question: Let $X$ and $Y$ be two simple objects in an abelian category with projective covers $P(X)$ and $P(Y)$. If $P(X)$ and $P(Y)$ are isomorphic, are $X$ and $Y$ necessarily isomorphic?

Answer: Yes


Question: Evaluate the integral of the function \( f(x,y,z) = 12xyz \) over the region \( D = \{ (x,y,z) \in \mathbb{R}^3 \mid 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1 \} \).

Answer: \dfrac{3}{2}


Question: Does there exist a continuous function that is not differentiable at any point?

Answer: Yes


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

Answer: \dfrac{3}{2} e^3


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

Answer: (-2, 0)


Question: Determine the number of roots (counting multiplicities) of the function \( f(z) = \cos(z) - 1 + \frac{z^2}{2} \) within the unit disk \( |z| < 1 \).

Answer: 4


Question: Evaluate the limit: \[ \lim \limits_{h\to \infty} \left(\frac{f(10+\frac{1}{h})}{f(10)}\right)^h \] given that \( f(10) = 5 \) and \( f'(10)=2 \).

Answer: e^{\frac{2}{5}}


Question: Find \( \lim_{n \to \infty} \sum_{k = 2n}^{10n} \frac{1}{k} \).

Answer: \ln 5


Question: Find the derivative of the expression \((Ax-b)^T(Ax-b)\) with respect to \(x\).

Answer: 2A^T(Ax - b)


Question: Consider the double integral \( \int_0^\infty \frac{2a^2}{x^2 + a^2} \left( \int_0^1 \cos(yx) \, dy \right) dx \). Can this integral be rewritten by changing the order of integration to \( \int_0^1 dy \int_0^\infty \frac{2a^2}{x^2 + a^2} \cos(yx) \, dx \)? Justify your answer based on the conditions for changing the order of integration.

Answer: Yes


Question: Determine whether the integral \( \int_{0}^{\infty} \frac{1}{e^x + e^{-x}} \, dx \) converges or diverges.

Answer: \dfrac{\pi}{4}


Question: Let $M$ be a nonsingular matrix over the field $\mathbb{F}_2$ with dimension $n$. If $MM^T = I$, where $I$ is the identity matrix, is it necessarily true that $M^TM = I$ as well? Provide a justification for your answer.

Answer: Yes


Question: Consider a polynomial $p : \mathbb R \to \mathbb R$ with $p'(x) > 0$ for all $x \in \mathbb R$. The function $p$ has exactly one real zero. Will the Newton method \[x_{n+1} = x_n - \frac{p(x_n)}{p'(x_n)}\] converge for all $x_0 \in \mathbb R$?

Answer: Yes


Question: Given that \( f(x) \) is a continuous function with \( f(0) = 1 \) and satisfies the equation \( f(7x) - f(x) = x \), find the value of \( f(42) \).

Answer: 8


Question: Determine whether there exist elements in the set \( B = \left\{ \frac{5x^2-y^2}{4} \mid x, y \text{ are odd positive integers} \right\} \) that are not in the set \( A = \left\{ \frac{5p^2-y^2}{4} \mid p, y \text{ are odd positive integers, } p \text{ is prime} \right\} \).

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\rightarrow\pi}\frac{x}{x-\pi}\int^x_\pi\frac{\cos t}{t} \, dt$$

Answer: -1


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \sum_{i=1}^{n}\sum_{j=1}^{i} \frac{j}{n^3}. \]

Answer: \dfrac{1}{6}


Question: Determine whether there exists a function that is holomorphic on the open unit disk \( \{z \in \mathbb{C} : |z| < 1 \} \) and continuous on the unit circle \( S^1 \), but not holomorphic on \( S^1 \).

Answer: Yes


Question: Determine the maximum size of a partition-free subset of the power set of \([n]\), denoted as \(2^{[n]}\). A subset is partition-free if it does not contain a partition of \([n]\).

Answer: 2^{n-1}


Question: Find the minimum value of the expression \(3x^2 - 2xy\) subject to the constraint \(\frac{x^2}{4} - y^2 = 1\), where \(x\) and \(y\) are real numbers.

Answer: 6 + 4\sqrt{2}


Question: Evaluate the integral \( \int \frac{2z-1}{z(z-1)} \, dz \) counterclockwise around the circle \( |z|=2 \).

Answer: 4\pi i


Question: Find the derivative of the trace function \( \text{Tr} \left( X|e^{itK} - X|^3 \right) \) with respect to \( t \) at \( t = 0 \), where \( K \) is a Hermitian matrix and \( X \) is a positive matrix.

Answer: 0


Question: In a finite Galois extension $L/K$ where $A$ is a discrete valuation ring (dvr) with fraction field $K$ of characteristic zero, and $B$ is the integral closure of $A$ in $L$, consider a prime $b$ of $B$. Let $e_b$ be the ramification index and $f_b$ the residue degree of $b$. Are $e_b$ and $f_b$ independent of the choice of $b$?

Answer: Yes


Question: Evaluate the integral \( \int_{\frac{1}{3\sqrt3}}^{3\sqrt3}\cfrac{1}{x^{4/3}+x^{2/3}}dx \).

Answer: \dfrac{\pi}{2}


Question: Let $C$ be a contour formed by the arc $y = x^2$ from $(0,0)$ to $(1,1)$, and a line segment from $(1,1)$ to $(0,1)$. Calculate the integral $\int_C f(z) \, dz$ where $f(z) = 2xy + i(-x^2 + y^2)$. What is the value of this integral?

Answer: -\dfrac{1}{3}


Question: Given a tetrahedron $ABCD$ with $D$ at the top, where $AB = 12$, $CD = 6$, and the shortest distance between the skew lines $AB$ and $CD$ is $8$. If the angle between these lines is $\frac{\pi}{6}$, find the volume of the tetrahedron $ABCD$. The answer is known to be $48$ cubic units.

Answer: 48


Question: Fifty identical dice are rolled. How many distinct outcomes are possible if each face (1 through 6) appears a positive even number of times?

Answer: 42504


Question: Calculate the limit: \[ \lim_{n\rightarrow\infty}\frac{n!}{\left(n+1/6\right)!} \]

Answer: 0