Question: Let $X$ and $Y$ be manifolds of dimensions $k$ and $l$ respectively, with $k > l$. Consider a smooth map $F : X \to Y$ and a regular value $y \in Y$. Does there exist a map $G : X \to \mathbb{R}^l$ such that $F^{-1}(y) = G^{-1}(0)$?

Answer: Yes


Question: A circle of radius 5 is centered at the point \( H(10,5) \). Tangents are drawn from the point \( A(0, 16) \) to the circle. Find the distance \( d \) between the x-axis intercepts of these tangents.

Answer: \dfrac{70}{3}


Question: Determine the most general form of the entire function $f(z)$ that satisfies the limit condition: \[ \lim_{z \to \infty} \frac{f(z)^2}{z^3} = 0. \]

Answer: f(z) = az + b


Question: Evaluate the integral \( \frac{1}{2\pi i}\int_C \frac{\sum_{n=0}^{15}z^n}{(z-i)^3}dz \), where \( C \) is the contour \(|z-i|=2\).

Answer: 52 + 60i


Question:
Consider the recurrence relation defined by:
\[ T(1) = 1 \]
\[ T(n+1) = T(n) + \lfloor \sqrt{n+1} \rfloor \text{ for all } n \geq 1 \]
Find the value of \( T(m^2) \) for \( m \geq 1 \). Choose the correct expression from the following options:

1. \( \frac{m}{6} (21m - 39) + 4 \)
2. \( \frac{m}{6} (4m^2 - 3m + 5) \)
3. \( \frac{m}{2} (m^{2.5} - 11m + 20) - 5 \)
4. \( \frac{m}{6} (5m^3 - 34m^2 + 137m - 104) + \frac{5}{6} \)

Answer: 2


Question: Evaluate the integral \( \int_0^\infty \int_x^\infty \frac{1}{y} e^{-y/2} \, dy \, dx \).

Answer: 2


Question: Let $G$ be a subgroup of $(\mathbb{R}^2,+)$ such that $\mathbb{R}^2/G \cong \mathbb{Q}$. Is $G$ dense in $\mathbb{R}^2$?

Answer: Yes


Question: Assume that the time between emissions from a radioactive source is independent and exponentially distributed with rate one. Each emission is detected by a Geiger counter with probability \( p \). Compute the density of the distribution of the time between detections of particles and identify the distribution by name.

Answer: p e^{-pt}


Question: Find \( \varepsilon > 0 \) such that the map \( F \colon NM \cap (\mathbb{R}^2 \times B_{\varepsilon}(0)) \rightarrow M_{\varepsilon} \colon (p,w) \mapsto p + w \) is a diffeomorphism, where \( M = \{(x,x^2) : x \in \mathbb{R}\} \) and \( M_{\varepsilon} = \bigcup_{p \in M} B_{\varepsilon}(p) \). The normal bundle is defined as \( N M = \{ (p,w) \in \mathbb{R}^2 \times \mathbb{R}^2 : p \in M \text{ and } w \in (T_{p} \, M)^{\bot} \} \).

Answer: \dfrac{1}{2}


Question: Evaluate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) where \( \mathbf{F}(x,y,z) = x\,\mathbf{i} + xy\,\mathbf{j} + x^2yz\,\mathbf{k} \) and \( C \) is the elliptical path given by \( x^2 + 4y^2 - 8y + 3 = 0 \) in the xy-plane, traversed once counter-clockwise.

Answer: \dfrac{\pi}{2}


Question: In a circle with center \(O\) and radius \(\sqrt{5} + 1\), a secant \(PBA\) and a tangent \(PE\) are drawn from an external point \(P\). Given that \(PE = AB = 2\), calculate the measure of \(\angle POB\).

Answer: 18^\circ


Question: Determine the value of $a$ such that the function \( f(z) = \int_{1}^{z} \left(\frac{1}{w} + \frac{a}{w^3}\right)\cos(w)\,\mathrm{d}w \) is single-valued in the complex plane.

Answer: 2


Question: Let $S$ be a semisimple linear algebraic group over a field $K$ with $\text{char } K = 0$. If $H$ is a closed subgroup of $S$ over $K$, is $H$ semisimple?

Answer: No


Question: Calculate \( \iint_D (x^2 + y^2) \, dx \, dy \) by changing variables, where \( D \) is the region in the first quadrant of the \( xy \)-plane bounded by the curves \( xy = 1 \), \( xy = 3 \), \( x^2 - y^2 = 1 \), and \( x^2 - y^2 = 4 \). Use the transformation \( (x, y) = T(u, v) \) defined by \( u = xy \) and \( v = x^2 - y^2 \) to map \( D \) to a rectangle \( E \) in the \( uv \)-plane with sides parallel to the axes. Compute the Jacobian \( \frac{\partial(x, y)}{\partial(u, v)} \) and evaluate the integral.

Answer: 3


Question: Two independent losses follow an exponential distribution with a mean of 1. Calculate the probability that either of the losses is more than twice the other.

Answer: \dfrac{2}{3}


Question: Find the area of the largest triangle that can be formed in the first quadrant by the x-axis, the y-axis, and a tangent line to the graph \(y = e^{-x}\).

Answer: \dfrac{2}{e}


Question: Determine the number of distinct equivalence classes in \( \mathbb{Z}_n \) under the equivalence relation defined by: for \(a, b \in \mathbb{Z}_n\), \(a \sim b\) if there exists \(k \in U_n = \mathbb{Z}_n^{\times}\) such that \(a = kb\).

Answer: \tau(n)


Question: Find the generating function for the sequence \( \binom{6}{1}, 2\binom{6}{2}, 3\binom{6}{3}, \ldots, 6\binom{6}{6}, \ldots \).

Answer: 6x(1 + x)^5


Question: Evaluate the limit \( \lim_{x \to 0} \frac{(x-\sin x)^2}{(e^{x^2}-1)x^4} \) using L'Hospital's rule.

Answer: \dfrac{1}{36}


Question: Consider a convex but non-smooth function \( f : \mathbb{R}^2 \to \mathbb{R} \). For a point in \( \mathbb{R}^2 \), if the function increases in the directions \((1,0)\), \((-1,0)\), \((0,1)\), and \((0,-1)\), is this point a local optimum? Justify your answer.

Answer: Yes


Question:
Are the following two forms of the Nelson Siegel formula equivalent?

\[ s_{m}(\beta )=\beta _{0}+\beta _{1}\frac{1-e^{\frac{-m}{\tau _{1}}}}{\frac{m}{\tau _{1}}}+\beta _{2}\left ( \frac{1-e^{\frac{-m}{\tau _{1}}}}{\frac{m}{\tau _{1}}}-e^{\frac{-m}{\tau _{1}}} \right) \]

\[ s_{m'}(\beta )=\beta _{0}+\beta _{1}\frac{1-e^{-m\tau _{1}}}{m\tau _{1}}+\beta _{2}\left ( \frac{1-e^{-m\tau _{1}}}{m{\tau _{1}}}-e^{-m\tau _{1}} \right) \]

Answer: Yes


Question: Let \( f : \mathbb{R}^{n+1} \to \mathbb{R}^n \) be a smooth function such that \( S = f^{-1}(\{0\}) \) is a path-connected 1-surface, meaning \( \text{rank} \, Df(a) = n \) for all \( a \in S \). Is it true that there exists a smooth function \( g: I \to \mathbb{R}^{n+1} \) such that \( g(I) = S \) and \( \text{rank} \, Dg(a) = 1 \) for all \( a \in I \), where \( I \) is an open connected subset of the real line?

Answer: Yes


Question: Evaluate the Legendre symbol \(\left(\frac{w}{p}\right)\), where \(w\) is a primitive root modulo \(p\).

Answer: -1


Question: Let $p$ be a prime number. Determine the number of incongruent solutions to the congruence \( x^{p^5} - x + p \equiv 0 \pmod{p^{2020}}.\)

Answer: p


Question: Calculate the flux of the vector field \( \mathbf{u} = x \hat{\imath} + y \hat{\jmath} + z \hat{k} \) through the surface of the ellipsoid \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \), where \( a > b > c > 0 \).

Answer: 4\pi abc


Question: Let $f$ be a differentiable function on $\mathbb{R}$ such that $f(0) = 0$ and $\lim_{x \to \infty} f(x) = 0$. Prove that there exists a point $x_0$ in $\mathbb{R}$ such that $f'(x_0) = 0$.

Answer: f'(x_0) = 0


Question: Find the limit of the sequence \( \frac{1}{2}+\frac{3}{2^{2}}+\frac{5}{2^{3}}+\cdots+\frac{2n-1}{2^{n}} \) as \( n \to \infty \).

Answer: 3


Question: Determine whether the statement is true or false: If $M$ and $N$ are cyclic $\mathbb{Z}$-modules, then $M \otimes_\mathbb{Z} N \cong \mathbb{Z}$.

Answer: B


Question: Determine the cardinality of the set \( X = \{ A \subseteq \mathbb{N} : A \text{ is finite} \} \).

Answer: \aleph_0


Question: Determine the value of \( \lim_{N \to \infty} \frac{f(N)}{N} \), where \( f(N) \) is the number of integers \( 1 \leq n \leq N \) such that the number of divisors of \( n \), denoted \( \tau(n) \), is greater than \( \ln{N} \).

Answer: 0


Question: Evaluate the surface integral \( \int_S \mathbf{F} \cdot \mathbf{n} \, dS \) where \( \mathbf{F}(x,y,z) = y^2 \hat{i} - y \hat{j} + xyz \hat{k} \) and \( S \) is the curved surface of the cylinder \( x^2 + y^2 = 4, \: 0 \leq z \leq 3 \). The vector \( \mathbf{n} \) is directed away from the \( z \)-axis.

Answer: -12\pi


Question: Determine the necessary and sufficient condition on \( \lambda \in R \) such that the ideal generated by the polynomials \( P_1(x) = x^2 - 1 \) and \( P_2(x) = x^3 + \lambda \) in \( R[x] \) is maximal.

Answer: \lambda = \pm 1


Question: Given two independent random variables $X$ and $Y$ with marginal probability density functions $f_X(x) = 2x$ for $0 \leq x \leq 1$ and $f_Y(y) = 1$ for $0 \leq y \leq 1$, calculate the probability $P\left(\frac{Y}{X} > 2\right)$. Provide your answer as a single numerical value.

Answer: \dfrac{1}{12}


Question: Determine the Euler characteristic of a Y-shaped pipe, which has three openings that converge together.

Answer: -1


Question: Is it possible to simulate from a joint distribution containing both continuous and discrete variables using Gibbs sampling?

Answer: Yes


Question:
Jenny visits a restaurant \(\frac{a}{2}\) times, spending \(b^2\) hundreds of dollars each time. Given the system of equations:
\[\begin{cases}
3^a - 2^{b^2} = 77 \\
3^{\frac{a}{2}} - 2^{\frac{b^2}{2}} = 7 \\
\end{cases}\]
Calculate the total amount of money Jenny spent. Choose the correct answer from the following options:
\[\begin{array}{ll}
1.&\text{600 dollars}\\
2.&\text{500 dollars}\\
3.&\text{400 dollars}\\
4.&\text{300 dollars}\\
\end{array}\]

Answer: C


Question: Determine the value of the p-adic limit: \[ \lim_{n \to \infty} p^{p^n} \] Is it 0 or 1?

Answer: 0


Question: Let \( f \) be a continuous function on the interval \([0,1]\) such that \( \int_0^1 f(x) \, dx = 0 \). Prove that there exists a point \( c \in (0,1) \) such that \( f(c) = \int_0^c f(x) \, dx \).

Answer: c


Question: Find the limit, if it exists: $$\lim_{x\to\infty } \left(\sqrt{e^{2x}+e^{x}}-\sqrt{e^{2x}-e^{x}}\right)$$

Answer: 1


Question: Given that $a, b, c, d$ are positive integers satisfying \( \binom{a+b}{a} = \binom{c+d}{c} \) and \( a = c \), do these conditions imply that \( b = d \)? Justify your answer.

Answer: Yes


Question: Does there exist a function \( f: [0,1] \to \mathbb{R} \) such that its set of points of discontinuity, \( D(f) \), is an uncountable set containing no rational numbers?

Answer: Yes


Question: Solve the inequality $n \leq 8\lg n$ and find the largest integer $n$ that satisfies it.

Answer: 43


Question: Determine whether there exists a prime number that cannot be paired with another prime such that their difference is a power of 2. If such a prime exists, provide an example.

Answer: 2


Question: Given the curve \( A \) defined by the equation \( 9x + 27y - \frac{10}{81} (x+y)^3 = 0 \), where \( A \) is the graph of an infinitely differentiable function \( f \) near \((0,0)\), find the value of \( f''(0) \).

Answer: 0


Question: Evaluate the infinite series: \[ S = \sum_{n=1}^{\infty} \left( \frac{2}{n} - \frac{4}{2n+1} \right) \]

Answer: 4 - 4\ln 2


Question: Find the set $A = \{x \in \mathbb{R} \mid x^2 + 3x - 9 + 2^{x^2 + 3x - 2} + \log_2{(x^2 + 3x - 2)} = 0\}$.

Answer: \{-4, 1\}


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a \( C^2 \) function. Consider the function \( F: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( F(x,y) = \left(xf\left(\sqrt{x^2+y^2}\right), yf\left(\sqrt{x^2+y^2}\right)\right) \). Determine whether there exists a function \( g: \mathbb{R}^2 \to \mathbb{R} \) such that \( \nabla g = F \).

Answer: Yes


Question: Calculate \( \lim\limits_{x\to\infty} \frac{\log(x!)}{x\log(x)} \). Assume the logarithm is base \( e \).

Answer: 1


Question: Determine the number of values of $x$ for which the function $f(x) = \cos x + \cos(\sqrt{2}x)$ attains its absolute minimum.

Answer: 0


Question: Consider a connected (possibly reducible) proper algebraic variety $X$ over $\mathbb{C}$ of pure dimension $n$. Determine if it is true that $H^{2n}(X,\mathbb{Q})\cong \mathbb{Q}^r$, where $r$ is the number of irreducible components of $X$, and the left-hand side is the singular cohomology with respect to the analytic topology.

Answer: Yes


Question: Calculate the work done to pump water out of a conical tank with a top radius of 3 ft and a height of 5 ft to a window located 10 ft above the top of the tank. The tank is completely full of water. Assume the density of water is 62.4 lb/ft³.

Answer: 10530\pi


Question: Determine the singular value of a Householder reflector, which is given by the matrix $I - 2ww^*$, where $w$ is a unit vector.

Answer: 1


Question: Given the joint density function \( f_{X,Y}(x,y) = \begin{cases} 8xy & \text{for } 0<y<x<1 \\ 0 & \text{otherwise} \end{cases} \), determine the conditional probability \( P\left( X>\frac{3}{4} \big| Y = \frac{1}{2}\right) \).

Answer: \dfrac{7}{12}


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

Answer: \dfrac{8}{5}


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

Answer: \infty


Question: Let $R$ and $T$ be linear operators on a Hilbert space $H$. Prove that if $(Rv,v) = (Tv,v)$ for all vectors $v$ in $H$, then $R = T$.

Answer: R = T


Question: Let \(\phi\) be an entire function such that \(|\phi(z)| \leq e^{|z|}\) for all complex numbers \(z\). Prove that there exists a constant \(c > 0\) such that \(|\phi'(z)| \leq c e^{|z|}\) for all \(z\).

Answer: e


Question: Let \( f: \Bbb R \to [0,\infty) \) be a continuous function such that \( g(x) = (f(x))^2 \) is uniformly continuous. Determine which of the following statements is always true: \( \text{A. } f \text{ is bounded} \), \( \text{B. } f \text{ may not be uniformly continuous} \), \( \text{C. } f \text{ is uniformly continuous} \), \( \text{D. } f \text{ is unbounded} \).

Answer: C


Question: Evaluate the integral \( \int_{[0,1]^n} \min(x_1,\ldots,x_n) \, dx_1\cdots dx_n \).

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


Question:
Let \( f:[0,1] \rightarrow \mathbb{R} \) be a continuous function such that \( \int_0^1 f(x) \, dx = 0 \). Compute the limit:
\[ \lim _{t\to 0} \frac{1}{t} \int_0^1 (f(x+t)-f(x))x \, dx \]

Answer: f(1)


Question: Let \( x_{1}, x_{2}, \ldots, x_{n} \) be independent and identically distributed (IID) random variables uniformly distributed on \([0, \theta]\). Find the probability density function (PDF) of the random variable \( \frac{x_{(n)}}{\theta} \), where \( x_{(n)} \) is the maximum order statistic.

Answer: n y^{n-1}


Question: Find all natural numbers $k$ such that $4957^k \equiv 1 \pmod{7007}$, where $7007 = 7 \cdot 7 \cdot 11 \cdot 13$. Use the Chinese Remainder Theorem to solve the problem.

Answer: 210


Question: Evaluate \( \lim_{x \to 0} \left( \frac{1}{{\sin^2 x}} - \frac{1}{x^2} \right) \).

Answer: \dfrac{1}{3}


Question: Given that $x + y + z = 1$ and $0 \le x, y, z \le 1$, find the maximum value of the expression $xy + yz + xz - 2xyz$. Provide your answer as a single number.

Answer: \dfrac{7}{27}


Question: Determine which expression is larger as \( x \to \infty \): \( x^{\log(x)} \) or \( (\log(x))^x \).

Answer: (\log(x))^x


Question: Consider a compactly supported function $f$ in a domain $\Omega \subset \mathbb{R}^3$. Suppose $u$ satisfies the Poisson equation $\Delta u = f$ in $\Omega$ and $D^{\alpha}u = 0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| \geq 0$. Is it necessary that $f$ is zero?

Answer: Yes


Question: Evaluate the double integral \( \iint |\cos(x+y)| \, dx \, dy \) over the region \( 0 \leq x \leq \pi \) and \( 0 \leq y \leq \pi \) using the transformation \( x = u-v \) and \( y = v \).

Answer: 2\pi


Question:
Calculate the Lebesgue integral of the function \( f:(0,\infty)\rightarrow [0,\infty) \) defined by:
\[
f(x)=\begin{cases}
|\sin(x)| & \text{if } x\in \mathbb{Q},\\
4x & \text{if } x\in[0,1]\backslash \mathbb{Q},\\
0 & \text{if } x\in (1,\infty)\backslash \mathbb{Q}
\end{cases}
\]
Determine \( \int_{(0,\infty)} f \, d\lambda \).

Answer: 2


Question: Determine whether the following statement is true or false: If $U \subset \mathbb{R}^m$ is an open set and $F: U \rightarrow \mathbb{R}^n$ is a smooth map with $m < n$, then $F(U)$ has measure zero. Justify your answer.

Answer: True


Question:
Consider the diffusion equation:
\[ \frac{\partial ^2 u}{\partial x^2} = \frac{\partial u}{\partial t},\; u = u(x,t),\; u(0,t) =0 = u( \pi ,t),\; u(x,0) = \cos x \sin 5x \]
Which of the following is a solution to the equation?

1. \( \frac{e^{-36t}}{2} [\sin 6x + e^{20t} \sin 4x] \)
2. \( \frac{e^{-36t}}{2} [\sin 4x + e^{20t} \sin 6x] \)
3. \( \frac{e^{-20t}}{2} [\sin 3x + e^{15t} \sin 5x] \)
4. \( \frac{e^{-36t}}{2} [\sin 5x + e^{20t} \sin x] \)

Answer: 1


Question: Find the last two digits of the number $2^{11212}(2^{11213}-1)$. What are the last two digits of this perfect number?

Answer: 36


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

Answer: 0


Question: Find the remainder when \(32^{32^{32}}\) is divided by 7.

Answer: 4


Question: Evaluate the series \( \sum_{k=1}^{\infty} e^{-\pi k^2} \left( \pi k^2 - \frac{1}{4} \right) \).

Answer: \dfrac{1}{8}


Question: Evaluate the limit: $$\lim_{x\to2}\dfrac{\sqrt{x^2+1}-\sqrt{2x+1}}{\sqrt{x^3-x^2}-\sqrt{x+2}}$$

Answer: \dfrac{4\sqrt{5}}{35}


Question: Evaluate the limit: \[ \lim_{x\to+\infty} \frac{3x^3+x\cos{\sqrt{x}}}{x^4\sin{\frac{1}{x}}+1} \]

Answer: 3


Question: Evaluate the double integral \( \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}e^{-5x^2-5y^2+8xy}dxdy \).

Answer: \dfrac{\pi}{3}


Question: Determine the values of \(x\) for which the power series \(\sum_{k=1}^\infty \frac{x^k}{k}\) converges.

Answer: [-1, 1)


Question: Let $N$ be a type $II_{1}$ factor. Determine whether there exists a diffuse abelian subalgebra of $N$ that is not maximal abelian.

Answer: Yes


Question: Determine if there exists a non-integral number X, not between -1 and 1, such that the sequence of distances from X^n to the nearest integer converges to 0 as n approaches infinity.

Answer: Yes


Question: Find another element of order 2 in the dihedral group \(D_{2p}\) besides the element \(b\), where \(p\) is a prime greater than 2.

Answer: ab


Question: Evaluate the limit: \[ \lim_{n\to\infty} \sum_{k=1}^{n/2} \left(\frac{en}{2k}\right)^{2k} \frac{1}{\sqrt{\pi k}}^{\frac{3n}{\log_2{n}}} \]

Answer: 0


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

Answer: 7


Question: Determine whether every simply connected open set in \(\mathbb{R}^2\) is homeomorphic to an open ball.

Answer: Yes


Question:
Consider \( \mathbb{R} \) equipped with the topology generated by the sets of the form \((a,b)\) and \((a,b) \cap \mathbb{Q}\). Determine which of the following statements is correct:

1. The space is normal.
2. The space is regular.
3. \( \mathbb{R} \setminus \mathbb{Q} \) is dense.
4. \( \mathbb{Q} \) is dense.

Answer: 4


Question: Suppose $f$ is a univariate polynomial of degree at most $2k-1$ for some integer $k \geq 1$. Let $f^{(m)}$ denote the $m$-th derivative of $f$. If $f$ and $f^{(m)}$ have $k$ distinct common roots, is it true that $f$ must be the zero polynomial? Assume $m < k$ is a positive integer.

Answer: Yes


Question: Let $\mathcal{L}$ be an $n$-dimensional real Euclidean space. Determine the maximum number $r$ such that there exist $r$ vectors $\beta_1, \beta_2, \ldots, \beta_r$ in $\mathcal{L}$ satisfying $$(\beta_i, \beta_j) < 0, \quad 1 \leq i < j \leq r.$$

Answer: n + 1


Question: Find the radius of convergence of the series \( \sum_{k=1}^{\infty} \frac{k^3 (x-3)^{k!}}{3^{k!}} \).

Answer: 3


Question: Let \( a \) be any odd positive integer and \( n \) be an integer greater than 5. Determine the smallest possible integer \( N \) such that \( a^N \equiv 1 \pmod{2^n} \).

Answer: 2^{n-2}


Question: Compute the work done by the vector field \( F(x,y,z) = xi + yj + zk \) on a particle moving along the curve \( C \) given by the parametric equations \( r(t) = (1+2\sin(t))i + (1+5\sin^2(t))j + (1+4\sin^3(t))k \) for \( 0 \leq t \leq \pi/2 \). The work done is given by the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \).

Answer: \dfrac{67}{2}


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a function in $C^{\infty}(\mathbb{R})$. Define the graph of $f$ as $\text{Graph}(f) := \{(x,f(x)):x\in \mathbb{R}\}$. Let $A(f) := \bigcup\limits_{n=0}^{\infty} \text{Graph}(f^{(n)})$. If $g = f^{(n)}$, then $A(g) \subset A(f)$. Is the converse true? That is, if $A(g) \subset A(f)$, does it imply that $g = f^{(n)}$ for some $n \geq 0$?

Answer: Yes


Question: Determine for which values of \( \alpha \) the series \( \sum_{n=1}^{\infty} \left(e - \left(1 + \frac{1}{n}\right)^n\right)^\alpha \) converges.

Answer: \alpha > 1


Question:
Compute the order of the abelian group given by the presentation:
\[ \langle x_{1}, x_{2}, x_{3} \mid x_{1}+3x_{2}, x_{1}-2x_{2}, -x_{1}+x_{3} \rangle. \]

Answer: 5


Question: Verify the calculation of \( E[E[Y|X]^2] \) given the joint probability density function \( f(x,y) = 2\exp(-x-y) \) for \( 0 < x < y < \infty \). The conditional expectation is \( E[Y|X] = x + 1 \). Calculate \( E[E[Y|X]^2] = \int_0^\infty 2(x+1)^2\exp(-2x)\,dx \). Is the value \( -7/2 \) correct?

Answer: \dfrac{5}{2}


Question: Evaluate the expression \( (\sqrt{i})^i \).

Answer: e^{-\pi/4}


Question: Consider a normed vector space $X$ and a closed subspace $Y$ of $X$. Suppose $Y$ is second countable and the quotient space $X/Y$ is also second countable. Is it necessarily true that $X$ is second countable? Provide a proof or counterexample to support your answer.

Answer: Yes


Question:
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0)=0 \) and \( f^{\prime}(x)+2 f(x)>0 \) for all \( x \in \mathbb{R} \). Which one of the following statements is true?\
\(\text{A. } f(x)>0, \text{ for all } x>0 \text{ and } f(x)<0, \text{ for all } x<0\)\
\(\text{B. } f(x)<0, \text{ for all } x \neq 0\)\
\(\text{C. } f(x)>0, \text{ for all } x \neq 0\)\
\(\text{D. } f(x)<0, \text{ for all } x>0 \text{ and } f(x)>0, \text{ for all } x<0\)

Answer: A


Question:
Solve for the constant \( C \) in the equation:
\[
1 = C\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+y^2)}dydx
\]

Answer: \dfrac{1}{\pi}


Question:
Consider the following Cartesian diagram of morphisms of algebraic varieties, where all varieties are proper surfaces and the maps are finite and flat:
$$\begin{array}{ccc}
A & \to^\alpha & B \\
\downarrow^\beta & & \downarrow^\gamma \\
C & \to^\delta & D
\end{array}$$
Suppose there is a curve $Y \subset C$ that is irreducible and nonsingular. Define $Z = \delta(Y)$, $X = \gamma^{-1}(Z)$, and $W = \beta^{-1}(Y)$. Is the following diagram, obtained by restricting all the morphisms in the previous diagram, also Cartesian?
$$\begin{array}{ccc}
W & \to^\alpha & X \\
\downarrow^\beta & & \downarrow^\gamma \\
Y & \to^\delta & Z
\end{array}$$

Answer: Yes


Question: Given the equation \(x^3 - x^2 - x + 0.5 = 0\), use the substitution \(x^3 = y\) to transform the equation and find the sum of the cubes of the roots \(\alpha^3 + \beta^3 + \gamma^3\) of the resulting equation.

Answer: \dfrac{5}{2}


Question:
Let $X$ and $Y$ be continuous random variables with the joint probability density function:
\[ f_{X,Y}(x,y) = \begin{cases} \frac{10-x y^2}{64} & \text{for } 2 \leq x \leq 10 \text{ and } 0 \leq y \leq 1 \\ 0 & \text{otherwise} \end{cases} \]
Compute the expected value of $Y$, denoted as $\mathsf{E}[Y]$. \( \mathsf{E}[Y] = \boxed{?} \)

Answer: \dfrac{7}{16}


Question: Verify the claim that for every power \(x \in \mathbb{Z}_+\), each \(n^{th}\) column \((r_n)\) in the factorial base representation of \(a^x\) has a repeating pattern with length \((n+1)!\).

Answer: (n+1)!


Question: Let \( G \) be a finite non-abelian group of order 39. Determine the number of subgroups of order 3 in \( G \).

Answer: 13


Question: Let \( f:[a,b]\to \mathbb{R} \) be a function such that for every \( x\in[a,b] \) there exists a \( y \in [a,b] \) such that \( |f(y)|<\frac{1}{2}|f(x)| \). What is the infimum of \( |f(x)| \) on \( [a,b] \)?

Answer: 0


Question: Suppose that $f$ is a continuous map from $\mathbb{R}$ to $\mathbb{R}$, which satisfies $f(f(x)) = x$ for each $x \in \mathbb{R}$. Does $f$ necessarily have a fixed point?

Answer: Yes


Question: Determine whether there exists a positive number $x$ such that the measure of goodness $G(x) = \infty$. A number is defined as "good" if $G(x) = \infty$. Provide a justification for your answer.

Answer: Yes


Question: Find all positive integer solutions \((x, y, p)\) such that \(x^2 - 1 = y^p\), where \(p\) is a prime number.

Answer: (3, 2, 3)


Question: Given a 10-digit panel and a door code of length $n$, determine the length of the shortest sequence of digits that tests all possible combinations of the code, considering that the code can start at any point in the sequence.

Answer: 10^n + n - 1


Question: Determine the number of zeros of the function \( f(z) = 2 + z^4 + e^{iz} \) within the set \( \{z : |z| \leq r, \Im z \geq 0\} \) for any \( r > 2 \).

Answer: 2


Question: Let $V$ be a finite-dimensional vector space over a field $K$ with $\dim_K(V) \geq 1$. Suppose $M$ is a non-empty proper subset of $V$, i.e., $\emptyset \neq M \subsetneq V$. Can $M$ exist such that its annihilator $M^0 = V$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}\,\ln(x-\ln(x))$$

Answer: \infty


Question: Find \( \lim_{n\rightarrow\infty} \left(e^{n+\frac{1}{n}}-e^n\right) \).

Answer: \infty


Question: Calculate the surface integral of the portion of the paraboloid \(x^2+y^2=2ay\) that is cut by the plane \(y=a\). Express your answer in terms of \(a\).

Answer: \dfrac{2\pi a^2}{3} \left(3\sqrt{3} - 1\right)


Question: Find the period of the function that satisfies the equation \( f(x-2) + f(x+2) = f(x) \).

Answer: 12


Question: Find the last two digits of the sum of the eighth powers of 100 consecutive integers.

Answer: 30


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables, each uniformly distributed on the interval $[0,1]$. Calculate the conditional expectation $\mathbb{E}[X_1 \mid X_1 + X_2 + \cdots + X_n = x]$. Express your answer in terms of $n$ and $x$.

Answer: \dfrac{x}{n}


Question: Find the minimal polynomial over \( \mathbb{Q} \) for \( \omega = \cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} + i\frac{1}{2} \).

Answer: x^4 - x^2 + 1


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

Answer: \infty


Question: Find the radius of convergence for the series \( \sum \frac{[1\cdot 3 \cdots (2n-1)]^2}{2^{2n}(2n)!}x^n \).

Answer: 4


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

Answer: \dfrac{4}{e}


Question: Evaluate the integral: $$\large \int_{2016}^{3\cdot 2016}\frac{\sqrt[5]{3\cdot 2016-x} }{\sqrt[5]{3\cdot 2016-x}+\sqrt[5]{x-2016}}\mathrm dx$$

Answer: 2016


Question: Evaluate the line integral \( \oint_C F \cdot n \, ds \) to find the flux through the unit circle \( C \), where \( C \) is oriented counterclockwise and the vector field is given by \( F = (x^2, xy) \).

Answer: 0


Question:
Determine the largest possible natural number N such that when 85! is expressed in base N, it ends with exactly 20 trailing zeros. Choose from the following options:
1. 80
2. 160
3. 240
4. 200
5. None of these.

Answer: E


Question: Evaluate the double integral \( \int_0^1 \int_0^2 x^2 y e^{xy} \, dx \, dy \).

Answer: 4


Question: Determine whether the function \( e^{3x^{2}} + \frac{1}{1+3x^2} - 2\cos(x^2) \) is \( o(x^3) \) as \( x \to 0 \).

Answer: Yes


Question: Compute the integral \( \int_S \omega \) over the set \( S = \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2=1, z\geq 0\} \), where the differential form \( \omega \) is defined as follows: \( \omega = \frac{dydz}{x} \) when \( x \neq 0 \), \( \omega = \frac{dzdx}{y} \) when \( y \neq 0 \), and \( \omega = \frac{dxdy}{z} \) when \( z \neq 0 \).

Answer: 2\pi


Question: Determine the integral closure of the ring $k[t^2, t^3]$ in the ring $k[t]$, where $k$ is a field.

Answer: k[t]


Question: Let $f_{ij} : \mathbb{R}^m \to \mathbb{R}$ be smooth functions for $1 \leq i, j \leq n$. Suppose that for any $x \in \mathbb{R}^m$, the matrix $M_x = (f_{ij}(x))$ is nonsingular. Is it true that the inverse matrix $M_x^{-1}$ is composed of smooth functions with respect to $x$?

Answer: Yes


Question: Given three lines $L, M, N$ in $\mathbb{P}^4$, which are not in one hyperplane and do not pairwise intersect, calculate the dimension of the intersection $\dim(\langle L,M \rangle \cap N)$. Use the dimension of intersection theorem for projective spaces to find your answer.

Answer: 0


Question: In the ring \( \mathbb{Z}_n \) where \( n = p^a q^b \) and \( p < q \) are primes, let \([x] \in \langle [p] \rangle \setminus \langle [pq] \rangle\) and \([y] \in \langle [q] \rangle \setminus \langle [pq] \rangle\). Is it true that \( \langle [x] \rangle + \langle [y] \rangle = \mathbb{Z}_n \)?

Answer: Yes


Question: Use the Hopf fibration \(\eta : S^3 \rightarrow S^2\) with fibre \(S^1\) to calculate \(\pi_{3} (S^2)\).

Answer: \mathbb{Z}


Question:
For a given integer \( n \), determine the integer \( m \) such that:
\[ 10 \uparrow \uparrow m < a(n) < 10 \uparrow \uparrow (m+1) \]
where the sequence \( a(n) \) is defined recursively by:
\[ a(1) = 3! \]
\[ a(n+1) = a(n)! \quad \text{for all } n \geq 1 \]

Answer: n - 1


Question: Find the cumulative distribution function (CDF) of the probability density function (PDF) given by \( f(x) = e^{(x-e^x)} \) for \( x \in \mathbb{R} \).

Answer: 1 - e^{-e^x}


Question:
In the context of solving the initial-value problem for Burgers' equation given by:
\[
\begin{cases} u_{t}+\left(\frac{u^{2}}{2}\right)_{x}=0 &\mbox{in } \mathbb{R}\times(0,\infty) \\
u=g & \mbox{on } \mathbb{R}\times\{t=0\}, \end{cases}
\]
with initial data
\[
g(x)=\begin{cases} 1 &\mbox{if } x\le 0 \\
1-x & \mbox{if } 0\le x\le 1, \\
0 & \mbox{if } x\ge 1. \end{cases}
\]
The weak solution for \(t\ge 1\) is given by:
\[
u(x,t)=\begin{cases} 1 &\mbox{if } x\le s(t) \\
0 & \mbox{if } x\ge s(t)\end{cases}
\]
where \(s(t) = \frac{1+t}{2}\). Using the Rankine-Hugoniot condition, verify that the constant \(c\) in \(s(t) = \frac{1}{2}t + c\) is \(\frac{1}{2}\).

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: If $q$ is a prime number and $w = \frac{1+q+q^2}{3}$ is an integer, then $w$ is composite if and only if $w$ has a prime divisor less than $\sqrt{w}$ and congruent to $1$ modulo $6$.

Answer: True


Question: Given that events occur as a Poisson process with parameter \( \lambda \) in a fixed size interval, and \( Y \) is a random variable defined as \( Y = \frac{\text{count of intervals with 0 events}}{\text{total count of intervals}} \), where \( E(Y) = e^{-\lambda} \). If an observed value of \( Y \) is \( y \), estimate \( \lambda \) as a function of \( y \).

Answer: -\ln(y)


Question: Calculate the Linear Minimum Mean Square Error (LMMSE) of $Y$ given $\theta$, where $\theta$ is uniformly distributed over $[0,\pi]$ and $Y = \cos(\theta)$. Compute the covariance $\text{Cov}(Y, \theta)$ and verify the result. The expected answer is $-\frac{2}{\pi}$. Provide a detailed calculation to support your answer.

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


Question: Consider two regular maps between affine algebraic sets over \(\mathbb{C}\): \(f:X\to Y\) and \(g:X\to Z\). Suppose \(Z\) is normal, \(g\) is onto, and for every \(x_1,x_2\), \(g(x_1)=g(x_2)\) implies \(f(x_1)=f(x_2)\). Does this imply that \(f\) factors through \(g\), i.e., \(f=hg\) for some \(h:Z\to Y\)? Justify your answer.

Answer: Yes


Question: Determine the Lebesgue measure of the set of numbers in the interval [0,1] such that each digit from 0 to 9 appears at least once in their decimal representation after the decimal point.

Answer: 1


Question: Given the matrix \( A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \\ -1 & 1 \end{bmatrix} \) and vectors \( u = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \) and \( v = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} \) with entries in \( \mathbb{R} \), where \( u_1^2 + u_2^2 = v_1^2 + v_2^2 + v_3^2 = 1 \), suppose \( Au = v\sqrt{2} \) and \( A^T v = u\sqrt{2} \). Find the value of \( |u_1 + 2\sqrt{2}v_1| \).

Answer: 3


Question: Suppose that three components in a system each function with probability \( p \) and fail with probability \( 1-p \), with each component operating or failing independently of the others. The probability \( p \) is a random variable uniformly distributed over the interval \((0,1]\). The system operates if at least two of the components operate. What is the probability that the system operates?

Answer: \dfrac{1}{2}


Question: Determine the number of real roots of the polynomial equation \(p_n(x) = 1 + 2x + 3x^2 + \, \ldots \, + (n+1)x^n\) where \(n\) is an odd integer.

Answer: 1


Question: Determine the number of possible combinations of percentages for five additives that can be mixed into a chemical. Each additive can be mixed in at a discrete percentage from 0.02% to 0.22% (i.e., 0.02%, 0.03%, 0.04%, ..., 0.22%). Each additive must be present at a minimum of 0.02%. The total sum of all percentages cannot exceed 0.3%. How many valid combinations are there?

Answer: 53130


Question: Does Polya's theorem imply that for a sequence of random variables $X_n$ with a continuous distribution function $X$, the following holds: $$\sup_x | F_n(x) - P(X \leq x) | \stackrel{a.s.}{\rightarrow} 0$$ where $F_n(x) = \frac{1}{n}\sum_{i=1}^n 1\{X_i \leq x\}$ is the empirical CDF?

Answer: Yes


Question: Let \( f(x) = \sqrt{x + \sqrt{0 + \sqrt{x + \sqrt{0 + \sqrt{x + \dots}}}}} \,.\) If \( f(a) = 4 \), find \( f'(a) \).

Answer: \dfrac{4}{31}


Question: Let \( P(x) \) be a monic 4th degree polynomial with real coefficients such that \( P(3i) = P(1 - i) = 5 \). Find \( P(2) \).

Answer: 31


Question: Evaluate the limit: $$\lim_{x \rightarrow \frac{\pi}{6}} (2+\cos {6x})^{\ln |\sin {6x}|}$$

Answer: 1


Question: How many ways are there to distribute 6 copies of one book, 7 copies of a second book, and 11 copies of a third book between two teachers if each teacher receives exactly 12 books and each teacher gets at least 2 copies of each book?

Answer: 12


Question: Find the volume of the solid enclosed by the surfaces \(x^2+y^2=9\) and \(x^2+z^2=9\).

Answer: 144


Question: Determine the cardinality of the set \( S = \{ f \in C^1(\mathbb{R}): f(0)=0, f(2)=2 \; \text{and} \; |f'(x)| \leq \frac{3}{2} \; \forall x \in \mathbb{R} \} \).

Answer: \mathfrak{c}


Question: Compute the limit \( \lim_{t\to\infty} \frac{\exp \{(1+t)^{1/\theta} \}}{\exp \{(1+2t)^{1/\theta} \} } \) where \(\theta > 0\).

Answer: 0


Question: Find the limit superior as \( x \to +\infty \) of the expression \( \cos\sqrt{x-2015} - \cos\sqrt{x+2015} \).

Answer: 0


Question: Given that $x^2 + 3x + 1 = 0$, determine if $x^{2048} + \frac{1}{x^{2048}}$ is divisible by 3.

Answer: No


Question: Given the equation \( {}^xC_4 - {}^yC_4 = 425 \), find the value of \( x + y \).

Answer: 20


Question: Let $k$ be a positive integer. Determine whether there exist distinct positive integers $a_1, a_2, \ldots, a_k$ such that for $1 \le j \le k$, the product $\prod_{i=1}^j a_i \equiv 0 \pmod{k}$ if and only if $j = k$. Provide a justification for your answer.

Answer: Yes


Question: Does there exist an infinite, imprimitive, and non-abelian group?

Answer: Yes


Question: Let $f(z)$ be an analytic function on $\mathbb{C}$, where $f$ is not identically zero. For each $\varepsilon>0$, define $U_\varepsilon = \{ z \in \mathbb{C} : |f(z)| < \varepsilon \}$. Does $m(U_\varepsilon) \to 0$ as $\varepsilon \to 0$, where $m$ is the Lebesgue measure on $\mathbb{C}$?

Answer: No


Question: Do the coefficients of the inverse of a cyclotomic polynomial always form a periodic sequence?

Answer: Yes


Question: Let $X$ be a finite-dimensional vector space over a field $K$, and let $T: X \rightarrow X$ be a linear transformation. Suppose $A = [T]_\alpha$ is the matrix representation of $T$ with respect to a basis $\alpha$ of $X$. If $A$ is similar to another $n \times n$ matrix $B$, where $n = \dim X$, is it true that there exists a basis $\beta$ for $X$ such that $B = [T]_\beta$? Justify your answer.

Answer: Yes


Question: Find the limit of \( \frac{3f(n)}{(n+1)(n+2)} \) as \( n \to \infty \), where \( f(n) = \cot^2\left(\frac{\pi}{n}\right) + \cot^2\left(\frac{2\pi}{n}\right) + \cdots + \cot^2\left(\frac{(n-1)\pi}{n}\right) \).

Answer: 1


Question: Find the smallest integer \(x\) such that \[ \frac {365!} {365^{x+1}}\cdot\sum\limits_{i=0}^x \frac{i}{(364-i)!} \ge 1. \]

Answer: 2


Question: On a genus 1 smooth projective curve $C$ over $\mathbb{C}$, if $p$ and $q$ are two points and there exists a rational function $f$ on $C$ such that the divisor of $f$ is $(f) = p - q$, must $p$ equal $q$? Provide a justification for your answer.

Answer: Yes


Question: Let \( a_2=\frac{1}{2} \), and for \( n>2 \), \( a_n=a_{n-1}^{\frac{1}{n}} \). Find \( \lim_{n\to\infty}a_n \).

Answer: 1


Question: Evaluate the surface integral \( \iint_S f(x,y,z) \, dS \) where \( S \) is the part of the cone \( z = \sqrt{x^2+y^2} \) above the circle \( x^2+y^2 \leq 2x \) and \( f(x,y,z) = x^2+y^2 + z^2 \).

Answer: 3\sqrt{2}\pi


Question: Given the function \( f(x) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n \cdot 3^n} (x-3)^n \), find \( f''(-2) \).

Answer: -\dfrac{1}{4}


Question: Find the exact sum of the series: \[ \sum_{n=0}^{\infty} \frac{(-1)^{n+1}{\pi}^{2n}}{2(2n)!} \]

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{-\infty}^\infty e^{-(x+ia)^2} \, \text{d}x \), where \( a \in \mathbb{R} \).

Answer: \sqrt{\pi}


Question: In how many ways can the sides of an icosahedron be numbered from 1 to 20, considering the sides indistinguishable?

Answer: \dfrac{20!}{60}


Question: Find the critical point(s) of the differential equation: \[ \frac{d^2x}{dt^2}+2a(x^2-1)\frac{dx}{dt}+kx=0. \]

Answer: (0, 0)


Question: Evaluate the limit: \[ \lim_{x\to 0} (\ln (x+e))^{\cot x} \] without using l'Hôpital's rule or series expansion.

Answer: e^{\frac{1}{e}}


Question: Determine the order of the subgroup of the symmetric group $S_5$ generated by the elements $(123)$ and $(345)$. Provide your answer as a single number.

Answer: 60


Question: Evaluate the integral \( \iint_D \left(\sqrt{a^2 - x^2 - y^2} - \sqrt{x^2 + y^2}\right)\,\mathrm{d}x\,\mathrm{d}y \), where \( D \) is the disk defined by \( x^2 + y^2 \leq a^2 \). Use polar coordinates to transform the integral and find the correct value.

Answer: 0


Question: Use Stokes' Theorem to evaluate the line integral \( \int_C \mathbf{F} \cdot \mathrm{d}\mathbf{r} \) for the vector field \( \mathbf{F}(x,y,z) = \langle 5y+\sin x, z^2+\cos y, x^3\rangle \), where \( C \) is the curve given by \( \mathbf{r}(t)=\langle \sin t, \cos t, 2\sin t\cos t\rangle, t\in[0,2\pi] \).

Answer: 5\pi


Question: Let $f:[0,1]^2\rightarrow \mathbb{R}$ be a twice continuously differentiable function such that for every $x\in [0,1]$, there exists an interval $I_x\subset [0,1]$ where $f(x,y)=0$ for all $y\in I_x$. Determine whether there must exist an open ball in $[0, 1]^2$ where $f$ is identically zero.

Answer: Yes


Question: Determine the conditional expectation \( \mathrm{E}(A|B) \) where \( B \) is the number of bats leaving a cave at the time of a nuclear explosion, following a geometric distribution with mean \( \lambda \). Each bat contributes an independent exponentially distributed amount of wind movement with mean \( \frac{1}{\mu} \). Let \( A \) be the total amount of wind movement at the time of the explosion. Assume no wind movement is accumulated if no bats are flying.

Answer: \dfrac{B}{\mu}


Question: Find all exact solutions to the equation \(\sqrt{x} + \ln(x) - 1 = 0\).

Answer: 1


Question: Let $f = u + iv$ be an entire function, where $u = \text{Re}(f(z))$ and $v = \text{Im}(f(z))$. If $u + v$ is bounded, is $f$ necessarily a constant function? Justify your answer.

Answer: Yes


Question: Determine the number of elements of order 6 in the group \( \text{Aut}(\mathbb{Z}_{105}) \).

Answer: 14


Question:
Given a real matrix \(A\) of order \(n \times n\), it is known that \(\langle Av, v \rangle = 0\) for any vector \(v\) in \(\mathbb{R}^n\). Which of the following must be true about \(A\)?

1. Every eigenvalue of \(A\) is real.
2. \(A\) is not invertible.
3. \(A\) is Hermitian.
4. \(A\) is not Hermitian.
5. All answers are incorrect.

Select the correct statement(s).

Answer: 5


Question:
Suppose \( x:[0,\infty)\to [0,\infty) \) is a continuous function with \( x(0)=0 \). If \( \left(x(t) \right)^2\leq 2+\int_{0}^{t}x(u)du,~~~\forall ~t\geq 0, \) determine which of the following is true for \( x(\sqrt{2}) \):

- \( x(\sqrt{2})\in [0,2] \)
- \( x(\sqrt{2})\in [0,\frac{3}{\sqrt{2}}] \)
- \( x(\sqrt{2})\in [\frac{5}{\sqrt{2}},\frac{7}{\sqrt{2}}] \)
- \( x(\sqrt{2})\in [10,\infty) \)

Answer: B


Question: Determine the values of \(x\) for which the series \(\sum_{n=1}^\infty \left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)x^n\) converges.

Answer: (-1, 1)


Question: For what value of \( p \) is the function \( w \) in the fractional Sobolev space \( H^{s}(\mathbb{R}) \) for some \( s > 3/2 \), given that \( w \) is analytic for \( x<0 \), behaves as \( e^{x} \) as \( x\rightarrow -\infty \), \( w \sim |x|^p \) as \( x\rightarrow 0^- \), and \( w=0 \) for \( x>0 \)?

Answer: p > 1


Question: Consider open sets \(O \subseteq \mathbb{R}^m\) and \(U \subseteq \mathbb{R}^n\), and a bijective function \(f: O \to U\) such that both \(f\) and its inverse \(f^{-1}\) are differentiable. Is it true that \(n = m\)?

Answer: Yes


Question: A player randomly selects digits, either 1 or 0, to form a sequence. The player wins if the last four digits of the sequence are '0010' and loses if they are '1001'. Determine the probability that the player wins.

Answer: \dfrac{5}{12}


Question: Determine whether the following statement is true or false: \[ \operatorname{Re} \int_a^b e^{-i\theta} f(x) \, dx = \int_a^b \operatorname{Re} (e^{-i\theta} f(x)) \, dx \] where \( f: \mathbb{R} \rightarrow \mathbb{C} \). Justify your answer.

Answer: True


Question: Let $A_1, A_2, \dots, A_d$ be $d \times d$ matrices that are strictly upper triangular. Prove that the product $A_1 A_2 \cdots A_d$ is the zero matrix.

Answer: 0


Question: Evaluate the limit \( \lim_{x\rightarrow 0}\frac{(1+x)^{\frac{1}{x}}+(1+2x)^{\frac{1}{2x}}+(1+3x)^{\frac{1}{3x}}-3e}{9x} \) without using series expansion or L'Hôpital's Rule.

Answer: -\dfrac{e}{3}


Question:
Find the constant $A$ such that the joint density function $f(x, y) = Axe^{-x(y+1)^2}$ for $x > 0$ and $y > 0$, and $0$ otherwise, integrates to 1. Evaluate the integral:
$$1 = \int_{0}^{\infty}\int_{0}^{\infty} Axe^{-x(y+1)^2} \, dx \, dy$$ and solve for $A$.

Answer: 3


Question: Evaluate the limit: \[ \lim_{n\to\infty} \left(0.\underbrace{99\ldots99}_{n\text{ nines}}\right)^{10^n}. \]

Answer: \dfrac{1}{e}


Question: Calculate the surface area generated by revolving the polar curve \( r = \sqrt{\cos 2\theta} \) around the y-axis, with the limits of integration from \( \theta = 0 \) to \( \theta = \frac{\pi}{4} \).

Answer: \pi \sqrt{2}


Question: Determine whether the following statement is true or false: If $f$ is continuous at $b$ and $f(b)=a$ and $\lim _{y\to a} g(y) = L$, then $\lim_{ x\to b} g(f(x))=L$. Provide a justification for your answer.

Answer: B


Question: Can a real operator \( \phi : L_2 (\mathbb{R}) \to L_2 (\mathbb{R}) \) have infinitely many linearly independent eigenvectors corresponding to a single eigenvalue, given that all its eigenvalues are real and the set of eigenvectors forms a basis for \( L_2 (\mathbb{R}) \)?

Answer: Yes


Question: Let $M^n$ be a topological manifold of dimension $n=3$ or $n=4$, which has the same homotopy type as the standard torus $T^n$. Is $M^n$ homeomorphic to the standard torus $T^n$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty}\left(\frac{(2n)!}{n!n^n} \right)^{1/n}.$$

Answer: \dfrac{4}{e}


Question: Evaluate the integral \( \int_{|z|= \pi/4}\left(1+z+\frac{1}{\tan z}\right)\,dz \) using the residue theorem.

Answer: 2\pi i


Question: Find the distance between the polynomial \( f = 2 \) and the subspace of polynomials in \( \mathbb{R}[x]_{\leq n} \) with zero constant term.

Answer: 2


Question: Let $K$ be an extension of $\mathbb{Q}$ with $[K:\mathbb{Q}]=m$, and let $\{e_i\}_{i=1}^m$ be an integral basis of $\mathcal{O}_K$. If $\mu \in \text{Gal}(K/\mathbb{Q})$, are the images $\{\mu(e_i)\}_{i=1}^m$ also an integral basis of $\mathcal{O}_K$?

Answer: Yes


Question: Let $G$ be the symmetric group $S_5$ of permutations of five symbols. Consider the set $J$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order $4$. Two subgroups $H$ and $K$ in $J$ are considered equivalent if they are conjugate, meaning there exists an element $g$ such that $gHg^{-1} = K$. How many equivalence classes are there in $J$?

Answer: 2


Question: Find the mass of a thin wire in the shape of a helix given by \(\overrightarrow{r}(t)=\langle \cos t, \sin t, t\rangle\) for \(0 \le t \le 6\pi\) with a density of \(z\) at the point \((x,y,z)\) on the wire.

Answer: 18\sqrt{2}\pi^2


Question: Find the maximum value of the expression \( \frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \) where \( x, y, z \) are real numbers satisfying the condition \( x + y + z = 1 \).

Answer: \dfrac{9}{10}


Question: Evaluate the integral \( \int_{-n}^{n} (-1)^{\lfloor x \rfloor} \, dx \).

Answer: 0


Question: Determine the values of \( x \) for which the series \( \sum_{n=1}^\infty \frac{1}{x^{\ln n}} \) converges.

Answer: (e, \infty)


Question: Determine the degree of the splitting field of the polynomial $x^4 + 1$ over $\mathbb{Q}$.

Answer: 4


Question: Determine the constant \( A \) such that the surfaces \( x^2 + y^2 + 2z^2 = A \) and \( xy + z^2 = 2 \) have a common tangent plane at the same point.

Answer: 4


Question:
Evaluate the limit of the sequence \((x_n)_{n \in \mathbb{N}^*}\) defined by \(x_1 = 2\) and the recurrence relation:
\[ \frac{1}{x_{n+1}^{\frac{1}{n+1}} - 1} - \frac{1}{x_{n}^{\frac{1}{n}} - 1} = 1 \]
Find \(\lim_{n \to \infty} x_n\).

Answer: e


Question: Consider finitely generated modules over an Artin algebra. Let $X$ be an indecomposable module and $f: X \longrightarrow X \oplus X$ be a monomorphism. Determine whether $f$ must always be a split monomorphism.

Answer: Yes


Question: Let $(R, \mathfrak{m}, k)$ be an Artinian local Gorenstein ring. Consider the category $\mathcal{F}(R)$ of finitely generated free $R$-modules and the category $\text{mod}(R)$ of finitely generated $R$-modules. Define the covariant functors $F, G: \mathcal{F}(R) \to \text{mod}(R)$ by $F(-) = \text{Hom}_R(k, -)$ and $G(-) = k \otimes_R -$. Given that $F(M) \cong G(M)$ for all $M \in \mathcal{F}(R)$, are the functors $F$ and $G$ naturally isomorphic?

Answer: Yes


Question: Determine whether there exist right ideals in the matrix ring \(M_n(k)\) that are not of the form of zeroing one or more rows at a time. Provide a justification for your answer.

Answer: Yes


Question: Let \(X\) be a continuous random variable with the probability density function \(f(x) = ke^{-x^2-7x}\). Find the expected value \(E[X^2]\).

Answer: \dfrac{51}{4}


Question:
Solve the following system of equations for real numbers \((x, y, z)\):
\[ x^2 + y + z = -1 \]
\[ x + y^2 + z = -1 \]
\[ x + y + z^2 = -1 \]

Answer: (-1, -1, -1)


Question: Evaluate the integral: \[ \int_{-\infty}^\infty e^{-\frac{x^2}{3}}\,\mathrm{d}x \]

Answer: \sqrt{3\pi}


Question: Calculate the determinant of the matrix expression \( |A_0 + A_0^2B_0^2 + A_0^3 + A_0^4B_0^4 + \ldots \) for 10 terms, where \( A_0 = \begin{bmatrix}2 & -2 & -4\\-1 & 3 & 4\\1&-2&-3\end{bmatrix} \) and \( B_0 = \begin{bmatrix}-4&-3&-3\\1&0&1\\4&4&3\end{bmatrix} \). Assume \( B_n = \text{adj}(B_{n-1}) \) for \( n \in \mathbb{N} \) and \( I \) is the identity matrix of order 3.

Answer: 0


Question: Calculate the volume of the region $G$ bounded by $z = -\pi$, $z = \pi$, and the surface $x^2 + y^2 - \cos^2 z = 1$. Determine the appropriate limits for $x$, $y$, and $z$, and evaluate the triple integral to find the volume of $G$. Express the volume as a single numerical value.

Answer: 3\pi^2


Question: Determine the value of \( \alpha \) such that the function \( F(x,y) = (x^3 - \alpha xy^2, \alpha x^2y - y^3) \) is a conformal map from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \).

Answer: 3


Question:
Determine whether the following inequality holds for any continuous function \( f(x) > 0 \):
\[
\int_0^{1} \left( (\sqrt{3}f(x))^{2} - 2(f(x))^{3} \right) dx \leq 1.
\]

Answer: 1


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{dx}{(x^2+4)(x^2+9)} \).

Answer: \dfrac{\pi}{60}


Question: In \(\Delta ABC\), the points \(P(1,1,2)\) and \(Q(4,7,8)\) satisfy \(\frac{AP}{AQ}=\frac{BP}{BQ}= \frac{CP}{CQ}=2\). Find the coordinates of the circumcenter of \(\Delta ABC\).

Answer: (5, 9, 10)


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

Answer: \infty


Question: Determine whether the limit \( \lim_{x\to \infty}{\sum_{i,h=1}^x \frac{1}{i^h} - x-\ln{x}} \) is equal to \( \gamma \), the Euler-Mascheroni constant.

Answer: \gamma


Question: Evaluate the double integral: \[ \int_0^1\int_{y}^{\sqrt{y}}\frac{\sin x}{x} \, dx \, dy \]

Answer: 1 - \sin 1


Question: Calculate the limit: \[ \lim_{x \to 0}(\cos x)^{1/x^4} \]

Answer: 0


Question: Calculate the remainder of \( 13^{13^{13}} \mod 22 \).

Answer: 19


Question: Determine whether the following statement is true or false: If a function $f$ is analytic and nowhere zero in the domain $D = \{z : \Re z < 2019 \}$, then $\log |f|$ is harmonic in $D$. Provide a justification for your answer.

Answer: A


Question: If \( \lim_{h\to 0} \frac{f(x_0 + h) - f(x_0 - h)}{2h} = f'(x_0) \) exists, is the function \( f \) differentiable at \( x_0 \)? Provide a justification for your answer.

Answer: No


Question: Convert the integral \( \iiint_E z(x^2+y^2+z^2)^{-3/2} \, dV \) to spherical coordinates and evaluate it, where \( E \) is the region defined by \( x^2+y^2+z^2 \leq 16 \) and \( z \geq 2 \).

Answer: \pi


Question: Evaluate the triple integral \( \iiint_D (x^2 + 2y^2)z \, dxdydz \) where the region \( D \) is defined by \( x^2 + y^2 + z^2 \leq 4 \) and \( z > \sqrt{x^2 + y^2} \).

Answer: 2\pi


Question:
Consider the wave equation given by:
\[ \partial_t^2 u(t,x) - \partial_x^2 u(t,x) = 0 \]
with initial conditions:
\[ u(0,x) = g(x) \]
\[ \partial_t u(0,x) = h(x) \]
If both \(g(x)\) and \(h(x)\) are monotone increasing functions, is it true that the solution \(u(t,x)\) is also monotone increasing? Justify your answer.

Answer: Yes


Question: Find the rate of change of the function \( f(x,y,z) = \frac{x}{z} + \frac{y}{z} \) with respect to \( t \) along the curve \( r(t) = \sin^2{t} \mathbf{i} + \cos^2{t} \mathbf{j} + \frac{1}{2t} \mathbf{k} \).

Answer: 2


Question: Given that $f(x)$ is a periodic function with period $t$ and satisfies the equation $f(2x+3) + f(2x+7) = 2$ for all $x \in \mathbb{R}$, find the period $t$.

Answer: 8


Question: Find the value of $x$ such that the infinite power tower $x^{x^{x^{\dots}}}=2$.

Answer: \sqrt{2}


Question: Suppose that $A \subseteq B \subseteq C$, where $A$ and $C$ are Lebesgue measurable sets with equal Lebesgue measure. Does $B$ have to be Lebesgue measurable as well?

Answer: Yes


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

Answer: -e^2


Question: Show that \( \max_{i \leq n} |X_i|^p/n \) converges to 0 in probability, where \( X_i \) are i.i.d. random variables and \( \mathbb{E}(|X_i|^p) = k < \infty \) for some \( p, k \).

Answer: 0


Question: If $A$ and $B$ are compact and connected subsets of the complex plane, and there is a bijection $f$ that maps $A$ onto $B$, is $f$ necessarily continuous?

Answer: No


Question: A room is illuminated by two light bulbs. The bulbs are replaced only when both have burned out. The lifetimes of the bulbs are independent and identically distributed, following an exponential distribution with parameter \( \lambda = 1 \). What fraction of the time is the room only half-lit?

Answer: \dfrac{2}{3}


Question: Determine whether there exists a non-linear transformation $T: V \rightarrow V$ on a vector space $V$ over the complex numbers $\mathbb{C}$ such that $T(x+y) = T(x) + T(y)$ for all vectors $x, y \in V$. Provide a justification for your answer.

Answer: Yes


Question:
Given the following equations:
\[ \alpha + \beta + \gamma = 14 \]
\[ \alpha^{2} + \beta^{2} + \gamma^{2} = 84 \]
\[ \alpha^{3} + \beta^{3} + \gamma^{3} = 584 \]
Find the value of \(\alpha^{4} + \beta^{4} + \gamma^{4}\).

Answer: 4368


Question:
Which of the following identities is correct when applying the secant method to $f(x)$ using the mean value theorem?

(a) $\xi - x_{k+1} = \xi - x_k - \left[\frac{f'(\eta_k)}{f'(\theta_k)}\right](\xi - x_k)$, where $\eta_k$ is between $x_k$ and $\xi$, and $\theta_k$ is between $x_k$ and $x_{k-1}$.

(b) $\xi - x_{k+1} = \xi - x_k + \left[\frac{f'(\eta_k)}{f''(\theta_k)}\right](\xi - x_k)$, where $\eta_k$ is between $x_k$ and $\xi$, and $\theta_k$ is between $x_k$ and $x_{k-1}$.

(c) $\xi - x_{k+1} = -\frac{(\xi - x_k)^2 f''(\eta_k)}{2f'(x_k)}$, where $\eta_k$ is between $x_k$ and $\xi$.

(d) $\xi - x_{k+1} = \xi - x_k + \left[\frac{f'(\eta_k)}{f'(\theta_k)}\right](\xi - x_k)^3$, where $\eta_k$ is between $x_k$ and $\xi$, and $\theta_k$ is between $x_k$ and $x_{k-1}$.

Select the correct identity.

Answer: a


Question: Find a function \( g(\theta) \) such that the quantity \( Q(\theta, W) = g(\theta) W \) follows a chi-squared distribution with \( k \) degrees of freedom, where \( W(X_1, \dots, X_n) = \sum_{i=1}^n \ln\left(\frac{X_i}{8}\right) \) and \( X_1, \dots, X_n \) are i.i.d. with probability density function \( f(x;\theta) = \theta \frac{8^\theta}{x^{\theta +1}} \) for \( x \geq 8 \) and \( \theta > 0 \).

Answer: 2\theta


Question: Given the expressions \(X = a + 2b + 3c + \ldots\) and \(Y = a + b + c + \ldots\), where all variables \(a, b, c, \ldots \geq 0\), and the constraint \(X - Y \geq N\) with \(N\) being a positive number, find the minimum value of \(X\).

Answer: N


Question: Suppose $A$ and $B$ are two Hermitian positive semi-definite unit trace matrices. Is the matrix $|A-B|$ positive semi-definite?

Answer: Yes


Question: For how many sets of natural numbers \((x, y, z)\) with \(1 < x < y < z\) is \((x-1)(y-1)(z-1)\) a factor of \(xyz-1\)?

Answer: 2


Question: Evaluate the definite integral \( \frac{105}{19}\int^{\pi/2}_0 \frac{\sin 8x}{\sin x}\,dx \).

Answer: 8


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^{1} e^{x^2} \sin(nx) \, dx \] without using the concept of Lebesgue integration.

Answer: 0


Question: Evaluate the integral \( \int_{C} \frac{z}{\cos(z)}\,dz \), where \( C \) is the path \( e^{jt} \) for \( 0 < t < 2\pi \). Determine if the integral equals zero, given that the poles of the function are at \( +\frac{\pi}{2} \) and \( -\frac{\pi}{2} \), both of which are outside the path of integration.

Answer: 0


Question: For how many integer values of \( b \) in the range \( 1 \leq b \leq 20 \) is the expression \( \frac{2a+b}{a-2} \) an integer for exactly four integer values of \( a \)?

Answer: 7


Question: Let $X$ be a metric space and $A \subset X$ be a closed set. Consider a function $f : A \rightarrow \mathbb{R}$ that is continuous and bounded. According to Tietze's extension theorem, $f$ can be extended to a continuous function $F : X \rightarrow \mathbb{R}$ such that $F(a) = f(a)$ for all $a \in A$. Can the extension $F$ be chosen to be bounded as well?

Answer: Yes


Question: Determine whether the expected value \( E\left[(X+Y-10)_+\right] \) is equal to \( E\left[(2X-10)_+\right] \), where \( X \) and \( Y \) are independent random variables uniformly distributed over \([0, 10]\).

Answer: No


Question: Determine whether the following statement is true or false: Let \(W\) be a two-dimensional subspace of \(\mathbb{R}^3\). Then two of the following three vectors span \(W\): \(X = [1,0,0]^t\), \(Y = [0,1,0]^t\), \(Z = [0,0,1]^t\).

Answer: B


Question: Determine the limit points of the set \( \{\sqrt{m} - \sqrt{n} \mid m, n \in \mathbb{N} \} \).

Answer: \mathbb{R}


Question: Compute the integral \( \int_{T} \frac{1}{z^2} \, dz \) along the contour \( T \).

Answer: 0


Question: Consider a function \( f:[0,1]\to[0,1] \) that is continuous on a co-meager set \( C\subset[0,1] \) and discontinuous on \( D=[0,1]\setminus C \). Suppose that \( D\cap I \) is uncountable for every open interval \( I\subseteq[0,1] \). Determine whether \( f \) can be of Baire class 1.

Answer: Yes


Question: Two irrational numbers in the interval $(0,1)$ are chosen at random, and their decimal parts are considered up to 2014 places. The probability that the smaller number can be subtracted from the larger one without borrowing is expressed as $p^k$, where $p$ is a rational number and $k > 1000$ is a positive integer. Given that the first decimal digit of the larger number is greater than that of the smaller number, find the last three digits of $\lfloor{100(p+k+1)}\rfloor$. Provide your answer as a three-digit number.

Answer: 455


Question: Determine the maximum value of the expression \(2m + 7n\) given that there exist distinct positive integers \(x_i\) \((1 \leq i \leq m)\) and \(y_j\) \((1 \leq j \leq n)\) such that the \(x_i\)'s are even, the \(y_j\)'s are odd, and \(\sum_{i = 1}^m x_i + \sum_{j=1}^n y_j = 1986\).

Answer: 322


Question: Evaluate the limit \( \lim_{x\to 0^+}\dfrac{\tan^2\left(\frac{1}{\sqrt{1-x^2}}-1\right)}{\left(1-\cos(\sqrt{2x})\right)^n}=a \). What is the value of \( a+n \)? Choose the correct answer: \(1)\frac{7}{4}\quad 2)\frac{9}{4}\quad 3)\frac{15}{4}\quad 4)\frac{17}{4}\).

Answer: D


Question:
Determine the correct expression for the asymptotic variance of the OLS estimator under the homoskedasticity assumption. Is it:

1. \( \widehat{Avar\left(\hat{\beta}\right)}=\hat{\sigma}^{2}\left(X^{\prime}X\right)^{-1} \)
2. \( \widehat{Avar\left(\hat{\beta}\right)}=\hat{\sigma}^{2}\left(\frac{1}{N}X^{\prime}X\right)^{-1} \)
3. \( \widehat{Avar\left(\hat{\beta}\right)}=\hat{\sigma}^{2}\left(\sum_{i=1}^{N}x_{i}^{\prime}x_{i}\right)^{-1} \)
4. \( \widehat{Avar\left(\hat{\beta}\right)}=\hat{\sigma}^{2}\left(\frac{1}{N}\sum_{i=1}^{N}x_{i}^{\prime}x_{i}\right)^{-1} \)

Choose the correct expression.

Answer: 4


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

Answer: (-1)^{k+1}


Question: Compute the volume of the solid bounded by the surface \( z = 3 \sqrt{x^2 + y^2} \), the plane \( z = 0 \), and the cylinder \( x^2 + (y-1)^2 = 1 \).

Answer: \dfrac{32}{3}


Question: Solve the partial differential equation \(x\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=0\) with the initial condition \(f(1,y)=e^{-2y}\) using the change of variables \(u=xe^y\) and \(v=xe^{-y}\).

Answer: x^2 e^{-2y}


Question: Let \(\omega\) be a complex number such that \(\omega^7 = 1\) and \(\omega \neq 1\). Define \(\alpha = \omega + \omega^2 + \omega^4\) and \(\beta = \omega^3 + \omega^5 + \omega^6\). Determine the ordered pair \((p, q)\) such that \(\alpha\) and \(\beta\) are roots of the quadratic equation \(x^2 + px + q = 0\).

Answer: (1, 2)


Question: Evaluate the limit \( S_0 = \lim_{x \to \infty} S(x) \) where \( S(x) = \sum_{k=1}^\infty \frac{\sin(kx)}{kx} \) and is given by the closed form \( S(x) = \frac{1}{x}\left(\frac{1}{2}i\left(\ln(1-\exp(ix)\right)-\ln\left(1-\exp(-ix)\right)\right) \).

Answer: 0


Question: Given a real periodic signal with period $T_0=2$ and Fourier coefficients $$X_k=\left [\frac{2}{3}, \ \frac{1}{3}e^{j\pi/4}, \ \frac{1}{3}e^{-i\pi/3}, \ \frac{1}{4}e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$, calculate the integral $\int_0^{T_0}x^2(t)\, dt$ using the Fourier coefficients.

Answer: \dfrac{217}{36}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{f(x) - e^x}{1 - \cos x} \] where \( f(x) = \sum_{n=0}^\infty \frac{x^n}{n! \, 2^\frac{n(n-1)}{2}} \).

Answer: -\dfrac{1}{2}


Question: Let $G$ be a connected topological group and $N$ a discrete normal subgroup of $G$. Is it true that $Z(G)/N = Z(G/N)$, where $Z(G)$ denotes the center of $G$?

Answer: Yes


Question: In triangle \( \triangle ABC \), if \( \frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} = \sqrt{3} \), prove that at least one of the angles measures \(60^{\circ}\).

Answer: 60^\circ


Question:
Find all surjective functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), the following equation holds:
\[
f(x+f(x)+2f(y))=f(2x)+f(2y)
\]

Answer: f(x) = x


Question: Find all negative values of $x$ such that \((1+x^a+x^b)^2 = 3(1+x^{2a}+x^{2b})\), where $a$ and $b$ are natural numbers.

Answer: -1


Question: Determine the value of $s(1)$, where $s(d)$ is defined as follows: For $n \ge 1$ and $0 \le k < n$, define $B_{n,k} = [k/n, k/n + 1/n)$. Fix an integer $d \ge 0$ and suppose $(x_1, x_2, ..., x_{s+d})$ is a sequence with $x_i$ belonging to $[0,1)$ and with $s = s(d)$ chosen to be maximal such that for each $r \le s$ and each $k < r$, $B_{r,k}$ contains at least one point of the subsequence $(x_1, x_2, ..., x_{r+d})$. It is known that $s(0) = 17$. What is the value of $s(1)$?

Answer: 34


Question: Evaluate the limit: $$\lim_{x \rightarrow \infty} \frac{1}{x}\log(e^{\pi x} -1).$$

Answer: \pi


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int^{\infty}_0 n\ln\left(1+\frac{e^{-x}}{n}\right) \, dx. \]

Answer: 1


Question: Given the elliptic curve \( y^2 \equiv x^3 + 17x + 5 \pmod{59} \) and the point \( P = (4, 14) \), calculate the point \( 8P \) on the curve.

Answer: (16, 40)


Question: Find the expectation of the stochastic integral \( \int_0^t \sqrt{s+B_s^2} \, dB_s \).

Answer: 0


Question: Determine the total number of automorphisms of the symmetric group $S_6$. Express your answer as a single integer.

Answer: 1440


Question: If \( D \) is a dense subset of a Hilbert space, is the orthogonal complement of \( D \) equal to \( \{ 0 \} \)? Provide a justification for your answer.

Answer: \{0\}


Question: Consider a Markov chain \( \{X_n\} \) on \( \mathbb{N}_0 \) with transition probabilities \( p(x,x+1) = p(x,0) = \frac{1}{2} \) for all \( x \in \mathbb{N}_0 \). Find the expected time \( E(T_0 \mid X_0 = 1) \), where \( T_0 = \inf\{k \geq 0 : X_k = 0\} \).

Answer: 2


Question: Find the derivative of the expression \(-a^T X^T y\) with respect to the vector \(a\).

Answer: -X^\top y


Question: Calculate the probability that a gambler, starting with $1, wins by reaching $n$ dollars before losing all his money, given that each bet has a probability of $1/2$ to win or lose $1.

Answer: \dfrac{1}{n}


Question: Determine whether the following statement is true or false: If $M$ is an even-dimensional smooth manifold, then there exists a nondegenerate 2-form on $M$. Provide a justification for your answer.

Answer: False


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

Answer: \sqrt{ab}


Question: Place 10 guards in a 17-walled art gallery such that the entire gallery is protected, but the removal of any one guard leaves some part of the gallery unprotected. Determine a configuration that satisfies this condition.

Answer: 10


Question: Identify the analytic function $f$ defined in the unit disk $\{z \in \mathbb{C}: |z| < 1 \}$, given that for any $n > 0$, the condition \(\left|f\left(\frac{1}{n}\right)\right| \leq e^{-n}\) holds. Find $f$.

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to 0}\frac{4^{\arccos^2 \left(\frac{1}{1 + x^2}\right)} - 1}{\log_4(1 + x)} \]

Answer: 0


Question: Find the formula for the summation \( S = \sum_{i=1}^{n} \frac{n}{i} \).

Answer: n H_n


Question: Determine the cardinality of the equivalence class of the function \( h(n) = n^2 \) under the equivalence relation \( R \) on \( \mathbb{N}^{\mathbb{N}} \), where \( R = \{(f,g) \in \mathbb{N}^{\mathbb{N}} \mid \exists b \in \mathbb{N}. \forall n > b, f(n) = g(n)\} \).

Answer: \aleph_0


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(\int_0^1 \frac{1}{1+x^n} \, dx\right)^n \]

Answer: \dfrac{1}{2}


Question:
Calculate the volume of the region defined by the curvilinear coordinates \((u, v, w)\) related to Cartesian coordinates \((x, y, z)\) by the transformations:
\[x=\frac{1}{2}(u^2-v^2), \quad y=uv, \quad z=w\]
with the limits \(0 \le u \le 1\), \(0 \le v \le 1\), \(0 \le w \le 1\). Use the scalar factors \(h_u = \sqrt{u^2+v^2}\), \(h_v = \sqrt{u^2+v^2}\), and \(h_w = 1\) to find the differential volume element \(dV = (u^2+v^2) \, du \, dv \, dw\). Evaluate the triple integral:
\[ \iiint (u^2+v^2) \, du \, dv \, dw \]
to find the volume of the region.

Answer: \dfrac{2}{3}


Question: Let $A_1A_2A_3A_4$ be a square, and let $A_5, A_6, A_7, \ldots, A_{34}$ be distinct points inside the square. Non-intersecting segments $\overline{A_iA_j}$ are drawn for various pairs $(i,j)$ with $1 \le i,j \le 34$, such that the square is dissected into triangles. Assume each $A_i$ is an endpoint of at least one of the drawn segments. How many triangles are formed?

Answer: 62


Question: Let $f$ be a power series with rational coefficients, $f \in \mathbb{Q}[[z]]$. Consider algebraic numbers $\alpha_1, \ldots, \alpha_s, b_1, \ldots, b_s$ that belong to a Galois field $K$ with Galois group $G$. Determine if the power series \[ \prod_{\sigma \in G} \left( \sum_{i=1}^s \sigma(b_i) f(\sigma(\alpha_i) z) \right) \] belongs to $\mathbb{Q}[[z]]$. Provide a justification for your answer.

Answer: Yes


Question: For any integer \( p \ge 3 \), determine the largest integer \( r \) such that \((x-1)^r\) is a factor of the polynomial \(2x^{p+1}-p(p+1)x^2+2(p^2-1)x-p(p-1)\).

Answer: 3


Question: Determine the smallest integer \( n \) such that \( |e^x - p_n(x)| < 10^{-15} \) for all \( x \) in the interval \([-1, 1]\).

Answer: 17


Question:
Determine whether the following condition implies the other:

1. For all unit vectors $\mathbf{x}$ and $\mathbf{y}$, if $\left|\left|\mathbf{x} + \mathbf{y}\right|\right| = 2$, then $\mathbf{x} = \mathbf{y}$.

2. For all vectors $\mathbf{x}$ and $\mathbf{y}$, if $\left|\left|\mathbf{x} + \mathbf{y}\right|\right| = \left|\left|\mathbf{x}\right|\right| + \left|\left|\mathbf{y}\right|\right|$, then $\mathbf{x}$ and $\mathbf{y}$ are colinear.

Does the first condition imply the second?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to\infty}\frac{d^n}{dx^n} (\exp(x)) \).

Answer: e^x


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

Answer: 2048


Question: Evaluate the limit using Taylor series: $$\lim_{x\to{}0}\frac{e^{3x}-\sin(x)-\cos(x)+\ln(1-2x)}{-1+\cos(5x)}$$

Answer: -\dfrac{6}{25}


Question: Let $D$ be a finite central division algebra over its center $Z$. Consider a maximum subfield $F$ of $D$, meaning there is no larger subfield $G$ of $D$ such that $G$ properly includes $F$. Is the dimension of $F$ over $Z$ an invariant, i.e., is $\dim_Z F$ the same for all maximum subfields of $D$?

Answer: Yes


Question: Given the joint probability density function \( f(x,k) = 2 \) for \( 0 \leq k \leq x \leq 1 \), find the cumulative probability distribution function \( F(k_0) = P(K \leq k_0) \).

Answer: 2k_0 - k_0^2


Question: Compute the mass of the solid tetrahedron bounded by the planes $x = 0$, $y = 0$, $z = 0$, and $x+y+z = 1$, given that the density of the tetrahedron is described by the function $y$. Provide your answer as an integral expression and evaluate it.

Answer: \dfrac{1}{24}


Question: A diagnostic test for a disease has a 95% chance of returning a true positive and a 10% chance of a false positive. The true proportion of diseased individuals in the population is \(d\). Everyone is screened, and those who test positive are tested again. Of these, 25% test positive on the second test, assuming the tests are independent. What is the value of \(d\)?

Answer: \dfrac{3}{136}


Question: Given that the function $f$ is differentiable and satisfies the equation \( \int^x_0(6f(t)+4t)dt=\sin(x) \), determine the value of \( f'\left(\frac{\pi}{6}\right) \).

Answer: -\dfrac{3}{4}


Question: In $\triangle ABC$, $AD$ bisects $\angle BAC$. $E$ is the midpoint of $AC$. $BP$ is the tangent from $B$ to the circumcircle of $\triangle DEC$. Given $AB=18$, $AC=24$, and $BP=12$, find $BC$ in the form $a\sqrt{b}$ and determine $a+b$. What is $a+b$?

Answer: 25


Question: Let $\mu$ and $\nu$ be finite measures on Borel subsets of $[0,1]$, and suppose $\nu \ll \mu$ (i.e., $\nu$ is absolutely continuous with respect to $\mu$). Is there always a constant $C \in \mathbb{R}$ such that for all Borel subsets $A \subseteq [0,1]$, we have $\nu(A) < C \mu(A)$? Determine whether this statement is true or false.

Answer: False


Question: Compute the line integral \( \int -4x^2y \, dx -(x^3+y^3) \, dy \) along the closed curve that starts at \((-2,0)\), follows the curve \(y=4-x^2\) to \((2,0)\), and returns to \((-2,0)\) along the curve \(y=x^2-4\), using Green's Theorem.

Answer: \dfrac{256}{15}


Question: Determine the maximum $L_1$ distance that can be achieved between a function $f: \mathbb{F}_2^n \to \mathbb{F}_2$ and the set of $k$-sparse parity functions $\text{SPF}_k$, given that for any subset $X$ of $\mathbb{F}_2^n$ with cardinality $m$, there exists a $k$-sparse parity function $h$ such that $f(x) = h(x)$ for all $x \in X$. What is this maximum distance?

Answer: m - 1


Question: Evaluate the limit: $$\lim\limits_{x\rightarrow 0} \left(\frac 14\left(\dfrac{e^2}{x} +3\right)\right)^{4x}.$$

Answer: 1


Question: Calculate the integral \( \int_{\gamma} \omega \), where \( \omega = \frac{x\,\text{d}y - y\,\text{d}x}{x^2 + y^2} \) is a 1-form on \( \mathbb{R}^2 \) and \( \gamma(t) = (\cos(t), \sin(t)) \) is the unit circle parameterized by \( t \in [0, 2\pi] \).

Answer: 2\pi


Question: Determine whether the integral \( \int_0^1 x(t) \int_0^t x(s) \, ds \, dt \) is positive for a general function \( x \in L_2[0,1] \).

Answer: No


Question: Find the supremum of the set of maximum eigenvalues of correlation matrices, where a correlation matrix is defined as a positive semidefinite matrix $B \in M_{n}(\mathbb{C})$ with the identity matrix $I$ as its diagonal.

Answer: n


Question: Evaluate the limit: \[ \lim_{k \to \infty} \int_{-\infty}^{\infty} \frac{1}{1+e^{-kx}} \frac{1}{1+x^2} \, dx \]

Answer: \dfrac{\pi}{2}


Question:
Determine whether there exists a function \( f: \mathcal{P}(\mathbb{N}) \rightarrow \{0,1\} \) such that:

1. \( f(\mathbb{N}) = 1 \)
2. \( f(A \cup B) = f(A) + f(B) \) for disjoint sets \( A \) and \( B \)
3. \( f(A) = 0 \) for any finite set \( A \)

Answer: Yes


Question: Given that $(a+b)$ and $(a-b)$ are relatively prime integers, find the greatest common divisor (gcd) of the expressions $2a+(1+2a)(a^2-b^2)$ and $2a(a^2+2a-b^2)$. Provide your answer as a single expression or number.

Answer: 1


Question: Let \( \alpha \) be a real number such that \( 0 \leq \alpha \leq \pi \). If the function \( f(x) = \cos x + \cos(x+\alpha) + \cos(x+2\alpha) \) takes a constant value \( c \) for any \( x \in \mathbb{R} \), find the value of \( \lfloor c + \alpha \rfloor \).

Answer: 2


Question: Consider two distinct irreducible factors $f$ and $g$ of $x^n-1$ over the $p$-adic integers $\\mathbb{Z}_p[x]$. Suppose their reductions modulo $p$, denoted as $\overline{f}$ and $\overline{g}$, are coprime in $\mathbb{F}_p[x]$. Does it follow that the ideals generated by $f$ and $g$ in $\mathbb{Z}_p[x]$ are comaximal?

Answer: Yes


Question: Find the minimum value of the expression \(a^2 + b^2\) given that \(a\) and \(b\) are non-zero real numbers satisfying the equation \(ab(a^2 - b^2) = a^2 + b^2\).

Answer: 4


Question: Is a group of order $16 \times 17$ solvable?

Answer: Yes


Question: Is there a finite-dimensional Lie algebra \( L \) such that there are infinitely many non-isomorphic compact connected Lie groups whose Lie algebras are isomorphic to \( L \)?

Answer: No


Question: Find the multiplicative inverse of $x + (f(x))$ in the ring $\mathbb{Q}[x]/(f(x))$, where $f(x) = x^3 - 2$. Express your answer in terms of the basis $\{1, x, x^2\}$.

Answer: \dfrac{1}{2}x^2


Question: Evaluate whether the following limit holds: $$\lim_{n\to \infty} \binom{2n}{n}(1-p)^{n}p^{n}=0$$ for $p\in[0,1]$ with $p\ne 1/2$. Provide a justification for your answer.

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\rightarrow0}\left[x\left(\frac{1+3x}{1+2x}\right)^{\frac {1}{x}}\right] \]

Answer: 0


Question: Evaluate the limit: $$\lim_{x\rightarrow0^{+}} x^{0.7}(\ln(e^{x} - 1))$$

Answer: 0


Question: Determine the radius of convergence of the power series for \( \frac{1}{1+x^2} \) centered at \( x=1 \) without using complex analysis.

Answer: \sqrt{2}


Question: Determine whether the function \( f(t) = t^{\alpha} \) is sub-additive for \( \alpha \in (0,1) \) and \( 0 \leq t \leq 1 \).

Answer: Yes


Question: Find an analytic expression for the covariance of a normally distributed random variable $X \sim N(\mu, \sigma^2)$ with its square $X^2$. Specifically, determine $\operatorname{Cov}(X, X^2)$.

Answer: 2\mu\sigma^2


Question:
Let $f$ be a continuous strictly increasing function from $[0,\infty)$ onto $[0,\infty)$ and $g=f^{-1}$ (that is, $f(x)=y$ if and only if $g(y)=x$). Let $a,b>0$ and $a\ne b$. Evaluate the expression:
\[ \int_0^a f(x) \, dx + \int_0^b g(y) \, dy \]
Choose the correct option:
(A) greater than or equal to $ab$,
(B) less than $ab$,
(C) always equal to $ab$,
(D) always equal to $\frac{af(a)+bg(b)}{2}$.

Select the correct answer.

Answer: A


Question: Given the functional equation \( f(x+f(y))=x+f(f(y)) \) for all \( x,y \in \mathbb{R} \), and the condition \( f(2)=8 \), calculate \( f(2005) \).

Answer: 2011


Question: Given \( L = \lim_{x\rightarrow 0}\left(\frac{1}{\ln(1+x)}-\frac{1}{\ln(x+\sqrt{x^2+1})}\right) \), find the value of \( L +\frac{153}{L} \).

Answer: \dfrac{613}{2}


Question: Compute the limit \( \lim_{x \rightarrow \infty} \left(1+\frac{1}{x} \right)^{\sqrt{ax^2 +bx+c}} \) for \( a \geq 0 \).

Answer: e^{\sqrt{a}}


Question: Suppose that \( f:\mathbb{R}\rightarrow\mathbb{R} \) is a continuous function such that the image \( f[G] \) is a subgroup of \( \mathbb{R} \) for every subgroup \( G \) of \( \mathbb{R} \). Must \( f \) be an endomorphism of \( (\mathbb{R},+) \), i.e., is it true that \( f(x)=ax \) for some real number \( a \)?

Answer: Yes


Question: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a bounded measurable function, and $g: \mathbb{R} \to \mathbb{R}$ be an absolutely continuous function. Determine whether it is true that if $x \in \mathbb{R}^n$ is a Lebesgue point of $f$, then $x$ is also a Lebesgue point of $g \circ f$. Assume the strong definition of Lebesgue points.

Answer: True


Question: A club has 11 boys and 12 girls. We need to choose a subset of kids such that the number of girls is one more than the number of boys in this subset. The subset must have at least 1 but at most 23 kids. Let N be the number of such subsets we can form. Find the sum of prime numbers that divide N.

Answer: 81


Question: In the alternating group \( A_7 \), consider the permutation \( \sigma = (1\; 2\; 3\; 4\; 5) \). How many elements in \( A_7 \) are conjugate to \( \sigma \)?

Answer: 504


Question: Given the equation \( \frac{3-\tan^2\frac{\pi}{7}}{1-\tan^2\frac{\pi}{7}}=\alpha \cos\frac{\pi}{7} \), where \( \alpha \) is a natural number, find the value of \( \alpha \).

Answer: 4


Question: Evaluate the expression \( A(1-\cos(\frac{\pi}{2018})) + B(\sin(\frac{\pi}{2018})) \), where \( A = \sum_{i=1}^{3027} \sin(\frac{\pi i}{2018}) \) and \( B = \sum_{i=1}^{3027} \cos(\frac{\pi i}{2018}) \).

Answer: -1


Question: Evaluate the Riemann-Stieltjes integral \( \int_{-1}^{1} x \, dg(x) \), where \( g(x) = \begin{cases} x+1, & \text{if } x < 0, \\ 0, & \text{if } x = 0, \\ x-1, & \text{if } x > 0. \end{cases} \)

Answer: 0


Question: Determine whether there exists a function $f$ such that $f$ has a finite value for each point $x$ in the interval $[0,1]$, but for any neighborhood of $x$, $f$ is unbounded.

Answer: Yes


Question: Determine the number of real values of $k$ for which the points where the lines $3x - 2y - 12 = 0$ and $x + ky + 3 = 0$ intersect the coordinate axes are concyclic.

Answer: 2


Question: Determine the cardinality of the set \( \{ X \subseteq \omega_1 \mid |X| = \aleph_0 \} \). Is there a bijection from this set to \( P(\omega) \), or is there a straightforward counting argument?

Answer: 2^{\aleph_0}


Question: Given the joint probability density function \( f_{xy}(x,y) = 8xy \) for \( 0 < y < x < 1 \), and the marginal densities \( f_x(x) = 4x^3 \) for \( 0 < x < 1 \) and \( f_y(y) = 4y(1-y^2) \) for \( 0 < y < 1 \), find the conditional probability \( P(Y \leq 1/2 \ | \ X > 1/2) \).

Answer: \dfrac{2}{5}


Question: Does there exist a finite nonoriented graph whose automorphism group is transitive but not generously transitive, meaning it is not true that each pair \((x,y)\) of vertices can be interchanged by some automorphism?

Answer: Yes


Question: Calculate the limit: \[ \lim_{n\rightarrow\infty} \sum_{k=0}^n {n \choose k} (-1)^{k}\frac{1}{k+2} \]

Answer: 0


Question: Evaluate the integral \( \oint_{C(0,2)} \frac{e^{2 \pi z}-1}{z(z-i)} \, dz \), where \( C(0,2) \) is the circle centered at 0 with radius 2, oriented in the positive direction.

Answer: 0


Question: Consider you have 100 identical-looking balls, but one of them is either heavier or lighter than the rest. Using a beam balance, determine the minimum number of weighings required to identify the odd ball.

Answer: 5


Question: In a complete lattice $(X, \leq)$ with least element $\perp$ and greatest element $\top$, let $f: X \to X$ be a monotonic function. If both $\top$ and $\perp$ are fixed points of $f$, is every element in $X$ a fixed point? In other words, is $X = \{x \in X : f(x) = x\}$?

Answer: No


Question: Is a finite-dimensional topological vector space over the real numbers locally compact?

Answer: Yes


Question: If $L/K$ is a finite extension of finite fields, is it true that the image of the norm map $N_{L/K}(L)$ is the entire field $K$?

Answer: Yes


Question: Two people start on opposite vertices of an octahedron. Each turn, they move to a random adjacent vertex. What is the probability that the person who moves first will land on the vertex occupied by the other person and win the game?

Answer: \dfrac{2}{5}


Question: Determine whether the integral \( \int_0^1 \frac{1}{t^{\beta}} \sin\left(\frac{\pi}{t}\right) dt \) converges for different values of \( \beta \).

Answer: \beta < 2


Question: How many distinct circular garlands can be made using 3 white flowers and 6m red flowers, where m is a natural number? Two garlands are considered the same if they can be rotated to match each other. Express your answer in terms of m.

Answer: 6m^2 + 3m + 1


Question: Find the value of \( u(1/2, 1/2) \) where \( u(x,t) \) is the solution to the partial differential equation \( u_{tt} - u_{xx} = 1 \) with initial conditions \( u(x,0) = 0 \) and \( u_t(x,0) = 0 \) for \( x \in \mathbb{R} \) and \( t > 0 \).

Answer: \dfrac{1}{8}


Question: Evaluate the limit: \[ \lim_{x\to \infty} \left( \ln(e^{\operatorname{LambertW}(x)}+1)(e^{\operatorname{LambertW}(x)}+1) - x - \ln(x) \right) \] where the Lambert W function is defined as the inverse of the function \( f(W) = We^W \).

Answer: -\infty


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function such that \(|f(x)-f(y)| \geq \frac{1}{2}|x-y|\) for all \(x, y \in \mathbb{R}\). Is \(f\) bijective?

Answer: Yes


Question:
Which matrix $A$ in real Jordan form, when paired with suitable matrices $B$ and $C$, results in a minimal continuous-time state-space model $(A,B,C)$ with the impulse response $h(t)=e^{2t}(t \sin(t)+5\cos(t))$? Choose from the following options:

$A) \quad A=\begin{bmatrix} 2&1\\-1&2 \end{bmatrix}$

$B) \quad A=\begin{bmatrix} 1&2&0&0\\-2&1&0&0\\0&0&1&2\\0&0&-2&1\end{bmatrix}$

$C) \quad A=\begin{bmatrix} 1&2&1&0\\-2&1&0&1\\0&0&1&2\\0&0&-2&1\end{bmatrix}$

$D) \quad A=\begin{bmatrix} 2&1&1&0\\-1&2&0&1\\0&0&2&1\\0&0&-1&2\end{bmatrix}$

$E) \quad A=\begin{bmatrix} 2+i&1&0&0\\0&2+i&0&0\\0&0&2-i&1\\0&0&0&2-i \end{bmatrix}$

Answer: D


Question:
Consider an antisymmetric 3-tensor $A$ over a sequence space, where $A_{ijk} = -A_{jik} = -A_{ikj} \in \mathbb{R}$ for $(i,j,k) \in \mathbb{N}^3$ and $\lVert A \rVert^2 = \sum_{i,j,k} |A_{ijk}|^2 < \infty$. Define the function
\[ f(A) = \sum_{i,j,k,l,m,n} A_{ijk} A_{imn} A_{ljn} A_{lmk}. \]
Is there a constant $C > 0$ such that $|f(A)| \leq C \lVert A \rVert^4$ for all antisymmetric 3-tensors $A$? Justify your answer.

Answer: Yes


Question: Determine the closure of the interval \((0,1)\) in the \(K\)-topology on \(\mathbb{R}\).

Answer: [0,1]


Question: Evaluate the expression \( \left\lfloor \frac{1}{\sqrt[3]{1}}+\frac{1}{\sqrt[3]{2^2}}+\frac{1}{\sqrt[3]{3^2}}+\cdots +\frac{1}{\sqrt[3]{(1000)^2}} \right\rfloor \), where \( \lfloor x \rfloor \) denotes the floor of \( x \).

Answer: 27


Question: Given the polynomial \( P(x) = x^3 + ax^2 + bx + c \) with distinct roots \( x_1, x_2, x_3 \), and a first-degree polynomial \( Q(x) \), find the value of \( \frac{Q(x_1)}{P'(x_1)} + \frac{Q(x_2)}{P'(x_2)} + \frac{Q(x_3)}{P'(x_3)} \).

Answer: 0


Question: Two people each take a multiple-choice test consisting of 30 questions. The questions are selected from a pool of questions of unknown size. Both individuals end up with 15 questions in common. What is the most likely size of the pool of questions?

Answer: 60


Question: John has 100 marbles and wants to split them into 4 groups such that the greatest common divisor of the number of marbles in all of the groups is 1. How many ways can John do this?

Answer: 137540


Question: Consider a non-abelian group $G$ of order 21. Determine the number of normal subgroups of $G$, excluding the identity subgroup $\{e\}$ and $G$ itself.

Answer: 1


Question: Evaluate the double integral \( \int_0^2 \int_{\frac{y}{2}}^1 e^{x^2} \, dx \, dy \).

Answer: e - 1


Question:
A device contains two circuits. The second circuit is a backup for the first, so the second is used only when the first has failed. The device fails when and only when the second circuit fails. Let $X$ and $Y$ be the times at which the first and second circuits fail, respectively. $X$ and $Y$ have the joint probability density function:
$$f_{X,Y}(x,y)=\begin{cases}
6e^{-x} e^{-2y} & 0<x<y<\infty \\
0 & \text{otherwise}
\end{cases}$$
What is the expected time at which the device fails?

Answer: \dfrac{5}{6}


Question: Determine the number of positive integers $x$ less than 2014 for which the expression \(3^x + 5^x + 7^x + 11^x + 13^x + 17^x + 19^x + 23^x + 25^x + 29^x + 31^x + 35^x + 37^x\) is a prime number.

Answer: 0


Question: Consider a sequence of $n$ independent discrete random variables $X(k)$, where $k$ is an integer from $1$ to $n$. Each $X(k)$ can take the values $-\sqrt{k}$ with probability $1/k$, $0$ with probability $1 - 2/k$, and $\sqrt{k}$ with probability $1/k$. Determine whether this sequence adheres to the weak law of large numbers.

Answer: Yes


Question: Let $A = (a_{ij})$ be an $n \times n$ matrix with all entries equal to 0 or 1, where $a_{ii} = 1$ for $i = 1, \cdots, n$ and $\det A = 1$. Determine whether all the eigenvalues of $A$ are equal to 1.

Answer: Yes


Question: Find the maximum value of \(|\sin z|\) on the region \([0,2\pi] \times [0,2\pi]\).

Answer: \cosh(2\pi)


Question: Determine the Lebesgue measure of the image of the function \( f: [0,1] \times [0,1] \to \mathbb{R}^2 \), defined by \( f(x,y) = (y, x^2y + x) \).

Answer: \dfrac{3}{2}


Question: Calculate the expectation \( \mathbb{E}\left[|\langle a, Cx \rangle|^2\right] \) where \( a \sim \mathcal{N}(0,I_m) \) is independent and \( C \in \mathbb{R}^{m \times n} \) has iid \( \mathcal{N}(0,1) \) entries. Fix \( x \in \mathbb{R}^n \).

Answer: m \|x\|^2


Question: Determine if there is more than one occurrence of a power of two between twin primes. If so, how many such occurrences are there?

Answer: 1


Question: Given that $0 \leq a,b,c \leq \frac{3}{2}$ and $a+b+c=3$, find the maximum value of the expression \[ N = a^3 + b^3 + c^3 + 4abc. \]

Answer: \dfrac{243}{32}


Question:
Consider the function \( F(x) = \int_1^x t^{\sqrt{t}} \, dt \) defined for every \( x \ge 0 \). Determine which of the following statements is correct:

(a) \( F(x) > 0 \) for every \( x > 0 \).
(b) \( F(x) > F(e) \) for every \( x > 1 \).
(c) \( x = e^{-2} \) is an inflection point.
(d) None of the above statements are correct.

Answer: C


Question: Given that $f(x,y) = u(x,y) + iv(x,y)$ is an entire function of $z = x + iy$ such that $f(0) = -1$, $\frac{\partial u}{\partial x} = (e^y + e^{-y})\cos x$, and $\frac{\partial u}{\partial y} = (e^y - e^{-y})\sin x$, find $f(\pi/3)$. Assume $u(x,y)$ and $v(x,y)$ satisfy the Cauchy-Riemann equations.

Answer: \sqrt{3} - 1


Question: Determine the number of distinct Celtic knots of size (2, 2), considering knots that differ only by rotations and reflections as identical.

Answer: 1


Question: If $A$ and $B$ are both lower triangular matrices in $M_{n \times n}$, are they simultaneously triangularizable?

Answer: Yes


Question: Consider the self-adjoint semigroup of operators $K_{t}=e^{-tK}$ for $t>0$, where $K$ is a self-adjoint operator. The semigroup extends as a $C_{0}$-semigroup with $||K_{t}||\leq 1$. Determine if the following implication is true: If $K_{t}$ is compact for every $t$, then $K$ has only discrete spectrum.

Answer: Yes


Question: Let $f:I\to\mathbb{R}$ be a nonnegative, bounded, integrable, and continuous function with $I\subseteq\mathbb{R}$ being an interval. If $\int_I f \, d\mu=1$, does $\mu(f>0)>0$ hold, where $\mu$ is the Lebesgue measure?

Answer: Yes


Question: Find the maximum value of the function \( f(x) = \int_0^x \sqrt{t^4+(1-x)^2(x-t)^2} \, \mathrm{d}t \) on the interval \([0,1]\).

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{\sin(1)+\sin\left(\frac{1}{2}\right)+\dots+\sin\left(\frac{1}{n}\right)}{\ln(n)} \]

Answer: 1


Question: Determine the asymptotic behavior of the sum \( \sum\limits_{k=2}^{m}\frac{1}{\ln(k!)} \) as \( m \to \infty \).

Answer: \ln \ln m


Question: What is the remainder when $34!$ is divided by $71$?

Answer: 69


Question: Evaluate the integral \( \int_T x^2 \, dV \), where \( T \) is the tetrahedron bounded by the planes \( x=0 \), \( y=0 \), \( z=0 \), and \( 4x+2y+z=4 \). Set up the region of integration correctly for the triple integral.

Answer: \dfrac{2}{15}


Question: Consider the function \( f(x) = \frac{1}{3}(x^2 + x + 1) \). Define \( f_0(x) = x \) and for all \( k \ge 1 \), \( f_k(x) = f(f_{k-1}(x)) \). Let \( a_k = f_k(0) - f_{k-1}(0) \). Determine whether the series \( \sum\limits_{k=1}^{\infty} k a_k \) is infinite.

Answer: \infty


Question: Consider probability spaces $(\Omega, \mathcal{G}, P)$ and $(\Omega, \mathcal{F}, Q)$ with $\mathcal{F}\subseteq \mathcal{G}$ and $P|_{\mathcal{F}}=Q$. Let $X$ be a $Q$-integrable and $\mathcal{F}$-measurable random variable. Prove or disprove that $E_P[X]=E_Q[X]$. Provide a justification for your answer.

Answer: E_P[X] = E_Q[X]


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{g(x)}{x-a} \delta(x-a) \, dx \), where \( \delta(x) \) is the Dirac delta distribution and \( g(x) \) is a nonsingular function.

Answer: g'(a)


Question: Calculate the probability \( P(XY + Z - XZ = 2) \) given that \( Z \sim N(0,1) \), \( Y \sim B(3, \frac{1}{2}) \), and \( X \sim B(1, \frac{1}{3}) \), where \( X, Y, Z \) are independent random variables.

Answer: \dfrac{1}{8}


Question: Determine whether the following statement is true or false: If \( \limsup\limits_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| > 1 \), then \( \sum a_n \) diverges.

Answer: B


Question: In a ticket counter, females and males arrive according to independent Poisson processes with rates of 30 and 20 customers per hour, respectively. Given that exactly two customers arrived in the first five minutes, calculate the probability that both customers arrived in the first three minutes.

Answer: \dfrac{9}{25}


Question: Consider an event with probability \( p \). What is the probability that this event never occurs over an infinite number of trials?

Answer: 0


Question: Determine the number of ways to paint $n$ sectors of a disk using $n$ different colors such that no two adjacent sectors have the same color.

Answer: (n-1)!


Question: Find the degree of the reflection map \( f: S^n \to S^n \) given by \( f(x_0, \dots, x_n) = (x_0, \dots, x_{n-1}, -x_n) \) using singular homology theory.

Answer: -1


Question: Consider two matrices: A, an n x m matrix, and B, an m x n matrix. Let AB be an n x n matrix and BA be an m x m matrix. Do the non-zero eigenvalues of AB and BA have the same algebraic multiplicity?

Answer: Yes


Question: Determine whether the following statement is true: Let \(D_N(x)=\frac{\sin [(N+(1/2))t]}{\sin (t/2)}\) be the Dirichlet kernel. Let \(x(N)\) be the number in \(0<x<\pi/N\) such that \(D_N(x)=1\). Is it true that \(\left|\int_{x(N)}^{\pi/N} D_N(t)\mathrm dt \right|=O\left(\frac1{N}\right)\)?

Answer: True


Question: Evaluate the limit \( \lim_{N \to \infty} \frac{C_{2N}^{N}}{2^{2N}} \), where \( C_{2N}^{N} = \frac{(2N)!}{N!N!} \). Determine if this limit is equal to zero.

Answer: 0


Question: Let $(M,d)$ be a compact metric space and $f: M \to M$ be a bijective function such that $d(f(x),f(y)) \leq d(x,y)$ for all $x, y \in M$. Is $f$ an isometry, i.e., does $d(f(x),f(y)) = d(x,y)$ for all $x, y \in M$?

Answer: Yes


Question: Determine whether the following statement is true for integer exponents in a group: \((ab)^x = b^x a^x\).

Answer: False


Question: Does every IA automorphism \( \varphi \in \mathrm{IA}_n \) have a fixed point in the complement of the commutator subgroup \( [F_n,F_n] \)?

Answer: Yes


Question: Does there exist a set $A \subseteq P(\{x_1, x_2, x_3, \ldots \})$ of cardinality $\aleph$ such that for every two sets $B, C \in A$, either $B \cap C = B$ or $B \cap C = C$?

Answer: Yes


Question: In a game where players A and B alternate writing one digit to form a six-digit number \( \overline{abcdef} \), with \( a, b, c, d, e, f \) being distinct digits and \( a \neq 0 \), player A wins if the number is composite, and player B wins if it is prime. Is there a strategy that guarantees a win for either player A or player B?

Answer: A


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{2^{n-1}-4^n}{3^n\sin n + 4^{n+2}}$$

Answer: -\dfrac{1}{16}


Question: Determine the structure of the quotient ring \( \Bbb{C}[X,Y]/(Y-X^2) \). Show that it is isomorphic to \( \Bbb{C}[X] \).

Answer: \Bbb{C}[X]


Question: In the topology of uniform convergence on compact sets, is the set \( X_\Omega = \{ f \in H^\infty(\Omega) : \sup_{z \in \Omega} |f(z)| \leq 1 \} \) dense in the set \( X_B = \{ f \in H^\infty(B) : \sup_{z \in B} |f(z)| \leq 1 \} \)?

Answer: No


Question:
Given that 2 is a root of the equation \(|A-xI|=0\) where \(A\) is a non-singular matrix, and \(\frac{|A|}{2}\) is a root of the equation \(|B-xI|=0\), determine which of the following could be the matrix \(B\):

(A) \(e\)
(B) \(\text{Adj}(A)\)
(C) \(\cos(\alpha)-i\sin(\alpha)\)
(D) NONE

Select the correct option.

Answer: B


Question: Find all entire functions $f$ such that \( \lim_{z\to \infty}\left|\frac{f(z)}{z}\right|=0 \).

Answer: f(z) = c


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

Let \(X\) be a topological space and \(A \subseteq X\) a subset. \(A\) is called sequentially compact if every sequence in \(A\) has a convergent subsequence with a limit in \(A\). A point \(x \in X\) is an \(\omega\)-accumulation point of \(A\) if every open neighborhood of \(x\) contains infinitely many points of \(A\).

The statement to evaluate is: \(A\) is sequentially compact if and only if every countably infinite subset \(B \subseteq A\) has an infinite subset \(B' \subseteq B\) that has an \(\omega\)-accumulation point \(x \in X\).

Answer: B


Question: Evaluate the product \( \prod_{i=1}^{89} \tan i^\circ = \tan 1^\circ \cdot \tan 2^\circ \cdots \tan 89^\circ \).

Answer: 1


Question: Evaluate the integral: \[ \int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm{d}x \]

Answer: 0


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

Answer: e


Question: Evaluate the limit: \[ \lim_{x \searrow 0} \frac{\frac{\pi}{6} - \frac{1}{3} \int_0^{1/x^3} \frac{\sin t}{t} \,\mathrm{d}t}{x} = 0. \]

Answer: 0


Question:
Determine whether the following limit converges to 0:
\[ \lim_{N \to \infty} e^N \int_{N}^{\infty} \frac{e^{-x}}{\log(x)} \, dx \]

Answer: 0


Question: Find all functions $g: \mathbb{N} \to \mathbb{Z}$ such that for all $m, n \in \mathbb{N}$, the expression $m+n$ divides $g(m) + g(n)$. Provide a complete characterization of such functions.

Answer: g(n) = cn


Question:
Given that $xyz = -1$, find the value of the determinant:
\[\begin{vmatrix} x & x^2 & x^3+1\\ y & y^2 & y^3+1 \\ z & z^2 & z^3+1 \end{vmatrix}\]

Answer: 0


Question: Consider the set TW of all transitive well-founded binary relations $R$ on the set of natural numbers. For a countable ordinal $\alpha$, let TW$(\alpha)$ denote the set of all relations $R$ in TW that have a rank smaller than $\alpha$. Is it true that TW$(\alpha)$ has a Borel rank of at most $\alpha$?

Answer: Yes


Question:
Is it true that the average value of the orders of all elliptic curves over the finite field \( \mathbb{F}_p \) is \( p + 1 \)? Specifically, for a prime \( p \), let \( \mathbb{F}_p \) be the field of \( p \) elements. Consider the set \( S = \{(a,b) \in \mathbb{F}_p \times \mathbb{F}_p : 4a^3 + 27b^2 \neq 0 \text{ in } \mathbb{F}_p\} \), where each \((a,b) \in S\) defines an elliptic curve \( E(a,b,p) = \{(x,y) \in \mathbb{F}_p \times \mathbb{F}_p : y^2 = x^3 + ax + b \text{ in } \mathbb{F}_p\} \cup \{\infty\} \). Is the following equation true?
\[ \frac{1}{|S|} \sum_{(a,b) \in S} |E(a, b, p)| = p + 1 \]

Answer: True


Question: Consider all the permutations of the 26 English alphabets that start with the letter 'z'. In how many of these permutations is the number of letters between 'z' and 'y' less than the number of letters between 'y' and 'x'?

Answer: 144 \times 23!


Question:
If \(x = (9+4\sqrt{5})^{48} = [x] + f\), where \([x]\) is the integer part of \(x\) and \(f\) is the fractional part, find the value of \(x(1-f)\). Choose the correct answer from the options below:

\(\text{a.) } 1 \\
\text{b.) less than } 1 \\
\text{c.) more than } 1 \\
\text{d.) between } 1 \text{ and } 2 \\
\text{e.) none of these}

Answer: A


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\int_{0}^{x^4}(e^{u^{10}}-1)du}{x^{44}} \]

Answer: \dfrac{1}{11}


Question: Consider the equation \(2x^3-3x^2+2y^3+3y^2-y=0\). Using the implicit function theorem, this equation defines a function \(y=f(x)\) in a neighborhood of \((0,0)\). Determine the limit of \(\frac{f(x)}{x}\) as \(x \to 0\).

Answer: 0


Question: Let \( f \) be a real-valued continuous function defined on \([-2, 2]\) that is three times differentiable on \((-2, 2)\). Given that \( f(2) = -f(-2) = 4 \) and \( f'(0) = 0 \), prove that there exists a point \( x \in (-2, 2) \) such that \( f'''(x) \ge 3 \).

Answer: 3


Question: Find all integers \( n > 1 \) such that \( \frac{2^n + 1}{n+1} \) is an integer.

Answer: \emptyset


Question: Two points are dropped at random onto the unit interval, creating three sub-intervals. What is the probability that the central sub-interval is two times shorter than the right sub-interval?

Answer: \dfrac{1}{3}


Question: Determine the complexity of the recurrence relation \( T(n) = \sqrt{2n}T(\sqrt{2n} )+ n^2 \).

Answer: O(n^2)


Question: For a fixed integer $k > 2$, can a word $w$ in the generators of the group $B_n^{(k)}$, defined by the relations of the braid group $B_n$ with additional relations $\sigma_i^k = 1$ for $i=1,\ldots,n-1$, be reduced to a normal form in polynomial time with respect to $n$ and $|w|$?

Answer: Yes


Question: Determine whether the sequence \( a_{n}=n\left(\prod\limits_{k=1}^{n} \sin\frac{1}{k}\right)^{1 / n} \) converges. If it converges, find its limit.

Answer: e


Question: Consider a sequence $X_{n} \in \ell^{1}$ such that $X_{n} \geq 0$ and $|X_{n}|_{1} = c$ for all $n$, and $X_{n} \to a \in \ell^{1}$. Define the sequence in $\mathbb{R}$ as $s_{m} = \lim_{n\to\infty}\sum_{i>m}X_{n,i}$. Determine whether $\lim_{m\to\infty}s_{m} = 0$.

Answer: 0


Question: Evaluate the integral: \[ \int_{-\infty}^{\infty} e^{-\lvert x-y \rvert} \, \mathrm{d}x. \]

Answer: 2


Question: Determine whether the equation $x(3y-5)=y^2+1$ has any integral solutions. If so, provide one such solution.

Answer: (-1, 1)


Question: Given the functional equation \( f(x-y) = f(x) \cdot f(y) - f(a-x) \cdot f(a+y) \) and the condition \( f(0) = 1 \), find the value of \( f(2a-x) \).

Answer: -f(x)


Question: Consider a linear hypergraph $H=(V,E)$ where $V$ is finite and each edge $e \in E$ satisfies $|e| \geq 3$. Does $H$ necessarily have an injective edge choice function?

Answer: Yes


Question: Determine the values of \( \alpha \) for which the generalized Gaussian probability density \( p_{\alpha}(t) \propto \exp (- |t|^\alpha) \) is infinitely divisible, given that \( 0<\alpha<\infty \).

Answer: (0, 2]


Question: Find the volume of the solid bounded below by the surface \( z = x^2 + 4y^2 \) and above by the plane \( z = 4x - 8y - 4 \).

Answer: 4\pi


Question: Find the smallest natural number $n$ such that \( \left(\frac{1}{5} + i \frac{\sqrt{3}}{5}\right)^n \) is a real number.

Answer: 3


Question: Estimate the number of terms needed to approximate the convergent series \( \sum_{n=1}^\infty\frac{(-1)^nn}{5^n} \) to within 0.0001. Determine the smallest integer \( n \) such that \( \frac{n}{5^n} < 0.0001 \).

Answer: 7


Question: Given the sequence defined by \( x_1 = \frac{1}{2} \) and \( x_{n+1} = x_n - x_n^2 \), determine if there exists a real number \( \alpha \) such that \( \lim_{n \to \infty} n^\alpha x_n \neq 0 \).

Answer: 1


Question: Determine the number of one-dimensional F-algebras up to isomorphism, where F is a field. An F-algebra is a vector space over F with a defined multiplication operation. Note that the algebra does not have to be unital.

Answer: 2


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{n^n}{n!}. \]

Answer: \infty