Question: Find the area bounded by the inverse of the function \( f(x) = x^3 + 3x + 1 \), the ordinates at \( x = -3 \), \( x = 5 \), and the x-axis.

Answer: \dfrac{9}{2}


Question: Determine how many equilateral triangles can be formed with pearls, starting with 2008 pearls and increasing the number of pearls by an arithmetic sequence where each term increases by 1 more than the previous increase. Specifically, after n rounds, the number of pearls is given by \( 2008 + \frac{n(n+1)}{2} \). How many such equilateral triangles can be formed as the number of pearls increases indefinitely?

Answer: 2


Question: Determine if there exists a value \( M \geq 0 \) such that the function \( g(x) = x f(x) + \sqrt{x} \) is increasing on the interval \([M, \infty)\) and \( g(x) \to \infty \) as \( x \to \infty \), given that \( f:[0,\infty)\to[0,\infty) \) is a decreasing function that decreases to 0 as \( x \to \infty \) and \( \int_1^\infty \frac{f(x)}{x} = \infty \).

Answer: Yes


Question: Let \( f: \bar{U} \rightarrow \mathbb{C} \) be a continuous function such that \( f|_U \) is holomorphic, where \( U \) is the square with vertices at \( 0, 1, i, i+1 \). Given that \( f(t) = \cos t + e^t \) for \( t \in [0,1] \), determine the function \( f \).

Answer: f(z) = \cos z + e^z


Question: Let $f$ be an entire function such that there exists a constant $A > 0$ with $|f(z)| \le A|z|^2$ for all $z \in \mathbb{C}$. Given that $f(1) = f(-1) = 1$, find the formula for $f(z)$. \(\text{Express your answer in terms of } z.\)

Answer: z^2


Question: Is a characteristic 0 field that is complete with a discrete valuation always non-Archimedean?

Answer: Yes


Question:
Given that \(a, b, c \in \mathbb{R}\) are distinct and satisfy the equation \(\det\begin{bmatrix}2a&b&c\\b&c&2a\\c&2a&b\end{bmatrix} = 0\), determine the location of the roots of the quadratic equation \(24ax^2 + 4bx + c = 0\). Choose the correct option:

(a) At least one root in \([0, \frac{1}{2}]\)
(b) At least one root in \([-\frac{1}{2}, 0)\)
(c) At least one root in \([1, 2]\)
(d) None of these

Answer: A


Question: Find the remainder when the number consisting of 1000 consecutive ones is divided by 13.

Answer: 6


Question: Evaluate the integral: $$\int_{0}^{4\pi} \ln|13\sin x+3\sqrt3 \cos x|\;\mathrm{d}x$$

Answer: 4\pi \ln 7


Question: Consider two identically distributed independent random vectors $X$ and $Y$ of dimension $n$, where $n$ is large. Both $X$ and $Y$ have non-negative integer entries less than or equal to $n$, and their entropies are $H(X) = H(Y) = n$. Determine the maximum possible value of $H(X + Y)$ and evaluate how close it can get to $2n$.

Answer: 2n


Question: Determine the number of solutions to the equation \( \ln(x) = \frac{x^2}{2} - 1 \).

Answer: 2


Question: Evaluate the integral \( \int_0^{\frac{\pi}{2}} \frac{\mathrm{d}x}{1+\tan^{2013}x} \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \left(e^{(1+\frac{k}{n})^2} - \frac{3e^{(1 + \frac{3k}{n})}}{2\sqrt{1 + \frac{3k}{n}}}\right). \]

Answer: 0


Question: Find the maximum positive integer \( n \) such that \( w(n) + \varphi(n) - \pi(n) = 1 \), where \( \varphi(n) \) is Euler's totient function, \( \pi(n) \) is the prime-counting function, and \( w(n) \) is the number of distinct prime factors of \( n \).

Answer: 30


Question: Show that \( \iint\limits_D \left(x\frac{\partial f }{\partial x}+ y \frac{\partial f}{\partial y}\right) \, dx \, dy = \frac{\pi}{2e} \), where \( f \in C^2(D) \) and \( D=\{(x,y)\in \mathbb R^2: x^2+ y^2 \le 1\} \), given that \( \frac{\partial^2 f }{\partial x^2} + \frac{\partial^2 f }{\partial y^2} = e^{-x^2 -y^2} \).

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


Question: Let $f(z)$ be a rational function on $\mathbb{C}$. If the residues of $f$ at $z=0$ and $z=\infty$ are both $0$, is it true that $\oint_{\gamma} f(z)\,\mathrm{d}z=0$ for any closed curve $\gamma$ in $\mathbb{C}$?

Answer: No


Question: Let $A$ and $B$ be square matrices such that $A + B^{-1} = I$ and $A^2 = 0$. Prove or disprove that $AB = BA$.

Answer: AB = BA


Question: If two continuous functions defined on metric spaces $X$ and $Y$ are equal on a dense subset of $X$, are they equal on all of $X$?

Answer: Yes


Question: Calculate the limit of the function \( f(x,y) = \frac{\ln(x+e^y)-x-y}{\sqrt{x^2+y^2}} \) as \((x,y)\) approaches \((0,0)\). Find \( \lim_{(x,y) \to (0,0)} f(x,y) \).

Answer: 0


Question: Determine the number of curly brackets in the expression \( \wp^5(\varnothing) \), where \( \varnothing \) is the empty set and counts as \( \{ \} \) (1 curly bracket).

Answer: 3211265


Question: Is the set \( \{c_1 a_1 + \cdots + c_n a_n \mid c_i \geq 0 \text{ for all } i\} \) closed in \( \mathbb{R}^d \) given vectors \( a_1, \ldots, a_n \in \mathbb{R}^d \)?

Answer: Yes


Question: Find \( f'(0) \) for the function \( f(x) = \int_{0}^{x} \frac{\sin(t)}{t} \, dt \).

Answer: 1


Question: Let $\mathcal{A}$ and $\mathcal{B}$ be maximal subsets of $\textbf{Mat}_n(\mathbb{R})$ consisting only of asymptotically nilpotent matrices, and closed under the Lie bracket operation. Is there a matrix $P \in \text{GL}_n(\mathbb{R})$ such that $P\mathcal{A}P^{-1} = \mathcal{B}$?

Answer: Yes


Question: Consider the function \( f(t) = 2 + \sin(t) + \sin(t\sqrt{2}) \). Determine if there exists a sequence \((t_n)_{n \geq 0}\) such that \( \lim_{n \to \infty} f(t_n) = 0 \).

Answer: Yes


Question: Let $(X, \Vert \cdot\Vert)$ be a normed space and $A \subset X$. If $x \in X$ is an accumulation point of $A$, does every open ball $B(x, r)$, where $r > 0$, contain infinitely many points of $A$? Justify your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^a \frac{f(x)}{f(x) + f(a-x)} \, dx \), where \( f \) is a continuous function on \([0, a]\) and \( f(x) + f(a-x) \neq 0 \) for all \( x \in [0, a] \).

Answer: \dfrac{a}{2}


Question: Given the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) with foci \(F_1\) and \(F_2\), where \(F_1 = (\sqrt{5}, 0)\) and \(F_2 = (-\sqrt{5}, 0)\), point \(P\) is on the ellipse such that \(|PF_1|:|PF_2| = 2:1\). Calculate the area of triangle \(\triangle PF_1F_2\).

Answer: 4


Question: Evaluate the integral \(\int_{-2}^{2}\int_{y^2-3}^{5-y^2}dxdy\) by changing the order of integration to \(dydx\).

Answer: \dfrac{64}{3}


Question: Find all derivations on the set of continuous functions $C(X)$, where $X = \{1, 2, \ldots, n\}$ and $C(X)$ is the set of all continuous functions from $X$ to $\mathbb{C}$. A derivation $\delta: C(X) \to C(X)$ satisfies the property $\delta(fg) = \delta(f)g + f\delta(g)$ for all $f, g \in C(X)$. Determine the derivations for $X = \{1, 2\}$.

Answer: 0


Question: For given integers $x$ and $d$, determine the largest number of $\mathbb{Z}^d$ lattice points that are mutually $x$-visible. Denote this number as $g(x,d)$. For instance, it is known that $g(0,2) = 4$ and $g(1,2) \ge 9$. What is $g(x,d)$?

Answer: (x + 2)^d


Question: For every positive integer \( n \), determine the greatest possible value of the quotient \[ \frac{1-x^n-(1-x)^n}{x(1-x)^n+(1-x)x^n} \] where \( 0 < x < 1 \).

Answer: 2^n - 2


Question: Determine if $c=0$ is the unique center of symmetry for the mixture density $f(x) = 0.5f_{-a}(x) + 0.5f_{a}(x)$, where $f_{-a}(x)$ is the probability density function of $N(-a, \sigma^2)$ and $f_{a}(x)$ is the probability density function of $N(a, \sigma^2)$. Specifically, is $f(x) = f(2c-x)$ for any $x$ only when $c=0$?

Answer: 0


Question: Given a finite poset \((P, <)\) with \(n\) elements chosen uniformly at random, determine a good upper bound on the expected height of \((P, <)\). The height of a poset is defined as the length of the longest maximal chain within it.

Answer: 3


Question: Evaluate the limit \( \lim_{n \to \infty} \int_0^1 \frac{\sin{nt}}{t} f(t) \, \mathrm{d}t \), where \( f(t) \colon [0,1] \to \mathbb{R} \) is a smooth function with \( f(0) = 1 \).

Answer: \dfrac{\pi}{2}


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be defined as follows: \[ f(x) = \begin{cases} \frac{e^x - 1}{x}, & x \neq 0 \\ 1, & x = 0 \end{cases} \] Determine the largest integer \( n \) such that \( f \) is \( n \)-times differentiable at \( x = 0 \).

Answer: \infty


Question: Evaluate the limit: $$\lim_{x\to0}\left(\frac {\csc(x)}{x^3} - \frac{\sinh(x)}{x^5}\right)$$

Answer: \dfrac{1}{90}


Question: Let $u(x,y)$ be a harmonic function on a domain $D \subset \mathbb{C}$. If $D \cap \mathbb{R}$ is non-empty, is the function $f(x) = u(x,0)$ real analytic on $D \cap \mathbb{R}$?

Answer: Yes


Question: Find the smallest prime number factor of \(2020^8 + 1\).

Answer: 17


Question: Let \( A \) be an \( n \times n \) Hermitian matrix such that \( A^5 + A^3 + A = 3I_n \). Determine whether it is true that \( A = I_n \).

Answer: A = I_n


Question: Let $X$ and $Y$ be Hilbert spaces with a continuous embedding $X \hookrightarrow Y$. Consider a family of uniformly bounded linear operators $T_n$ such that \( \exists c>0 : \|T_n\|_{\mathcal{L}(X,Y)} \leq c \) for all \( n \in \mathbb{N} \). Assume the operators converge pointwise, i.e., \( T_n x \to x \) in \( Y \) for all \( x \in X \). Given a bounded sequence \( \{x_m\}_{m} \subset X \), does \( \max_{m \in \mathbb{N}} \|x_m - T_n x_m\|_Y \to 0 \) as \( n \to \infty \) hold?

Answer: No


Question: Determine the radius of convergence of the power series \( \sum_{n=0}^\infty 3^{-\frac{n(n+1)}{2}} z^{n^2} \).

Answer: \sqrt{3}


Question: Find the Hensel lift of the monic polynomial \( g(x) = x^3 + x + 1 \) over \( \mathbb{Z}_8 \), given that \( g(x) \) has no multiple roots and \( x \nmid g(x) \) in \( \mathbb{F}_2[x] \).

Answer: x^3 + x + 1


Question: Does there exist a bounded linear operator \(T: \ell^{2}(\mathbb{N}) \rightarrow \ell^{2}(\mathbb{N})\) which is normal but not self-adjoint? A bounded linear operator \(T\) on a Hilbert space \(H\) is normal if \(TT^{*} = T^{*}T\).

Answer: Yes


Question: Given a triangle ABC with an inscribed circle centered at O, three smaller triangles are formed: \( \triangle OA_1B_1 \), \( \triangle OB_1C_1 \), and \( \triangle OC_1A_1 \). Define points \( O_1, O_2, O_3 \) as the centers of the circumscribed circles around \( \triangle OB_1C_1 \), \( \triangle OC_1A_1 \), and \( \triangle OA_1B_1 \) respectively. If the area of \( \triangle ABC \) is 24, find the area of \( \triangle O_1O_2O_3 \).

Answer: 6


Question:
Determine whether the following claim is true for a Riemannian manifold $\M$ and a point $p$ on it:
\[
\inf_{r \geq 0} \sup_{u,v \in \mathbb{B}(p,r)}
\frac{\|\exp_p^{-1}u - \exp_p^{-1}v\|_p}
{d(u,v)}=1.
\]

Answer: True


Question: Compute the limit of the sequence: \[ \lim_{n\rightarrow \infty}(2-2^{1/2})(2-2^{1/3})\cdots(2-2^{1/n}). \]

Answer: 0


Question: Find the remainder when $10^{5^{102}}$ is divided by 35.

Answer: 10


Question: Determine whether the following sum approaches zero as \( n \to \infty \): \[ \sum_{k=1}^{n}\frac{k^{1-n}}{n(n-1)}. \]

Answer: 0


Question: Let \( f \in L^1(\mathbb{R}^d) \) be a function that is continuous at 0. Prove or disprove that \( f(0) < \infty \).

Answer: f(0) < \infty


Question: Find all integer solutions to the equation \( \left\lfloor x \right\rfloor^3 + 2x^2 = x^3 + 2\left\lfloor x \right\rfloor^2 \), where \( \left\lfloor t \right\rfloor \) denotes the largest integer not exceeding \( t \).

Answer: \mathbb{Z}


Question: Calculate the fundamental group \(\pi_1(\mathbb{R}^3 - T)\), where \(T\) is a solid torus. Describe the group and identify its generator.

Answer: \mathbb{Z}


Question: Find the volume of the region bounded by the equation \((x^2 + y^2 + z^2)^2 = x\) using spherical coordinates.

Answer: \dfrac{\pi}{3}


Question: Given the polynomial \( p(x) = x^4 + ax^3 + bx^2 + cx + d \) with the conditions \( p(1) = 10 \), \( p(2) = 20 \), and \( p(3) = 30 \), compute \( \frac{p(12) + p(-8)}{10} \).

Answer: 1984


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

Answer: a = e^{-b}


Question: Find the value of the sum \( \sum_{j = 0}^{n}{n \choose j}(-1)^{j}(n - j)^k \) when \( k < n \).

Answer: 0


Question: Find the smallest integer \( n \) such that a square can be decomposed into \( n \) non-overlapping triangles, all of whose angles are acute.

Answer: 8


Question: Does there exist a sequence \( \{a_n\} \) in the real numbers such that for every element \( l \) in an infinite countable set \( L \), there is a subsequence of \( \{a_n\} \) that converges to \( l \)?

Answer: Yes


Question: Find the value of \([2^{(2^{403})}]\) in \(\mathbb{Z}_{23}\), where \(0 \le a < 23\).

Answer: 3


Question: Evaluate the infinite series: $$ \sum_{n=1}^{\infty}\frac{n}{1+n^2+n^4} $$

Answer: \dfrac{1}{2}


Question:
Let \(X_1, X_2, \ldots\) be independent random variables with the following distributions:
\[ P(X_n = n^2-1 )= \frac{1}{n^2} \]
\[ P(X_n = -1 )= 1-\frac{1}{n^2} \]
Define \(S_n = X_1 + X_2 + \ldots + X_n\). Prove that \(\frac{S_n}{n} \to -1 \ a.e.\)

Answer: -1


Question: Calculate the determinant of the following 4x4 matrix: $$\begin{vmatrix} a^2 & (a+1)^2 & (a+2)^2 & (a+3)^2 \\ b^2 & (b+1)^2 & (b+2)^2 & (b+3)^2 \\ c^2 & (c+1)^2 & (c+2)^2 & (c+3)^2 \\ d^2 & (d+1)^2 & (d+2)^2 & (d+3)^2 \end{vmatrix}$$

Answer: 0


Question: Find the value of \( g(x) = \lim_{n \to \infty} \frac{\{x\}^{2n} - 1}{\{x\}^{2n} + 1} \), where \( \{x\} \) is the fractional part of \( x \).

Answer: -1


Question: Compute the Laplacian of the scalar field \( \dfrac{(\mathbf{\mu} \cdot \mathbf{r})}{r^3} \), where \( r = \Vert \mathbf{r} \Vert \).

Answer: 0


Question: Calculate the line integral \( \int_C \vec{B} \cdot d\vec{r} \), where \( \vec{B} = \frac{\cos(\alpha)}{p^2}e_p + \left(\frac{\sin(\alpha)}{p^2}+\frac{1}{ap}\right)e_\alpha + 0e_z \) and the curve \( C \) is parameterized by \( \vec{r} = (b\cos(s),b\sin(s),b) \) with \( b>0 \) and \( s: 0\rightarrow2\pi \).

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


Question: Find the residues of the function \( f(z) = \frac{2z+1}{z(z+1)} \) at the isolated singular points \( z=0 \) and \( z=1 \). Use these residues to evaluate the integral \( \int_C f(z) \, dz \), where \( C \) is the circle of radius 2 centered at the origin, using the Residue Theorem.

Answer: 4\pi i


Question: Evaluate the limit: \[ \lim_{x\to{0}^+}\frac{\exp[-(\log(x))^2]}{x} \]

Answer: 0


Question: Let $M$ be a connected, non-compact, non-orientable topological manifold of dimension $n$. Is the top singular cohomology group $H^n(M, \mathbb{Z})$ zero?

Answer: 0


Question: Let $G$ be a group containing a free subgroup $F = F_2 = \langle a, b \rangle$. Define $H = \langle a, b^2 \rangle$, and assume the centralizer of $F$ in $G$ is trivial, i.e., $C_G(F) = \{e\}$. Suppose $e \neq g \in G$ and $gag^{-1}, gb^2g^{-1} \in H$. Is it true that $gbg^{-1} \in F$?

Answer: Yes


Question: Given a homomorphism \( h : \{a, b\}^* \to \{0, 1\}^* \) defined by \( h(a) = 01 \) and \( h(b) = 10 \), and a regular language \( L = L((00 + 1)^*) \), determine the preimage \( h^{-1}(L) \).

Answer: (ba)^*


Question: Use Green's Theorem to evaluate the line integral \( \int_C y^3 \, dx + x^3 \, dy \), where \( C \) is the ellipse \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \).

Answer: -180\pi


Question: Find the coefficient of \(x^{1397}\) in the expansion of \((x^3 + x^4 + x^5 + \, \ldots)^6\).

Answer: \dbinom{1384}{5}


Question: Determine whether there exists a constant \( C \) such that for any integrable function \( f : [a, b] \rightarrow \mathbb{R} \), the inequality \( \left|\int_a^b f(t) \, dt\right| \geq C \int_a^b |f(t)| \, dt \) holds.

Answer: 0


Question: A windmill has 5 wings, each symmetrically connected to the axis and consisting of two parts. If 4 parts are colored black, 3 parts are colored red, and 3 parts are colored orange, how many distinct ways can the windmill be colored, considering rotations? Use Pólya's enumeration theorem and the cycle index for the group of rotations of a regular 10-polygon, given by \( \frac{1}{10}\sum_{d\,|\, 10} \varphi(d)t_d^{\frac{10}{d}} \), where \( \varphi(d) \) is Euler's totient function. Replace \( t_j \) with \( b^j + r^j + o^j \) and find the coefficient of \( b^4 \cdot r^3 \cdot o^3 \).

Answer: 420


Question: Given that \( x^{1/4} + \frac{1}{x^{1/4}} = 1 \), find the value of \( x^{1024} + \frac{1}{x^{1024}} \).

Answer: -1


Question: Is there a Banach space that is not isomorphic to $L^p(X)$ for any $1 \leq p \leq \infty$ and any measure space $X$?

Answer: Yes


Question: Calculate the discriminant of the number field \( \mathbb{Q}(\sqrt{-13}) \).

Answer: -52


Question: Does a $C^{1}$ function map a set of measure zero into a set of measure zero?

Answer: Yes


Question: Let \(X\) and \(Y\) be independent random variables, each following a standard exponential distribution. Define \(T = X + Y\). Calculate the conditional expectation \(E(X \mid T)\).

Answer: \dfrac{T}{2}


Question: Determine whether the following statement is true or false: For a function $f: \mathbb{R} \rightarrow \mathbb{R}$, is it true that $f \circ f = f$ if and only if $f \upharpoonright \text{rng}(f) = \text{id}_{\text{rng}(f)}$? Justify your answer.

Answer: True


Question: Calculate the sum \( \sum_{k=0}^\infty \frac{1}{k!} e^{ikx} \).

Answer: e^{e^{ix}}


Question: Let \( f(x) \) be a function defined on the set of integers such that \( f(92+x) = f(92-x) \), \( f(1748+x) = f(1748-x) \), and \( f(1992+x) = f(1992-x) \). Determine the maximum number of distinct elements that can be present in the range of \( f(x) \).

Answer: 5


Question: Find the limit of the decimal part of \(\sqrt{4n^2+38n}\) as \(n\) approaches infinity.

Answer: \dfrac{1}{2}


Question: In \(\triangle ABC\), let \(D\) and \(E\) be the feet of the perpendiculars from \(A\) to the angle bisectors of angles \(B\) and \(C\), respectively. The line through \(D\) and \(E\) intersects \(AC\) at \(X\) and \(AB\) at \(Y\). Determine the fractional part of the area of \(\triangle ABC\) that is occupied by the region \(XYBC\).

Answer: \dfrac{3}{4}


Question: Calculate the volume of the region that lies inside the ellipsoid \( \frac{x^2}{49} + y^2 + z^2 = 1 \) but outside the sphere \( x^2 + y^2 + z^2 = 1 \).

Answer: 8\pi


Question: Determine the values of \(a\) for which the improper integral \(I = \int_{0}^\infty x^a \ln{\left(1+\frac{1}{x^2}\right)} \, dx\) converges.

Answer: (-1, 1)


Question: The temperature outside a building is stable at $0^{\circ}$ Celsius. At time $t = 0$, the temperature inside the building is $18^{\circ}$ Celsius. The building has a time constant of 4 hours. At $t = \frac{1}{3}$ hour, a heat source is turned on, generating heat at an intensity of 60000 BTU/hr for 1/2 hour. Given that the heat capacity $\gamma$ of the building is $\frac{1}{6}$ degree per thousand BTU, find the temperature $T(t)$ inside the building at $t = 1$ hour. Use the equation $\frac{dT(t)}{dt} = K(M(t) - T(t)) + \gamma H(t)$, where $H(t)$ describes the production of heat, and express $H(t)$ using unit step functions.

Answer: 18.53


Question: Let $U$ be an open set in $\mathbb{C}$ and $a \in U$. Suppose $r > 0$ is such that the closed disk $\overline{D}(a,r) = \{z \in \mathbb{C} : |z-a| \leq r\}$ is contained in $U$. Can we conclude that there exists $R > r$ such that $\overline{D}(a,r) \subset D(a,R) \subset U$, where $D(a,R) = \{z \in \mathbb{C} : |z-a| < R\}$?

Answer: Yes


Question: Given that $f \cdot g$ and $f$ are differentiable at a point $a$, under what condition on $f$ is $g$ also differentiable at $a$?

Answer: f(a) \neq 0


Question: Determine the number of ordered pairs of positive integers \((m, n)\) such that the fraction \(\frac{n^2+1}{mn-1}\) is a positive integer.

Answer: 6


Question: Determine the maximum number of pairwise non-similar linear transformations on a three-dimensional vector space, each having the characteristic polynomial \(x^3\).

Answer: 3


Question:
Find the limit of the product series as \( n \to \infty \):
\[
\lim\limits_{n\to\infty} \left(1+\frac{ 1 }{ a_{ 1 } }  \right) \left( 1+\frac { 1 }{ a_{ 2} }  \right)\cdots\left( 1+\frac { 1 }{ a_{ n } }  \right)
\]
where \( a_1=1 \) and \( a_n=n(1+a_{n-1}) \) for all \( n \geq 2 \).

Answer: e


Question: Determine the radius of convergence of the power series expansion of the rational function \( f(z) = \frac{z^2 - 1}{z^3 - 1} \).

Answer: 1


Question: Is there a prime number for every positive integer \( n \) that can be expressed as the sum of \( n \) distinct powers of 2?

Answer: Yes


Question: Determine the characteristic of a unital ring $R$ where for each element $r \in R$, the equation $r^3 + 2r^2 + r = 0$ holds.

Answer: 2


Question: Determine whether there exists a function \( f:[0,\infty) \to \{-1, 1\} \) such that the integral \( \int_0^\infty{f(x)\,dx} \) is well-defined.

Answer: Yes


Question: Is every radical ideal in the ring of algebraic integers a finite intersection of prime ideals?

Answer: No


Question: Evaluate the definite integral: \[ \int_0^\infty \frac{r^3}{(r^2+d^2)^3} \, dr \]

Answer: \dfrac{1}{4d^2}


Question: Let \( Y_1, Y_2, \ldots, Y_n \) be i.i.d. random variables with mean \( \mu \) and variance \( \sigma^2 \). Define the sample mean as \( \bar{Y}_n = \frac{1}{n} \sum_{i=1}^n Y_i \). Consider the function \( g(z) = z^2 \). To what value does \( g(\bar{Y}_n) \) converge in probability as \( n \to \infty \)?

Answer: \mu^2


Question: If \( \{f_{n}\} \) is a sequence of orthogonal functions in a Hilbert space \( H \), and \( \sum_{n}\|f_{n}\|_{H}^{2}<\infty \), does the series \( \sum_{n}f_{n} \) belong to the space \( H \)?

Answer: Yes


Question: Find the degree of the polynomial \( p(x) \) that satisfies the equation \( (x)p(x - 1) = (x - 15) p(x) \).

Answer: 15


Question: Calculate the value of \(\sum_{(x,y,z) \in S} g(x,y,z)\), where \(S\) is the set of integer triplets \((x, y, z)\) satisfying \(|x| + |y| + |z| = 2021\) and \(g(x, y, z) = x + y + z\).

Answer: 0


Question: Given the infinite series \( \alpha = \frac{5}{2!\cdot 3} + \frac{5\cdot 7}{3!\cdot 3^2} + \frac{5\cdot 7 \cdot 9}{4!\cdot 3^3} + \cdots \), find the value of \( \alpha^2 + 4\alpha \).

Answer: 23


Question: Determine the dimension of the vector space consisting of all $C^2$ functions $f: [a,b] \to \mathbb{C}$ that satisfy the boundary conditions $\alpha_1f(a) - \alpha_2f'(a) = 0$ and $\beta_1f(b) - \beta_2f'(b) = 0$, where $\alpha_1, \alpha_2, \beta_1, \beta_2$ are pre-specified real numbers.

Answer: \infty


Question: Find the number of elements in the intersection $A_7 \cap N_G(H)$, where $G = S_7$ and $H = \langle (1\ 2\ 3\ 4\ 5\ 6\ 7) \rangle$. Given that $|N_G(H)| = 42$ and $|A_7| = 2520$, use the internal product formula to determine $|A_7 \cap N_G(H)|$. Provide your answer as a single integer.

Answer: 21


Question: Evaluate the limit \( \lim_{n \to \infty} (n+1)\int_0^1x^{n}f(x)dx \), where \( f:\Bbb{R} \rightarrow \Bbb{R} \) is a differentiable function with a continuous derivative.

Answer: f(1)


Question: In a category with zero morphisms, kernels, and cokernels, consider a kernel morphism $m$ and a cokernel morphism $e$ such that the composition $e \circ m$ is defined. If $e \circ m$ has an image, is this image always a normal monomorphism in the category?

Answer: Yes


Question: Determine if there exists a function that is differentiable everywhere on a set $[a,b]$ and can be expressed as the limit of a sequence of continuous piecewise linear functions, other than linear or constant functions.

Answer: Yes


Question: Evaluate the integral \( \int_0^1 ((1-x)^8 x^{11} - (1-x)^{11} x^8) \, dx \).

Answer: 0


Question: Consider a locally compact Hausdorff topological group \( G \) with a left Haar measure \( \mu \) that is not right-invariant. Does there exist a Borel subset \( B \) of \( G \) and an element \( g \) of \( G \) such that the closure of \( B \) is compact, and for \( C = \{ h \in G : g^{-1} h g \in B \} \), the conditions \( C \subset B \) and \( \operatorname{interior}(B - C) \neq \emptyset \) are satisfied?

Answer: Yes


Question: Is it possible for a module \( M \) to be expressed as the sum of two nonzero submodules \( M = K + L \) where \( K, L \neq 0, M \), but not as the direct sum of any two nonzero submodules?

Answer: Yes


Question: Let \(2 \leq n \leq \omega\) and \(F\) be a set of size \(\leq n\). Let \(\mathbb{P}\) be a poset and \(Q \subseteq \mathbb{P}\) an \(n\)-linked subset. If \(\dot{a}\) is a name for a member of \(\mathbb{P}\) such that \(\Vdash \dot{a} \in F\), is there a \(c \in F\) such that for all \(p \in Q\), \(p \nVdash \dot{a} \neq c\)?

Answer: Yes


Question: Calculate the expected value of the function \( F(x) = x^2 1_{(0,1/2)}(x) + \frac{1}{2} 1_{(1/2,2)}(x) + 1_{[2, \infty)}(x) \) using the formula \( E(X) = \int_{0}^{\infty} (1 - F(x)) \, dx \).

Answer: \dfrac{29}{24}


Question: Consider a Hydra represented as a tree $T = (V,E)$ with root $r \in V$. The root represents the torso, and the leaves represent the heads. For any node $v \in V - \{r\}$, $p(v)$ denotes its parent node. When Hercules cuts a head $\ell \in V$ (where $p(\ell) \neq r$), the subtree $T_\ell$ rooted at $p(\ell)$ (after removing $\ell$) is duplicated and added to $p(p(\ell))$. The Hydra is defeated when only the torso remains. Is it possible for Hercules to defeat any Hydra?

Answer: Yes


Question: Determine the variance-covariance matrix of the weighted least squares estimator \(\hat{\beta}\) given by \(\hat{\beta} = (X^T W X)^{-1} X^T W y\), where \(W_{ii} = 1 / \sigma^2_i\). Is it the same as in the ordinary least squares (OLS) case?

Answer: (X^T W X)^{-1}


Question: Is it possible for an increasing function on the interval $[0,1]$ to be discontinuous at every irrational number?

Answer: No


Question: If \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + (a^4 + 4a^2 + 1)x = x^2 + a^2 \), find the minimum value of the expression \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} + \frac{\gamma}{\alpha} + \frac{\alpha}{\gamma} + \frac{\beta}{\gamma} + \frac{\gamma}{\beta} \).

Answer: 3


Question: Given a trapezoid $ABCD$ with $AD \parallel BC$ and $AB = CD$, the diagonals $AC$ and $BD$ intersect at point $P$, and the lines $AB$ and $CD$ intersect at point $Q$. Points $O_1$ and $O_2$ are the centers of the circumcircles of triangles $ABP$ and $CDP$, respectively, with radius $r$. Construct the trapezoid $ABCD$ using a straightedge and compass, given the segment $O_1O_2$, the line $PQ$, and the radius $r$. How many distinct solutions exist for this construction problem?

Answer: 2


Question: A group has exactly 24 elements of order 6. Determine the number of subgroups of order 6 in this group.

Answer: 12


Question: Determine the minimum sample size required to estimate the probability of a binary outcome with a margin of error of 1%. Assume a computer program randomly generates a "yes" or "no" answer, and you do not know the probability distribution of these outcomes. How many answers should you request from the computer to achieve this margin of error?

Answer: 9604


Question: Evaluate the integral \( \int_{\gamma} \frac{dz}{z+\frac{1}{2}-\frac{i}{3}} \) where \( \gamma(t) = i + e^{it} \) for \( t \in [0, 2\pi] \).

Answer: 2\pi i


Question: Determine the fundamental group of a surface \( S_{g,m} \) with genus \( g \) and \( m \) punctures.

Answer: F_{2g + m - 1}


Question: Let $A$ be a non-zero $4 \times 4$ complex matrix such that $A^2 = 0$. What is the largest possible rank of $A$?

Answer: 2


Question: Evaluate the limit: \[ \lim \limits_{x \to \pi} \frac {\tan^2(4x)}{(x - \pi)^2} \] without using L'Hôpital's Rule.

Answer: 16


Question: Determine the radius of convergence for the power series \( \sum_{n=0}^\infty n!x^{n^2} \).

Answer: 1


Question: Determine the interval of convergence for the power series representation of the function \( f(x) = x^{0.6} \) centered at \( a = 4 \).

Answer: [0, 8]


Question: Evaluate the integral \( \int_{0}^{\pi}\ln(5-4\cos x)\,dx \).

Answer: 2\pi \ln 2


Question: Find the exact y-coordinate of a circle with radius \( \frac{1}{4} \) that makes it tangent to the function \( y = e^{-x^2} \).

Answer: \dfrac{3}{4}


Question: Evaluate the limit: \[ \lim_{x\to1^{-}} (1-x)\left(\sum_{i=0}^{\infty} x^{i^2}\right)^{2} \]

Answer: \dfrac{\pi}{4}


Question: If $X$ is a Hilbert space and $A$ is an unbounded self-adjoint operator on $X$, is $A^k$ necessarily self-adjoint for all positive integers $k$?

Answer: Yes


Question: Determine the ratio of the circumference to the diameter of a circle on a sphere, given that the circle is defined by an angle \( \theta \) from the pole. The radius of the sphere is \( r \). Express the ratio \( R(\theta) \) in terms of \( \theta \).

Answer: \dfrac{\pi \sin \theta}{\theta}


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

Answer: 0


Question: Consider two non-zero $p$-dimensional column vectors $a$ and $b$, where $p \ge 2$. Determine the number of non-zero distinct eigenvalues of the $p \times p$ matrix $ab^T + ba^T$. Provide your answer as a single integer.

Answer: 2


Question:
Let $A$ be a square matrix of order $n$ such that there exists $m \in \mathbb{N}$ for which $A^m = 0$. Determine which of the following statements is true:

1) The only eigenvalue (complex) of $A$ is $0$ and then $\text{tr}(A)=0$.
2) $\text{tr}(A) = 0$ but $A$ can have complex eigenvalues $ \ne 0$.
3) We can deduce that $\det(A)=0$ but not $\text{tr}(A)=0$.

Select the correct statement.

Answer: 1


Question: Let $f(x)$ be a function with a second continuous derivative such that $f(0) = f'(0) = 0$. Define a function $g(x)$ by \[ g(x) = \begin{cases} \frac{f(x)}{x}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \] Determine which of the following statements is true: \(a\) $g(x)$ has a first continuous derivative at $0$. \(b\) $g(x)$ has a first derivative at $0$ which is not continuous. \(c\) $g(x)$ is discontinuous at $0$. \(d\) $g(x)$ is continuous but fails to have a derivative at $0$. Choose the correct option.

Answer: a


Question: Given the matrix \(A = \begin{pmatrix} 0 & 0 & 3 \\ 1 & 0 & 1 \\ 0 & 1 & -3 \end{pmatrix}\) over the field \(\mathbb{F}_2\), find the geometric multiplicity of the eigenvalue \(-1\). Determine if the matrix \(A\) is diagonalizable over \(\mathbb{F}_2\).

Answer: 1


Question: Find all analytic functions on the disc \( \{z:|z-1|<1 \} \) that satisfy the conditions \( f(1)=1 \) and \( f(z)=f(z^2) \).

Answer: 1


Question: Solve the partial differential equation \( u_{t}(x,y,t) + u_{x}(x,y,t) - u_{y}(x,y,t) = \sin(t) \) for \((x,y,t) \in \mathbb{R}^{2} \times (0,+\infty)\) with the initial condition \( u(x,y,0) = 1 \).

Answer: 2 - \cos(t)


Question: Evaluate the limit: $$\lim_{x\rightarrow 1}\ln(1-x)\cot\left({{\pi x}\over2}\right)$$

Answer: 0


Question: Suppose you and I each have a box of 600 marbles. In my box, 4 of the marbles are black, while 3 of your marbles are black. We each draw 300 marbles with replacement from our own boxes. Using Poisson approximation, calculate the probability that you and I draw the same number of black marbles.

Answer: 0.216


Question: Given that \( \sum_{k=1}^{\infty} kx^k = 30 \), find the value of \( x \).

Answer: \dfrac{5}{6}


Question: Find the minimum value of \(a\) such that the inequality \(2|x-y| \leq a|2x-3y^2-3|\) holds for all real numbers \(x\) and \(y\) with \(|x| < y\).

Answer: \dfrac{1}{2}


Question: Calculate \( \mathbb{E}[N(X)] \), where \( N(\cdot) \) is the cumulative distribution function (CDF) of the standard normal distribution, and \( X \) is a standard normal random variable.

Answer: \dfrac{1}{2}


Question: Evaluate the sum of the series: $$\sum_{n=0}^\infty \frac{(-1)^n \pi^{2n}}{9^n(2n)!}$$

Answer: \dfrac{1}{2}


Question: Evaluate the limit \( \lim_{n \to \infty} \sin(\pi(2+\sqrt{3})^n) \) where \( n \in \mathbb{N} \).

Answer: 0


Question: Let $A \subset \mathbb{R}^n$ be a compact set with positive Lebesgue measure. Does there exist an open set $B \subset \mathbb{R}^n$ such that $B \subset A$?

Answer: No


Question: Evaluate the limit \( \lim_{n \rightarrow \infty} \sum_{k=1}^{\infty} k^{1/2} n^{-1} (1 - \cos(n^2 k^{-2})) \) and determine if it equals infinity.

Answer: \infty


Question: Let $A \subset \mathbb{R}^n$ be a Lebesgue null set, and let $\mu$ be a positive $\sigma$-finite measure such that $\mu(A^c) = 0$ and $\mu(A + r) = \mu(A)$ for all $r \in \mathbb{R}^n$. Does this imply that $\mu = 0$?

Answer: 0


Question:
Determine the standard deviation of the insurance payment for an automobile with a damage deductible of $250, where repair costs are modeled by a uniform random variable on the interval $(0, 1500)$. Let $X$ denote the damage, and $Y$ the insurance payment, defined as:

$$Y = \begin{cases} 0 & \text{if } X < 250 \\ X - 250 & \text{if } 250 \le X \le 1500 \end{cases}$$

Calculate the standard deviation of $Y$. Use the formula $\text{var } Y = E(Y^2) - (EY)^2$.

Answer: \dfrac{625\sqrt{15}}{6}


Question: Let $X$ be a random variable with the moment generating function $$M_X(t) = \frac{6}{\pi^2}\sum_{n\geq 1}\frac{e^{\frac{t^2}{2n}}}{n^2}.$$ Determine the probability that $X$ is a rational number, $P(X \in \mathbb{Q})$. Choose the correct answer: \(A\) 0, \(B\) \(\frac{1}{4}\), \(C\) \(\frac{1}{2}\), \(D\) \(\frac{3}{4}\).

Answer: A


Question: Consider the space $X = \mathbb{R}^2 \setminus \{0\}$ and define a partition of $X$ by taking each ray emanating from the origin as an element in the partition. Determine the topological space that is topologically equivalent to the quotient space resulting from this partition.

Answer: S^1


Question: Let $(M, \omega)$ be a connected closed symplectic manifold of dimension $2n$. Assume there exist smooth covering maps $\phi_k:M\to M$ such that $\phi_k^* \omega=\sqrt[n]{k}\omega$ for all $k\geq 1$. Determine if $n=1$.

Answer: 1


Question: Benny plans to visit one of 8 different zoos each week for the next 13 weeks. Determine the number of ways Benny can schedule his visits such that he visits each zoo at least once and at most three times.

Answer: 61751289600


Question:
Let $f = u + iv$ be an entire function. If the Jacobian matrix \( J_a = \begin{bmatrix} u_x(a) & u_y(a) \\ v_x(a) & v_y(a) \end{bmatrix} \) is symmetric for all \( a \in \mathbb{C} \), which of the following is true?

(A) $f$ is a polynomial
(B) $f$ is a polynomial of degree $\leq 1$
(C) $f$ is necessarily a constant function
(D) $f$ is a polynomial of degree $> 1$

Choose the correct option.

Answer: B


Question: Let $K$ be a number field and $(\rho,V)$, $(\rho',V')$ be two Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/K)$. Suppose that for some positive integer $n$, $\mathrm{Sym}^n\rho\cong\mathrm{Sym}^n\rho'$. Does this imply the existence of a finite algebraic extension $L/K$ such that $\rho_{|\mathrm{Gal}(\overline{\mathbb{Q}}/L)}\cong\rho'_{|\mathrm{Gal}(\overline{\mathbb{Q}}/L)}$? Provide a justification for your answer.

Answer: Yes


Question: Consider two $k$-isogenous elliptic curves defined over a number field $k$. Fix a place $v$ of good reduction. Are the reduced curves modulo $v$ isogenous?

Answer: Yes


Question: In the decomposition of a semi-simple Lie algebra \( L \) into simple Lie algebras \( L = L_1 \oplus \ldots \oplus L_n \), are the components \( L_i \) necessarily ideals of \( L \)?

Answer: Yes


Question:
Given the polynomials \( f(x) = x^3 + x^2 + x + 1 \) and \( g(x) = x^3 - x^2 + x - 1 \) in \( \mathbb{Q}[x] \), determine which of the following is true for the sum of the ideals \( \langle f(x) \rangle + \langle g(x) \rangle \):

(a) \( \langle f(x) \rangle + \langle g(x) \rangle = \langle x^3 + x \rangle \)

(b) \( \langle f(x) \rangle + \langle g(x) \rangle = \langle f(x) \cdot g(x) \rangle \)

(c) \( \langle f(x) \rangle + \langle g(x) \rangle = \langle x^2 + 1 \rangle \)

(d) \( \langle f(x) \rangle + \langle g(x) \rangle = \langle x^4 - 1 \rangle \)

Select the correct option.

Answer: C


Question: Determine the minimum number of equally spaced subintervals, N, required such that the difference between the Upper and Lower Riemann sums for the integral \( \int_{1}^{17} \frac{1}{x^3} \, dx \) is less than \( \frac{1}{100} \).

Answer: 1600


Question: Suppose $a, b, c \in \mathbb{R}_{\geq 0}$ and $p, q \in \mathbb{Q} \cap [0,1]$ are fixed with $p + q = 1$. Determine if it is necessarily true that $a \leq b^p c^q$ implies $(a + \varepsilon) \leq (b + \varepsilon)^p (c + \varepsilon)^q$ for any $\varepsilon > 0$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the path integral \( \int_{\partial K(0,1)} \frac {dz} {(z-a)(z-b)} \) along the unit circle \( \partial K(0,1) \), where \(|a|,|b| < 1\) and \(a \neq b\).

Answer: 0


Question: Let \( t_1, t_2, t_3 \) be three points on the unit circle \( |t|=1 \). If \( x_1, x_2, x_3 \) are the arguments of these points, find the value of \( \cos (x_1-x_2) + \cos (x_2-x_3) + \cos (x_3-x_1) \).

Answer: -\dfrac{3}{2}


Question: For the complete graph \(K_4\) as a strongly regular graph, determine the parameter \(\mu\). Is it possible for \(\mu\) to be 2, even though Mathworld states that \(\mu = 0\) for this graph?

Answer: 0


Question: Let $\kappa$ be an infinite cardinal and let $X$ be a compact Hausdorff space of size $2^\kappa$. Let $U$ be a dense open subset of $X$. Determine a lower bound for the cardinality of $U$.

Answer: 2^\kappa


Question: Determine whether there exists a proper smooth rigid-analytic variety over an algebraic closure of the p-adic rationals, $K$, whose étale homotopy type is not isomorphic to the étale homotopy type of a proper smooth scheme over $K$. Provide a justification for your answer.

Answer: Yes


Question: Let $z_1, z_2, \ldots, z_7$ be the vertices of a regular heptagon inscribed in the unit circle centered at the origin in the complex plane. Calculate the magnitude of the sum $w = \sum_{1 \leq i < j \leq 7} z_i z_j$. What is $|w|$?

Answer: 0


Question: Let $T$ be a nonzero bounded linear operator in $B(H)$, where $H$ is an infinite dimensional Hilbert space, and let $\|T\|$ be the norm of $T$. Compute the norm $\|I-T\|$, where $I$ is the identity operator.

Answer: 1 + \|T\|


Question: Find the limit: \[ \lim_{n\to \infty}\frac{n}{\sqrt[n]{n!}} \]

Answer: e


Question: Seventeen people are seated at the vertices of a large circular table, with a constant distance between each pair of neighbors. After an hour, they are rearranged randomly. Choose any two people. Let \( X \) be the positive difference between their distances before and after the rearrangement. Calculate the expected value of \( X \). Note that the maximum distance is 8.

Answer: \dfrac{21}{8}


Question: In ZFC set theory, determine whether there exists a non-empty set $X$ such that every strict subset $Y$ of $X$ (i.e., $Y \neq X$) is a member of $X$. Provide a proof or disproof of the existence of such a set.

Answer: \{\emptyset\}


Question: Evaluate the double integral \( \int _{ 0 }^{ \frac { \pi }{ 2 } } \int _{ 0 }^{ x }{ e^{ \sin(y) } \sin(x)\,dydx } \).

Answer: e - 1


Question: Let \(x_n\) be the \(n\)th positive solution to the equation \(x = \tan(x)\). Determine \(\lim_{n\to\infty}(x_n - x_{n-1})\).

Answer: \pi


Question: In an M/M/2 queue, the stationary distribution is given as \( \pi \approx (0.20675105, 0.29535865, 0.05907173, 0.23628692, 0.1350211, 0.06751055) \). Calculate the throughput of the system using \( \sum_{i \in S} i\pi_i \) where \( S = \{0,1M,1J,2,3,4\} \) and \( 1J=1M=1 \). If the arrival rate is \( \lambda = 1 \), determine if the calculated throughput being greater than the arrival rate is erroneous.

Answer: Yes


Question:
Determine the maximum natural number \( n \) such that all polynomials of degree \( n \) are integrated exactly using the formula:
\[
\int_a^b f(t) \approx \frac {b-a} {4} (f(a) +3f(\frac {a+2b} {3}))
\]

Answer: 2


Question: Evaluate the limit: $$\lim\limits_{n\to +\infty}\sqrt[n]{|\sin(n+1)|+|\sin(n)|}$$

Answer: 1


Question: Determine the set of points in the complex plane where the function \( f(z) = ze^{\overline{z}} \) is differentiable.

Answer: \{0\}


Question: Evaluate the integral \( \oint_{\gamma}\frac{f(z)}{z^{4}} \, dz \), where \( \gamma(t) = \cos(t) + i\sin(t) \) for \( t \in [0, 2\pi] \) and \( f(z) = \frac{e^{z} + e^{-z}}{2} \).

Answer: 0


Question: Find the roots of the equation: \[ 8\lambda + 2\text{Trace}(Q\Sigma Q^\top) - \sum_i \frac{2 M_i}{1 - 2\lambda M_i} = 0 \] where \( M \) is a diagonal matrix with the eigenvalues of \( Q\Sigma Q^\top \).

Answer: 0


Question: Is the underlying space of a simplicial complex second countable? Consider a simplicial complex $X$ with its underlying space $|X|$ as a subset of $\mathbb{R}^n$, formed by the union of simplices of $X$. Determine if $|X|$ is second countable.

Answer: Yes


Question: Determine the domain of the integral function \( F(x) = \int_0^{x} \frac{1}{\ln(2-t^2)} \, dt \).

Answer: (-1, 1)


Question: Determine the probability that the determinant of a $2 \times 2$ matrix with arbitrary integer elements is odd.

Answer: \dfrac{3}{8}


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

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


Question: Given the joint probability density function $f(x,y) = 6y$ with boundaries $0 \leq y \leq x \leq 1$, find the expected value of the random variable $X$.

Answer: \dfrac{3}{4}


Question: Determine whether the set \( \mathcal{S} = \{(\mathbf{x}, \mathbf{y}) \in \mathbb{C}^{n} \times \mathbb{C}^{n} \mid \mathbf{x}^{H}\mathbf{y} = ||\mathbf{y}||^{2}_{2}\} \) is a set of measure zero in \( \mathbb{C}^{n} \times \mathbb{C}^{n} \).

Answer: Yes


Question: Evaluate the line integral \( \oint_{\partial R} \left((x^2-2xy)dx+(x^2y+3)dy\right) \) around the boundary of the region contained by \( y^2=8x \) and \( x=2 \) using Green's Theorem. Determine the value of the integral.

Answer: \dfrac{128}{5}


Question: Find the number of ordered pairs \((p, q)\) such that \(p\) and \(q\) are coprime positive integers, \(\frac{p}{q} \in (0,1)\), and \(pq = 20!\).

Answer: 128


Question: Let $X$ be a random variable with a distribution function $F(x)$. Use Chebyshev's Inequality to show that \( \lim_{x\rightarrow \infty} x(1-F(x)) = 0 \).

Answer: 0


Question: What is the smallest number of elements in a non-planar poset or lattice, where a non-planar poset is defined as one that admits no Hasse diagram in which no two edges intersect?

Answer: 6


Question: Determine whether the limit \( \lim_{n\to\infty} \left(\frac{\log(p_{n+1})}{\log(p_n)}\right)^n \) exists, and if it does, find its value.

Answer: 1


Question: Let $X$ be a locally compact Hausdorff space and let $\mu$ be a positive Borel measure that is finite on compact sets, outer regular with respect to open subsets, and inner regular with respect to compact subsets for each open set and for each Borel set with finite measure. Is it true that for every compact set $F$, there exists an open $\sigma$-compact set $G$ such that $F \subset G$ and $G$ has finite measure?

Answer: Yes


Question: Let $\phi: A \to B$ be a ring homomorphism where $B$ is finitely generated as an $A$-module. Let $\mathfrak{p} \subseteq A$ be a prime ideal and $K$ be the field of fractions of $A/\mathfrak{p}$. Determine whether the ring $B \otimes_A K$ has only finitely many prime ideals.

Answer: Yes


Question: How many pairs of natural numbers \((x, y)\), where both are not greater than 100, satisfy the conditions that the difference \(x - y\) is a prime number and the product \(xy\) is a perfect square?

Answer: 7


Question:
Consider a death benefit of a life insurance policy that can be obtained in four ways, all having the same present value:

1. A perpetuity of $300 at the end of each quarter.
2. Annuity payments of $600 at the end of each quarter for n years, with the first payment one quarter after death.
3. A lump sum payment of $60,000 at the end of n years after death.
4. A lump sum payment of $B at the moment of death.

Calculate the value of $B.

Answer: 30000


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

Answer: \dfrac{1}{2e}


Question: Find the natural number solutions \((m, n)\) for the equation \(m^2 - 3mn + (np)^2 = 12p\), where \(p\) is a prime number.

Answer: (6, 2)


Question:
Let \( f: \mathbb{C} \to \mathbb{C} \) be a holomorphic function. Find the value of the following limit superior:
\[ \varlimsup_{a \to 0, a \in \mathbb{C}} \left| \frac{|f(z+a)| - |f(z)|}{|a|} \right| \]

Answer: |f'(z)|


Question: Let $p$ be a polynomial of degree greater than 1. Prove that there exists a real number $R > 0$ such that for all $z$ with $|z| > R$, the iterates $p_n(z)$ tend to infinity as $n$ approaches infinity.

Answer: R


Question: Given independent normal distributions $U$, $V$, and $W$ with mean $0$ and unit variance, let $X = U + W$ and $Y = U + V$. Calculate $E[XY]$.

Answer: 1


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

Answer: 1


Question: Let \( \{a_n\} \subseteq \mathbb{C} \) be a discrete set with no limit point. For every sequence \( \{z_n\} \) of complex numbers, determine whether there exists an entire function \( f: \mathbb{C} \to \mathbb{C} \) such that \( f(a_n) = z_n \) for all \( n \in \mathbb{N} \).

Answer: Yes


Question: Given the group \(G=\mathbb{Z}_6 \times \mathbb{Z}_3\), the element \(a=(4,1)\) of \(G\), the subgroup \(H=\langle a\rangle\) of \(G\), and the group \(G_1=G/H\), find the order \(k\) of the class of \((3,1)\) in \(G_1\) and the order \(\ell\) of the class of \((2,2)\) in \(G_1\). What is \((k+\ell) \mod{5}\)?

Answer: 2


Question: For a symmetric and positive definite matrix \( A \in \mathbb{R}^{2 \times 2} \) with distinct eigenvalues \( \lambda_1 > \lambda_2 > 0 \), determine the number of points \( x \in \mathbb{R}^2 \) at which the maximum of the Rayleigh quotient \( \max_{\|x\|_2=1} \overline{x}^T A x \) is reached.

Answer: 2


Question: Determine the domain of convergence for the power series \( \sum_{n\ge1}{(\cos{\frac{1}{n}})^{\frac{n^2+2}{n+2}}x^n} \).

Answer: (-1, 1)


Question: Let $T$ be a compact operator on a Hilbert space $H$. Prove that for every sequence $(x_n)_n \subset H$ which converges weakly to $0$, the sequence $(Tx_n)_n$ converges to $0$ in norm.

Answer: 0


Question:
Given the joint moment generating function of random variables $X$ and $Y$:
\[ M(t_1,t_2) = \frac{1}{2}e^{t_1+t_2} + \frac{1}{4}e^{2t_1+t_2} + \frac{1}{12}e^{t_2} + \frac{1}{6}e^{4t_1+3t_2} \]
find the probability $P(X < Y)$. Provide your answer as a simplified expression or numerical value.

Answer: \dfrac{1}{12}


Question:
Determine if the square of the constant \( E \) is equal to the constant \( D \), where:

\[ E = \frac{1}{2}\sqrt{6}\exp{\sum_{j=1}^\infty2^{-j-1}\ln\left[1+(2e_j-1)^{-2}\right]} \]

and

\[ D = \ln(3/2)+\sum_{n \ge 0} \ln(1+(2a_n+1)^{-2}) \]

with the sequences defined as:

\[ a_n = a_{n-1}^2 + a_{n-1} = \left\lfloor D^{2^n} - \frac{1}{2} \right\rfloor, \quad a_0 = 1 \]

\[ e_n = e_{n-1}^2 - e_{n-1} + 1 = \left\lfloor E^{2^{n+1}} + \frac{1}{2} \right\rfloor, \quad e_0 = 2 \]

Answer: E^2 = D


Question: Calculate the Legendre symbol \( \left( \frac{2}{27} \right) \). Note that 27 is not a prime number, so consider how to approach this calculation.

Answer: -1


Question: Does there exist a sequence \(a_n \in \mathbb{R}_+\) such that \(\lim_{n\rightarrow \infty} a_n = 0\) and \(\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{i=1}^{n}a_i = c\) for some constant \(c > 0\)?

Answer: No


Question: Calculate the sum of the series \( \sum^{\infty}_{n=2}\frac{(-1)^{n+1}}{n} \). Verify if the solution \( \ln{2} - 1 \) is correct.

Answer: \ln 2 - 1


Question: Calculate the sum of the infinite series: \[ S = \sum_{n=1}^{\infty} \frac{n^3}{4^n} \]

Answer: \dfrac{44}{27}


Question: Determine whether for every entire function $f$, there exists an entire function $g$ such that the real part of $f$ is equal to the natural logarithm of the absolute value of $g$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of isomorphisms from the symmetric group $S_n$ to itself for $n = 20$. Provide a brief explanation of your reasoning.

Answer: 20!


Question: Find the value of \( \lim_{x\rightarrow \infty}\left[\left(x^5+7x^4+2\right)^c-x\right] \) for a certain value of \( c \) such that the limit is finite and non-zero.

Answer: \dfrac{1}{5}


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0}\frac{1-\cos (\sin ^5(x))}{(e^{x^4}-1)(\sin(x^2)-x^2)} \]

Answer: -3


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=1}^{2n}(-1)^k\left(\frac{k}{2n}\right)^{100} \]

Answer: \dfrac{1}{2}


Question: Find the value of \( c \) that makes the function \( f(x) = \begin{cases}\left(\sin\frac{\pi}{2+x}\right)^{1/x^2},& x \neq 0\\c,& x = 0\end{cases} \) continuous at \( x = 0 \).

Answer: e^{-\pi^2/32}


Question: Evaluate the integral \( \int_A \left(\frac{xe^{x^2+y^2}}{x^2+y^2} - 1\right)\mathrm{d}x\mathrm{d}y \) over the region \( A = \{(x,y) \in \mathbb{R}^2 : 1 \le x^2 + y^2 \le 4 \} \) using polar coordinates.

Answer: -3\pi


Question: Let $d$ be a positive integer with $d \geq 2$. By Dirichlet's theorem, there is an infinite set $\mathcal{P}$ of primes congruent to $1$ modulo $d$. Consider the set $N$ of integers $n = 1 + kd$ where all prime divisors of $k$ belong to $\mathcal{P}$. Can we expect that $N$ contains infinitely many primes, assuming $d$ is even?

Answer: Yes


Question: What is the expected number of flips of a fair coin before seeing both the sequences 'HH' and 'TT'? Assume the coin is unbiased, with the probability of heads and tails both being \( \frac{1}{2} \).

Answer: 9


Question: The line that is normal to the curve \( x^2 + xy - 2y^2 = 0 \) at the point \((4,4)\) intersects the curve at another point. Find this other point of intersection.

Answer: (16, -8)


Question: Determine the radius of convergence for the series \( \sum\limits_{n=1}^{\infty}\cfrac{{(4x-5)}^{2n+1}}{n^{3/2}} \).

Answer: \dfrac{1}{4}


Question: Simplify \( \frac{1}{x^2 + x} \) in \( \mathbb{Z}_2[x]/(x^3 + x + 1) \) to a polynomial of degree less than 3.

Answer: x + 1


Question: Assume that the random variable $Y$ has the same distribution as $\frac{Y_1 + Y_2}{\sqrt{2}}$, where $Y_1$ and $Y_2$ are independent and identically distributed with $Y$. Given that $E[Y^2] = 3$, find $\text{Var}[Y]$. Provide your answer as a single value.

Answer: 3


Question: Find all positive integers $n$ such that \( \left\lfloor\frac{n}{2}\right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor = n^2 \).

Answer: 24


Question: Minimize the value of the function \( P = a^2 + b^2 + c^2 + 2\sqrt{3abc} \) given that \( a, b, c \) are positive real numbers such that \( a + b + c = 1 \). Find the minimum value of \( P \).

Answer: \dfrac{1}{2}


Question:
Evaluate the limit of the following product series as \( n \to \infty \):
\[ \lim_{n\to \infty} \left(1- \frac{ 2}{3}\right)^\frac{3}{n} \left(1- \frac{ 2}{4}\right)^\frac{4}{n} \left(1- \frac{ 2}{5}\right)^\frac{5}{n}...\left(1- \frac{ 2}{n+2}\right)^\frac{n+2}{n} \]

Answer: e^{-2}


Question: Determine the fundamental period of the function \( f(x) \) that satisfies the functional equation \( f(x-1) + f(x+1) = f(x) \).

Answer: 6


Question: Find the general solution for \(x\) in the congruence \(2^x + 8 \equiv 0 \pmod{27}\).

Answer: x \equiv 12 \pmod{18}


Question:
Calculate the value of \( t \) at which the velocity of the Bezier curve defined by the control points \( P_0 = (0, 0) \), \( P_1 = (0.4, 0) \), \( P_2 = (0.2, 1) \), and \( P_3 = (1, 1) \) is maximized. The Bezier curve is given by the equations:
\[ X(t) = 1.6t^3 - 1.8t^2 + 1.2t \]
\[ Y(t) = -2t^3 + 3t^2 \]
Find the value of \( t \) where the rate of change of \( Y \) with respect to \( t \) is maximized.

Answer: \dfrac{1}{2}


Question: Let $V$ be a representation of a Lie algebra $\mathfrak{g}$ that decomposes into a direct sum of countably many distinct finite-dimensional irreducible representations. Determine whether it is true that if $v \in V$ is a weight vector whose weight $\lambda$ is a dominant integral element, then $v \in V_{\lambda}$. Provide a justification for your answer.

Answer: True


Question: Let $P \subset \mathbb{R}^n$ be a compact rational polytope, where the affine space spanned by $P$ may not be $\mathbb{R}^n$. Let $G \subset \mathbf{GL}(n,\mathbb{Z})$ be an arithmetic group. Define $C = \mathrm{Conv}(\bigcup_{g \in G} g \cdot P)$ as the convex hull. Does there exist a compact rational polytope $Q \subset C$ such that $\bigcup_{g \in G} g \cdot Q = C$?

Answer: No


Question: Determine whether all derivatives of the sinc function, defined as \( \text{sinc}(x) = \frac{\sin(x)}{x} \), are bounded on the real axis. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\sqrt[3]{1+\frac{x}{3}} - \sqrt[4]{1+\frac{x}{4}}}{1-\sqrt{1-\frac{x}{2}}} \] without using L'Hospital's rule, power expansions, or derivatives.

Answer: \dfrac{7}{36}


Question: Suppose that \(X_1\) and \(X_2\) are independent random variables with a common probability density function (pdf) \(f(x)\) and cumulative distribution function (cdf) \(F(x)\). Let \(Y_1 = \max\{X_1, X_2\}\) and \(Y_2 = \min\{X_1, X_2\}\). Find the joint pdf of \((Y_1, Y_2)\).

Answer: 2f(y_1)f(y_2)


Question: Determine the derived set of \( S = \{ m + \sqrt{2}n : m, n \in \mathbb{Z} \} \).

Answer: \mathbb{R}


Question: Calculate the result of the following differentiation: \( \frac{\partial n_{1,2}}{\partial n_{2,3}} \), where \( n_{1,2} = \frac{\partial n_1}{\partial x_2} \) and \( n_{2,3} = \frac{\partial n_2}{\partial x_3} \). Assume that \( n_1, n_2, \) and \( n_3 \) are differentiable functions of \( x_1, x_2, \) and \( x_3 \).

Answer: 0


Question:
Determine the radius of convergence for the power series solution of the differential equation \(y'' - 5xy' + y = 0\), given by:
\[
y(x) = a_{0}\left(1-\frac{1}{2!}x^{2}-\sum_{k=2}^{\infty}\frac{9\cdot19\cdot\ldots\cdot(10k-11)}{(2k)!}x^{2k}\right)+a_{1}\left(x+\sum_{k=1}^{\infty}\frac{4\cdot14\cdot\ldots\cdot(10k-6)}{(2k+1)!}x^{2k+1}\right)
\]

Answer: \infty


Question: Calculate the limit: \[ \lim_{x\rightarrow 0} (\sin(2\phi x)-2\phi x)\cot(\phi x)\csc^2(\phi x) \] where \( \phi \) is a real number.

Answer: -\dfrac{4}{3}


Question: Calculate the surface area of the portion of the cylinder \( S_1: x^2 + (y-1)^2 = 1 \) that lies between the paraboloid \( S_2: z = 4 - x^2 - y^2 \) and the plane \( S_3: z = 4 \).

Answer: 4\pi


Question: Determine the value of $k$ such that the differential form \((y^3 + k xy^4 - 2x) \, dx + (3xy^2 + 20 x^2y^3) \, dy\) is exact.

Answer: 10


Question: Let \( u_n \) be a sequence defined by \( u_0 = 1 \) and \( u_{n+1} = 1 + \frac{n}{u_n} \). Determine whether the sequence \( u_n \) diverges to \( +\infty \).

Answer: +\infty


Question: Determine the number of possible configurations for an \(m \times n\) table where each cell contains either \(1\) or \(-1\), and the product of all the cells in any given row and the product of all cells in any given column equals \(-1\). Assume that the parity of \(m\) and \(n\) is the same. What is the total number of such tables?

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


Question: Determine if the function \( f(x, y) = \frac{\sin^4(x+y)}{e^{\cos(x-y)}} \) is differentiable at the point \((0,0)\).

Answer: Yes


Question: Find the derivative of the expression \( c^T X^T X c \) with respect to the matrix \( X \), where \( X \) is a matrix and \( c \) is a vector with real entries.

Answer: 2 X c c^T


Question: Determine whether it is possible for $n \times n$ complex matrices $A$ and $B$ that $ABA - BAB = I$, where $I$ is the identity matrix.

Answer: Yes


Question: Prove that a non-empty set in \( \mathbb{R} \) that is both open and closed must be equal to \( \mathbb{R} \).

Answer: \mathbb{R}


Question: In triangle $ABC$, the area of the triangle is $70$. Points $E$ and $D$ are on side $AC$ such that $BE = ED = DA$, and points $F$ and $G$ are on side $AB$ such that $BF = FG = GC$. Quadrilateral $HIJK$ is formed within the triangle. What is the area of quadrilateral $HIJK$?

Answer: 10


Question: Given the probability density function \( f(x,y) = \frac{1}{2}xy \) for \( 0 \leq x \leq y \leq 2 \) and \( f(x,y) = 0 \) otherwise, compute the conditional probability \( P\left(x \geq \frac{1}{2} \mid y = 2x\right) \).

Answer: \dfrac{7}{8}


Question: Let X and Y be two random variables with a common variance \(a^2\) where \(a > 0\). Given that \(E(Y | X) = X + 1\), determine the correlation coefficient between X and Y.

Answer: 1


Question:
Suppose that $\tau_1$ and $\tau_2$ are topologies on $X$ induced by metrics $d_1$ and $d_2$, respectively, such that $\tau_1 \subseteq \tau_2$. Determine which of the following statements is true:

(a) Every open ball in $(X,d_1)$ is an open ball in $(X,d_2)$.
(b) The map $x \to x$ from $(X,d_1)$ to $(X,d_2)$ is continuous.
(c) If a sequence converges in $(X,d_2)$ then it converges in $(X,d_1)$.
(d) If a sequence converges in $(X,d_1)$ then it converges in $(X,d_2)$.

Select the correct option.

Answer: C


Question:
Let \( g : \mathbb{R}^{3} \to \mathbb{R}^{3} \) be a function defined by \( g(x,y,z) = (3y+4z, 2x-3z, x+3y) \). Define the set \( S = \{(x,y,z) \in \mathbb{R}^{3} : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1 \} \).

If \[ \int\int\int_{g(S)} (2x+y-2z) \, dx \, dy \, dz = \alpha \int\int\int_{S} z \, dx \, dy \, dz, \] find the value of \( \alpha \).

Answer: 75


Question: Evaluate the expression \( \log_{2}\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{\frac{2\pi iab}{2015}}\right)\right) \).

Answer: 13725


Question: What is the cardinality of the set of hyperreal numbers?

Answer: 2^{\aleph_0}


Question: Determine whether the edges of the directed graph $D$, formed by directing each edge of a $d$-dimensional hypercube in both directions, can be partitioned into $d$ directed Hamiltonian cycles. Consider the case for $d=3$.

Answer: Yes


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

Answer: e^{\frac{8}{3}}


Question: A wall 8 feet high is located 1 foot away from a house. Determine the length \( L \) of the shortest ladder that can be placed over the wall to reach the house. Assume the ladder forms a right triangle with the wall and the ground, where the height is \( y \), the base is \( 1 + x \), and the hypotenuse is \( L \).

Answer: 5\sqrt{5}


Question: Find the radius of convergence of the series \( \sum\limits_{n=0}^{\infty} \dfrac{(n!)^k\cdot x^n}{(kn)!} \), where \( k \) is a positive integer.

Answer: k^k


Question: Find the smallest integer value of \( n \) such that a labeled complete graph with \( n \) vertices has more than one million spanning trees. Use Cayley's formula, which states that the number of spanning trees in a complete graph with \( n \) vertices is \( n^{n-2} \).

Answer: 9


Question:
In a sequence $a_1, a_2, \dots$ of real numbers, it is observed that $a_p=\sqrt{2}$, $a_q=\sqrt{3}$, and $a_r=\sqrt{5}$, where $1\leq p<q<r$ are positive integers. Determine which of the following statements is true:

(A) $a_p$, $a_q$, $a_r$ can be terms of an arithmetic progression.

(B) $a_p$, $a_q$, $a_r$ can be terms of a harmonic progression.

(C) $a_p$, $a_q$, $a_r$ can be terms of an arithmetic progression if and only if $p$, $q$, and $r$ are perfect squares.

(D) $a_p$, $a_q$, $a_r$ can be terms of neither an arithmetic progression nor a harmonic progression.

Choose the correct option.

Answer: D


Question: Compute the value of \( \omega^{3n/2 + 1} + \omega \) where \( \omega \) is an \( n^{th} \) root of unity and \( n \) is a multiple of 4.

Answer: 0


Question: Let $G$ be a group with $|G| = 123456789$. Let $g$ be an element of $G$, and let $M = 2^{2^{2524820221019}}$. Determine whether $g^M = e$ implies $g = e$ for all $g \in G$. Justify your answer.

Answer: Yes


Question: Find the length of the projection of the point \( p = (4, 1, 2) \) onto the subspace spanned by the vectors \( (1, 2, 3) \) and \( (1, -3, -2) \). The basis is not orthogonal, and you should find the length of the projection without using the Gram-Schmidt process.

Answer: 3\sqrt{2}


Question: Given that \( \lim_{x\to1}\frac{f(x)\cos{\frac{\pi x}{2}}}{\sqrt[3]{x}-1}=3 \), find \( \lim_{x\to1}f(x) \).

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


Question: Two persons $A$ and $B$ flip two fair coins $n$ and $n+1$ times, respectively. Let $X \sim \text{Bin}(n,1/2)$ and $Y \sim \text{Bin}(n+1,1/2)$ be the number of heads for $A$ and $B$, respectively. Find the probability $P(X < Y)$. Express your answer as a fraction.

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{S_r} \frac{e^{iz}-1}{z^2} dz \) as \( r \rightarrow 0 \), where \( S_r = re^{it} \) and \( t \in [0, \pi] \).

Answer: -\pi


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\log(|\sin(\frac{1}{n})|)}{\log(n+1)} \]

Answer: -1


Question: Determine whether the function \( \frac{x}{e^x-1} \) is differentiable at \( x = 0 \), given that the limit of the function as \( x \to 0 \) is 1, the derivative exists at the origin and is 0, and the limit of the second derivative is positive.

Answer: Yes


Question: Consider an invertible linear map $A: \mathbb{C}^n \rightarrow \mathbb{C}^n$ where $n \geq 1$. Is every $A$-invariant subspace of $\mathbb{C}^n$ a direct sum of basic $A$-invariant subspaces, given that each Jordan block of $A$ provides a chain of nested $A$-invariant subspaces?

Answer: No


Question: Determine whether the Fourier transform $F: S(\mathbb{R}^n) \rightarrow S(\mathbb{R}^n)$, which maps a function $f$ to its Fourier transform $\hat{f}$, is a bijection in the Schwartz space.

Answer: Yes


Question: Evaluate the sum of the Fourier coefficients \( \sum_{n=-\infty}^\infty \hat{g}(n) \) for the function \( g(x) = \cos(\pi (x + 2010)) \), where \( \hat{g}(n) \) is the \( n \)-th Fourier coefficient of \( g(x) \).

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac {\cos 1 \cdot \arccos \frac{1}{n}+\cos\frac {1}{2} \cdot \arccos \frac{1}{(n-1)}+ \cdots +\cos \frac{1}{n} \cdot \arccos{1}}{n}$$

Answer: \dfrac{\pi}{2}


Question: Evaluate the integral \( \frac{1}{5} \int_0^5 (4 + 2^{0.1t^2}) \, dt \).

Answer: 6.06


Question: Given a symmetric graph $G(V,E)$ with $n$ vertices and $m$ edges, and a $2\text{-factor}$ with edge set $F$, represented as an unordered sequence $\big( (u_1,v_1),\,\dots,\,(u_n,v_n)\big)$, determine the complexity of finding a permutation of the $n$ vertices that has the same cycles as $F$. Specifically, store the vertices in an array $\boldsymbol{a}$ such that $(i,\boldsymbol{a}[i])\in F$. What is the complexity of this task?

Answer: O(n)


Question: Let $X$ be a finite dimensional space such that $X \subset H^1$. Consider a sequence of non-negative functions $f_n \in X$ for $n \geq 1$ and a function $f \in H^1$ such that \( \|f_n - f\|_{L_2} \to 0 \) as \( n \to \infty \). Is it sufficient to conclude that $f \geq 0$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0^+} \frac{e^{-1/x}}{x^k}, \text{ where } k \in \mathbb{N} \cup \{0\}. \]

Answer: 0


Question:
Find the determinant of the matrix:
\[
\begin{bmatrix}
283 & 5 & \pi & 347.86\times10^{15^{83}} \\
3136 & 56 & 5 & \cos(2.7402) \\
6776 & 121 & 11 & 5 \\
2464 & 44 & 4 & 2
\end{bmatrix}
\]
without using a calculator. Hint: The answer is 6.

Answer: 6


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

Answer: 0


Question: Solve for $w$ in the equation: $$\sin(x) + \sin(2^{\frac{n+12}{12}}x) = \sin(x+w) + \sin(2^{\frac{n+12}{12}}(x+w)).$$

Answer: 0


Question: Calculate the sum of the infinite series: $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$

Answer: \dfrac{65}{288}


Question: Determine whether there exists a set $S \subseteq \mathbb{N}$ such that $S$ is meager, the sum-graph $G_S = (\mathbb{N}, E)$ is connected, and the diameter of $G_S$ is infinite.

Answer: Yes


Question: Given that $f(x) + f(1 - \frac{1}{x}) = \ln|x|$ for all $x \in \mathbb{R} \setminus \{0, 1\}$, find the value of $\int_0^1 f(x) \, dx$. \(f: A \to \mathbb{R}\) where $A$ is the set of all real numbers except 0 and 1.

Answer: 0


Question: Calculate \( \lim \limits_{x \to 0} (\tan (\frac{\pi}{4}-x))^{\cot x} \) without using L'Hospital's Rule or Taylor expansion.

Answer: e^{-2}


Question: Let G be an infinite group. Suppose that the commutator subgroup [G,G] of G and the abelianization of G are finitely generated. Does this imply that G is finitely generated? Provide a justification for your answer.

Answer: Yes


Question:
Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be a function such that \( f(0) = 1 \) and for any \( x, y \in \mathbb{R} \), the following holds:
\[ f(xy+1) = f(x)f(y) - f(y) - x + 2 \]
Find the function \( f(x) \).

Answer: x + 1


Question: Determine the degree of the splitting field of the polynomial \( f(x) = x^7 - 3 \) over \( \mathbb{Q} \).

Answer: 42


Question: Calculate $23! \times 7! \bmod 29$ using Wilson's Theorem.

Answer: 13


Question: Determine the number of distinct colorings for a regular pentagon using four colors, considering both rotational and reflectional symmetries. Use the counting theorem to find the number of distinct colorings. What is the result?

Answer: 136


Question: Given the differential equation \( \frac{dx}{dt} = -x + x^7 \) with the initial condition \( x(0) = \lambda \), where \( x = x(t, \lambda) \), find \( \frac{ \partial x(t, \lambda)}{\partial \lambda} \bigg| _{\lambda = 0} \).

Answer: e^{-t}


Question: Let $D_1$ and $D_2$ be probability distributions on $\mathbb{R} \times \mathbb{R}$ with identical marginal distributions. Define $e_1 = \mathbb{E}_{D_1} [x - y]$ and $e_2 = \mathbb{E}_{D_2} [x - y]$. Does $e_1$ necessarily equal $e_2$?

Answer: Yes


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

Answer: 1


Question: Does there exist an infinite set $X$ and distinct topologies $\sigma \subseteq \tau$ on $X$ such that the interval $[\sigma, \tau)$ does not contain a maximal element?

Answer: Yes


Question: Let \(p\) and \(q\) be prime numbers such that \(p \not\equiv 1 \pmod{q}\). Let \(b \in \mathbb{F}_p\). Determine the possible number of solutions \(x\) to the equation \(x^q = b\) in \(\mathbb{F}_p\). Does this number of solutions depend on \(b\)?

Answer: 1


Question: Find the probability density function of the random variable $Z = XY$, where $(X,Y)$ is uniformly distributed over the region defined by $0 < X < 1$ and $0 < Y < X^2$.

Answer: -\ln z


Question: Determine the number of solutions to the inequality \( \frac{p_{n+1} - p_n}{2} \ge \pi(p_n) \), where \( p_n \) and \( p_{n+1} \) are consecutive primes and \( \pi(n) \) is the prime counting function.

Answer: 0


Question: Calculate the area of the region bounded by the curves $\sqrt{x} + \sqrt{y} = \sqrt{a}$ and $x + y = a$.

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


Question: Find the limit of the sequence \(a_n = n \sqrt{n}(\sqrt{n + 1} - 2\sqrt{n} + \sqrt{n - 1})\) as \(n\) approaches infinity, given that the sequence is convergent.

Answer: -\dfrac{1}{4}


Question:
Consider a sequence of Poisson random variables $X_n$ with parameter $n$. Determine which of the following statements is true:

1. $\lim_{n \to \infty} \mathbb{P}(X_n > n + \sqrt{n}) = 0$
2. $\lim_{n \to \infty} \mathbb{P}(X_n \le n) = 1$
3. $\lim_{n \to \infty} \mathbb{P}(X_n \le n) = \frac{1}{2}$

Select the correct statement.

Answer: 3


Question: Find a particular solution to the nonhomogeneous recurrence relation \(a_n = 2 a_{n-1} + 3 a_{n-2} + 8n - 4\) for \(n \geq 2\), given the initial conditions \(a_0 = 1\) and \(a_1 = 3\).

Answer: 3^{n+1} + (-1)^n - 2n - 3


Question: Determine the radius of convergence of the complex power series \( \sum_{n=0}^\infty a_n z^n \) given that \( \sum_{n=0}^\infty 2^n a_n \) converges while \( \sum_{n=0}^\infty (-1)^n 2^n a_n \) diverges.

Answer: 2


Question: Three circles are given by the equations: \( S_1: x^2+y^2+4y-1=0 \), \( S_2: x^2+y^2+6x+y+8=0 \), and \( S_3: x^2+y^2-4x-4y-37=0 \). These circles touch each other. Let \( P_1, P_2, P_3 \) be the points of contact, and \( C_1, C_2, C_3 \) be the centers of the circles. Find the ratio \( \frac{\Delta P_1P_2P_3}{\Delta C_1C_2C_3} \).

Answer: \dfrac{2}{5}


Question: Find the explicit Cartesian equation for the surface obtained by revolving the curve \( y = \frac{1}{x} \) about the line \( y = x \).

Answer: 2xy - z^2 = 2


Question: Convert the integral \( \int^{2}_{-2}\int^{\sqrt{4-x^2}}_{-\sqrt{4-x^2}}\int^{4}_{x^2+y^2}\;x\;dz\;dy\;dx \) to cylindrical coordinates and evaluate it. Verify if the conversion and evaluation are done correctly.

Answer: 0


Question:
Determine the order of a $p$-Sylow subgroup of the group $GL_{50}(\mathbb{F}_p)$, where $p$ is a prime number. Choose the correct answer from the following options:

- $p^{50}$
- $p^{125}$
- $p^{1250}$
- $p^{1225}$

Answer: D


Question: Evaluate the improper integral \( \int_{0}^{+\infty}\frac{\sin(x^3)}{x} \, dx \).

Answer: \dfrac{\pi}{6}


Question: Let X and Y be joint normal random variables with a common mean of 0, a common variance of 1, and a covariance of 1/2. Calculate the probability that \( X + Y \leq \sqrt{3} \).

Answer: 0.8413


Question: Determine whether a finite volume Lie group is necessarily compact, assuming the Lie group has the Haar measure.

Answer: Yes


Question: Determine whether the perimeter of the Mandelbrot set is finite or infinite.

Answer: infinite


Question: Evaluate the limit: \[ \lim_{x \to 0} (1-x)^{\frac{\log\left(1+\frac{x^2}{2}\right)}{\sin^5 x}} \] without using L'Hôpital's rule or asymptotic approximations.

Answer: 0


Question: Evaluate the double sum: $$\sum _{ m=1 }^{ \infty  }{ \sum _{ n=1 }^{ \infty  }{ \frac { { m }^{ 2 }n }{ { 3 }^{ m }(n{ 3 }^{ m }+m{ 3 }^{ n }) }  }  } $$

Answer: \dfrac{9}{32}


Question: Solve for \( r \) in the equation \( \int_{0}^{13} \frac{230}{(t+1)^{r} } \, dt = 375 \).

Answer: \dfrac{7}{5}


Question: Evaluate whether the integral \( \int_{-R}^R \frac{\cos(x)}{x} \, dx \) equals zero, given that \( f(x) = \frac{\cos(x)}{x} \) is an odd function but discontinuous at \( x = 0 \).

Answer: 0


Question: Evaluate the contour integral \( \int_C \frac{1}{(z+1)(z+2)(z+3)\ldots(z+100)} \, dz \), where \( C \) is the circle \( |z| = 150 \) oriented counterclockwise.

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n!}{n^{1000}} \]

Answer: \infty


Question: Let $(G,+,\leq)$ be a divisible totally ordered additive abelian group and let $g_{1}, g_{2} \in G$. If for every integer $n > 1$, it holds that $g_{1} \geq \left(1 - \frac{1}{n}\right)g_{2}$, determine whether $g_{1} \geq g_{2}$ must be true.

Answer: No


Question: Solve the exponential equation: \[ 2^{2x} = 7 \cdot 2^{x+\sqrt{x-1}} + 8 \cdot 2^{2\sqrt{x-1}} \] subject to the condition \( x - 1 \geq 0 \).

Answer: 5


Question: Can a uniformly continuous function have a weak derivative? In other words, can the space of uniformly continuous functions, $C_{unif.~cont.}$, be continuously embedded in the Sobolev space $W^{1,2}(\Omega)$?

Answer: No


Question: Determine whether the following statement is always true: If \( M \) is a smooth manifold without boundary such that the tangent bundle of \( M \) is trivial, then \( M \) is orientable.

Answer: True


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

Answer: \dfrac{2\pi}{y^2 + 4}


Question: Compute \( \lim \limits_{x\to 0}\frac{\ln(1+x^{2018} )-\ln^{2018} (1+x)}{x^{2019}} \).

Answer: 1009


Question: Evaluate the double integral \( \iint\limits_H \exp(-(x+y)^2) \, dx \, dy \), where \( H \) is the region in the first quadrant of the Cartesian plane, i.e., \( x, y \geq 0 \).

Answer: \dfrac{1}{2}


Question: Determine all the points of continuity for the function defined as follows: \[ f(x) = \begin{cases} x, & \text{if } x \text{ is rational} \\ 1-x, & \text{if } x \text{ is irrational} \end{cases} \]

Answer: \dfrac{1}{2}


Question:
Consider the sequence of functions \( f_n(x) = x^{1/n} \) for \( x \in [0,1] \). Determine which of the following statements is true:

1. \( \lim_{n\to\infty} f_n(x) \) exists for all \( x \in [0,1] \).
2. \( \lim_{n\to\infty} f_n(x) \) defines a continuous function on \([0,1]\).
3. \( f_n \) converges uniformly on \([0,1]\).
4. \( \lim_{n\to\infty} f_n(x) = 0 \) for all \( x \in [0,1] \).

Answer: A


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\sum^{n}_{r=1}\frac{r}{n^2+n+r} \]

Answer: \dfrac{1}{2}


Question: Arrange the functions \( \tan x^{\tan x} \), \( \tan x^{\cot x} \), \( \cot x^{\tan x} \), and \( \cot x^{\cot x} \) in descending order for \( x \in \left(0, \frac{\pi}{4}\right) \).

Answer: \cot x^{\cot x} > \cot x^{\tan x} > \tan x^{\tan x} > \tan x^{\cot x}


Question: Find the largest value of the expression \( P = \frac{3x+2y+1}{x+y+6} \) for real numbers \( x, y \) that satisfy the condition \( x > 0, y > 0 \) and \[ \log _{\sqrt{3}}\left(\frac{x+y}{x^2+y^2+xy+2}\right)=x\left(x-3\right)+y\left(y-3\right)+xy. \]

Answer: 1


Question: Let $X_1, X_2, X_3$ be independent random variables, each following an exponential distribution with parameter $λ$. Calculate the probability that $X_1 < X_2 < X_3$. Express your answer as a single probability value.

Answer: \dfrac{1}{6}


Question: Determine the number of ways to obtain a total of 15 points by rolling four different six-sided dice.

Answer: 140


Question: Given that \( \lim_{x\to \infty} x f(x^2+1) = 2 \), find the value of \( \lim_{x\to 0} \dfrac{2f'(1/x)}{x\sqrt{x}} \).

Answer: -2


Question: Find the smallest integer \( n \) such that the product \((n + 16)! (n + 20)!\) ends with a number of zeros divisible by 2016.

Answer: 4024


Question: Given a vector space $V$ of dimension $n$ and a basis $\{f_1, \ldots, f_n\}$ of its dual space $V^*$, is it possible to construct a basis $\{x_1, \ldots, x_n\}$ of $V$ such that $f_i(x_j) = \delta_{ij}$ for all $i, j$?

Answer: Yes


Question: Let $F$, $G$, and $H$ be linear functionals on a real vector space $V$. Assume there exists $x \in V$ such that $G(x)<0$ and $F(x) \geq 0$. Furthermore, assume that for any $x,y \in V$, if $G(x)<0$, $G(y)<0$, and $F(x)/G(x)=F(y)/G(y) \leq 0$, then $H(x) \geq 0$ if and only if $H(y) \geq 0$. Does this imply that $H = aF + bG$ for some constants $a$ and $b$?

Answer: Yes


Question: Determine the total number of real solutions to the equation \(x^2 = 2^{-x}\).

Answer: 3


Question: Prove that if \( \lim_{x \to \infty} \left(f(x) + \frac{f'(x)}{x}\right) = 0 \), then \( \lim_{x \to \infty} f(x) = 0 \).

Answer: 0


Question: Find the natural number \(a\) such that \(a^4 + 4\) is a prime number.

Answer: 1


Question: Let \( F(x,y,z) = (y+\sin(x),z^2+\cos(y),x^3) \) and \( \mathcal{C} \) be the curve parametrized by \( r(t)=(\sin(t),\cos(t),2\sin(2t)) \) with \( t \) ranging in \([0,2\pi)\). Use the curl theorem to evaluate \( \int_\mathcal{C} F \cdot ds \).

Answer: \pi


Question: Prove by induction that \( \sum \limits_{k=1}^{n}k\binom{n}{k}=n2^{n-1} \) for all natural numbers \( n \).

Answer: n2^{n-1}


Question: Solve the exponential equation: \[ 4^x - 3^{x-\frac{1}{2}} = 3^{x+\frac{1}{2}} - 2^{2x-1} \]

Answer: \dfrac{3}{2}


Question: Determine the distribution of the random variable \( \frac{\sqrt{n}(\bar{X} - \mu)}{\sigma s_Z} \), where \( X_i \sim N(\mu, \sigma^2) \) for \( i = 1, \ldots, n \) and \( Z_i \sim N(0,1) \) are independent random variables, and \( s^2_Z \) is the sample variance of \( Z_1 , \ldots, Z_n \).

Answer: t_{n-1}


Question: In an equilateral triangle $\triangle ABC$, there is a point $D$ inside the triangle such that $DB^2 + DC^2 + BC^2 = 100$ and the area of $\triangle DBC$ is $5 \sqrt{3}$. Find $AD^2$.

Answer: 20


Question: Compute the limit \( \lim_{n \to \infty} \frac{f(x)e^{nx} + g(x)}{e^{nx} + 1} \), where \( x > 0 \) and \( g(x) \) is a bounded function.

Answer: f(x)


Question: If \((i^n a_n)_{n=1}^{\infty}\) converges and \(a_n\in\mathbb{R}\) for all \(n\in\mathbb{N}\), does \((a_n)_{n=1}^{\infty}\) necessarily converge to 0? Justify your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{a \to 0} \frac{\tan(x+a)-\tan x}{a} \]

Answer: \sec^2 x


Question: Calculate the limit: \[ \lim_{n\to +\infty}n\int^1_0 x^nf(x)dx \] where \( f:[0,1]\to\mathbb{R} \) is a continuous function.

Answer: f(1)


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

Answer: -\infty


Question: Does there exist a distance-preserving isomorphism from \(\mathbb{R}^3\) to \(\mathbb{R}^2\) with respect to their Euclidean metrics? More generally, can such an isomorphism exist from \(\mathbb{R}^n\) to \(\mathbb{R}^2\)?

Answer: No


Question: Let \(X \sim \text{Exp}(\lambda)\) be an exponential random variable and define \(Y = X^a\). For what values of \(a \in \mathbb{R}\) is the expected value \(E[Y] < \infty\)?

Answer: a > -1


Question: Consider $N$ independent Bernoulli trials with probabilities of success $p_i$ such that $\sum_{i=1}^N p_i = 1$. Determine the values of $p_i$ that minimize the probability of having at least one success. Additionally, find the minimum probability as $N$ approaches infinity.

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


Question: Determine the radius of convergence for the power series with coefficients given by \( \frac{n!}{n^n} \).

Answer: e


Question: Let \( X = \begin{pmatrix}\cos\frac{2\pi}{5} & -\sin\frac{2\pi}{5}\\\sin\frac{2\pi}{5} & \cos\frac{2\pi}{5}\end{pmatrix} \) and \( Y = \begin{pmatrix}1 & 0\\0 & -1\end{pmatrix} \). Determine the number of distinct matrices in the set \( Z \), which consists of all matrices that can be formed by taking products of \( X \) and \( Y \), such as \( X^3YX^4Y \).

Answer: 10


Question: Given the polynomial $P(X) = X^4 + X^3 + X^2 + X + 1$, where the roots are the primitive 5th roots of unity, find the value of $x_1^{8} + x_2^{18} + x_3^{28} + x_4^{38}$, where $x_1, x_2, x_3, x_4$ are the roots of $P(X)$. Express your answer in terms of $x^3$.

Answer: -1


Question: Given the differential equation \((1-x^2)y''+2xy'-2y = 0\) for \(-1 < x < 1\), and a known solution \(y_1(x) = x\), use the reduction of order method to find a second linearly independent solution \(y_2(x)\).

Answer: 1 + x^2


Question: Calculate the limit \( \lim_{n\to\infty} n^2 \log \left(\cos \frac 1n\right) \) without using l'Hospital's rule, Taylor series, or derivatives.

Answer: -\dfrac{1}{2}


Question:
Given a decreasing sequence of Borel subsets $(A_n)_{n=1}^\infty$ of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$, and a positive number $\epsilon > 0$, determine whether there exists an open set $U \subseteq [0,1]$ such that:

1. $\lambda(A_n \setminus U) = 0$ for some $n$,
2. $\lambda(U) \leq \epsilon$, and
3. $\lambda(\partial U) = 0$.

Here, $\lambda(\cdot)$ denotes the Lebesgue measure and $\partial U$ is the boundary of $U$. Provide a proof or counterexample.

Answer: Yes


Question: Let $G$ be a finite simple group. Suppose $A$ and $B$ are proper subgroups of $G$ such that $G = AB$ and $A$ is an Abelian group. Is it true that $A \cap B = 1$?

Answer: Yes


Question:
Let $a, b, c$ be positive real numbers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = a + b + c$. Find the maximum value of the expression:
\[ \frac{1}{(2a+b+c)^2} + \frac{1}{(2b+c+a)^2} + \frac{1}{(2c+a+b)^2}. \]

Answer: \dfrac{3}{16}


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

Answer: 0


Question: Show that for the linear operator \( g_B: H \to H \) defined by \( A \mapsto BAB^+ \), where \( H \) is the subspace of Hermitian matrices in \( \mathbb{C}^{n \times n} \) and \( B \) is an arbitrary matrix in \( \mathbb{C}^{n \times n} \), the determinant \( \det{g_B} = |\det{B}|^{2n} \).

Answer: |\det B|^{2n}


Question: Consider a game where a number $u$ is drawn from a uniform distribution on $[0,1]$. On one side of a piece of paper, $u$ is written, and on the other side, $u^2$ is written. One side of the paper is randomly drawn, and you win one dollar if you correctly guess whether the observed number is $u$ or $u^2$. Determine the expected value of this game.

Answer: \dfrac{5}{8}


Question: Let $A$ be a nonempty subset of $\mathbb{R}$ that is bounded above. Suppose $s \in \mathbb{R}$ satisfies the condition that for all $n \in \mathbb{N}$, $s + \frac{1}{n}$ is an upper bound for $A$ and $s - \frac{1}{n}$ is not an upper bound for $A$. Prove that $s = \sup A$.

Answer: s


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

Let $(\mathcal{F}_0,\Vert\cdot\Vert_\infty)$ be a metric space of continuous functions that is dense in the space $\mathcal{F}$. For every $f\in\mathcal{F}$, there exists a sequence $(f_n)_{n\in\mathbb{N}}$ in $\mathcal{F}_0$ that converges uniformly to $f$. Can we demand uniform convergence?

Answer: A


Question: Let $y$ be implicitly defined by the equation \( \sin(x-y) - e^{xy} + 1=0 \) with the condition \( y(0) = 0 \). Find the value of \( y''(0) \).

Answer: -2


Question: Suppose $M$ is a smooth manifold and $S$ is a smooth embedded submanifold of $M$. Does there exist an open neighborhood $U$ of $S$ and a smooth map $U \to S$ which is a retraction?

Answer: Yes


Question: Are there infinitely many prime numbers of the form \( k \cdot 2^n + 1 \)?

Answer: Yes


Question: Given a prime ideal $P$ in a Dedekind domain $R$, determine if there always exists an element $r \in R$ such that $P$ divides the ideal $(r)$ but $P^{2}$ does not.

Answer: Yes


Question: Find the probability density function of the random variable \( V = XY \), where \( X \) and \( Y \) are independent random variables uniformly distributed over the interval \([0, 1]\).

Answer: -\ln v


Question: Evaluate the limit: \[ \lim_{x \to 0} \left(\frac{2+\cos x}{x^3\sin x}-\frac{3}{x^4}\right) \]

Answer: \dfrac{1}{60}


Question: Compute the limit: $$\lim_{n \to \infty}\left( \frac{1^a+2^a+3^a+\ldots+n^a}{n^a}-\frac{n}{a+1} \right)$$ where $a \in \mathbb{R}$ and $a > 1$. Provide your answer in terms of $a$.

Answer: \dfrac{1}{2}


Question: Let \(\Sigma\) be the portion of the surface \(z=16-x^2-y^2\) that lies inside the cylinder \(r=2\cos\theta\) and is oriented upwards. Find the rate at which the fluid described by the vector field \(\mathbf{F} = x\mathbf{j}\) flows through \(\Sigma\).

Answer: 0


Question: Suppose that \( A \) and \( B \) are any two square matrices of equal dimension with complex entries, and \( A^{\dagger}A \geq B^{\dagger}B \). Can we conclude that \(|\text{Tr}[A]| \geq |\text{Tr}[B]|\)? Justify your answer.

Answer: No


Question: Suppose $f: B \to C$ is a homomorphism between Boolean algebras such that $\ker(f) = 0$. Is $f$ necessarily a monomorphism?

Answer: Yes


Question: Let $x$, $y$, and $z$ be non-negative real numbers such that $x+y+z=1$. The expression $x^5y + y^5z + z^5x$ attains its maximum value, which can be expressed as $\frac{a^b}{c^d}$, where $a$ and $c$ are not perfect powers, and $a$, $b$, $c$, and $d$ are positive integers. Find the value of $a+b+c+d$. \( \text{Note: } a, b, c, \text{ and } d \text{ are positive integers.} \)

Answer: 22


Question: Solve the equation \(5 \log_{4}a + 48\log_{a}4 = \frac{a}{8}\) for \(a\).

Answer: 256


Question: Calculate the area of the region defined by \( M = \{(x,y) : (x^2 + y^2)^2 - 2xy = 0\} \).

Answer: 1


Question: Consider an open bounded set $U$ in $\mathbb{R}^n$ with a $C^1$ boundary. Let $1 \leq p < \infty$. Suppose $\{u_n\}_{n=1}^\infty$ is a sequence bounded in the Sobolev space $W^{2,p}(U)$. Is this sequence also bounded in $W^{1,p}(U)$?

Answer: Yes


Question: Determine whether every Polish space \(E\) has a countable \(\pi\)-system \(\mathcal{D} \subset \mathcal{B}(E)\) such that \(\sigma(\mathcal{D}) = \mathcal{B}(E)\).

Answer: Yes


Question: Determine whether both the set \( \mathcal{C} = \{A \subseteq \mathbb{N} : \lambda(\mathbb{N} \setminus A) = 1 - \lambda(A)\} \) and its complement in the power set, \( \mathcal{P}(\mathbb{N}) \setminus \mathcal{C} \), have cardinality \( 2^{\aleph_0} \).

Answer: Yes


Question: Determine whether there exists a constant \( C \) such that for any function \( f \in L^2(\mathbb{R}^d) \) and a real \( d \times d \) matrix \( A \) with \( \det A \neq 0 \), the inequality \( \sum_{n \in \mathbb{Z}^d} |\hat{f}(x - An)|^2 \leq C \|f\|_{L^2} \) holds for all \( x \in \mathbb{R}^d \).

Answer: No


Question:
Minimize the objective function:
\[ 2x_1 + 3x_2 - x_3 + x_4 \]
subject to the constraints:
\[ x_1 + 2x_2 - x_3 - 5x_4 = 10 \]
\[ -2x_1 + 3x_3 - 2x_4 = 5 \]
\[ x_1 \geq 0 \]

Answer: -\infty


Question: Determine whether a graph that contains a circuit, which includes every edge an odd number of times, must also contain an Euler circuit.

Answer: Yes


Question: Given the sequence \( \{a_n\} \) defined by \( a_1 = 1 \) and \( a_{n+1} = (-1)^n \left( \frac{1}{2} \right) \left( |a_n| + \frac{2}{|a_n|} \right) \) for \( n \ge 1 \), find the value of \( \sup \{a_n\} \).

Answer: \dfrac{17}{12}


Question: Evaluate the line integral \( \int_C \mathbf{f} \cdot d\mathbf{r} \) where \( \mathbf{f} = (\sin y, x\cos y) \), and the curve \( C \) is any closed circle.

Answer: 0


Question: Determine the number of distinct Jordan normal forms for a matrix with the characteristic polynomial \( p(A) = (\lambda-2)^2(\lambda-1)^4 \).

Answer: 10


Question: Determine the order of the ring $\mathbb{Z}[i]/(1+i)$. Provide your answer as a single integer.

Answer: 2


Question: Determine whether the improper integral \( \int_{-\infty}^{\infty} x e^{-x^2} \, dx \) converges.

Answer: 0


Question:
Determine whether there exists a function \( f(z) \) that satisfies the following properties:

- \( f \) is analytic in some region containing \(|z| \leq 1\).
- The only zero of \( f \) in \(|z| \leq 1\) occurs at \( \frac{1}{2} + \frac{i}{2} \), and it has order 3.
- \(|f(z)| = 1\) on \(|z| = 1\).
- \( f'(0) = \frac{3}{4} \).

Provide a justification for your answer.

Answer: Yes


Question: Find the area enclosed by the curve \( \lfloor x + y \rfloor + \lfloor x - y \rfloor = 5 \) for \( x \ge y \) and \( x, y \ge 0 \). Here, \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \).

Answer: \dfrac{3}{2}


Question: Find the limit: \[ \lim_{x\to 1} \frac{\ln x - x + 1}{e^{\pi(x-1)} + \sin (\pi x) -1} \] Choose the correct answer from the following options: (a) \(\frac{-1}{\pi}\) (b) \(\frac{-1}{\pi-1}\) (c) \(\frac{-1}{\pi^2}\) (d) undefined.

Answer: C


Question: Find the area of the region enclosed by the curve \( \sqrt[4]{\frac{x}{a}} + \sqrt[4]{\frac{y}{b}} = 1 \) and the coordinate axes, where \( a > 0 \) and \( b > 0 \).

Answer: \dfrac{ab}{70}


Question: Determine the sharp lower bound for the entrywise 1-norm \( \|A\|_1 = \sum |A_{i,j}| \) of a real-valued \( m \times n \) matrix \( A \) (where \( m > n > 1 \)) such that \( A^T A = I \).

Answer: n


Question: Evaluate the integral \( \int_{|z|=1}\frac{z^2-z+1}{z-1}dz \) using Cauchy's Integral Formula.

Answer: \pi i


Question:
Let $G$ be a finite solvable group of Fitting length $n$ with the upper Fitting series:
\[1 = U_0G \leq U_1G \leq \cdots \leq U_nG = G.\]
Is every normal subgroup of Fitting length $i$ contained in $U_iG$? Justify your answer.

Answer: Yes


Question: If you roll 10 six-sided dice, how many ways can you obtain a total sum of 25 from the top faces?

Answer: 831204


Question: Does there exist a matrix \( A \in \mathbb{F}_2^{m \times n} \) such that when interpreted as a matrix \( B \in \mathbb{F}_p^{m \times n} \) for some \( p > 2 \), the rank of \( A \) is different from the rank of \( B \)?

Answer: Yes


Question: Find the arc length of the curve defined by \( y = \int_{1}^{x} \sqrt{25t^3-1} \, dt \) from \( x = 1 \) to \( x = 4 \).

Answer: 62


Question: Evaluate the integral \( \int_C \ln |x - y| \, dy \) over the unit circle \( C \), where \( x, y \in \mathbb{R}^2 \) and both are on the unit circle. Show that this integral equals zero.

Answer: 0


Question: Consider a connected (undirected) graph $G$ with vertex set $V$ of size at least 2. Assign a real number $x_v$ to each vertex $v \in V$. The constraint is that for any subset $W \subseteq V$ such that the induced subgraphs on both $W$ and $V \setminus W$ are connected, it holds that \( \left| \sum_{v \in W} x_v \right| \le 1 \). Maximize \( \sum_{v \in V} |x_v| \). Is it true that for any maximizing solution, the sum of the $x_v$'s is 0?

Answer: No


Question: Consider the finite field $GF(p^n)$ as a vector space over $GF(p)$. Can $GF(p^n)$ be considered an elementary abelian group?

Answer: Yes


Question: Let $F$ be a field, and $c$ be algebraic over $F$ with minimum polynomial $p(x)$. If $d$ is any other root of $p(x)$, is $d$ necessarily in the field $F(c)$? Answer yes or no and justify your answer.

Answer: No


Question: Compute the line integral \( \oint_{\partial R} \mathbf{F} \cdot \mathrm{d}\mathbf{r} \) for the vector field \( \mathbf{F}(x,y) = (1-x^2y, xy^2+\exp(y^2)\cos(y)) \) over the region \( R \) enclosed by the curves \( C_1 = \{(x,y) \in \mathbb{R}^2 : x \in [-1,1], y=0\} \) and \( C_2 = \{(x,y) \in \mathbb{R}^2 : x^2+y^2=1, y=0\} \). Use Green's Theorem to evaluate the integral, considering the positive (counter-clockwise) orientation of the boundary \( \partial R = C_1 \cup C_2 \).

Answer: \dfrac{\pi}{4}


Question: Find the limit of the sequence given by the difference \( \left(\sum\limits_{n=1}^{10^{x+1}} \arctan\left(\frac{1}{n}\right)\right) - \left(\sum\limits_{n=1}^{10^x} \arctan\left(\frac{1}{n}\right)\right) \) as \( x \to \infty \). What is the number that this sequence approaches?

Answer: \ln 10


Question:
Determine the values of \(a\) for which the function \(f(x)\) is primitivable, where:
\[
f(x) =
\begin{cases}
\frac{1}{x}\sin \frac{1}{x^p}\sin \frac{1}{x^q}, & x \neq 0 \\
a, & x=0
\end{cases}
\]
and \(a \in \mathbb{R}\), \(p, q\) are natural numbers with \(p \geq q+2\).

Answer: 0


Question: Calculate \( \iint \nabla \times \vec{F} \cdot d\vec{s} \) for the vector field \( \vec{F} = (y, 3xy, x^2z) \) over the surface defined by \( z=\frac{1}{\sqrt{3}}\sqrt{x^2+y^2} \) for \( 0 \le z < 1. \)

Answer: -3\pi


Question:
Let $a, b, c$ be three consecutive terms of both an arithmetic progression and a geometric progression, with $a \neq 0$. Find the value of the expression:
\[ \frac{a^3b + 2b^2c^2 - c^4}{a^2b^2 + b^4 - 3c^4} \]

Answer: -2


Question: Determine the order of the expression \( \prod_{p\leq n} (1-1/p)^{-1} - \sum_{i=1}^n 1/i \). Is it \( O(\log n) \)?

Answer: O(\log n)


Question: Let $B$ be a faithfully flat $A$-algebra, and let $I \subset A$ be an ideal. Prove that $IB \cap A = I$. This is the second item of Exercise 2.6, Chapter 1, of Qing Liu's book "Algebraic Geometry and Arithmetic Curves."

Answer: IB \cap A = I


Question: Given the ordinary differential equation $\dot{\vec{x}} = A\vec{x}$, where $A$ is an $n \times n$ matrix, and the first coordinate of the solution is $x_1(t) = 5t^2\cos(3t) + 2\sin(3t) - e^{2t}\sin(3t) + 4t$, determine the minimum dimension $n$ of the matrix $A$.

Answer: 10


Question: Evaluate the limit: $$\lim_{z\to \infty} \frac{\sinh(2z)}{\cosh^2(z)}$$

Answer: 2


Question: Evaluate the limit: \( \lim_{x \to 0} x^{x^x} \)

Answer: 0


Question:
Determine under which of the following conditions the area enclosed between the x-axis and the curve is finite:

1. $a=0, \quad b=\infty,\quad f(x)=e^{-5x^5}$
2. $a=-\infty,\quad b=\infty,\quad f(x)=e^{-5x^5}$
3. $a=-7,\quad b=\infty,\quad f(x)=1/x^4$
4. $a=-7,\quad b=7,\quad f(x)=1/x^4$

Answer: 1


Question: Is it true that for every dense subset $S$ of $\mathbb{R}^3$ and every convex polyhedron $P$, there exists a combinatorially equivalent polyhedron whose vertices all belong to $S$?

Answer: Yes


Question: Calculate the volume enclosed by the paraboloids \(z = x^2 + (y-1)^2\) and \(z = \frac{5}{2} - x^2 - y^2\) for \(x \geq 0\). Use a triple integral to find the volume.

Answer: \dfrac{\pi}{2}


Question: Find the arc length of the curve given by \( x = \frac{y^2 + 2y}{8} - \ln(y+1) \) for \( 0 \leq y \leq 2 \).

Answer: 1 + \ln 3


Question: Determine the number of 3x4 matrices with non-negative integer entries such that each row sums to 3 and no column consists entirely of zeros.

Answer: 4380


Question: Let $A_n$ be a sequence of bounded linear operators on a Banach space $V$ such that $\lim_{n \to \infty} A_n = A$. Prove that $\lim_{n \to \infty} \left(I + \frac{A_n}{n}\right)^n = \exp(A)$, where $I$ is the identity operator on $V$.

Answer: \exp(A)


Question: Solve the congruence equation: \( x^{59} \equiv 604 \pmod{2013} \).

Answer: 10


Question: Determine whether the derived set of irrational numbers, denoted as $A'$, is an empty set. Consider the properties of closure, interior, and boundary in your reasoning.

Answer: \mathbb{R}


Question: Given the irreducible transition probability matrix (PTM) \( P = \begin{bmatrix} 0.25 & 0.25 & 0.25 & 0.25 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \) and the stationary distribution \([0.4, 0.3, 0.2, 0.1]\), find the expected number of steps \( E[T|X_0=1] \) it takes to return to state 1, where \( T = \min\{n > 0 : X_n = 1\} \).

Answer: \dfrac{5}{2}


Question: Evaluate the limit algebraically without using L'Hôpital's rule: $$\lim_{x\rightarrow 0}\frac{\sqrt[3]{1+x^2}-\sqrt[4]{1-2x}}{x+x^2}$$

Answer: \dfrac{1}{2}


Question: Find the fixed point of the integral operator \( T(f) = \int_0^x f(y) \left(\frac{1}{2} + \frac{y}{2}\right) dy \).

Answer: 0


Question: A polynomial function $P(x)$ of degree 5 with leading coefficient 1 increases in the intervals $(-\infty, 1)$ and $(3, \infty)$, and decreases in the interval $(1, 3)$. Given that $P(0) = 4$ and $P'(2) = 0$, find the value of $P'(6)$. Assume that $1, 2, 3$ are roots of $P'(x)$. Determine $P'(6)$. \( \boxed{} \)

Answer: 1200


Question:
For every \(\varepsilon > 0\), determine if there exists a polynomial of the form \(p(x^4) = a_1 x^4 + a_2 x^8 + \cdots + a_n x^{4n}\) such that
\[\|p(x^4)x^2 - x\| < \varepsilon\]
for every \(x \in [0,1]\).

Answer: Yes


Question: Evaluate the line integral of the vector field \(\mathbf{f}(x,y) = (x^2 - 2xy)\mathbf{i} + (y^2 - 2xy)\mathbf{j}\) from \((-1, 1)\) to \((1, 1)\) along the parabola \(y = x^2\).

Answer: -\dfrac{14}{15}


Question: Let $X_k$ be i.i.d. random variables with $X_k \sim \text{Exp}(1)$. Define $\xi_n = \max_{1\leq k\leq n} X_k$ and $\eta_n = \xi_n/\ln(n)$. Determine the limit of $\eta_n$ as $n \to \infty$. Provide your answer as a single value or expression.

Answer: 1


Question:
Solve the following recurrence relation for the sequence \(a_i\):

Given:
\[ a_1 = ra_2 + sa_n \]
\[ a_i = ra_{i+1} + sa_{i-1}, \quad i = 2, \ldots, n-1 \]
\[ a_n = ra_1 + sa_{n-1} \]

with the conditions \( \sum_{i=1}^n a_i = 1 \) and \( r + s = 1 \).

Express the sequence \(a_i\) such that it satisfies the above conditions.

Answer: a_i = \dfrac{1}{n}


Question: For a large integer $a$ and a small integer $d$, consider the expression $a^2 + d$. Determine the best lower bound for the sum $l + m$, where $l$ and $m$ are integers such that $lm = a^2 + d$. The trivial bound is $2a + 1$. Can you find a better bound?

Answer: 2a


Question: Find the greatest common divisor (GCD) of all numbers of the form \(p^8 - 1\), where \(p\) is a prime number greater than 5.

Answer: 480


Question: Consider a smooth projective variety $M$ of dimension $n$ over the complex numbers, where the cotangent bundle $T^*M$ forms a completely integrable system with $n$ independent regular functions $f_1, \ldots, f_n$ in involution. The moment map is given by $\mu: T^*M \to \mathbb{C}^n$, defined by $f_1, \ldots, f_n$. Let $Y \subset \mathbb{C}^n$ be the set of points $p \in \mathbb{C}^n$ such that the fiber $\mu^{-1}(p)$ is reducible. Determine the co-dimension of $Y$ and verify if it is always $\ge 2$. \( \text{Provide your answer in the form of a co-dimension value.} \)

Answer: 2


Question: Find the function \( f(x) \) that satisfies the following conditions: \( f(0) = 1 \), \( f'(0) = 1 \), and \( f(x+y) = f(x)f(y) \) for all real numbers \( x \) and \( y \). Use the definition of the derivative to determine \( f(x) \).

Answer: e^x


Question: Consider an event such that the time between consecutive events is exponentially distributed with a mean of 1 time unit. After each event, there is a 1/2 chance of no more events occurring. Find the variance of the time it takes until the final event occurs.

Answer: 4


Question: Let $k$ be a positive integer, and let $S$ be a set of $k$ nonzero complex numbers such that for any $a, b \in S$, the product $ab \in S$. Prove that for all $a \in S$, $a^k = 1$, and find the sum of the elements of $S$.

Answer: 0


Question: Find all functions \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x^2+f(y))=(f(x))^2+y \) for all \( x, y \in \mathbb{R} \).

Answer: f(x) = x


Question: Consider the space $X = [0,1] \times \{1,...,n\}$ with the topology $\tau \times \tau_D$, where $\tau$ is the usual topology on $[0,1]$ and $\tau_D$ is the discrete topology on $\{1, ..., n\}$. Define the equivalence relation $(x,n) \sim (x',n')$ if and only if both points are equal or $x = x' = 0$. Is the quotient space $X/\sim$ a Hausdorff ($T_2$) space?

Answer: Yes


Question: Given a natural number \( n \), determine whether there exists a critical covering \( \mathcal{U} \) of the set of natural numbers \( \omega \) such that every set in \( \mathcal{U} \) contains at least \( n \) elements.

Answer: Yes


Question: Consider a bounded domain $\Omega \subset \mathbb{R}^{n}$ that contains the origin and is star-shaped with respect to the origin. In spherical coordinates, is it true that the boundary $\partial \Omega$ can be expressed as $r = r(\omega)$, where $r$ is a function of the angle $\omega$ only? Answer true or false.

Answer: True


Question: Determine all real constants $a$ such that the existence and uniqueness theorem applies to the initial value problem \( x' = \frac{3}{2}|x|^{\frac{1}{3}}, \ x(0) = a \).

Answer: a \neq 0


Question: Given the differential equation \((1-x^2)\dfrac{d^2y}{dx^2}-2x\dfrac{dy}{dx}+\sec x y=0\), let \(y_1(x)\) and \(y_2(x)\) be two solutions with Wronskian \(W(x)\). If \(y_1(0)=1\), \(\left(\dfrac{dy_1}{dx}\right)_{x=0}=0\), and \(W\left(\frac{1}{2}\right)=\frac{1}{3}\), find the value of \(\left(\dfrac{dy_2}{dx}\right)_{x=0}\).

Answer: \dfrac{1}{4}


Question: Suppose $0 \neq X_n \in \mathbb{M}_{k(n)}(\mathbb{C})$. If $\lim_{n \to \infty} \text{tr}(X_n^*X_n) = 0$, can we conclude that $\lim_{n \to \infty} \|X_n^*X_n\| = 0$? Here, $\text{tr}$ denotes the standard trace on complex matrices, and $\|\cdot\|$ denotes the matrix norm.

Answer: Yes