Question: Given an idempotent matrix \(X_{m \times m}\) with \(\text{rank}(X) = n < m\), determine the rank of the matrix \(I_m - X\).

Answer: m - n


Question: If there exists a rational map $R$ from the extended plane $\hat{\mathbb{C}}$ to itself, and a Jordan curve $J$ on the plane, such that $R$ has no critical value on the curve, is the inverse image $R^{-1}(J)$ a finite union of disjoint Jordan curves?

Answer: Yes


Question: Find the decomposition of the module \( M = \langle x,y \rangle / \langle 2x-3y \rangle \) over the integers \( \mathbb{Z} \) into its invariant factors.

Answer: \mathbb{Z}


Question: Find the value of $i^j$ where $i$ and $j$ are quaternions.

Answer: -k


Question: Evaluate the limit using real analysis methods: $$\lim_{ n \rightarrow +\infty} \frac{1}{\Gamma(n)} \int_0^n x^{n-1}e^{-x}\, {\rm d}x$$ where \(\Gamma(n)\) is the gamma function.

Answer: \dfrac{1}{2}


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

Answer: 1


Question: Calculate the value of \( z = (1+i)^8(1-i\sqrt{3})^{-6} \) using the trigonometric form of complex numbers.

Answer: \dfrac{1}{4}


Question: Evaluate the integral \( \oint_C \frac{1}{z-a} \, dz \), where \(|a| < 1\) and \(C\) is the unit circle defined by \(e^{it}\) for \(t \in [0, 2\pi]\).

Answer: 2\pi i


Question: Calculate the residue of the function \( \sec(1/z) \) at \( z=0 \).

Answer: 0


Question:
Evaluate the limit \(\lim_{x \to 0} \frac{x - \sin x}{x^3}\) without using advanced techniques such as de l'Hôpital's Rule or Taylor series. Determine if the following solution is rigorous:

Since
\[\begin{align*}
\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}&=\lim\limits_{x \to 0} \dfrac{2x-\sin 2x}{(2x)^3}\\[4pt]&=\lim\limits_{x \to 0} \dfrac{2x-2\sin x\cos x}{(2x)^3}\\[4pt]&=\frac{1}{4}\lim\limits_{x \to 0} \dfrac{x-\sin x+\sin x-\sin x\cos x}{x^3}\\[4pt] &=\frac{1}{4}\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}+\frac{1}{4}\lim\limits_{x \to 0} \dfrac{\sin x(1-\cos x)}{x^3}\\[4pt] &=\frac{1}{4}\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}+\frac{1}{4}\lim\limits_{x \to 0} \dfrac{\sin x}{x}\cdot\lim_{x \to 0}\frac{1-\cos x}{x^2}\\[4pt] &=\frac{1}{4}\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}+\frac{1}{4}\cdot 1\cdot \frac{1}{2}\\[4pt] &=\frac{1}{4}\lim\limits_{x \to 0} \dfrac{x-\sin x}{x^3}+\frac{1}{8},\end{align*}\]
which sets up an equation on the limit. By solving the equation, we obtain that
\[\lim_{x \to 0} \frac{x-\sin x}{x^3}=\frac{1}{6}.\]

Is this solution rigorous? Provide your reasoning.

Answer: \frac{1}{6}


Question:
Find all functions \( f(x) \) that satisfy the differential equation:
\[(2f(x)xy)'_x - (f(x)(2y^2+3x))'_y = 0.\]

Answer: f(x) = Cx


Question: Find the radius of convergence of the series \( \sum_{n=0}^{\infty} \frac{w^{2^n}}{2^n} \).

Answer: 1


Question: Find the minimum value of the expression \[ \frac{a^{10}+ b^{10}}{a^{7}+ b^{7}} +\frac{b^{10}+ c^{10}}{b^{7}+ c^{7}} +\frac{c^{10}+ a^{10}}{c^{7}+ a^{7}} \] given that \( a, b, c \in (0, \infty) \) and \( a+b+c=1 \).

Answer: \dfrac{1}{9}


Question: Let $f$ be a function mapping positive reals between $0$ and $1$ to reals, defined by the equation \( f\left( \frac{x+y}{2} \right) = (1-a)f(x) + af(y) \) for $y > x$, where $a$ is a constant. Given that $f(0) = 0$ and $f(1) = 1$, find the value of $f\left( \frac{1}{7} \right)$. \( \)

Answer: \dfrac{1}{7}


Question: Given the joint density function \( f(x, y) = 3\min(x,y) \) for \( x \) and \( y \) between 0 and 1, find the covariance \( \operatorname{cov}(x, y) \).

Answer: \dfrac{3}{320}


Question: Determine whether the set \( \{f \in C^{2}[0,1] : f(0) = f(1) = 0\} \) is dense in \( L^{2}[0,1] \).

Answer: Yes


Question: Determine the number of zeros of the polynomial \( z^{10} - 6z^7 + 3z^3 + 1 \) inside the unit disk \( |z| < 1 \).

Answer: 7


Question: What is the maximum number of distinct positive integers whose squares sum up to 2002?

Answer: 17


Question: Given a monomorphism of abelian groups \( f: H \longrightarrow G \) where \( G \) is injective, determine whether the quotient group \( G/f(H) \) is divisible.

Answer: Yes


Question: Evaluate the integral \( \oint_C \frac{z^2+z+1}{(z-i)^2}dz \), where \( C \) is the positively oriented rectangle with vertices at \(-1, 1, 1+2i, -1+2i\).

Answer: -4\pi + 2\pi i


Question: Let \( S = \{ 1, 2, 3, \ldots, 2^n \} \) for some \( n \ge 2 \). Determine the maximum cardinality of a set \( S' \) consisting of \( 2^{n-1} \)-element subsets of \( S \) such that every pair of subsets in \( S' \) has exactly \( 2^{n-2} \) elements in common.

Answer: 2^n - 1


Question: Given the equation \[ \frac{\cos{x}\cos{\frac{y}{2}}}{\cos{(x-\frac{y}{2})}} + \frac{\cos{y}\cos{\frac{x}{2}}}{\cos{(y-\frac{x}{2})}} = 1, \] find the value of \( \cos{x} + \cos{y} \).

Answer: 1


Question:
Determine whether the following limit exists as \( z \) approaches \(-3\):
\[ \lim_{z\rightarrow -3}\frac{\log(z+3)}{e^{\cos(z)}}(z+3) \]
where \( \log(z) \) is the complex logarithm.

Answer: 0


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

Answer: 03


Question: Determine whether the Laurent series of \( \frac{1}{(1+x)^3} \) is the same as its Maclaurin series.

Answer: Yes


Question: Given a vector $x$, determine the total number of $n \times n$ matrices $A$ such that $Ax = Ix$, where $I$ is the $n \times n$ identity matrix.

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


Question:
Let \( X \) be a random variable with the density function:
\[ f(x) = \frac{1}{\sqrt{18\pi}} e^{a(c-x)^2} \]
where \(-\infty < x < \infty\). Given that \( \mathbb{E}[X^2] = 9 \), find \( \mathbb{E}[X^8] \).

Answer: 688905


Question: Calculate the complex path integral of \( \frac{1}{z} \) along the path \( y(t) = e^{it} \) for \( t \in [0, \pi] \). Verify if the result is \( i\pi \).

Answer: i\pi


Question: Find the value of the following limit: $$\lim_{x\rightarrow \infty}  {x  (1-\Phi(x))}$$ where $\Phi$ is the standard normal cumulative distribution function.

Answer: 0


Question:
Let \(f\) be a function on the nonnegative integers defined by the following properties:
\[ f(2n) = f(f(n)) \]
and
\[ f(2n + 1) = f(2n) + 1 \]
Determine all nonnegative integers \(k\) such that \(f(0) = 2^k\).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to \pi/4} \frac{x}{4x-\pi} \int_{\pi/4}^x \frac{\tan^2\theta}{\theta^2} \, d\theta \]

Answer: \dfrac{1}{\pi}


Question: Determine whether the following statement is true or false: Every permutation of a basic sequence is equivalent to the entire sequence, where a sequence \((x_n)\) in a Banach space \(X\) is called basic if it is a basis of \([x_1, x_2, x_3, \ldots]\) (its closed span).

Answer: False


Question: Calculate the flow of the vector field \( F(x,y,z) = (xz, yz, -z^2) \) out of the surface \( S = \{(x,y,z) \in \mathbb{R}^3 : z^2 = 1 + x^2 + y^2, 2 \leq z \leq 3\} \). Use the definition of surface integrals to find the result.

Answer: 60\pi


Question: Calculate the expected number of coin flips needed to achieve 5 successes, given the following rules: You start with 5 blank spots. The first spot is filled automatically. For each subsequent spot, you flip a coin with a 50% chance of success. If successful, you fill the next blank. If unsuccessful, you lose a filled spot, but you cannot have fewer than 0 filled spots. What is the expected number of coin flips needed to fill all 5 spots?

Answer: 28


Question: Determine if there exists a constant $c$ such that for sufficiently large $n$, the inequality \( \binom{2n}{n+10} \geq c\cdot\frac{2^{2n}}{\sqrt{\pi n}} \) holds true.

Answer: Yes


Question: Given that \( P_n(0) = -\frac{5}{16} \), find the value of \( \int_{-1}^{1} P_n^2(x) \, dx \).

Answer: \dfrac{2}{13}


Question: Consider the number field extension $L = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of the polynomial $p(X) = X^3 + 2X + 1$. The ring of integers of $L$ is $\mathbb{Z}[\alpha]$. Given that the discriminant of $L$ is $-59$, determine whether the prime number 3 remains prime in the field $L$. Assume that $p(X) \equiv 1 \pmod{3}$. Provide a justification for your conclusion.

Answer: 3


Question: Find a function $f(n)$ such that if $t = O(f(n))$, then the product \( \frac{t}{n} \times \frac{t-1}{n} \times \cdots \times \frac{2}{n} \sim e^{-an} \) is exponentially small in $n$. For example, if $t = n/2$, the product is exponentially small. Can you find a function $f(n)$ that satisfies this condition?

Answer: n


Question: Let $X$ be a random variable following a geometric distribution with parameter $p = \frac{2}{3}$, i.e., $X \sim \text{Geo}(\frac{2}{3})$. Define $Y = \left\lfloor \frac{X}{2} \right\rfloor + 1$. Find the expected value $\mathbb{E}[Y]$. \( \text{Express your answer as a single number.} \)

Answer: \dfrac{11}{8}


Question: Evaluate the line integral \( \int_C \mathbf{F} \cdot \, \mathrm{d} \mathbf{r} \) where \( \mathbf{F} = (1+xy)e^{xy} \mathbf{i} + x^2e^{xy} \mathbf{j} \) and \( C \) is parameterized by \( \mathbf{r}(t) = \cos t \mathbf{i} + 2\sin t \mathbf{j} \), for \( t \in [0, \frac{\pi}{2}] \).

Answer: -1


Question: Given the function $F(x,y,z,t)$ with partial derivatives $F_x(3,9,18,3)=1$, $F_y(3,9,18,3)=-2$, $F_z(3,9,18,3)=3$, and $F_t(3,9,18,3)=4$, find the derivative $\frac{dF}{dt}$ at time $t=3$ on the space curve $r(t) = t\vec{i} + t^2 \vec{j} + \frac{2}{3}t^3\vec{k}$. The answer should be $47$. Determine how this value is obtained.

Answer: 47


Question: Find all positive integer values of $a$ such that \( \lfloor \sqrt{a} \rfloor + \lfloor \sqrt{a+1} \rfloor + \lfloor \sqrt{a+2} \rfloor + \lfloor \sqrt{a+3} \rfloor + \lfloor \sqrt{a+4} \rfloor = 2022 \).

Answer: 164022


Question:
Let $a,b,c,d$ be real numbers with $a<c<d<b$. Consider the ring $C[a,b]$ with pointwise addition and multiplication. If $S=\{f\in C[a,b]:f(x)=0 \text{ for all } x\in [c,d]\}$, determine which of the following is true:

(A) $S$ is NOT an ideal of $C[a,b]$.
(B) $S$ is an ideal of $C[a,b]$ but NOT a prime ideal of $C[a,b]$.
(C) $S$ is a prime ideal of $C[a,b]$ but NOT a maximal ideal of $C[a,b]$.
(D) $S$ is a maximal ideal of $C[a,b]$.

Select the correct option.

Answer: B


Question:
Given the function \( V(x,y) = x^4 - x^2 + 2xy + y^2 \), consider the system of differential equations:
\[ \frac{\mathrm{d}x}{\mathrm{d}t} = -\frac{\partial V}{\partial x}, \quad \frac{\mathrm{d}y}{\mathrm{d}t} = -\frac{\partial V}{\partial y} \]
If \( x = 1 \) and \( y = -\frac{1}{2} \) at \( t = 0 \), determine the point to which the solution tends as \( t \to \infty \).

Answer: (1, -1)


Question:
Let $C$ be a category with all small colimits, and let $F: I \to C$ be a diagram with the following properties:

1. There exists an object $i_0 \in I$ such that for every other object $i$, there exists an arrow $f_i: i_0 \to i$.
2. For each object $i$ of $I$, there exists an arrow $g_i: i \to i_0$ such that $g_i \circ f_i = \text{id}_{i_0}$.
3. The diagram is filtered.

Prove or disprove that $\operatorname{colim} F = F(i_0)$. Why?

Answer: \operatorname{colim} F = F(i_0)


Question:
Given a sequence of square matrices $A_n$ with 3's on the diagonal, 2's on the diagonal above, and 1's on the diagonal below, where:
\[A_1=\begin{pmatrix} 3 \end{pmatrix}, A_2=\begin{pmatrix} 3 & 2 \\ 1 & 3\ \end{pmatrix}, A_3=\begin{pmatrix} 3&2&0\\ 1&3&2\\ 0&1&3\\ \end{pmatrix},... \]
The determinant of $A_n$, denoted as $D_n$, is defined recursively by:
\[D_n=3D_{n-1}-2D_{n-2}\]
with initial conditions \(D_1 = 3\) and \(D_2 = 7\).

Find the value of $D_5$ using the given recursive relation.

Answer: 63


Question: Determine whether the following statement is true: "Each Hilbert space of holomorphic functions on \(\mathbb{C}\) is a reproducing kernel Hilbert space." A Hilbert space of holomorphic functions on \(\mathbb{C}\) is defined as a Hilbert space \(H \subseteq O(\mathbb{C})\) such that the inclusion mapping \(H \hookrightarrow O(\mathbb{C})\) is continuous. Specifically, if \(H \subseteq O(\mathbb{C})\) is a Hilbert space of holomorphic functions, then the point evaluation map, \(f \mapsto f(z)\), is continuous for all \(z \in \mathbb{C}\).

Answer: True


Question: Determine whether the limit \( \lim_{(x,y)\to (0,0)}\frac{x^5+y^5}{x^2+y^4} \) exists.

Answer: 0


Question:
Find the eigenvalues of the following $4 \times 4$ matrix $A$:
\[
\begin{bmatrix}
7 & 1 & 2 & 2 \\
1 & 4 & -1 & -1 \\
-2 & 1 & 5 & -1 \\
1 & 1 & 2 & 8
\end{bmatrix}
\]

Answer: 6


Question: Determine all values of \( M \) such that the expression \( \langle u, v \rangle = ac - 3ad - 3bc + Mbd \) defines an inner product on \( \mathbb{R}^2 \), where \( u = \langle a, b \rangle \) and \( v = \langle c, d \rangle \).

Answer: M > 9


Question: Let $R$ be a noetherian domain and let $\mathcal{O}$ be an $R$-algebra that is finitely generated and projective as an $R$-module. Is the group of invertible fractional ideals of $\mathcal{O}$ always abelian?

Answer: Yes


Question: Let $A = K[x_1, x_2, \ldots, x_n]$ be a polynomial ring over a field $K$. Consider the ideal $m = \langle x_1 - a_1, x_2 - a_2, \ldots, x_n - a_n \rangle$ where $a_1, a_2, \ldots, a_n \in K$. Prove that the ideal $m$ is maximal in $A$.

Answer: m


Question: Evaluate the limit \( \lim_{n\to \infty}\left(\frac{1}{n^3}\sum^{n}_{k=1}k^2a_k\right) \) given that \( \lim_{n\to \infty}a_n = A \).

Answer: \dfrac{A}{3}


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

Answer: \dfrac{1}{e}


Question:
Determine the limit of the expression as \( x \to 0 \):
\[ \lim_{x \rightarrow 0} \frac{\cos(x \sqrt{2}) - \frac{1}{1+x^2}}{x^4}. \]

Answer: -\dfrac{5}{6}


Question: Calculate the volume of the solid enclosed by the surface \((x^2+y^2+z^2)^2=x\) using spherical coordinates.

Answer: \dfrac{\pi}{3}


Question: Let $k = \mathbb{F}_{2^{10}}$ and $K = \mathbb{F}_{2^{20}}$ be a Galois extension of degree 2. Calculate the number of elements in $K^\times = K - \{ 0 \}$ such that the norm $N_{K/k}(x) = \sigma(x)x = 1$, where $\sigma : x \mapsto x^{2^{10}}$ is the Frobenius automorphism.

Answer: 1025


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

Answer: \infty


Question: Determine the value of $a$ such that the series \( \sum_{n=1}^{\infty} n^{\frac{1}{3}} \left| \sin\left(\frac{1}{n}\right) - \frac{1}{n^a} \right| \) converges.

Answer: 1


Question: Determine the value of the constant \( \lambda > 0 \) such that the equation \( 2\lfloor{\lambda n}\rfloor = 1 - n + \lfloor\lambda \lfloor\lambda n \rfloor\rfloor \) holds true for all positive integers \( n \).

Answer: 1 + \sqrt{2}


Question: Find the limit \( \lim_{n\to\infty} ((n!)^{1/n}) \).

Answer: \infty


Question: Let \( P \in \mathbb{C} \) and consider a sequence of holomorphic functions \( f_n : D(P,r)\setminus \{P\} \to \mathbb{C} \) that converges uniformly on every compact subset of \( D(P,r)\setminus \{P\} \) to a holomorphic function \( f: D(P,r)\setminus \{P\} \to \mathbb{C} \). If \( P \) is a removable singularity for each \( f_n \), is \( P \) also a removable singularity for \( f \)?

Answer: Yes


Question: Evaluate the integral \( \frac{1}{\pi} \int_{0}^{\pi} e^{2\cos{\theta}} \, d\theta \).

Answer: I_0(2)


Question: Determine the value of \( h \) such that the matrix \( \begin{pmatrix} h & h+1 & 0 \\ 2h & 1 & h-1 \\ 2h-2 & 0 & 1 \end{pmatrix} \) can be diagonalized by an orthogonal matrix.

Answer: 1


Question: Calculate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \frac{1}{n+3} + \cdots + \frac{1}{2n} \right) \]

Answer: \ln 2


Question: Find all polynomials \( p(x) \) such that \( p(x+c) = p(x) + c \) for any constant \( c \).

Answer: x + k


Question: Given two independent and identically distributed exponential random variables X and Y with rate parameter \( \lambda = 4 \), find the probability density function of the random variable \( Z = \frac{X}{X+Y} \).

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to \infty} \sum \limits_{k=1}^{n}\frac{k^2 }{n^3 +n^2 +k}$$ without using integration.

Answer: \dfrac{1}{3}


Question: Determine whether the limit \( \lim_{(x,y)\to(0,0)}\frac{x^3y^3}{x^8+y^4} \) exists.

Answer: 0


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

Answer: \dfrac{1}{2}


Question: Determine the values of \(\alpha\) for which the improper integral \(\iint_{\Bbb R^2}\frac{dxdy}{(1+x^2+xy+y^2)^\alpha}\) converges.

Answer: \alpha > 1


Question: Solve the equation: \[ \frac{\ln (x)}{\ln (1-x)} = \frac{1}{x} - 1 \]

Answer: \dfrac{1}{2}


Question: Calculate the expected number of rolls needed to obtain two consecutive sixes when rolling a die, given that all rolls must be even numbers.

Answer: 12


Question: Find the limit: \[ \lim_{n \to \infty} \frac{1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}}{1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n+1}}. \]

Answer: 2


Question: Let $c_0=\{ (x_n) : x_n \in \mathbb{R}, x_n \to 0 \}$ and $M=\{(x_n) \in c_0 : x_0+x_2+ \cdots +x_{10}=0 \}$. Determine the dimension of the quotient space $c_0/M$.

Answer: 1


Question: Find the order of the normalizer of the permutation $s = (12)(34)$ in the symmetric group $S_6$. Provide your answer as a single integer.

Answer: 16


Question: Determine all integers \( n \geq 6 \) for which the number of permutations with \( n-2 \) fixed points and one 2-cycle is equal to the number of permutations with \( n-6 \) fixed points and three 2-cycles.

Answer: 6


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \frac{\cot(x)(1-\cos^2(3x))}{x^2 + 5x} \right) \]

Answer: \dfrac{9}{5}


Question: Evaluate the vector line integral \( \int_{\gamma} x^2y\,dx - xy\,dy \) where \( \gamma \) is the curve with equation \( y^2 = x^3 \), from \((1,-1)\) to \((1,1)\). Use the parametrization \( \vec{r}(t) = (t^2, t^3) \) for \(-1 \le t \le 1\).

Answer: \dfrac{4}{9}


Question: Let \( X \sim \text{Poisson}(\lambda) \), where \( \lambda > 0 \) is unknown. If \( \delta(X) \) is the unbiased estimator of \( g(\lambda)=e^{-\lambda}(3\lambda^2+2\lambda+1) \), find the value of \( \sum_{k=0}^\infty \delta(k) \).

Answer: 9


Question: Given that a vowel (v) is followed by a consonant (c) 75% of the time and a consonant is followed by a vowel 65% of the time, determine the most likely sequence of vowels and consonants when three successive letters are chosen.

Answer: cvc


Question: Find the value of \( \lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}\frac{\sqrt[k]{k!}}{k} \).

Answer: \dfrac{1}{e}


Question: Find a generator of the ideal \( I = \langle x^4 + x^3 - 3x^2 - 5x - 2, x^3 + 3x^2 - 6x - 8 \rangle \subseteq \mathbb{Q}[x] \).

Answer: x^2 - x - 2


Question: Evaluate the expression \( \bigcap_{k=0}^{\infty} \bigcup_{n=k}^{\infty} A_n \), where \( A_n = \{ m \in \mathbb{N} : \exists k \in \mathbb{N} \text{ such that } km = n \} \).

Answer: \mathbb{N}


Question: Let $G$ be a finite group and $H, K$ be subgroups of $G$. Define $HK := \{hk: h \in H, k \in K \}$. Let $\phi: H \times K \rightarrow HK$ be defined by $(h,k) \mapsto hk$. Prove that $|\phi^{-1}(g)| = |H \cap K|$ for any $g \in HK$. Provide a bijection between $\phi^{-1}(g)$ and $H \cap K$ to demonstrate this equality.

Answer: |H \cap K|


Question: Find a nontrivial example of a function \( f \) defined on a neighborhood of \( z \in \mathbb{C} \) and a path from \( z \) to \( z \) such that the analytic continuation of \( f \) along the path is \( f' \). Ensure that \( f \neq f' \).

Answer: f(z) = \sqrt{z}


Question: Does there exist an irrational number $a$ such that for a given increasing sequence of positive integers $(n_k)_{k\in \mathbb{N}}$, the sequence $\{an_k\} := (a n_k) \mod 1$ converges to $1/2$ as $k \rightarrow \infty$?

Answer: Yes


Question: Calculate the arc length of the polar curve given by \( r = 4(1 - \sin \phi) \) over the interval \( -\frac{\pi}{2} \leq \phi \leq \frac{\pi}{2} \). Use the formula for arc length in polar coordinates: \[ l = \int_{\alpha}^{\beta} \sqrt{r^2 + (r')^2} \, d\phi \] where \( r' = \frac{dr}{d\phi} \).

Answer: 16


Question: Does there exist a function that has an uncountable number of continuity points and an uncountable number of discontinuity points in any neighborhood contained in the interval \((0,1)\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\sin 5x-5\sin 3x+10\sin x}{\sin (\sin x)+\tan x-2x} \]

Answer: \dfrac{480}{7}


Question: Find all prime numbers \( p \) such that the last \( p \) digits of \( p^p \) form a prime number. It is known that \( p = 433 \) satisfies this condition. Are there any other such prime numbers?

Answer: 433


Question: What is the smallest value of \( n \) such that the final digit of \( 13^n \) is one more than the digit adjacent to it?

Answer: 14


Question: Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}$. The minimum value of $x$ can be expressed in the form $a^c b^d$, where $a, b, c, d$ are positive integers. Compute $a + b + c + d$. Provide your answer as a single integer.

Answer: 31


Question: Find an analytic function \( g(z) \) that is continuous in the closure of the region \( D \) such that \( \ln|g(z)| = u(x, y) \), where \( f(z) = u(x, y) + iv(x, y) \) is an analytic function in the region \( D \) and continuous in the closure of \( D \).

Answer: e^{f(z)}


Question: Given the sequences \(a_n\) and \(b_n\) such that \(a_n > 0\), \(b_n > 1\), \(\lim_{n \rightarrow \infty} a_n = 0\), and \(\lim_{n \rightarrow \infty} b_n = 1\), define the set \(A_n = \{x: a_n \leq x < b_n\}\). Determine \(\limsup_{n \rightarrow \infty} A_n\) and \(\liminf_{n \rightarrow \infty} A_n\).

Answer: (0, 1]


Question: Determine the minimum size of a subset \( M \) of the digits \( \{0,1,\ldots,9\} \) such that every number from 1 to 99999999 can either be formed directly from the digits in \( M \) or as a sum of two numbers formed from the digits in \( M \).

Answer: 5


Question:
Determine which of the following sequences \( \{f_n\} \in C[0,1] \) must contain a uniformly convergent subsequence:

1. \(|f_n(t)| \le 3 \forall t \in [0,1], \forall n\)
2. \(f_n \in C^1[0,1], |f_n(t)| \le 3, |f_n'(t)| \le 5 \forall t \in [0,1], \forall n\)
3. \(f_n \in C^1[0,1], \int_{0}^{1}|f_n(t)|dt \le 1, \forall n\)

Answer: 2


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

Answer: e^{-3}


Question: Consider a Simple Random Walk (SRW) that starts at position 1 and terminates when it reaches position 0. The probability of moving up or down by 1 is \( \frac{1}{2} \) each. What is the probability that the SRW reaches a value \( K \in \mathbb{N} \) before it terminates at 0? Express your answer in terms of \( K \).

Answer: \dfrac{1}{K}


Question: Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a one-to-one function such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. Determine the smallest possible value of $f(999)$. \( \text{(Note: The answer is given as 24.)} \)

Answer: 24


Question:
Determine whether the function \( f: \mathbb{R}^d \to \mathbb{R}^d \) defined by \( f(x) = \exp(-\|x - \mu\|) (\mu - x) \) is Lipschitz continuous. Specifically, find if there exists a constant \( D \in \mathbb{R} \) such that for all \( x, y \in \mathbb{R}^d \), the inequality
\[
| \exp(-\|x - \mu\|) (\mu - x) - \exp(-\|y - \mu\|) (\mu - y) | \leq D\|x-y\|
\]
holds, where \( \| \cdot \| \) denotes the Euclidean norm.

Answer: Yes


Question: Let $K$ be a field of characteristic $p$. Consider the automorphism $\sigma \in \operatorname{Aut}(K(x)/K)$, where $x$ is transcendental over $K$ and $\sigma(x) = x + 1$. Find a primitive element of the fixed field of $\langle \sigma \rangle$. Provide your answer as a specific element or expression.

Answer: x^p - x


Question: Determine the cardinality of the set of equivalence classes \( P(\mathbb{N} \times \mathbb{N})/_R \), where \( R \subseteq P(\mathbb{N} \times \mathbb{N})^2 \) is defined by the relation \( \langle r,s\rangle \in R \Leftrightarrow (\exists \pi : \mathbb{N} \to \mathbb{N} \forall x,y \in \mathbb{N} (\langle x,y\rangle \in r \leftrightarrow \langle\pi(x),\pi(y)\rangle \in s)) \). Here, \( \pi \) is a bijection, and \( A/_R \) denotes the set of equivalence classes of \( A \) under \( R \).

Answer: 2^{\aleph_0}


Question: Let \(f\colon (M,\partial M)\to (M',\partial M')\) be a map between two compact connected oriented \(n\)-manifolds such that \(f\big|\partial M\to \partial M'\) is a homeomorphism. Prove that \(\deg(f) = \pm 1\).

Answer: \pm 1


Question: Suppose the quadratic polynomial \(p(x) = ax^2 + bx + c\) has positive coefficients \(a, b, c\) in arithmetic progression in that order. If \(p(x) = 0\) has integer roots \(\alpha\) and \(\beta\), find the value of \(\alpha + \beta + \alpha \beta\).

Answer: 7


Question: Consider the function \( f: \overline{\mathbb{R}} \to [-1,1] \) defined by \( f(x) = \frac{x}{1+|x|} \) for \( x \in \mathbb{R} \), \( f(\infty) = 1 \), and \( f(-\infty) = -1 \). The metric \( d(x,y) = |f(x) - f(y)| \) is defined on the extended real line \( \overline{\mathbb{R}} \). For a given radius \( r > 0 \), determine if the open ball \( B(\infty; r) = \{ x \in \overline{\mathbb{R}} : 1-r < \frac{x}{1+|x|} < 1+r \} \) can be expressed as an interval.

Answer: Yes


Question: Let $G$ be a noncyclic p-group of order $p^n$ with $n \ge 2$. Determine whether there exists a surjective group homomorphism from $G$ to $\mathbb{Z}_p \times \mathbb{Z}_p$. Provide a justification for your answer.

Answer: Yes


Question: Find the smallest positive integer \( q \) such that \( \binom{2n}{n} < q^n \) for all positive integers \( n \).

Answer: 4


Question: Let $G:[0,\infty)\to \mathbb{R}^n$ be a continuous and differentiable function such that $G(0) = 0\in\mathbb{R}^n$. Suppose there exists a constant $c>0$ such that for all $t>0$, the inequality $$||DG(t)||\leq c||G(t)||$$ holds, where $DG(t)$ is the derivative of $G$ and $||\cdot||$ denotes the Euclidean norm. Does this imply that $G(t) \equiv 0$ for all $t \geq 0$?

Answer: G(t) \equiv 0


Question: Let $G$ be a graph with $n$ vertices that does not contain a simple cycle $v_1 \ldots v_k$ with two crossing chords $v_iv_j$ and $v_pv_q$, where $i<p<j<q$. Prove or disprove that the number of edges in $G$ does not exceed $2n-3$.

Answer: 2n - 3


Question: Evaluate the double integral \( \iint_M \left|\cos(x+y)\right|\,dx\,dy \), where \( M = [0,\pi] \times [0,\pi] \).

Answer: 2\pi


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

Answer: 1


Question: For the logarithmic spiral \(\gamma(t) = (e^t \cos t, e^t \sin t)\), determine the angle between the curve and its tangent vector at any point \(\gamma(t)\). Is this angle \(\pi/4\) or \(3\pi/4\)?

Answer: \dfrac{\pi}{4}


Question: Determine the Galois group of the polynomial \( f(x) = x^3 + x^2 - 2x - 1 \) over \( \mathbb{Q} \).

Answer: A_3


Question: Determine the value of \(a\) such that the range of the function \(f(x) = \frac{1}{3x^4 - 8ax^3 + 12a^2 + x^2 + a}\) is completely contained within the interval \([0, \frac{1}{3}]\).

Answer: \frac{1}{2}


Question: Find the 1000th decimal place of \( \sqrt{\underbrace{111\ldots111}_{1998 \text{ times}}} \).

Answer: 1


Question: Find all non-negative integer pairs $(m, n)$ such that the Ackermann function $A(m, n)$ satisfies $A(m, n) = m + n$. The Ackermann function $A(m, n)$ is defined by the recursion: $$\begin{cases} A(0, n) = n + 1, \\ A(m + 1, 0) = A(m, 1), \\ A(m + 1, n + 1) = A(m, A(m + 1, n)). \end{cases}$$

Answer: \emptyset


Question: Let $G$ be a Lie group acting smoothly and effectively on a smooth manifold $M$. Consider the quotient map $\pi: M \to M/G$. Determine if there exists a point $p \in M$ such that an open neighborhood of $\pi(p)$ in $M/G$ is smooth.

Answer: No


Question: Given the equation \(2\arctan\left(\frac{y}{x}\right)=\ln(x^2+y^2)\), find the value of \(\frac{d^2y}{dx^2}\) at the point where \(x=1\). Assume that \(y\) satisfies the equation at this point.

Answer: 2


Question: Find the number and multiplicity of the intersections between the polynomials \( p = x_0x_1^2 + x_1x_2^2 + x_2x_0^2 \) and \( q = -8(x_0^3 + x_1^3 + x_2^3) + 24x_0x_1x_2 \) in the projective plane \( \mathbb{P}^2(\mathbb{K}) \), where \( \mathbb{K} \) is an algebraically closed field and \([x_0, x_1, x_2]\) are the homogeneous coordinates. Use Bézout's theorem and elimination theory to determine the intersections.

Answer: 9


Question:
Estimate the upper bound of the number of solutions \(N_{\sqrt[3]{2}}(1)\) for the inequality:
\[ \left \lvert \sqrt[3]{2} - \frac{p}{q} \right \rvert < \frac{1}{q^{3}} \]

Answer: 2


Question: Determine whether the following statement is true: If $\lim_{t\to\infty}\varphi(t)=x_0$, does this imply that $\lim_{t\to\infty}\varphi'(t)=0$? Assume $\phi:\mathbb{R} \to \mathbb{R}^n$ and $\lim_{t \to \infty} \phi(t) = X_0$, where $X_0$ is a constant in $\mathbb{R}^n$. Then, does $\lim_{t\to \infty} \phi'(t) = 0$ hold true?

Answer: False


Question: Calculate the 100th derivative of the function \( \frac{1}{x^2+3x+2} \) at \( x=0 \) with 10% relative error.

Answer: 100!


Question: Let \( m > 1 \), and \( n \), \( k \) be positive integers. Determine whether there exists a positive integer \( l > m-1 \) such that \( \binom{l}{m} \equiv k \pmod{2^n} \).

Answer: Yes


Question: In a connected metric space $(X,d)$, a sequence $\{x_n\}_n \subseteq X$ is said to be T-convergent to $x \in X$ if for any $a,b \in X$ with $d(a,x) < d(x,b)$, there exists an $N$ such that for all $n \geq N$, $d(a,x_n) < d(x_n,b)$. Determine whether T-convergence implies standard convergence in $(X,d)$. Provide a justification for your answer.

Answer: Yes


Question: Find the integer \( m \) such that \( m^7 \equiv 169 \pmod{1643} \).

Answer: 524


Question: Find the number of integer solutions to the equation \(a+b+c+d+e+f=20\) subject to the constraints \(2 \leq a,b,c,d,e,f \leq 6\).

Answer: 951


Question: Calculate the limit: \( \lim_{x\to1}\frac{\cos x \ln x}{\ln(e^x - e)} \)

Answer: 0


Question: Let $K/\mathbb{Q}$ be an algebraic extension, and let $E_1$ and $E_2$ be elliptic curves defined over $\mathbb{Q}$. Is it possible for the Mordell-Weil rank of $E_1(K)$ to be finite while the rank of $E_2(K)$ is infinite?

Answer: Yes


Question: Evaluate the integral \( \int \int \int_A \frac{dxdydz}{(x+y+1)^2} \) where \( A=\{(x,y,z):0<x<1, 0<x+y<1, 0<z(x+y+1)<1\} \).

Answer: \dfrac{3}{8}


Question: Evaluate the integral \( 4\pi R \int_{-r}^{r}\sqrt{r^2-y^2} \, dy \) to find the volume of a torus.

Answer: 2\pi^2 R r^2


Question: Calculate the surface area of the surface \( M = \{(x,y,z) \in \mathbb{R}^3 : x^2+y^2<1 \text{ and } z=3-2x-2y\} \).

Answer: 3\pi


Question: Find the power series solution for the differential equation \((x-1)y'' - xy' + y = 0\) with initial conditions \(y(0) = -2\) and \(y'(0) = 6\).

Answer: -2e^x + 8x


Question: Given a complete balanced ternary tree where every edge has length 1, determine if there exists a height $h$ such that for any equal coloring of the tree's edges, one color has more than $n$ connected components. Provide your answer for a fixed positive integer $n$.

Answer: Yes


Question: Determine the number of real solutions to the equation \( \log_{\frac{1}{16}}(x) = \left(\frac{1}{16}\right)^x \).

Answer: 2


Question: Let $A$ be a measurable subset of $[0,1]$ such that for every interval $[a,b] \subseteq [0,1]$, the measure $m(A \cap [a,b]) \leq \frac{b-a}{2}$. Prove that the measure of $A$, $m(A)$, is zero.

Answer: 0


Question: Find the maximum value of the expression \( \sin{x} + \sin{y} - \sin{(x+y)} + \sqrt{3}(\cos{x} + \cos{y} + \cos{(x+y)}) \).

Answer: 3\sqrt{3}


Question: Find the largest eigenvalue of the matrix \(\begin{bmatrix} 1 & 4 & 16 \\ 4 & 16 & 1 \\ 16 & 1 & 4 \end{bmatrix}\).

Answer: 21


Question: Evaluate the surface integral \( \int_S f\; dA \) where \( f(x,y,z)=2y(x^2+1)^{-1}(1+4z)^{-1/2} \) and \( S=\{z=x^2+y^2, |y|<1\} \). Determine the appropriate limits of integration and compute the integral.

Answer: 0


Question: Calculate the limit: \[ \lim _{(x,y)\to (0,0)} (xy) \ln (x^2+y^2) \]

Answer: 0


Question: Let \( F: B(0,1) \to \mathbb{R} \) be a differentiable function such that \( |F| \leq 1 \) on the unit ball \( B(0,1) \) in \( \mathbb{R}^d \). Prove that there exists a point \( \xi \in B(0,1) \) such that \( |\nabla F(\xi)| \leq 2 \).

Answer: 2


Question: Calculate the number of elements of order 24 in the group \( C_{24} \times C_{40} \).

Answer: 96


Question: Does there exist a basis for the space of all real \( n \times n \) matrices, \( M_n(\mathbb{R}) \), that consists entirely of diagonalizable matrices?

Answer: Yes


Question: Evaluate the integral \( \int_0^{2 \pi} \frac{2}{\cos^{6}(x) + \sin^{6}(x)} \, dx \).

Answer: 8\pi


Question: Determine the maximum number of distinct sets that can be generated from a single set \( B \subseteq \mathbb{R} \) by taking a finite number of closures and interiors.

Answer: 7


Question: In how many ways can one choose a composition \( \alpha \) of \( n \), and then choose a composition of each part of \( \alpha \)?

Answer: 3^{n-1}


Question: Find the sum of the maximum and minimum curvature of the ellipse given by the equation \(9(x-1)^2 + y^2 = 9\).

Answer: \dfrac{28}{9}


Question: Determine whether the actions on a $3 \times 3 \times 3$ Rubik's Cube form a group, considering each state of the cube as an element and the solved cube as the identity element.

Answer: Yes


Question: For what value of \( p \) is the series \( \sum_{n=1}^{\infty} (-1)^{n-1}\frac{(\ln(n))^p}{n} \) convergent?

Answer: p \in \mathbb{R}


Question: Let $p$ be a prime and $n$ a positive integer dividing $p-1$. Suppose $a^{(p-1)/n} = 1$ in the finite field $\mathbb{F}_p$. Does there exist an element $b \in \mathbb{F}_p$ such that $a = b^n$?

Answer: Yes


Question: Determine the number of possible values of \( x \) that satisfy the equation \( \frac{x}{100} = \sin(x) \).

Answer: 63


Question: Consider a $(-1)$-curve $E \subseteq S$ inside a non-singular complex projective surface, where $E \cong \mathbb{P}^1_\mathbb{C}$ and $E^2 = -1$. Let $\sigma \in \text{Aut}(\mathbb{C})$ be a field automorphism, and define the fibered product $E^\sigma := E \times_{\text{Spec } \mathbb{C}} \text{Spec } \mathbb{C}$. Is it true that ${(E^\sigma)}^2 = -1$? In other words, is the auto-intersection preserved by the action of $\text{Aut}(\mathbb{C})$ on $E$?

Answer: -1


Question: Determine the radius of convergence of the series \( \sum_{n=0}^\infty \frac{a_n}{n!}z^n \), given that the series \( \sum_{n=0}^\infty a_n z^n \) has a radius of convergence \( R_1 > 0 \).

Answer: \infty


Question: Evaluate the integral \( \int_{1}^{100} \frac{f(x)}{x} \, dx \) given that \( f(x) = f\left(\frac{100}{x}\right) \) for all positive real \( x \) and \( \int_{1}^{10} \frac{f(x)}{x} \, dx = 5 \).

Answer: 10


Question: Let $R$ be a commutative ring, and let $P$ be a matrix in $M_n(R)$ such that $\det(P)$ is a zero divisor in $R$. Must $P$ be a zero divisor in $M_n(R)$? Justify your answer.

Answer: Yes


Question: Let $f:[0,1]\to\mathbb{R}$ be a Lebesgue integrable function. Prove that $\lim_{n\to\infty} n\lambda(\{x:|f(x)|\geq n\})=0$, where $\lambda$ denotes the Lebesgue measure.

Answer: 0


Question: Consider a line of $n$ bugs, each with a unique size, moving in the same direction. If a larger bug is behind a smaller bug, it will eat the smaller bug. What is the expected number of bugs remaining after a long enough time?

Answer: H_n


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{\binom{3n}{n}}{\binom{2n}{n}} \right)^{1/n} \]

Answer: \dfrac{27}{16}


Question: Let $(f_n)$ be a sequence of measurable functions defined on a measurable set $E$. Is the set $A = \{x \in E \mid \lim_{n \to \infty} f_n(x) \text{ exists finitely}\}$ measurable?

Answer: A


Question: Determine the number of natural numbers \( n \) in the interval \([1005, 2010]\) for which the polynomial \(1 + x + x^2 + \cdots + x^{n-1}\) divides the polynomial \(1 + x^2 + x^4 + \cdots + x^{2010}\).

Answer: 0


Question: Find the absolute maximum of the function \( f(x) = \frac{\sqrt{4+32x^{2}+x^{4}}-\sqrt{4+x^{4}}}{x} \) for \( x \in \mathbb{R} \) and \( x \neq 0 \). Determine the pair \((x_0, M)\) where \( f(x_0) = M \) is the absolute maximum.

Answer: (\sqrt{2}, 4)


Question: Find the set of limit points of the sequence \( \{s_n\} \) where \( s_n = \cos n \).

Answer: [-1, 1]


Question: Given that \( \log_{a}b + \log_{b}c + \log_{c}a = 8 \) and \( \log_{b}a + \log_{c}b + \log_{a}c = 13 \), find the value of \((\log_{a}b + 1)(\log_{b}c + 1)(\log_{c}a + 1)\).

Answer: 23


Question: Determine whether the following statement is true or false: There are infinitely many continuous functions $f$ such that \( \int_0^1 f(x)(1-f(x)) \, dx = \frac{1}{4} \).

Answer: False


Question: Let $G$ be a finite group with no element of order $p^2$ for any prime $p$, and suppose that $|G| \neq p$ for each prime $p$. Does there always exist an automorphism $\phi$ of order 2 such that for at least one subgroup $H$ of $G$, we have $\phi(H) \neq H$? Provide a justification for your answer.

Answer: No


Question: Can a compact operator on the Hilbert space $l^2$ have both infinite rank and infinite kernel? Provide a justification for your answer.

Answer: Yes


Question: Suppose $L$ and $L'$ are line bundles over a compact Riemann surface $C$. Is it true that $h^0(C, L \otimes L') = h^0(C, L) + h^0(C, L')$, where $h^0(V)$ denotes the complex dimension of the global sections of a vector bundle $V$?

Answer: No


Question: Calculate the circulation of the vector field \( \vec{F}(x, y, z) = \langle \sin(x^2+z)-2yz, 2xz + \sin(y^2+z), \sin(x^2+y^2)\rangle \) along the circle parameterized by \( \vec{r}(t)=\langle\cos(t), \sin(t), 1\rangle \) for \( t \in [0, 2\pi] \). Use Stoke's Theorem to evaluate the integral.

Answer: 4\pi


Question:
Evaluate the integral \( \int_{0}^{1}\sqrt{1+\frac{1}{3x}}\text{ d}x \) and determine which of the following is closest to its value:

(A) 1 \quad (B) 1.2 \quad (C) 1.6 \quad (D) 2 \quad (E) The integral doesn’t converge.

Answer: C


Question: Evaluate the integral \( \int_S \vec{V} \cdot \hat{n} \, dS \) using the Divergence Theorem, where \( \vec{V} = x^2z\hat{i} + y\hat{j} - xz^2\hat{k} \). The surface \( S \) is the boundary of the region bounded by \( x^2 + y^2 = z \) and \( z = 4y \).

Answer: 8\pi


Question: Find the value of the tenth derivative of the function \( f(x) = e^{-x^2} \) at \( x = 0 \), denoted as \( f^{(10)}(0) \).

Answer: -30240


Question: Let \( a \) be a given complex number satisfying \( 0 < \left\vert a\right\vert < 2 \) and let \( \gamma \) be the unit circle of radius 3 oriented in the positive sense. Determine the value of the contour integral \( \oint_{\gamma}{\Re\left(z\right) \over z - a}\,\mathrm{d}z \) in terms of \( a \).

Answer: \pi i a


Question: In a normed vector space, is it true that compact sets, which are closed and bounded, must be finite dimensional?

Answer: No


Question: Evaluate the integral: $$\int_{-0.5}^{0.5}\cos(x)\ln\frac{1+x}{1-x}dx$$ and show that it equals 0.

Answer: 0


Question: Determine whether there exists a topological space $X$ such that $X$ is connected, the boundary $\partial X$ is connected, but the interior $\text{int}(X)$ is not connected.

Answer: Yes


Question: Is the space $H^1_0(\mathbb{R}^n)$ dense in $H^1(\mathbb{R}^n)$?

Answer: Yes


Question: In a box, there are eight real coins and two fake coins, with the fake coins having a different weight than the real ones. You randomly select a pair of coins without replacement from the ten coins, and then select another pair from the remaining eight coins. What is the probability that all four coins selected are real, given that the combined weight of the first pair is the same as the second pair?

Answer: \dfrac{15}{19}


Question: Calculate the intersection multiplicity of the curves defined by $f(y, x) = y^2 - x^3$ and $g(y, x) = y^3 - x^7$ at the point $(0,0)$. Provide your answer as a single integer.

Answer: 9


Question: Determine whether the function \( f(x) = \begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \) is differentiable more than once at \( x = 0 \).

Answer: Yes


Question: Suppose $T$ is a compact operator on a Hilbert space, and there exists a sequence of finite rank operators $T_n$ converging to $T$. If there exists an $N$ and a scalar $\lambda$ such that $\lambda$ is an eigenvalue of $T_n$ for all $n > N$, can we conclude that $\lambda$ is also an eigenvalue of $T$?

Answer: No


Question: In the context of Kripke-Platek set theory (KP), determine whether the formula \(\forall w(w\in x \leftrightarrow\forall y\exists z F(w,y,z))\) is classified as \(\Pi_3\) when \(F(w,y,z)\) is a \(\Delta_0\) formula.

Answer: \Pi_3


Question: Consider the group \( G = \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} : a, b, c \in \mathbb{R} \right\} \) and its action \( \pi \) on \( \mathbb{R}^2 \) defined by \( \pi_g(x) = gx \) for all \( x \in \mathbb{R}^2 \). Determine the number of distinct orbits of this action on \( \mathbb{R}^2 \).

Answer: 3


Question: Determine whether the following statement is true or false: If $G$ is a group and $H \triangleleft G$, then $G/H \cong G$ implies $H = \{1\}$.

Answer: B


Question: Find the minimum positive integer value of \( q \) such that there exists a fraction \( \frac{p}{q} \) with \( p \) and \( q \) as positive integers, satisfying the inequality \( \frac{95}{36} > \frac{p}{q} > \frac{96}{37} \).

Answer: 5


Question:
Given that \(g(n) \in \mathbb{C}\) satisfies the following conditions:

- \(g(n) \neq 0\)
- \(\lim\limits_{n \to \infty}{g(n)} = 0\)
- \(\sum\limits_{n=2}^{\infty}{ n^\alpha g(n)} = -1\)
- \(\sum\limits_{n=2}^{\infty}{ n^{-\alpha} g(n)} = -1\)

Determine if it is necessarily true that \(\alpha = 0\).

Answer: \alpha = 0


Question: Are the slices \( \{b\} \times F \subset B \times F \) homeomorphic to \( F \) when considering the continuous projection \( B \times F \rightarrow B \)?

Answer: Yes


Question: Given a function \( f: \mathcal{F} \times \mathcal{G} \to \mathcal{F} \) that is bijective with respect to its first argument, and an inverse bundle function \( g: \mathcal{F} \times \mathcal{G} \to \mathcal{F} \) such that \( g(f(x, a), a) = x \) for all \( x \in \mathcal{F} \), suppose for all \( x \in \mathcal{F} \) and \( a, b \in \mathcal{G} \), there exists \( p \in \mathcal{G} \) such that \( f(f(x, a), b) = f(x, p) \). It follows that \( f(f(f(x,a),b),c) = f(x,q) \). Is it then also the case that \( f(g(f(x,a),b),c) = f(x,r) \) for some \( r \in \mathcal{G} \)?

Answer: Yes


Question: Evaluate the derivative \( \frac{d}{dx}\int_{0}^{x} \sin(1/t) \, dt \) at \( x = 0 \). Determine if the derivative exists at this point, and if so, find its value.

Answer: 0


Question: Let $X$ and $Y$ be finite-dimensional vector spaces. Suppose $u \in X$ satisfies $Tu = 0$ for every linear operator $T: X \rightarrow Y$. Prove that $u = 0$.

Answer: 0


Question: Evaluate the double integral \( \iint_R x \, dx \, dy \) using the change of variables \( x = \sqrt{u+v} \) and \( y = v \), where the region \( R \) is in the first quadrant of the \( xy \)-plane and is bounded by \( y = 0 \), \( y = 16 \), \( y = x^2 \), and \( y = x^2 - 9 \).

Answer: 72


Question: Consider a 2D random walk on a lattice grid starting from the origin. Determine the average waiting time for the walk to first reach the line $y = 1 - x$. Provide your answer as an expected value.

Answer: \infty


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

Answer: 2^{\aleph_0}


Question: Find the angle \( \theta \) in the range \( 0 \leq \theta < 360 \) degrees that minimizes the x-coordinate of the Archimedean spiral given by \( r = 1 - \frac{0.5}{2\pi}\theta \).

Answer: 174


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

Answer: \pi^2


Question: Let \(a, b, c\) be non-zero real numbers satisfying the equation \[ \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} - \frac{a^3+b^3+c^3}{abc} = 2. \] Evaluate the expression \[ M = [(a+b)^{2018} - c^{2018}][(b+c)^{2018} - a^{2018}][(a+c)^{2018} - b^{2018}]. \]

Answer: 0


Question:
Given a topological space \( X \) and a subset \( A \subseteq X \), define \( \hat{A} = A \cup \{ \text{connected components of } X-A \text{ which are relatively compact in } X \} \). Determine which of the following properties hold for every \( A \subseteq X \):

1. \( \hat{A} \) is compact.
2. \( \hat{A} = \hat{\hat{A}} \).
3. \( \hat{A} \) is connected.
4. \( \hat{A} = X \).

Consider the example where \( X = \mathbb{R} - \{1,2,3\} \) with the metric topology and \( A = (-\infty,1) \). Then \( \hat{A} = (-\infty,1) \cup (1,2) \cup (2,3) \). Based on this example, determine which options are false and which option is correct.

Answer: 2


Question: Calculate the limit: \[ \lim_{(x,y)\to(1,2)} [\ln(5-(x^2+y^2))]\sqrt{y^2-4}. \]

Answer: 0


Question: Evaluate the total mass of a wire that has the shape of a curve obtained by the intersection of the portion of the sphere $x^2+y^2+z^2=4$, $y\geq 0$, with the plane $x+z=2$. The density at each point of the wire is given by $f(x,y,z)=xy$. The curve is parameterized by $\alpha(t)=\left(t,\sqrt{4t-2t^2},2-t\right)$. Calculate the total mass of the wire.

Answer: 4


Question: Find the set of values of \( \lambda \) for which the perpendicular distance from the point \( P \) on the line \( l \), given by \( r = 3i + 8j + 2k + \lambda(4i + 6j + 5k) \), to the plane \( \Pi_1 \) with equation \( 2x + 3y + 6z = 14 \) is not greater than 4.

Answer: [-1, 0]


Question: Evaluate the surface integral \( \int_{M}(F\cdot N)\,dS \) where \( F=(e^{\cos z},x^2,z) \) and \( M=\{(x,y,z):z=1-(x^2+y^2),x^2+y^2\leq 1\} \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the integral: \[ \int_0^\infty \frac{\sin(2x)}{e^x} \, dx \]

Answer: \dfrac{2}{5}


Question: Let $K$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$ with ring of integers $O_K$. Is the field of fractions of $O_K$ equal to $K$? Justify your answer.

Answer: Yes


Question: Evaluate the triple integral \( \iiint_R (x^2+y^2+z^2)^{-2}\,dx\,dy\,dz \) where \( R \) is the region in the first octant outside the sphere \( x^2+y^2+z^2 = 1 \).

Answer: \dfrac{\pi}{2}


Question: Find the least positive residue of \(463! \mod 479\).

Answer: 478


Question: Given that $f'(x) = \sqrt{1 + x^3}$ and $f(1) = 0.5$, find the value of $f(4)$. Use the Fundamental Theorem of Calculus to solve the problem.

Answer: 13.372


Question: Evaluate the limit \( I = \lim_{n \to 0} \lim_{m \to 0} (m^2 - 2n)^n \).

Answer: 1


Question: Determine the cardinality of the set $S$, which is the collection of closed intervals on the real line with positive rational lengths. Justify your answer.

Answer: \mathfrak{c}


Question: Find the length of the curve \( y = \frac{3}{4}x^\frac{4}{3} - \frac{3}{8}x^\frac{2}{3} + 7 \) from \( x = 1 \) to \( x = 8 \).

Answer: \dfrac{99}{8}


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

Answer: -\dfrac{e}{2}


Question: Given the equations \( \frac{1}{\omega+a}+\frac{1}{\omega+b}+\frac{1}{\omega+c} = 2\omega^2 \) and \( \frac{1}{\omega^2+a}+\frac{1}{\omega^2+b}+\frac{1}{\omega^2+c} = 2\omega \), where \( \omega \) is a complex cube root of unity, find the value of \( \frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c} \).

Answer: 2


Question: A teaching assistant (TA) needs to keep his computer running continuously for the next 7 days (168 hours) using batteries, as his power cable is missing. He has 100 batteries available, each with an average lifetime of 2 hours and a standard deviation of 30 minutes. What is the minimum number of batteries he should use to ensure the computer runs for 168 hours with at least a 95% probability?

Answer: 88


Question: Find the Jordan measure of the set \( M = \{(x,y) \in [0,1]^2 : y^2 \leq x \} \).

Answer: \dfrac{2}{3}


Question: Find the first common value of the sequences defined by the expressions \(m^2 + 928m + 687\) and \(n^2 + 54n + 729\).

Answer: 11514148416


Question: Determine the values of \( \alpha \) for which the integral \( \int_0^1 \frac{1-x^\alpha}{1-x} \, \mathrm{d}x \) converges.

Answer: \alpha > -1


Question:
The kernel \(k(x,y)=\frac{y}{y^2+x^2}\) is a solution of which equation? Choose the correct option:

(A) Heat equation
(B) Wave equation
(C) Laplace equation
(D) Lagrange equation

Answer: C


Question: A manufacturing process produces a large number of components each day, and most of these components will meet all specifications. Let $p$ be the probability that a component will meet all specifications, and assume that one component meeting specifications is independent of others. The value of $p$ varies from day to day according to the prior distribution: \( \pi(p) = 9p^8 \quad 0\le p\le 1 \). A sample of \( n = 5 \) components from one day's production was inspected, and it was found that 4 of the 5 components met all specifications. Find the Bayes estimator of $p$ for this day.

Answer: \dfrac{13}{15}


Question:
Given the joint density function of the 2-dimensional random variable (X, Y):

\[f_{X,Y}(x,y) = \begin{cases} \frac{x^2}{2y^3}\cdot e^{-\frac{x}{y}},&  0<x<\infty, 0<y<1 \\\ 0,& \text{otherwise.} \end{cases}\]

Calculate the variance of the random variable X, denoted as Var(X).

Answer: \dfrac{7}{4}


Question: Evaluate \( \frac{\partial z}{\partial x}(1, 2) \) where \( z \) is a function of \( (x, y) \) defined implicitly by the equation \( z^2f + zg + x - y = 0 \). Given that \( f(1, 2) = 2 \), \( g(1, 2) = 1 \), and at \( (1, 2) \), \( df = -2dx + 5dy \), \( dg = 4dx - 3dy \), and \( z(1, 2) = -1 \). What is \( \frac{\partial z}{\partial x}(1, 2)? \)

Answer: -\dfrac{5}{3}


Question: Given the joint probability distribution function \( F(x,y) = kxye^{-(x+y)} \) for \( x, y > 0 \) and 0 elsewhere, where \( k = 1 \), calculate the conditional probability \( P(A|B) \) where \( A \) is the event \( x < y \) and \( B \) is the event \( x < 2y \).

Answer: \dfrac{27}{40}


Question: Find the continuous and bounded function $f: \mathbb{R} \to \mathbb{R}$ such that for any $x \in \mathbb{R}$, the equation \[ x \int_x^{x+1} f(t)\,dt = \int_0^x f(t)\,dt \] holds. Describe the nature of the function $f$.

Answer: f(x) = C


Question: Identify the topological group that is isomorphic to the quotient group $\mathbb{R}/\mathbb{Z}$.

Answer: S^1


Question: Let $A$ be a positive definite matrix representing an inner product in $\mathbb{C}^n$ such that $\langle v, w \rangle = v^* A w$. Given a new basis represented by the matrix $Q$, where the columns of $Q$ are the new basis vectors, express the matrix $B$ that represents the original inner product in the new basis in terms of $Q$.

Answer: Q^* A Q


Question: Given that \(\det\begin{pmatrix}a&1&d\\ b&1&e\\ c&1&f\end{pmatrix}=4\) and \(\det\begin{pmatrix}a&1&d\\ b&2&e\\ c&3&f\end{pmatrix}=3\), find \(\det \begin{pmatrix}a&-1&d\\ b&-3&e\\ c&-5&f\end{pmatrix}\).

Answer: -2


Question: Find the exact value of the sum: \( \cos 1^{\circ} + \cos 2^{\circ} + \cos 3^{\circ} + \ldots + \cos 358^{\circ} + \cos 359^{\circ} \).

Answer: -1


Question: Evaluate the integral \( \int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta}\,\mathrm{d}\theta \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the triple integral \( \int_{0}^{2} \int_{0}^{\sqrt{2x - x^2}} \int_{0}^{1} z \sqrt{x^2 +y^2} \, dz\,dy\,dx \) by converting it to cylindrical coordinates.

Answer: \dfrac{8}{9}


Question: Evaluate the integral: $$\int_{-\infty}^{\infty} \left|2Af_0 \frac{\sin(2\pi f_0 t)}{2\pi f_0 t}\right|\,dt$$

Answer: \infty


Question: Calculate the number of 10-letter words that do not contain all five different vowels (A, E, I, O, U).

Answer: 140948727706936


Question: Compute the group of homomorphisms from the additive group of rational numbers \( \mathbb{Q} \) to the additive group of integers \( \mathbb{Z} \), denoted as \( \operatorname{Hom}_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Z}) \).

Answer: 0


Question: Solve the differential equation \( x''(t) + tx'(t) - (t+1)x = 0 \) using the Laplace transform, given the initial conditions \( x(0) = 1 \) and \( x'(0) = 1 \). Verify that the solution is \( x(t) = e^t \).

Answer: e^t


Question: Given the polynomial \(2x^4 + ax^3 + 17x^2 + 11x + 15 = 0\), where \(x_1\) and \(x_2\) are roots such that \(x_1, x_2 < 0\), find the value of \(a\) if the negative roots are \(-2.5\) and \(-3\).

Answer: 11


Question: Consider a sequence of random variables \(X_n = X^n\), where \(X\sim\mathrm{Uniform}(0,1)\). Determine the random variable \(Y\) to which \(X_n\) converges in distribution as \(n\) approaches infinity.

Answer: 0


Question: Evaluate the improper integral \( \int_{0}^{\infty} \frac{\sin(x)}{x^3} \, dx \). Verify if the result is \( -\frac{\pi}{4} \).

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


Question:
An analytic function $f(z)$ satisfies $\text{Re}(f'(z)) = 2y$ and $f(1+i) = 2$. Determine the imaginary part of $f(z)$ from the following options:

$A.$ $-2xy$

$B.$ $x^{2}-y^{2}$

$C.$ $2xy$

$D.$ $y^{2}-x^{2}$

Given that $f(1+i) = 2$, options $A$ and $C$ are incorrect. Choose between $B$ and $D$.

Answer: D


Question: Determine whether the value of \( \lim_{x \to 0^-} x^x \) is equal to 1.

Answer: 1


Question: Find a line that is tangent to the curve \( y = x^4 - 4x^3 \) at two distinct points.

Answer: y = -8x - 4


Question: Given that \( xf(x) - yf(y) = (x-y)f(x+y) \) for a differentiable function \( f \) defined on \( \mathbb{R} \), find the value of \( f'(2x) \). Choose from the following options: \( f'(x), 2f'(x), xf'(x), 4f'(x), \) or \( 8f'(x) \).

Answer: f'(x)


Question: Find the greatest common divisor (gcd) of the elements \(2-7\) and \(11\) in the ring \(\mathbb{Z}[w]\), where \(w=\frac{1+\sqrt{-7}}{2}\). Use the Euclidean algorithm to determine the gcd.

Answer: 1 + 2w


Question: Evaluate the sum \( \sum\limits_{i,j,k=0}^n{n \choose i+j}{n \choose i+k}{n \choose k+j} \).

Answer: 4^n


Question: Is there a nonstandard model $M$ of Peano arithmetic and a nonstandard $c \in N$ such that the operations $\oplus_{<c}$ and $\otimes_{<c}$ are computable?

Answer: No


Question: Determine whether the following claim is true or false: If a sequence of real-valued upper semicontinuous functions is a decreasing sequence and converges to 0 pointwise, then the convergence is uniform.

Answer: False


Question: Determine the values of $a \in \mathbb{R}$ for which the set \( M_a = \{(x,y) \in \mathbb{R}^2 \mid y^2 = x^3 + a\} \) is a submanifold of \( \mathbb{R}^2 \).

Answer: a \neq 0


Question: Consider a presheaf $F$ from a topological space $X$ to a category of sets or abelian groups. Suppose $F$ satisfies the following unique gluing condition: for any open covering $\{U_i\}$ of an open set $U$, if for each $i$ there is a section $s_i \in F(U_i)$ such that for each pair $U_i, U_j$ of the covering sets, the restrictions of $s_i$ and $s_j$ agree on the overlaps, i.e., $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$, then there exists a unique section $s \in F(U)$ such that $s|_{U_i} = s_i$ for all $i$. Is it true that $F$ is a sheaf?

Answer: Yes


Question:
If the Fourier transform of the function $f(x, y)$ is $F(m, n)$, what is the Fourier transform of the function $f(2x, 2y)$? Choose the correct option:

1. $\cfrac{1}{4} F(m/2,n/2)$
2. $\cfrac{1}{4}F(2m,2n)$
3. $\cfrac{1}{4} F(m,n)$
4. $\cfrac{1}{4} F(m/4,n/4)$

Answer: 1


Question: Determine whether the following statement is true: \( \sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \mathbb{Q} \iff \exists k\in \mathbb{N}^+ \text{ such that } a^2-b=k^2 \). Here, \(a\) and \(b\) are positive integers.

Answer: True


Question: Using Lagrange's multiplier method, find the shortest distance between the line \( y = 10 - 2x \) and the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \).

Answer: \sqrt{5}


Question: Given the expansion \((1+x+x^2)^{1061} = a_0 + a_1x + a_2x^2 + \ldots + a_{2122}x^{2122}\), find the value of the expression \(1 - a_1^2 + a_2^2 - a_3^2 + \ldots\) in terms of a single coefficient \(a_n\).

Answer: a_{1061}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=n}^{5n}{k-1\choose n-1}\left(\frac{1}{5}\right)^{n}\left(\frac{4}{5}\right)^{k-n} \]

Answer: \dfrac{1}{2}


Question: Find the limit \( \lim_{n\to\infty}\frac{K_{n+1}}{K_n} \), where \( K_n \) is the least integer \( k \) such that the partial sum of the harmonic series \( S_k = 1 + \frac{1}{2} + \cdots + \frac{1}{k} \geq n \).

Answer: e


Question: Evaluate the integral \( \int^{\frac{91\pi}{6}}_0 |\cos(x)| \, \mathrm{d}x \).

Answer: \dfrac{61}{2}


Question: Determine all real numbers $x$ such that the series \( \sum_{n=2}^{\infty} \frac{\sin(nx)}{\log n} \) converges.

Answer: \mathbb{R}


Question: Determine the number of permutations \(\sigma\) in \(S_6\) such that \(\gamma ' = \sigma \gamma \sigma ^{-1}\), where \(\gamma = (123)(45)(6)\) and \(\gamma ' = (1)(23)(456)\).

Answer: 6


Question:
Consider the homogeneous linear differential equation:
\[ y'' + p(x)y' + q(x)y = 0 \]
where \( p(x) \) and \( q(x) \) are continuous on an interval \( I \). Let \( f \) and \( g \) be solutions to this equation. If the Wronskian of \( f \) and \( g \) is zero for every \( x \) in \( I \), are \( f \) and \( g \) necessarily linearly dependent? Justify your answer.

Answer: Yes


Question: Given that $f''(x) = -f(x)$, where $f(x)$ is a continuous twice differentiable function, and $g(x) = f'(x)$, define $F(x) = \left(f\left(\frac{x}{2}\right)\right)^2 + \left(g\left(\frac{x}{2}\right)\right)^2$. If $F(5) = 5$, find $F(10)$.

Answer: 5


Question: Determine whether the group $SO(3)$ has any 2-dimensional Lie subgroups.

Answer: No


Question: Find the limit of the sequence \( a_n = \ln\left(\frac{2n+1}{2n}\right) \sum_{k=1}^n \sqrt[k]{k} \).

Answer: \dfrac{1}{2}


Question: Find all entire functions \( f \) such that for all \( z \in \mathbb{C} \), \( |f(z)| \ge \frac{1}{|z|+1} \).

Answer: f(z) = c


Question: Evaluate the limit: \[ \lim_{x\to1^-}\frac{1}{x^2-1}\,\exp\left(\frac{1}{x^2-1}\right) \]

Answer: 0


Question: Find the radius of the largest circle that is internally tangent to the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) at the point \((3,0)\), such that the circle intersects the ellipse only at this point.

Answer: \dfrac{4}{3}


Question: Evaluate the limit: \[ \lim_{x \to e} (1-\log x)\log (x-e) \] without using L'Hôpital's Rule.

Answer: 0


Question: Let $H$ be a subgroup of a group $G$. Determine the stabilizer of the coset $aH$ for the action of $G$ on the set $G/H$ by left multiplication.

Answer: aHa^{-1}


Question: Determine the number of roots of the equation \( z^4 = \omega \overline{z} \) for \( z \in \mathbb{C} \), where \( \omega = \left(\frac{3+4i}{3-4i}\right)^5 \).

Answer: 6


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the integral \( \int_{\mathbb{R}^2} \frac{1}{(1+x^2+xy+y^2)^\alpha} \, dx \, dy \) converges.

Answer: \alpha > 1


Question: Evaluate the integral \( \int_0^{2\pi} \frac{1}{13-12\cos(t)}\;dt \) by converting it to a contour integral along the unit circle in \( \mathbb{C} \).

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


Question: Determine the order of the center of the general linear group \( GL_n(\mathbb{C}) \). Express your answer as \( |Z(GL_n(\mathbb{C}))| = \boxed{?} \).

Answer: \infty


Question: Let \( p \) be a prime. Determine the number of subgroups of \( \Bbb Z_{p^2} \oplus \Bbb Z_{p^2} \) that are isomorphic to \( \Bbb Z_{p^2} \).

Answer: p(p + 1)


Question: In the representation theory of finite groups, the Mackey test for irreducibility of an induced representation is stated for the field $\mathbb{C}$. Is this test also valid for representations over an arbitrary field whose characteristic is zero or co-prime to the order of the group?

Answer: Yes


Question: Given a $9 \times 9$ matrix $A$ over the field $\mathbb{Z}/5$, the number of solutions to the equation $Ax=0$ is between 40 and 150. Determine the rank of the matrix $A$.

Answer: 6


Question: A plane region $R$ is bounded by a piecewise smooth Jordan curve $C$. The moments of inertia of $R$ about the $x$- and $y$-axes are $a$ and $b$, respectively. Compute the line integral \( \oint_C \nabla(r^4)\cdot \mathbf{n} \,ds \) in terms of $a$ and $b$, where $r = ||x\mathbf{i} + y\mathbf{j} ||$, $\mathbf{n}$ is the unit outward normal of $C$, and $s$ is the arc length. The curve is traversed counterclockwise. Express your answer as a function of $a$ and $b$.

Answer: 16(a + b)


Question: Find all polynomials $p(x)$ such that $p(p(x)) + p(x) = x^4 + 3x^2 + 3$ for all $x \in \mathbb{R}$. Assume $p(x)$ is a quadratic polynomial and determine its coefficients.

Answer: x^2 + 1


Question: Find the second partial derivatives $f_{x}^{\prime}(0,0)$ and $f_{y}^{\prime}(0,0)$ for the function \[ f(x,y)=\begin{cases} \frac{x^{4}y^{2}}{x^{4}+y^{2}}, & x^{2}+y^{2}\neq0 \\ 0, & x^{2}+y^{2}=0 \end{cases} \] at the point $(0, 0)$. Provide a detailed explanation of your solution.

Answer: 0


Question: Let $X$ be a smooth projective curve in $\mathbb{P}^2(\mathbb{C})$ of degree $4$ and $p, q, r \in X$. Compute $\dim L(p+q+r)$.

Answer: 1


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty}\int_0^1 \int_0^1 \cdots\int_0^1 {n \over x_1+x_2+\cdots+x_n} \, dx_1 \, dx_2 \cdots dx_n$$

Answer: 2


Question: Is the norm closed unit ball of $c_0$ weak* dense in the norm closed unit ball of $l^{\infty}(\mathbb{N})$?

Answer: Yes


Question: Determine whether the set of polynomials \( \{ P \in \mathbb{Q}(X) \mid P(1) = P(3) = 0 \} \) forms an ideal in \( \mathbb{Q}[X] \). If it does, find the monic generator of this ideal.

Answer: X^2 - 4X + 3


Question: Let $G$ and $H$ be two finite $p$-groups, and let $f: \mathbb{F}_p[G] \to \mathbb{F}_p[H]$ be a ring homomorphism of mod $p$ group rings. Is it always true that $f(I_G) \subset I_H$, where $I_G$ and $I_H$ are the respective augmentation ideals? Justify your answer.

Answer: Yes


Question: Determine the values of \( x \) for which the series \( \sum_{n=1}^{\infty} \frac{(x-4)^{n^{2}}}{n!} \) converges.

Answer: [3, 5]


Question: Let $(X,d)$ be a complete separable metric space (Polish space) with a Borel probability measure $\mu$. Given $\varepsilon_1, \varepsilon_2 > 0$, determine if there exists a finite set of disjoint open balls $B_1,\ldots, B_n$ of diameter $< \varepsilon_1$ such that $\mu(B_1) + \ldots + \mu(B_n) > 1-\varepsilon_2$. Provide a justification for your answer.

Answer: Yes


Question: Four runners, A, B, C, and D, are organized into two teams: Team 1 consists of runners A and B, and Team 2 consists of runners C and D. Each runner's time to complete a race is exponentially distributed with means \( \mu_A = 9 \), \( \mu_B = 10 \), \( \mu_C = 8 \), and \( \mu_D = 12 \) minutes, respectively. Assuming all running times are independent, calculate the probability that the fastest runner is from Team 1.

Answer: \dfrac{76}{151}


Question: Calculate the area of the half-annulus with inner radius $r_1 = 1$ and outer radius $r_2 = 5$, bounded by the angles $\theta_1 = -\frac{3\pi}{4}$ and $\theta_2 = \frac{\pi}{4}$, using integration.

Answer: 12\pi


Question: Determine whether the linear operator \( A: l^2 \rightarrow l^2 \) defined by \( Ax = ((1+\frac{1}{2})x_1, (1+\frac{1}{2}+\frac{1}{4})x_2, \ldots, (1+\frac{1}{2}+\frac{1}{4}+\ldots+\frac{1}{2^n})x_n, \ldots) \) is bounded. If it is bounded, find its norm.

Answer: 2


Question:
Let $f:[0,1]\to\mathbb{R}$ be defined as \[ f(x) = \max_{0 \le y \le 1} \frac{|x-y|}{x+y+1} \] for $0 \le x \le 1$. Determine which of the following statements is correct:

1. $f$ is strictly increasing on $[0,1/2]$ and strictly decreasing on $[1/2,1]$.
2. $f$ is strictly decreasing on $[0,1/2]$ and strictly increasing on $[1/2,1]$.
3. $f$ is strictly increasing on $[0,(\sqrt{3}-1)/2]$ and strictly decreasing on $[(\sqrt{3}-1)/2,1]$.
4. $f$ is strictly decreasing on $[0,(\sqrt{3}-1)/2]$ and strictly increasing on $[(\sqrt{3}-1)/2,1]$.

Choose the correct statement.

Answer: D


Question: Find all limit points of the sequence \(a_n = (-n)^{(-n)^n}\) for \(n \in \mathbb{N}\).

Answer: 0


Question: Find a root of the polynomial \( f(x) = x^4 + 10x^2 + x + 3 \) modulo \( 3^4 \).

Answer: 15


Question: Find the limit of the sequence of functions \( g_n(x) = \sqrt[n]{x^n + \tan^n x} \) as \( n \to \infty \), where \( x \in \left[0, \frac{\pi}{2}\right) \).

Answer: \tan x


Question: Evaluate the limit: \( \lim_{x \to \infty} \frac{x^x}{e^x x!} \)

Answer: 0


Question: Determine whether the function \( f \) satisfying the relation \( \forall x,y \in \mathbb{R}: f(x+f(y))=f(x)+y \) is one-to-one.

Answer: Yes


Question: Evaluate the integral: \[ \int_0^{2\pi} - \frac{\cos t \; ( 2 (\sin t)^2 + (\cos t)^2)}{(\cos t)^4 + (\sin t)^2} \, dt \]

Answer: 0


Question: Determine the number of nonabelian groups of order 2009.

Answer: 0


Question: Find the maximum value of \( \int^{3\pi/2}_{-\pi/2} \sin(x)f(x) \, dx \) subject to the constraint \(|f(x)| \leq 5\).

Answer: 20


Question: For any integer \( M \), determine if there exists a finite sequence of positive integers \((n_i)_{i \in I}\) such that \( \sum_{i \in I} \frac{1}{n_i} = M \) and no subsequence \((n_i)_{i \in J}\) of \((n_i)_{i \in I}\) satisfies \( \sum_{i \in J} \frac{1}{n_i} \) being an integer.

Answer: Yes


Question: Can a measurable function on \(\mathbb{R}^2\) be essentially unbounded on every measurable rectangle of positive measure?

Answer: Yes


Question: Evaluate the limit of the sequence: $$\lim_{n \rightarrow + \infty } \left(\frac{n^3}{4n-7}\right)\left(\cos\left(\frac{1}{n}\right)-1\right).$$

Answer: -\dfrac{1}{8}


Question: Given that \(x, y, z > 0\) and \(xyz = 1\), find the minimum value of the expression \[ \frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y}. \]

Answer: \dfrac{3}{2}


Question: Evaluate the double integral \( \iint_D (x + y)\, dx \, dy \) where \(D\) is the trapezoidal region with vertices \((0,0), (5,0), (5/2, 5/2), (5/2, -5/2)\), using the transformations \( x = 2u + 3v \) and \( y = 2u - 3v\).

Answer: \dfrac{125}{4}


Question: Evaluate the triple integral \( \iiint_{x^2+y^2+z^2\le 2x}  (x + z) \,dV \) by converting to spherical coordinates. The region of integration is a sphere centered at \((1,0,0)\).

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


Question: Determine the chromatic number of the graph $G$ with vertex set $\{n\}^{(r)}$, where two vertices are adjacent if and only if their coordinates differ by one.

Answer: 2


Question: Given that the function $f$ satisfies the equation $f(x + y) = f(x) + f(y) + xy$ and that $\lim_{h \to 0} \frac{f(h)}{h} = 2$, find the value of $f'(1)$. Provide your answer as a single number.

Answer: 3


Question: Calculate the volume between the surfaces defined by $x^2 + y^2 + z = 1$ and $z = x^2 + (y-1)^2$.

Answer: \dfrac{\pi}{16}


Question: Find the inverse of the polynomial \( g = x^4 + 3x^3 + x^2 + 1 \) in the quotient ring \( \mathbb{Z}_5[x]/(x^3 + x + 1) \).

Answer: 4x^2 + 3x


Question: Provide an example of an infinite field with characteristic \( p \neq 0 \).

Answer: \mathbb{F}_p(x)


Question: Find the maximum value of \(|z|\) given that \(|z^2 + 2z \cos \alpha| \leq 1\) for all real values of \(\alpha\).

Answer: \sqrt{\sqrt{5} - 2}


Question: Consider the sets $A = \{1, 2, 3, 4, 5\}$ and $B = \{1, 2, 3\}$. Determine the number of surjective functions $f: A \to B$ such that $f(1) \neq 1$, $f(2) \neq 2$, and $f(3) \neq 2$. Provide your answer as a single integer.

Answer: 45


Question: Find the volume of the solid in \(\mathbb{R}^{3}\) bounded by the cylinders \(y^{2}+z^{2}=1\), \(y^{2}+z^{2}=4\) and the planes \(x=-1\), \(x+z=3\).

Answer: 12\pi


Question: Consider the integral \( I_n = \int_{\mathbb{R}} \frac{\sin(x^2+n^2)}{x^2+n^2} \, d\lambda \) for every \( n \geq 1 \). Show that \( I_n \) is well-defined for each \( n \), and compute \( \lim_{n \to \infty} I_n \).

Answer: 0


Question: Does there exist a non-unital $C^*$ algebra that has uncountably many tracial states?

Answer: Yes


Question: Evaluate the double integral \( \iint \frac{1}{x+y} \, dx \, dy \) over the region bounded by \( x=0, y=0, x+y=1, \) and \( x+y=4 \) using the transformation \( T(u,v) = (u - uv, uv) \).

Answer: 3


Question:
Find the number of integer solutions to the equation \(x_1 + x_2 - x_3 + x_4 - x_5 = 3\) given the constraints:
\[
\begin{aligned}
&0 < x_1 \le 6, \\
&-8 \le x_2 < -2, \\
&x_3 \le 1, \\
&3 < x_4, \\
&2 \le x_5 \le 8.
\end{aligned}
\]

Answer: 2016


Question: Calculate the Legendre symbol \( \left(\frac{211}{307}\right) \).

Answer: -1


Question: Determine the number of Pythagorean triples \((a, b, c)\) where \(c\) is the hypotenuse, such that the inequality \(10 < a + b - c < 18\) holds.

Answer: 13


Question: Determine whether the expression \( \frac{(n^2 - n)!}{n! ((n-1)!)^n} \) is an integer for all positive integers \( n \).

Answer: Yes


Question: Evaluate the triple integral \( \iiint_K (x^2 + y^2) \, dx \, dy \, dz \) over the region \( K \) defined by the inequalities \( 0 \leq x^2 + y^2 \leq z^2 \) and \( 0 \leq z \leq 1 \).

Answer: \dfrac{\pi}{10}


Question: Determine whether there are infinitely many $2 \times 2$ matrices $A$ with complex entries that satisfy the equation $A^3 = A$. Is the statement true or false?

Answer: True


Question: Evaluate the limit \(\lim_{(x,y)\to(0,0)}\frac{-16x^3y^3}{(x^4+2y^2)^2}\).

Answer: 0


Question: Find the determinant of the linear operator \( T: K[x]_{\leq n} \to K[x]_{\leq n} \) defined by \( T(f(x)) = f(x) + x \cdot f'(x+1) \).

Answer: (n+1)!


Question: Find all possible solutions of the form \(x = \frac{m}{n}\) (with \(m, n\) coprime) for the equation: \[ x + \lfloor x \rfloor = 2013x\cdot\lfloor x \rfloor + 2013^{-1} \] where \(\lfloor x \rfloor\) is the integer part of \(x\).

Answer: \dfrac{1}{2013}


Question: Given the recurrence relation $a_{n+1} = \frac{10}{a_n} - 3$ with the initial condition $a_1 = 10$, find $\lim_{n \to \infty} a_n$. \(\text{(Express your answer as a single number or expression.)}\)

Answer: -5


Question: Calculate the last two digits of the number $9^{9^{9^{9}}}$ using modular arithmetic.

Answer: 89


Question: Determine the number of possible solutions for the equation \( \int_{0}^{x} (t^2 - 8t + 13) \, dt = x \sin\left(\frac{a}{x}\right) \).

Answer: 1


Question: Let $X$ be a countable $T_B$ space, where every compact subset is closed. Consider an infinite subset $D \subseteq X$ such that every infinite subset of $D$ has an infinite number of accumulation points in $X$. Suppose $n_0 \in \omega$ is the smallest integer such that $x_{n_0}$ is an accumulation point in $D$. If $V$ is a neighborhood of $x_{n_0}$ and $D - V$ is finite, does every enumeration $\{ y_n : n \in \omega \}$ of $D$ converge to $x_{n_0}$?

Answer: Yes


Question: Calculate the sum \( \sum_{k \in S(m,n)} \varphi(k) \), where \( S(a,b) = \{ k \in \mathbb{Z} \mid a \equiv a \pmod{k} + b \equiv b \pmod{k} \ge k \} \) and \( \varphi \) is Euler's totient function.

Answer: mn


Question: Calculate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) where \( \mathbf{F} = (e^y + ye^x, xe^y + e^x) \) and \( C \) is the path from \((a,0)\) to \((-a,0)\) over the upper half of the ellipse \( b^2x^2 + a^2y^2 = a^2b^2 \). The answer is known to be \(-2a\).

Answer: -2a


Question: Determine the number of straight lines contained on the hyperboloid of one sheet defined by the equation \(x^2 + y^2 - z^2 = 1\).

Answer: \infty


Question: Find the minimum value of the expression \( \sin \alpha + \sin \beta + \sin \gamma \), where \( \alpha, \beta, \gamma \in \mathbb{R} \) and \( \alpha + \beta + \gamma = \pi \). Choose the correct option: (a) positive, (b) negative, (c) 0, (d) -3.

Answer: B


Question: Suppose $(x_1, x_2, x_3) \sim N(\mu, \Sigma)$ where $\mu \in \mathbb{R}^3$ and $\Sigma$ is a $3 \times 3$ covariance matrix. Are the variables $A = x_1 + x_2$ and $B = x_2 + x_3$ distributed according to a multivariate normal distribution? Specifically, do there exist $\nu \in \mathbb{R}^2$ and a $2 \times 2$ covariance matrix $M$ such that $(A, B) \sim N(\nu, M)$?

Answer: Yes


Question:
Let $m \leq n \leq q$ be three positive integers. Given a $m \times q$ matrix $A$ and a $q \times n$ matrix $B$ with $B$ of rank $n$ and $AB$ of rank $m$, determine if there exists a $n \times q$ matrix $X$ such that:
\[ ABX = A \]
and
\[ XB = I_n. \]

Answer: Yes


Question: Determine the minimum size of an edge cut that separates the complete graph $K_{3n}$ into three components. Assume the components are complete graphs with orders $x$, $y$, and $z$, where $x + y + z = 3n$. Find the maximum of $\frac{(x-1)x}{2} + \frac{(y-1)y}{2} + \frac{(z-1)z}{2}$ to solve the problem.

Answer: 6n - 3


Question: Let $X^n$ be a family of continuous stochastic processes such that $E[ \sup_{t \in [0,T]} |X^n_t|^2 ] < \infty$ for all $n$. Assume that \( \lim_{n \to \infty} E\left[ \int_0^T |X^n_t|^2 \, dt \right] = 0. \) Does this imply that \( \lim_{n \to \infty} E\left[ \sup_{t \in [0,T]} |X^n_t|^2 \right] = 0? \)

Answer: No


Question: Find the radius of convergence for the power series: \[ \sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}(x+1)^{2n+1} \]

Answer: 1


Question: Compute the squared length of the highest root \( \theta \) of the classical Lie algebra \( \mathfrak{sp}(2r) \) using the Killing form. The highest root is given by \( \theta = 2(\alpha_{1} + \ldots + \alpha_{r-1}) + \alpha_{r} \), where \( \alpha_{i} \) are the simple roots. Express your answer as \( ||\theta||^2 = \kappa(\theta, \theta) \).

Answer: 2


Question: Determine whether the Gaussian integer $a + 2bi$, where $a, b \in \mathbb{Z}$, can always be expressed as a sum of three squares.

Answer: Yes


Question: Suppose $X$ and $Y$ are Banach spaces and $T$ is a continuous operator between them. If for any sequence $(x_n)_{n=1}^{\infty}$ with $\|x_n\|=1$, it holds that $\|Tx_n\| \nrightarrow 0$, must $T$ be bounded below? A bounded below operator satisfies $\|Tx\| \geq C \|x\|$ for some positive constant $C$.

Answer: Yes


Question: Is it always possible to find a minimal integral Weierstrass equation of the form \( y^2 = x^3 + Ax + B \) with \( A, B \in \mathbb{Z} \) for an elliptic curve over \( \mathbb{Q} \) that has good reduction at 2 and 3?

Answer: No


Question: Evaluate the limit \( \lim_{n\to \infty} \{(1+x)(1+x^2)(1+x^4)\cdots(1+x^{2^n})\} = f(x) \) for \( |x|<1 \). What is the value of \( f\left(\frac{1}{2}\right) \)?

Answer: 2


Question: A frog jumps for \(T\) steps starting from 0, with each step \(X_i\) being uniformly distributed over \((0,1)\). The frog continues jumping until the sum of its steps, \(S_T = \sum_{i=1}^T X_i\), exceeds 1. Compute the expected number of steps, \(\mathbb{E}[T]\).

Answer: e


Question:
Determine the set of values of \( k \) for which all solutions of the system of equations \( Y'(x) = AY(x) \), where \( A = \begin{pmatrix} -3 & 1 \\ k & -1 \end{pmatrix} \), tend to zero as \( x \to \infty \). Choose the correct option:
(a) \( \{k : k \leq -1\} \)
(b) \( \{k : k \leq 3\} \)
(c) \( \{k : k < 3\} \)
(d) \( \{k : k < -1\} \)

Answer: C


Question: Prove that for a non-square matrix $A$ with dimensions $(n \times m)$ where $n > m$, and a square matrix $B$ with dimensions $(m \times m)$, the determinant of the matrix product $ABA^T$ is zero, i.e., $\det(ABA^T) = 0$. Provide a detailed proof.

Answer: 0


Question: Evaluate the integral \(\int_0^1 ((1-x^a)^\frac{1}{a}-x)^2 \, dx\) for a real number \(a\).

Answer: \dfrac{1}{3}


Question: Determine the number of ways to distribute 10 Kit-Kat bars and 15 Mr. Big bars to four different people, such that no person receives more than five of each type of bar. Use generating functions to find the solution.

Answer: 8176


Question: If \(a, b, c, d, e\) are the roots of the polynomial \(6x^5 + 5x^4 + 4x^3 + 3x^2 + 2x + 1\), find the value of \((a+1)(b+1)(c+1)(d+1)(e+1)\).

Answer: \dfrac{1}{2}


Question: Let $X_k$ be exponentially distributed with rate parameter $\lambda = \sqrt{k}$. Determine the distribution to which $(X_k+1)^2$ converges as $k \to \infty$. Calculate the limit: \( \lim_{n \to \infty} P((X_n+1)^2 \leq 2) \). Use characteristic functions to aid in your solution.

Answer: 1


Question: Determine the values of \( p \) for which the integral \( \int_{|x|\leq 1} \frac{1}{|x|^p} \,dx \) is finite, where \( d\mathbf{x} = dx_1 \cdots dx_n \) denotes Lebesgue measure.

Answer: p < n


Question: If $\omega^{1997} = 1$ and $\omega \neq 1$, evaluate the expression \( \frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} + \dots + \frac{1}{1 + \omega^{1997}} \) in the form \( m/n \), where \( m \) and \( n \) are relatively prime positive integers. Find the remainder when \( m + n \) is divided by 1000.

Answer: 999


Question: Let $f_n : X \times Y \rightarrow \mathbb{R}$ be a sequence of measurable functions that converge almost everywhere to $f$, with respect to the measure $\mu \otimes \nu$. Is it true that for almost every $x \in X$, the sequence $f_n(x,y)$ converges to $f(x,y)$ for almost every $y \in Y$?

Answer: Yes


Question:
Suppose $R$ is a unique factorization domain (UFD) and $a, b \in R$ are distinct irreducible elements. Determine which of the following statements is true:

1. The ideal $\langle 1+a \rangle$ is a prime ideal.
2. The ideal $\langle a+b \rangle$ is a prime ideal.
3. The ideal $\langle 1+ab \rangle$ is a prime ideal.
4. The ideal $\langle a \rangle$ is not necessarily a maximal ideal.

Answer: 4


Question: Thor, Captain America, and Spiderman are to split 2001 candies among themselves. Thor must receive more candies than Captain America, who must receive more candies than Spiderman, and Spiderman must receive at least one candy. How many ways can this distribution be done?

Answer: 332667


Question: Determine whether the following statement is true or false: For a matrix $A \in \mathbb{M}_n$, is $e^{\text{tr}(A)} = \text{tr}(e^A)$?

Answer: False


Question: Find the number of 3-subsets \( S \) of the set \( \{2, 3, \, \ldots, 50, 53, 55\} \) such that for every pair of distinct elements \( x, y \in S \), the condition \( |x - y| > 2 \) holds.

Answer: 16352


Question:
Solve for \(x\) in the equation:
\[ 2^{\lfloor \log_2{(x)} + \frac{1}{2} \rfloor} = 2^{\lfloor \log_2(x-2^{\lfloor{ \log_2{(\frac{x}{2})} + \frac{1}{2}}\rfloor}) + \frac{1}{2}\rfloor}  + 2^{\lfloor \log_2{(\frac{x}{2})+\frac{1}{2}}\rfloor} \]

Answer: 4


Question:
Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables, each following an exponential distribution with mean $\frac{1}{\lambda}$. Define $S_n = X_1 + X_2 + \cdots + X_n$ and let $N = \inf\{n \geq 1 : S_n > 1\}$. Determine the variance of $N$, $\operatorname{Var}(N)$. Choose the correct answer from the following options:

1. 1
2. $\lambda$
3. $\lambda^2$
4. $\infty$

Answer: 2


Question: Determine whether there exists a constant \( M < 1 \) such that for any analytic map \( f \) from the unit disk \( \mathbb{D} \) to \( \mathbb{D} - \{1/2, 1/3\} \), the inequality \( |f'(0)| \leq M \) holds.

Answer: Yes


Question:
Consider the set $S$ of integers from 1 to 2014. For two distinct subsets $U$ and $V$ of $S$, we define $U < V$ if the minimum element in the symmetric difference of the two sets is in $U$. Determine which of the following statements is correct:

$S_1$: There is a subset of $S$ that is larger than every other subset.
$S_2$: There is a subset of $S$ that is smaller than every other subset.

Choose the correct option:
A) Both $S_1$ and $S_2$ are true
B) $S_1$ is true and $S_2$ is false
C) $S_2$ is true and $S_1$ is false
D) Neither $S_1$ nor $S_2$ is true

Answer: A


Question: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function in an open set around some $c \in \mathbb{C}$, excluding $c$. Assume that the Laurent series for $f$ around $c$ converges. If for $|x| > |c|$, the limit \( \lim_{x \rightarrow c, |x| > |c|} f(x)(x-c) = a \) exists, does it follow that $f$ has at most a finite order pole at $c$?

Answer: Yes


Question: Evaluate the double integral \( \iint_D (x + y) \, dx \, dy \) over the region \( D = \{(x,y) \mid 0 \leq y \leq x, 0 \leq x \leq 1\} \) by making the change of variables \( x = u + v \), \( y = u - v \). Verify your result by evaluating the integral directly using an iterated integral.

Answer: \dfrac{1}{2}


Question: Determine the fundamental group of the topological space \( X = \mathbb{R} \cup \{\infty_1, \ldots, \infty_n\} \), where the neighborhood of each \( \infty_i \) is given by \( A \cup \{\infty_i\} \) with \( A^c \) being a compact set in \( \mathbb{R} \).

Answer: F_n


Question:
Let \(f\) be an analytic function on \(\bar{D} = \{z \in \mathbb{C}: |z| \le 1\}\). Assume that \(|f(z)| \le 1\) for each \(z \in \bar{D}\). Which of the following is NOT a possible value of \((e^{f})''(0)\)?

\((A)\ 2\)
\((B)\ 6\)
\((C)\ \frac{7}{9}e^{\frac{1}{9}}\)
\((D)\ \sqrt{2} + \sqrt{2}\)

Answer: B


Question: Eight identical dice are rolled simultaneously. How many possible outcomes are there in which each of the six numbers appears at least once?

Answer: 191520


Question: Evaluate the limit: \[ \lim_{T \to \infty} \int_{0}^{T} t^2 e^{-t^2} \, dt \] given that \[ \lim_{T \to \infty} \int_{0}^{T} e^{-t^2} \, dt = \frac{\sqrt{\pi}}{2}. \] Choose the correct answer from the following options: \(\text{(A) } \frac{\sqrt{\pi}}{4}, \text{(B) } \frac{\sqrt{\pi}}{2}, \text{(C) } \sqrt{2\pi}, \text{(D) } 2\sqrt{\pi}.\)

Answer: A


Question: Determine the number of isomorphism classes of $\mathbb{Z}[i]$-modules that have exactly 5 elements.

Answer: 2


Question: For which value of $y_0$ does the recursive sequence $y_{n+1} = \sqrt{\frac{n+3}{n+1}} y_{n}$ converge?

Answer: 0


Question: If $R$ and $S$ are homeomorphic closed Riemann surfaces with genus $g$, does there always exist an orientation-preserving homeomorphism $h: R \to S$?

Answer: Yes


Question: Find an example of a constructible angle that cannot be quintsected using an unmarked straightedge and compass.

Answer: 72^\circ


Question: Let $f(x)$ and $g(x)$ be polynomials in $\mathbb{Q}[x]$ that share a common real root $\alpha$. Must $f(x)$ and $g(x)$ have a common divisor $h(x) \in \mathbb{Q}[x]$ with $\deg(h) \geq 1$? Justify your answer.

Answer: Yes


Question: Let $A$ be the set of nonnegative sequences $(a_n)$ such that $\sum_{n=1}^{\infty}a_n=1.$ Determine the range of the map $P:A\to \mathbb{R}$ defined by \[P((a_n))= \prod_{n=1}^\infty(1+a_n).\]

Answer: [2, e)


Question: Let $S$ be a distribution with compact support on $\mathbb{R}$. If $\langle S, x^n \rangle = 0$ for all $n \in \mathbb{N}$, prove that $S$ is the zero distribution, i.e., $S = 0$. Provide a justification for your answer.

Answer: S = 0


Question: Find the value of $r$ such that a Möbius transformation maps the annulus $\{z:r<|z|<1\}$ to the region between the circles $\{z:|z-1/4|=1/4\}$ and $\{z:|z|=1\}$.

Answer: 2 - \sqrt{3}


Question: Determine whether there exist functions \(f, g: [0, \infty) \to [0, \infty)\) such that \(\int_{0}^\infty g(x)dx < \infty\) and the expression \[ \frac{\int_0^a f(x)g(x)dx}{f(a)\int_a^\infty g(x)dx} \] grows exponentially with respect to \(a\).

Answer: Yes


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

Answer: -e^2


Question:
Suppose $f$ and $g$ are entire functions and $g(z) \neq 0, \forall z \in \mathbb{C}$. If $|f(z)| \leq |g(z)|$, which of the following statements is true?

(a) $f(z) \neq 0, \forall z \in \mathbb{C}$
(b) $f$ is a constant function
(c) $f(0) = 0$
(d) For some $c \in \mathbb{C}, f(z) = c g(z)$

Select the correct option.

Answer: D


Question: Determine whether \( \limsup_{n\rightarrow\infty} \frac{f(n)}{n} > 0 \), where \( f(n) =  |\{m : m\le n,  \exists k \text{ s.t. }\phi(k) = m\}| \) and \( \phi \) is the Euler's totient function.

Answer: No


Question: Determine the number of integral values of \(k\) for which the equation \(2x^3 + 3x^2 + 6x + k = 0\) has exactly three real roots.

Answer: 0


Question: Given a positive integer S, determine the maximum value of N such that the sum of all elements in a 3D array A of dimensions N x N x N, where each element A[i][j][k] is defined as ik(i+j+k) for 1 <= i, j, k <= N, does not exceed S. Constraints: 3 <= S <= 10^{16}. For example, if S = 720, the maximum value of N is 3.

Answer: 3


Question: Evaluate the limit: \[ L = \lim_{n \rightarrow \infty} \prod_{r=3}^{n} \frac{r^3 - 8}{r^3 + 8} \]

Answer: \dfrac{2}{7}


Question: Find a value for \( b \neq 0 \), independent of \( a \), such that the Newton Iterative Method for the function \( \phi(x) = x^b(x^2-a) \) locally converges cubically to \( \sqrt{a} \).

Answer: -\dfrac{1}{2}


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

Answer: \infty


Question: Compute the value of the expression \(2\arctan{2} + \arcsin{\frac{4}{5}}\).

Answer: \pi


Question: Calculate the flux of the vector field \( F = \langle z, y, zx \rangle \) through the surface of the tetrahedron enclosed by the coordinate planes and the plane \( \frac{x}{4}+\frac{y}{2}+\frac{z}{5}=1 \). Use the divergence theorem to find the correct flux value.

Answer: \dfrac{40}{3}


Question: Does there exist a separable metric compactification \( \overline{X} \) of a separable metric space \( X \) such that \( X \) is open in \( \overline{X} \)?

Answer: Yes


Question: Determine whether the function \( f(x,y) = -(x^2 - 1)^2 -(x^2y -x -1)^2 \) has only two critical points, and if they are local maximum points.

Answer: Yes


Question: Consider a square matrix $A = (a_{i,j})$ where $0 \leq a_{i,j} < 1$ for all $i,j$, and the sum of the entries in each row is less than 1, i.e., $\sum_{j=1}^{n} a_{i,j} < 1$ for each row $i$. Is it true that $|\det(A)| < 1$?

Answer: True


Question: Compute \( \lim_{n\rightarrow\infty}\int^n_0\frac{1}{n} \sin(nx)e^{-x}dx \).

Answer: 0


Question: Evaluate the contour integral \( \oint_{C} \frac{z^{2}}{z-4} \, dz \), where \( C \) is the circle \( |z| = 1 \) oriented in the counterclockwise direction.

Answer: 0


Question: Given the recurrence relation \(a_{n+1} = 2a_n - n^2 + n\) with the initial condition \(a_1 = 3\), find the value of \(\frac{|a_{20} - a_{15}|}{18133}\).

Answer: 28


Question: If \(A\) is an \(n \times n\) Hermitian matrix, and \(x_i\) is the eigenvector associated with the eigenvalue \(\lambda_i\) of \(A\) such that \(A x_i = \lambda_i x_i\) for \(i=1,\ldots,n\), is it true that \(\operatorname{span}\{x_1,\ldots,x_n\} = \mathbb{C}^n\)?

Answer: Yes


Question: Evaluate whether the integral \( \int_0^{2\pi}\cos(mt+\theta)\sin(nt+\phi)\,\text{d}t \) equals zero for arbitrary phases \( \theta \) and \( \phi \), where \( m \) and \( n \) are integers.

Answer: 0


Question: Evaluate the surface integral \( \int \mathbf{r} \cdot  dS \) over the surface of a sphere with radius \( a \) centered at the origin. Use the parameterization \( \mathbf{r} = a\sin\theta \cos\phi \mathbf{i} + a\sin\theta \sin\phi \mathbf{j} + a\cos\theta \mathbf{k} \) and the surface element \( dS = \left|\frac {\partial \mathbf{r}}{\partial \theta} \times \frac {\partial \mathbf{r}}{\partial \phi}\right|d\theta d\phi \).

Answer: 4\pi a^3


Question: Find the distance from the vector \((1,0,0,1)\) to the subspace \(L = (\text{Sp}\{(1,0,2,1),(3,1,4,4),(1,1,0,2)\})^{\perp}\).

Answer: \dfrac{\sqrt{14}}{3}


Question: A fixed parabola is given by the equation \( y^2 = 4ax \). A variable parabola, with its axis parallel to the x-axis, touches this fixed parabola. Find the equation of the locus of the vertex of the variable parabola, assuming the two parabolas are equal.

Answer: y^2 = 8ax


Question: Let \(X_1, X_2, \dots, X_n\) be a random sample from a distribution with probability density function (p.d.f.): \[ f(x, \theta) = e^{-(x-\theta)}; \quad \theta < x < \infty; \quad -\infty < \theta < \infty. \] Find the sufficient statistic for the parameter \(\theta\) using the factorization theorem.

Answer: X_{(1)}


Question: Calculate the work done by the vector field \( \vec{F}(x,y,z)=(xz,yz+x^2yz+y^3z+yz^5,2z^4) \) along the edge of the surface \( \Sigma \), where \( \Sigma \) is defined by \( (x^2+y^2+z^4)e^{y^2}=1, x\geq 0 \) and oriented such that its normal at the point \( (1,0,0) \) is \( \vec{N} = (1,0,0) \). The edge is oriented with respect to \( \Sigma \)'s orientation.

Answer: 0


Question:
Given two decision problems A and B, where A is NP-Complete and B can be solved in $O(n^2 \log^4 n)$ time, and it is known that $B \leq_p A$ (i.e., each problem of B can be converted to a problem of A in polynomial time), determine which of the following statements can be inferred:

1. P = NP and each NP problem can be solved in $O(n^3)$.
2. P = NP and some NP problems need times more than $O(n^3)$ to solve.
3. P ≠ NP.
4. None of the above.

Select the correct inference.

Answer: 4


Question: Is the algebraic Grothendieck group of a weighted projective space finitely generated?

Answer: Yes


Question: Determine whether the function \( f(x,y) = \frac{(1+x^2)x^2y^4}{x^4+2x^2y^4+y^8} \) for \((x,y) \neq (0,0)\) and \( f(0,0) = 0 \) admits directional derivatives at the origin \((0,0)\).

Answer: Yes


Question: A number coincides with the sum of the prime positive divisors of the number of trailing zeros in the product of the first 784 integers when expressed in base 15. Find this number minus 60.

Answer: 39


Question: Consider a scenario with $n$ people, each having a unique ball. They place their balls into a box, and the balls are mixed. Each person then draws a ball from the box. If a person draws their own ball, they leave; otherwise, they participate in the next round. What is the expected number of rounds until everyone has retrieved their own ball?

Answer: n


Question: Determine whether the operator \( F(x) = (f'(u) + \sum_{i=1}^m \lambda_i g_i'(u), -g_1(u), \ldots, -g_m(u)) : \Re^n \rightarrow \Re^n \) is monotone in the set \( G = \{x = (u, \lambda) \in \Re^n : \lambda \geq 0\} \), given that \( f \) and \( g_i \) are convex functions belonging to \( C^1 \). Specifically, is the condition \((F(x_1) - F(x_2))^T(x_1 - x_2) \geq 0\) always satisfied in \( G \)?

Answer: Yes


Question: Evaluate the contour integral \( \oint_C \frac{z^2+1}{e^{\frac {z}{10}}-1}dz \), where \( C \) is the unit circle traversed counterclockwise.

Answer: 20\pi i


Question: Let $H$ be a separable Hilbert space, and let $Q: B(H) \rightarrow B(H)/B_0(H)$ be the natural quotient map, where $B(H)$ is the algebra of bounded operators and $B_0(H)$ is the algebra of compact operators on $H$. Consider the orthonormal basis of $H$ and let $P_n$ be the projection onto the span of the first $n$ elements of this basis. Is it true that for the quotient norm, \( \|Q(T)\| = \lim_{n \to \infty} \|T(I - P_n)\| \) for any operator $T \in B(H)$?

Answer: Yes


Question: Given that \( \int_{0}^\infty e^{-x^2}\,dx=\alpha \), find the value of \( \int_{0}^{1}\sqrt{|\ln x|}\,dx \) in terms of \( \alpha \).

Answer: \alpha


Question: Given the differential equation \( ax(t) - \frac{d}{dt}x(t) = -\exp(at) \) with initial conditions \( x(0) = 0 \) and \( x(t_0) = \frac{1}{a} \), find the real constant \( t_0 \).

Answer: \dfrac{W(1)}{a}


Question:
Consider an undirected, infinite, connected graph \( \Gamma = (G,E) \) with no multiple edges or loops, equipped with edge weights \( \pi_{xy} \) and vertex weights \( (\theta_x)_{x\in G} \). The generator of a continuous time random walk on \( \Gamma \) is given by:
\[
(\mathcal{L}_\theta f)(x) = \frac{1}{\theta_x}\sum_{y\sim x}\pi_{xy}(f(y)-f(x)).
\]
The heat kernel of the random walk is defined as \( p_t(x,y) = \frac{\mathbb{P}^x(X^\theta_t=y)}{\theta_y} \). Given that \( u(x) = p_t(x_0,x) \) is always in \( L^2(\theta) \), is \( \mathcal{L}_\theta u \) also always in \( L^2(\theta) \)?

Answer: Yes


Question: Estimate the surface integral \( \iint_H f(x,y,z) \, dS \) over the hemisphere \( H \) defined by \( x^2 + y^2 + z^2 = 50 \) for \( z \geq 0 \), where \( f \) is a continuous function with the following values: \( f(3, 4, 5) = 7 \), \( f(3, -4, 5) = 8 \), \( f(-3, 4, 5) = 9 \), and \( f(-3, -4, 5) = 12 \). Divide \( H \) into four patches and use these values to estimate the integral.

Answer: 900\pi


Question: Determine the order of $E[||\sigma||_p]$ with respect to $m$, where $\sigma=(\sigma_1,...,\sigma_m)$ are i.i.d. uniform binary 0-1 valued variables. Given that Jensen's inequality provides an upper bound of $(m/2)^{1/p}$, find a lower bound of the same order.

Answer: \Theta(m^{1/p})


Question: Let $f(x)$ be a monic cubic polynomial such that $f(0) = -2$ and $f(1) = -5$. If the sum of all solutions to $f(x+1) = 0$ and $f\left(\frac{1}{x}\right) = 0$ are the same, find $f(2)$. \( \text{(Assume the polynomial is of the form } f(x) = x^3 + ax^2 + bx - 2 \text{)} \)

Answer: -6


Question: Calculate the residue of \( \cot^2(z) \) at each of its poles.

Answer: 0


Question:
Compute the determinant of the matrix \(D\alpha^{tr}D\alpha\) for the parametrization of the \(n\)-sphere \(S^n(a)\) given by:

\[D\alpha^{tr}D\alpha = \begin{bmatrix}
r^2 & 0 & 0 & \cdots & 0 \\
0 & \frac{x_3^2}{r^2}+1 & \frac{x_3x_4}{r^2} & \cdots & \frac{x_3x_{n+1}}{r^2} \\
0 & \frac{x_4x_3}{r^2} & \frac{x_4^2}{r^2}+1 & \cdots & \frac{x_4x_{n+1}}{r^2} \\
\vdots & \vdots & \vdots & \ddots &\vdots \\
0 & \frac{x_{n+1}x_3}{r^2} & \frac{x_{n+1}x_4}{r^2} & \cdots & \frac{x_{n+1}^2}{r^2}+1
\end{bmatrix}.\]

Verify if the determinant is indeed \(a^2\).

Answer: a^2


Question: A right-angled triangle with vertices \((0,−b),(3,0),(0,a)\), where \(a\) and \(b\) are positive numbers, is rotated about the line \(x=4\). Given that \(ab=9\), find the smallest possible volume of the solid formed by this rotation.

Answer: 54\pi


Question: Determine the limit of the function \( u(x_1,x_2) = \frac{1}{n^2}\sin(nx_1)\sinh(nx_2) \) as \( n \to +\infty \), where \((x_1,x_2) \in \mathbb{R}^2\).

Answer: 0


Question: Along a road 1 mile long, 3 people are distributed at random. What is the probability that no two people are less than a distance of \(d\) miles apart, given that \(d \leq \frac{1}{2}\)?

Answer: (1 - 2d)^3


Question: Let $X = [a,b]$ where $0 < a < b$. Find a subset $Y \subseteq \mathbb{R}$ such that $X * Y = X$, where $X * Y = \{ z \in \mathbb{R} \mid \text{there exist } x \in X, y \in Y \text{ such that } z = xy \}$.

Answer: \{1\}


Question: Compute the line integral of the vector field \( F(x,y,z) = \langle x^2, y^2, z^2 \rangle \) along the curve \( C \), which is the intersection of the plane \( z = x + 1 \) and the cylinder \( x^2 + y^2 = 1 \). The curve is traversed from the lowest point \((-1, 0, 0)\) to the highest point \((1, 0, 2)\), counterclockwise when viewed from above. Express the curve in cylindrical coordinates and set up the line integral in terms of a parameter \( t \). Evaluate the integral.

Answer: \dfrac{10}{3}


Question: Calculate the limit \( \lim_{n \to \infty} \frac{(2n-1)x_1+(2n-3)x_2+\dots +3x_{n-1}+x_n}{n^2} \) given that \( x_n \to x \).

Answer: x


Question: Evaluate the limit: \[ \lim_{u\to\infty} \frac{-bu^{a-1}e^u}{e^{be^u}} \] where \( a \) and \( b \) are constants.

Answer: 0


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

Answer: \dfrac{4}{e}


Question: Given a group $G$ of order 24 with 8 elements of order 3, determine the number of subgroups of $G$.

Answer: 30


Question: Evaluate the residue of the function \( \frac{\cot z}{z^4} \) at \( z = 0 \).

Answer: -\dfrac{1}{45}


Question: Find a solution to the differential equation \( ty'' - (t+1)y' + y = 0 \) that is not a constant multiple of the solution \( y(t) = e^t \).

Answer: t + 1


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{\ln(x)^{\ln(x)^{\ln(x)}}}{x^x} \]

Answer: \infty


Question: Evaluate the limit: \[ \lim_{n\to\infty} n^{1/2}\int_0^\infty \left( \frac{2x}{1+x^2} \right)^n \, dx \]

Answer: \sqrt{2\pi}


Question: Find the complex number $z$ that satisfies the equation \( \frac{z+2i}{z-2i}=\frac{7-6i}{5} \).

Answer: -3 + 3i


Question: Using the delta-epsilon definition of a limit, evaluate \( \lim_{x \to 0} x \) for some limiting value \( L \).

Answer: 0


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

Answer: -1


Question: Determine the number of ways to distribute five red balls and five blue balls into three distinct boxes such that no box is empty.

Answer: 336


Question: A circle C has a radius of 5 cm, and two smaller circles C1 and C2 have radii of 3 cm and 2 cm, respectively. Circles C1 and C2 touch each other externally and both touch circle C internally. Another circle C3 touches circles C1 and C2 externally and touches circle C internally. Determine the radius r of circle C3.

Answer: \dfrac{30}{19}


Question: Let \(S\) be the sum of all the real coefficients in the expansion of \((1+ix)^{2020}\). Calculate \(\log_{2}|S|\).

Answer: 1010


Question:
Does there exist a sequence \((a_n)_{n \in \mathbb{N}}\) of real numbers in the range \([-1, 1]\) such that the sequence of their arithmetic means \((\alpha_n)_{n \in \mathbb{N}}\), given by
\[\alpha_n = \frac{1}{n}\sum_{k=1}^n a_k,\quad n \in \mathbb{N}\]
has a dense image in \([-1, 1]\)?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty} \int_0^1 \frac{\ln x}{\sqrt{x+t}} \, dt \]

Answer: 0


Question: An urn contains the numbers 1, 2, 3, ..., 2019. Four numbers are drawn at random, without replacement, and in order from the urn, denoted as \(a, b, c, d\). What is the probability that the system of equations \((S)\): \(ax + by = ab\) and \(cx + dy = cd\) has a strictly internal solution (i.e., not on the axes) in the first quadrant?

Answer: \dfrac{1}{2}


Question: Evaluate the improper integral \( \int_2^\infty \frac{3x-2}{x^2(x-1)} \, dx \) and verify that it equals \( 1 - \ln 2 \).

Answer: 1 - \ln 2


Question: The sequence \(a_n\) is defined such that \(a_n\) and \(a_{n+1}\) are the roots of the quadratic equation \(x^2 - c_n x + \frac{1}{3^n} = 0\), with the initial condition \(a_1 = 2\). Determine the value of \(\sum_{n=1}^{\infty} c_n\).

Answer: \dfrac{9}{2}


Question:
Find the area of a petal of the polar curve given by \( r^2 = 3\sin{3\theta} \) using the parametrization \( \alpha(t) \) and the formula:
\[ \frac{1}{2}\int_a^b \begin{vmatrix} \alpha_1 & \alpha_2 \\ \alpha_1' & \alpha_2' \end{vmatrix} dt \]
Use the parametrization \( \theta = t \) and \( r = \sqrt{3\sin{3t}} \) for \( 0 \leq t \leq \frac{\pi}{3} \). Calculate the area of one petal.

Answer: 1


Question: Given the graph of the function \( y = \sqrt{3}x \), its inverse, and the line \( x = 4 \), find the area of the black square formed by these lines. The point of intersection between the line \( y = \sqrt{3}x \) and the line \( x = 4 \) is \( (4, \frac{4}{\sqrt{3}}) \). Determine the exact coordinates of the other intersection point on the blue line and calculate the area of the square.

Answer: \dfrac{64}{3}