Question: Let \(A\) be an infinite set in \(\mathbb{R}\) with a single accumulation point in \(A\). Must \(A\) be compact?

Answer: No


Question: Describe the ring obtained by adding an inverse of the element \((2, 0)\) to the ring \(\Bbb{R \times R}\).

Answer: \mathbb{R}


Question: Find the minimum value of $p$ for which the series \( \sum_{n=1}^{\infty}(1n^{27}+8)^{27}(1+27n^{29}+8n^{30})^p \) is divergent.

Answer: -\dfrac{73}{3}


Question: Find a closed form solution for the derivative \( \frac{\partial \lambda_i}{\partial E} \), where \( E \) is a real, positive-definite 3x3 symmetric tensor, and \( \lambda_i \) are the eigenvalues obtained from the eigendecomposition \( E_p = \sum_{i=1}^{3} \lambda_i N_i \otimes N_i \).

Answer: N_i \otimes N_i


Question: If $X$ and $Y$ are independent random variables such that $X + Y$ has the same distribution as $X$, is it always true that $P(Y = 0) = 1$?

Answer: Yes


Question: Determine the values of $x$ in the interval $(-3, 17)$ for which the series \( \sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n[\log (n+1)]^2} \) converges.

Answer: [-1, 1]


Question: Show that from the premises $\exists x : (P(x) \land x = c)$ and $\forall x(P(x) \implies Q(x))$, it follows that $Q(c)$.

Answer: Q(c)


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

Answer: \dfrac{1}{2}


Question:
Find the length of the curve defined parametrically by the equations:
\[ x = a \cos^3 t \]
\[ y = a \sin^3 t \]
for \(0 \leq t \leq 2\pi\).

Answer: 6a


Question: Calculate the integral \( \int_\gamma \frac{1}{\sin(z)} \, dz \) where \( \gamma(t) = 4e^{2 i \pi t} \) for \( t \in [0,1] \). Use the residue theorem to find the value of the integral.

Answer: -2\pi i


Question: Let $m$ and $n$ be positive integers such that $m(n-m) = -11n + 8$. Find the sum of all possible values of $m-n$.

Answer: 18


Question: If a complex function $f: D \to \mathbb{C}$ is continuous and one-to-one on an open and connected set $D$, is its inverse function $f^{-1}$ also continuous?

Answer: Yes


Question: Given the conditions \(\int_a^\infty \int_{-\infty}^a f(x,y) dy dx = \int_{-\infty}^a \int_a^\infty f(x,y) dy dx = 0\) for every \(a \in \mathbb{R}\), simplify the integral \(\int_{\mathbb{R}^2} f(x,y) dy dx\).

Answer: 0


Question: Determine the dimension and a basis for the extension field \(\mathbb{Q} (\sqrt[3]{2}, \sqrt{3})\) over \(\mathbb{Q}\).

Answer: 6


Question: Given the function \( f(x) = 4\arcsin(\sqrt{x}) + 2\arcsin(\sqrt{1-x}) \) with its derivative \( f'(x) = \frac{1}{\sqrt{x-x^2}} \), find the maximum value of \( f^{-1}(x) \).

Answer: 1


Question: Suppose we have a group ring \( \mathbb{Z}[G] \), where \( G \) is a torsion-free group, and a bounded chain complex \( M_{\bullet} \) of \( \mathbb{Z}[G] \)-modules such that each \( M_{i} \) is a finitely generated free \( \mathbb{Z}[G] \)-module. If the homology of \( M_{\bullet}\otimes_{\mathbb{Z}[G]}\mathbb{Z} \) is trivial, i.e., \( H_{n}(M_{\bullet}\otimes_{\mathbb{Z}[G]}\mathbb{Z})=0 \) for all \( n \in \mathbb{Z} \), does it imply that \( H_{n}(M_{\bullet})=0 \) for all \( n \in \mathbb{Z} \)?

Answer: No


Question:
Evaluate the surface integral \( \iint_S \mathrm{curl}\,\mathbf{F} \cdot d\mathbf{S} \) over the surface \( S \) of the cone \( z = 2 - \sqrt{x^2 + y^2} \) above the \( xy \)-plane, where the vector field \( \mathbf{F} \) is given by:
\[ \mathbf{F} = (x-z)\mathbf{i} + (x^3+yz)\mathbf{j} - (3xy^2)\mathbf{k} \]

Answer: 12\pi


Question: Evaluate the limit: $$\lim_{(x,y)\to(0,0)} \tan\left(x^{2}+y^{2}\right)\arctan\left(\frac{1}{x^{2}+y^{2}}\right)$$

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{e^{iyx}}{1+y^2} \, dy \) for \( x \ge 0 \) without using Fourier transform methods.

Answer: \pi e^{-x}


Question: Consider a game where you start with a bit string of $l$ zeroes. At each step, you randomly choose a bit from the string and flip it. The game ends when the bit string returns to its initial state of all zeroes. Determine the closed-form expected value of the length of the game as a function of $l$.

Answer: 2^l


Question: Given that \(x = 1\) is a root of the polynomial \(p(x) = x^4 + ax^3 + bx^2 + ax + 1\), find the range of values of \(a\) such that \(p(x)\) has complex (non-real) roots.

Answer: (-4, 0)


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x} \] without using derivation or L'Hôpital's rule.

Answer: 0


Question: Calculate the limit \( \lim_{h\to0}\frac{f(a+h^2)-f(a+h)}{h} \) given that the function \( f \) is differentiable at \( a \).

Answer: -f'(a)


Question: Find the entire function $f(z)$ such that $f(0) = 1$ and for all $z = x + iy$, the real part of $f(z)$ is given by $\operatorname{Re}(f(z)) = x^2 - y^2 + e^{-x} \cos y$. Provide the expression for $f(z)$. \( \text{Note: An entire function is analytic on all } \mathbb{C}. \)

Answer: z^2 + e^{-z}


Question:
Let $V = C^1[0,1]$, $X = (C([0,1]), ||\cdot||_\infty)$, and $Y = (C[0,1], ||\cdot||_2)$. Determine which of the following statements is correct:

(a) $V$ is dense in $X$ but not in $Y$.
(b) $V$ is dense in $Y$ but not in $X$.
(c) $V$ is dense in both $X$ and $Y$.
(d) $V$ is dense in neither $X$ nor $Y$.

Select the correct option.

Answer: C


Question: In a group of 6 kids, the average weight of any 2 kids is a distinct natural number. The minimum weight of any kid is 11 kg. Determine the minimum possible sum of the weights of the heaviest and the second heaviest kid in the group.

Answer: 60


Question: Calculate the integer part of the sum: \[ y = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} + \ldots + \frac{1}{\sqrt{1000000}} \]

Answer: 1998


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

Answer: 2^{\aleph_0}


Question: Determine the number of zeros of the function \( f(z) = e^z - 2i \).

Answer: \infty


Question: Calculate the moment of inertia of a cube with side length $a$ and uniform density $\rho$ about an axis that passes through two opposite corners using direct integration. Show that the moment of inertia is given by \( I = \frac{ma^2}{6} \).

Answer: \dfrac{ma^2}{6}


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

Answer: \dfrac{a}{2}


Question: Use continued fractions to find a rational number that approximates \(\sqrt{11}\) to within \(10^{-4}\).

Answer: \dfrac{199}{60}


Question: Let \( M \) be the set of Borel integrable functions \( f:[0,\pi]\to \mathbb{R} \) such that \( \int_0^\pi f(x) \sin(x) \, dx = \int_0^\pi f(x) \cos(x) \, dx = 1 \). Find the minimum value of \( \int_0^\pi f^2(x) \, dx \) for \( f \in M \).

Answer: \dfrac{4}{\pi}


Question: Determine the maximum value of \(|z^3 - z + 2|\) for all complex numbers \(z\) such that \(|z| = 1\).

Answer: \sqrt{13}


Question: Evaluate the contour integral \( \int_{\Gamma} \frac{1}{z^2+1} \, dz \) where \( \Gamma \) is the circle \( |z| = 2 \) oriented counterclockwise.

Answer: 0


Question: Evaluate the sum: \( \sum_{r=0}^{100} (-1)^{r} \binom{100}{r} r^{50} \)

Answer: 0


Question: Determine the value of \( c \in \mathbb{R}^+ \) for which the Banach fixed-point theorem is satisfied for the function \( g(x_1, x_2) = \frac{1}{6} \begin{pmatrix} x_1x_2+cx_2-1 \\ x_1^2-x_2+1 \end{pmatrix} \) on the set \( M = [-1,1]\times [-1,1] \).

Answer: 4


Question: Evaluate the limit: \[ \lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!} \]

Answer: \dfrac{1}{2}


Question: Evaluate the triple integral \( \int \int \int x \, dx \, dy \, dz \) over the region \( D = \{x^2 - 2x + y^2 \le 0, -x^2 \le z \le 2 - x - y\} \).

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


Question: Evaluate the limit: \[ \lim_{k\to 2}\frac{2^{2^k}-2^{k^2}}{2^k-k^2} \]

Answer: 16 \ln 2


Question: In an ordered ring $R$ with unity $1$, consider any $r \\in R$ such that $0 < r < 1$. Determine whether the sequence $(r^n)$ converges to zero.

Answer: No


Question:
Consider the linear operator \( T \) from the \( l_1 \) space to itself, defined by \( T(x_1, x_2, \ldots) = (x_2 - x_1, x_3 - x_2, \ldots) \). Determine which of the following statements about the norm \( \|T\| \) is true:

1. \( \|T\| = 1 \).
2. \( \|T\| > 2 \).
3. \( 1 < \|T\| \leq 2 \).
4. None of the above.

Answer: 3


Question: Evaluate the triple integral \( \iiint_{E} xy \, dV \) where \(E\) is the solid tetrahedron with vertices \((0,0,0), (8,0,0), (0,8,0), (0,0,9)\).

Answer: \dfrac{1536}{5}


Question: Evaluate the limit: \[ \lim_{z \to 0} \frac{z^5}{1+z-e^z} \]

Answer: 0


Question: Evaluate the triple summation \( \sum_{1 \le i < j \le k \le 20} a_i a_j a_k \) where \( a_i = 1 \) for \( 1 \le i \le 10 \) and \( a_i = 2 \) for \( 11 \le i \le 20 \). Find the value of the summation.

Answer: 4585


Question: Determine the dimension of the set of $n \times n$ matrices $A$ such that $Ae_{ij} = e_{ij}A$ for all $i, j$, where $e_{ij}$ is the matrix with a $1$ in the $(i,j)$ position and $0$ elsewhere.

Answer: 1


Question: Evaluate the limit: \[ \lim_{m \to \infty} \frac{\int_{0}^{m}t^{m-1}e^{-t}dt}{(m-1)!} = \frac{\gamma(m,m)}{\Gamma(m)}, \qquad m \in \mathbb{Z^{+}} \] where \(\gamma(s,x)\) is the lower incomplete gamma function, and \(\Gamma(s)\) is the ordinary gamma function. Verify that this limit approaches \(\frac{1}{2}\).

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{\arcsin x-\sin x}{x^3} \] without using series expansion or L'Hospital's rule.

Answer: \dfrac{1}{3}


Question: Given a function \( f \) that satisfies \( f(3x) = 3f(x) \) for any positive real number \( x \), and \( f(x) = 1 - |x-2| \) for all real \( x \) in the interval \([1,3]\), find the value of \( f(2001) \).

Answer: 186


Question: Let $X$ and $Y$ be two Banach spaces over $\mathbb{C}$, and let $V$ be a dense subspace of $X$. Suppose $T : V \to Y$ is a bounded linear operator that can be uniquely extended to a bounded linear transformation $S : X \to Y$ such that $\| T \| = \| S \|$. If $T$ is injective, is $S$ also injective?

Answer: Yes


Question: Identify a Banach space that is isometric to the quotient space $C[0,1]/M$, where $M=\{f\in C[0,1]:f(0)=0\}$ and $C[0,1]$ is equipped with the supremum norm.

Answer: \mathbb{R}


Question:
Let $X$ be a topological space and let $\emptyset \neq A \subset X$ be such that $A = \partial A$. Which one of the following statements is guaranteed to be true?

(i) $\exists x \in A: \forall N \in N(x): N \cap (X \backslash A) = \emptyset$

(ii) $\text{int}(A) \neq \emptyset$

(iii) $A = \bar{A}$, where $\bar{A}$ is the closure of $A$ (boundary points + interior points).

Answer: C


Question: Find the volume of a generalized tetrahedron in \(\mathbb{R}^n\) bounded by the coordinate hyperplanes and the hyperplane \(x_1 + x_2 + \ldots + x_n = 1\).

Answer: \dfrac{1}{n!}


Question: Find the value of \( \lim_{n\to\infty}(\sin^4x+\frac{1}{4}\sin^4(2x)+\cdots+\frac{1}{4^n}\sin^4(2^nx)) \). Choose the correct option: \(\text{(A) } \sin^4x \), \(\text{(B) } \sin^2x \), \(\text{(C) } \cos^2x \), \(\text{(D) does not exist} \).

Answer: B


Question: Let $g(x)$ be a bounded measurable function such that \(\lim_{n \rightarrow \infty} \int_E g(nx)dx = 0\) for any measurable set $E$ with finite measure. Given $f \in L^1(\mathbb{R})$, determine whether \(\lim _{n \rightarrow \infty} \int _{\mathbb{R}} f(x)g(nx)dx = 0\).

Answer: 0


Question: Calculate \( \lim_{n\to\infty} f(n) \sin\left(\frac{1}{n}\right) \), where \( f(x) = \int_{x}^{x^2} \left(1+\frac{1}{2t}\right)^t \sin\left(\frac{1}{\sqrt{t}}\right) dt \) for \( x > 0 \).

Answer: 2\sqrt{e}


Question: Evaluate the integral: \[ \int_{0}^{\pi} \sin (\sin x) \, dx \]

Answer: \pi \mathbf{H}_0(1)


Question: Evaluate the integral \(\int_{0}^{2\pi} (\cos(z))^6 \, dz\).

Answer: \dfrac{5\pi}{8}


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f'(x) > x^2 \) for every \( x < 0 \). Determine \( \lim_{x \to -\infty} f(x) \).

Answer: -\infty


Question: Let \(f : D(0, 1) \to \mathbb{C}\) be a holomorphic function. Suppose there exists a sequence \(z_n\) of distinct points in \(D(0, 1)\) such that \(z_n \to 0\) and \(f(z_n) = \sin z_n\) for all \(n\). Prove that \(f(z) = \sin z\) for all \(z \in D(0, 1)\).

Answer: \sin z


Question: In triangle $\triangle ABC$, point $D$ is on $AC$ such that $D$ divides $AC$ in the ratio $1:2$. Point $E$ is on $AB$ such that line $CE$ divides $BD$ in the ratio $2:1$. Determine the ratio $AB:EB$. \(\text{Express your answer as a simplified ratio.}\)

Answer: 7:4


Question: Let $(X, Y, Z)$ have a multinomial distribution with parameters $n = 3$, $p_1 = \frac{1}{6}$, $p_2 = \frac{1}{2}$, and $p_3 = \frac{1}{3}$. Find $\text{cov}(X, Y)$. Provide your answer as a single numerical value.

Answer: -\dfrac{1}{4}


Question:
Consider 100 sets $A_1, A_2, \ldots, A_{100}$, each a subset of $\mathbb{R}$. For each set $A_i$, the complement in $\mathbb{R}$ is countable. Let $A = A_1 \cap A_2 \cap \ldots \cap A_{100}$ and $B$ be the complement of $A$. Determine the cardinality of $B$. Choose from the following options:

1. $0$
2. A finite number but not $0$
3. $\aleph_0$
4. $c$

Answer: 3


Question:
Compare the errors for the following methods of computing \( \ln 2 \) using Taylor series approximations. Determine which method provides the smallest error.

1. \( P_{n,1}(2) \)
2. \( -P_{n,1}(0.5) \)
3. \( P_{n,1}(\frac{4}{3}) - P_{n,1}(\frac{2}{3}) \)

The Taylor series for \( \ln x \) centered at \( a=1 \) is given by:
\[ P_{n,1}(x) = (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} + O(x^5). \]

The error for the \( n \)th term is given by:
\[ |f(x) - P_{n,1}(x)| = \frac{|(x-1)|^{n+1}}{n+1}. \]

Which method provides the smallest error for \( n \geq 1 \)?

Answer: 3


Question: Find the equation of the plane $P$ that is parallel to the vector $\vec{v} = \langle 1,0,-1 \rangle$ and contains the line $L_1 : (1-2t)\vec{i}+(4+3t)\vec{j}+(9-4t)\vec{k}$.

Answer: x + 2y + z = 18


Question: Find an asymptotic upper bound for the recurrence relation \( T(n) \le 2T(n−1)+n \) with the initial condition \( T(1) = 1 \), using big-O notation.

Answer: O(2^n)


Question: Determine the generator of the ideal \( \{p \in \mathbb{R}[X]; p(0)=p(1)=p'(0)=0 \} \) in the ring \( \mathbb{R}[X] \), which is a principal ideal domain (PID). Is the generator the polynomial of lowest degree, specifically \( X^3 - X^2 \)?

Answer: X^3 - X^2


Question: If \(a, b, c\) are in geometric progression and \(a-b, c-a, b-c\) are in harmonic progression, find the value of \(a+4b+c\).

Answer: 0


Question: Suppose that \(\lim\limits_{n\rightarrow \infty} \left|\frac{a_{n+1}}{a_n}\right| = \frac{1}{\alpha}\) and \(\lim\limits_{n\rightarrow \infty} \left|\frac{b_{n+1}}{b_n}\right| = \frac{1}{\beta}\) with \(\alpha > \beta\). Determine if \(\lim\limits_{n\rightarrow \infty} \left|\frac{a_n}{b_n}\right| = 0\).

Answer: 0


Question: Determine the number of 3x3 orthogonal matrices over the integers, where an orthogonal matrix A satisfies the condition \( AA^t = I \).

Answer: 48


Question: Given that \( f \) is continuously differentiable on \([0,1]\) with \( f(0) = f(1) = 0 \) and \( \int_0^1 f^2(x) \, dx = 1 \), calculate the value of \( \int_0^1 x f(x) f'(x) \, dx \).

Answer: -\dfrac{1}{2}


Question: Find the sum of the digits of the smallest positive integer $k$ such that the sum of the coefficients $a_0 + a_1 + \ldots + a_{k-1}$ in the expansion of the polynomial \((1 + x)(1 + 2x)(1 + 3x) \cdots (1 + kx) = a_0 + a_1x + a_2x^2 + \cdots + a_kx^k\) is divisible by 2005.

Answer: 5


Question:
Determine the infimum of the energy of the hydrogen atom given by the functional:
\[ \mathcal{E}(u) = \frac{9}{4} \left(\int_{\Bbb{R}^3}\Vert\mathbf{x}\Vert^2\ \vert u(\mathbf{x}) \vert^{2}\ d^3\mathbf{x}\right)^{-1} - \int_{\mathbb{R}^{3}} \frac{ \vert u(\mathbf{x}) \vert^{2}}{\Vert\mathbf{x}\Vert}\ d^3\mathbf{x} \]
where \( u \in C_{c}^{1}(\mathbb{R}^3) \) and \( \int_{\mathbb{R}^{3}} \vert u(\mathbf{x}) \vert^{2} d^3\mathbf{x} = 1 \).

Answer: -\dfrac{1}{3}


Question:
Let \( f : [1, ∞) \to \mathbb{R} \) be a real-valued function such that \( f(1) = 1 \) and for all \( x \in [1, ∞) \), \( f'(x) = \frac{1}{x^2 + f(x)^2} \). Determine \( \lim_{x \to \infty} f(x) \). Choose the correct option:

(a) The limit exists and is equal to zero.
(b) The limit does not exist.
(c) The limit exists and lies in the interval \([1, 1 + \pi/4]\).
(d) The limit exists and lies in the interval \((1 + \pi/4, \infty)\).

Answer: C


Question: Consider a sequence defined by the recursion \(a_{n+1} = r_n a_n\), where \(a_1 = 1\) and each \(r_n\) is a real number between 0 and 1. Determine the limit of this sequence as \(n\) approaches infinity.

Answer: 0


Question: Determine the number of rotational symmetries of the 120-cell.

Answer: 7200


Question: Does the space $L^2([0,1])$, with the inner product \( \langle f,g \rangle = \int_0^1 \overline{f(x)}g(x) \, dx \), have an orthonormal basis consisting solely of even-degree polynomials of the form \( \left\{f_n(x):=\sum\limits_{k=0}^n a_k x^{2k} \mid n \ge 0\right\} \)?

Answer: Yes


Question: Given that $a^2 + b^2 = ab = 4$, find the value of $a^3 + b^3$. Provide a justification for your answer.

Answer: 0


Question: Find the order of the element \((3, 1) + \langle(0, 2)\rangle\) in the quotient group \(\mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(0, 2)\rangle\).

Answer: 4


Question: Evaluate the limit: $$ \lim_{x\to 0}\frac{\sqrt{1+2x}-\sqrt[\Large3]{1+5x}}{\sqrt[\Large5]{1+x}-\sqrt[\Large5]{1+2x}} $$

Answer: \dfrac{10}{3}


Question: Evaluate the limit: \[ \lim_{h \to 0} \frac{1}{h} \int_1^{1+h} \sqrt{1+t^2} \, dt \]

Answer: \sqrt{2}


Question: Determine whether the series \( \sum_{n=0}^\infty \int_{[-1,1]^2} \left(1-\frac{x^2+y^2}{2}\right)^{n/2}\lambda_2(d(x,y)) \) is finite, where \( \lambda_2 \) is the 2-dimensional Lebesgue measure.

Answer: \infty


Question: Determine whether the function \( \frac{e^{-\frac{1}{x}}}{x} \) is uniformly continuous on the interval \((0, 1)\).

Answer: Yes


Question: What is the minimal number of cuts necessary to find the center of a cylindrical cake using a knife, given that you can only score the surface of the cake? Is it possible to determine the center with fewer than 7 cuts?

Answer: 3


Question: Evaluate the limit: $$ \lim_{x \to 0}\frac{1-\log x}{x^2} $$

Answer: \infty


Question: Determine the number of partitions of the set \( \{0, 1, 2, 3, 4, 5, 6, 7\} \) that have an even number of parts. What is the cardinality of this set of partitions?

Answer: 2095


Question: Consider the polynomial ring $A = k[x_1, \ldots, x_n]$ and let $e_1, \ldots, e_n$ be $n$ $k$-algebraically independent elements of $A$. Define $C = k[e_1, \ldots, e_n]$, so $C \simeq A$. Suppose there is another $k$-subalgebra $B$ of $A$ such that $A \supset B \supset C$. Given that $A$ is integral over $B$, $B$ is integral over $C$, and the field of fractions of $B$ is equal to the field of fractions of $C$, must it be the case that $B \simeq A$? Justify your answer.

Answer: B \simeq A


Question: Determine whether the limit \( \lim_{x \to \infty} \frac{\ln(x!)}{x} \) is convergent. If it is, find its value.

Answer: \infty


Question: Evaluate the infinite series: \[ \sum_{n=1}^{\infty} \frac{2^n}{n(n+2)} \cdot \frac{5n+18}{3^n} \]

Answer: 8


Question: Evaluate the sum \( \sum_{k=1}^{n} (2k-1) {n \choose k} \) using calculus, specifically by utilizing the binomial expansion of \((1-x)^n\).

Answer: (n - 1)2^n + 1


Question: Calculate the greatest common divisor (GCD) of the numbers \(3^{100!} - 1\) and 116.

Answer: 116


Question: Determine the probability that the interval \( \left[\frac{Y}{124.34},\frac{Y}{77.93}\right] \) contains \( \sigma^2 \), where \( Y = \sum_{i=1}^{100}(X_i-\mu)^2 \) and \( \mu \) is known for a normally distributed population of 100 individuals.

Answer: 0.90


Question: Evaluate the complex contour integral \( \oint \limits_{C} \frac{2}{z^3+z}dz \) without using Residue Theorems, where \( C: \big|~z~-~\frac{i}{2}~\big|=1 \) is positively oriented.

Answer: 2\pi i


Question: Let $g(x) = e^{f(x)}$ be a differentiable function on $(0,\infty)$ such that $g(x+1) = (x+1)g(x)$. For $n = 1, 2, 3, \dots$, find the value of $f'(n+1) - f'(1)$. \( \)

Answer: H_{n+1} - 1


Question: Given a power series $f(z) = \sum_{n = 0}^\infty c_n z^n$ with $|z| < R$, if there exists at least one coefficient $c_n$ such that \( |c_n| = \frac{\max_{|z| = r} |f(z)|}{r^n} \) for some $0 < r < R$, prove that $f(z)$ must be of the form $f(z) = c_n z^n$.

Answer: f(z) = c_n z^n


Question: Determine whether there exists an algebraic modular lattice \((X, \vee, \wedge)\) and elements \(a, b \in X\) with \(a \le b\) such that the function \(f: X \to X\) defined by \(f(x) = (x \vee a) \wedge b\) is not a \(\vee\)-homomorphism.

Answer: Yes


Question: Let $(G,+,\le)$ be a partially ordered group with identity $0$. Suppose that for each positive $g \in G$, there exists $g' \in G$ such that $0 < g' < g$. Determine whether there exists a sequence $(g_n)_{n \ge 1}$ of positive elements of $G$ such that $\inf \{g_n\} = 0$. Provide a proof or counterexample.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{2\cos(x) - 2 + x^{2}}{x^{4}}$$

Answer: \dfrac{1}{12}


Question: Evaluate the double integral: \[ \int_{0}^{\pi}\int_{0}^{x}\frac{\sin y}{\pi-y}\,dy\,dx \]

Answer: 2


Question: Evaluate the integral \( \int_0^1 \frac{f(x)\,dx}{f(x)+f(1-x)} \) where \( f:[0,1] \to \mathbb{R} \) is a continuous function and \( f(x) > 0 \) for all \( x \in [0,1] \).

Answer: \dfrac{1}{2}


Question: Find the equation of a straight line that passes through the point \((3, 6)\) and is orthogonal to the curve \(y = \sqrt{x}\).

Answer: y = -4x + 18


Question: Determine the exponential type of the entire function $\sin z$.

Answer: 1


Question: Find the extremal of the functional \( J[y]=\int\limits_1^2 (y'^2+2yy'+y^2)\,dx \) given the boundary condition \( y(1)=1 \) and \( y(2) \) is arbitrary.

Answer: e^{1 - x}


Question: Find the number of \(2 \times 2\) matrices \(A\) with entries in \(\mathbf{Z}/3\mathbf{Z}\) such that \(A^{50} = I\).

Answer: 14


Question: Evaluate the limit \( L = \lim_{n\rightarrow \infty }(a_{0}\sqrt[3]{n}+a_{1}\sqrt[3]{n+1}+\ldots+a_{k}\sqrt[3]{n+k}) \) given that \( a_{0},a_{1},\ldots,a_{k} \) are real numbers and \( a_{0}+a_{1}+\ldots+a_{k}=0 \). What is the value of \( L \)?

Answer: 0


Question: Determine the value of \( n \) such that the center of the alternating group \( A_n \), denoted as \( \textbf{Z}(A_n) \), is greater than 1.

Answer: 3


Question: Find \((f^{-1})'(1)\) for the function \(f(x) = x - \frac{2}{x}\) where \(x < 0\).

Answer: \dfrac{1}{3}


Question: Evaluate the flux of the vector field \( \vec{F} = -9\hat{j} - 3\hat{k} \) across the surface \( z = y \) bounded by the sphere \( x^2 + y^2 + z^2 = 16 \).

Answer: 48\sqrt{2}\pi


Question: For which values of \( n \geq 1 \) is the ring \( R_n = \begin{bmatrix} \frac{\mathbb{C}[x]}{((x-\alpha)^n)} & \frac{\mathbb{C}[x]}{((x-\alpha)^n)} \\ \frac{\mathbb{C}[x]}{((x-\alpha)^n)} & \frac{\mathbb{C}[x]}{((x-\alpha)^n)} \end{bmatrix} \) semisimple, where \( \alpha \in \mathbb{C} \)?

Answer: 1


Question: Let $f(z)$ be a Hardy function in $H^{2+}$ on the upper half complex plane, with a zero of order $m$ at $\omega$, where $\operatorname{Im}(\omega) > 0$. Determine whether the function \( \frac{f(z) - f(\omega)}{(z - \omega)^n} \) belongs to $H^{2+}$ for all $n \leq m$. Provide a justification for your answer.

Answer: Yes


Question: For a 21-sided regular polygon $A_1A_2A_3 \ldots A_{21}$ inscribed in a circle with center $O$, determine the number of triangles $A_iA_jA_k$ (where $1 \le i < j < k \le 21$) that contain the point $O$ in their interior.

Answer: 385


Question: Compute the integral of curvature \( \int_C k \) for the curve \( C \) defined by \( C := \{ (x,y) \in \mathbb{R}^2 \mid x^4 + y^4 = 1 \} \).

Answer: 2\pi


Question: Evaluate the contour integral \( \int_\gamma \frac{1}{z^2 - 2z} \, dz \), where \( \gamma \) is the circle of radius 1 centered at 2, traveled once counterclockwise.

Answer: \pi i


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{(1+x^5)^{10} -1}{(\sqrt{1+x^3}-1)(\sqrt[5]{1+x^2}-1) } \]

Answer: 100


Question: Let $f$ be an entire function such that $f(0) = 1$ and $|f(z)| \geq \frac{1}{3}|e^{z^3}|$ for all $z$. Show that $f(z) = e^{z^3}$ for all $z$. Can this be shown using the Cauchy integral formula?

Answer: e^{z^3}


Question: Given that $x_1, x_2, \ldots, x_{2019}$ are the roots of the polynomial $P(x) = x^{2019} + 2019x - 1$, determine the value of the sum $$\sum_{i=1}^{2019}\frac{x_i}{x_i-1}.$$

Answer: 2017


Question: Let $F$ be the free group on two generators, and let $n > 1$ be an integer. Consider the characteristic quotient $F/F''F^n$. Is this group finite?

Answer: Yes


Question: Does there exist a non-rational irreducible algebraic curve that contains infinitely many rational points?

Answer: Yes


Question: Let $(a_n)$ be a non-increasing sequence of positive real numbers such that the series $\sum a_n$ converges. Determine whether it is true that $\lim_{n \to \infty} n a_n = 0$. Justify your answer.

Answer: 0


Question: Determine the positive values of $x$ for which the series \( \sum_{n=1}^{\infty}x^{\sqrt{n}} \) is convergent.

Answer: (0, 1)


Question: Compute the integral \( \int_{\mathbb{R}^2} e^{-(4x^2 + 4xy + 5y^2)} \, d(x,y) \).

Answer: \dfrac{\pi}{4}


Question: Calculate the triple integral \( \iiint_R y \, dxdydz \) where \( R \) is the region defined by the cube portion \( 0 \le x,y,z \le 1 \) that lies under the plane \( x+y+z=2 \) and above the plane \( y+z=1 \).

Answer: \dfrac{5}{24}


Question: Let \(a_n\) and \(b_n\) be two positive real sequences. Suppose that \(\lim_{n\to\infty}\frac{a_n}{b_n}=1\) and \(b_n\leq \frac{c}{n}\) for some \(c>0\) and all sufficiently large \(n\). Does it follow that \(a_n\leq \frac{C}{n}\) for some \(C>0\) and all sufficiently large \(n\)?

Answer: Yes


Question: Let $K$ be a number field where the rational prime $2$ splits as $(2)\mathcal{O}_K = \mathfrak{P}_1\mathfrak{P}_2$. Determine the intersection $K_{\mathfrak{P}_1} \cap K_{\mathfrak{P}_2}$, where $K_{\mathfrak{P}_i}$ denotes the completion of $K$ at the prime $\mathfrak{P}_i$.

Answer: \mathbb{Q}_2


Question:
Given the system of differential equations:
\[ y'_1 = y_1 + y_2 + 1 \]
\[ y'_2 = -y_1 + y_2 + 1 \]
with initial conditions \( y_1(0) = 1 \) and \( y_2(0) = -1 \), find the value of \( y_1(\pi) \).

Answer: -e^{\pi}


Question: If the fundamental group of a compact $n$-dimensional manifold $M$ is trivial, i.e., $\pi_1(M)=0$, is $M$ necessarily orientable? Provide a justification for your answer.

Answer: Yes


Question: Let $V$ be a finite-dimensional real vector space, and let $\omega$ be a skew-symmetric nondegenerate bilinear form (i.e., a symplectic form) on it. Consider a map $f: V \to V$. If $f(0) = 0$ and $\omega(fu, fv) = \omega(u, v)$ for all $u, v \in V$, is it true that $f$ is linear?

Answer: Yes


Question: Let $P$ and $Z$ be two $n \times n$ real positive-semidefinite symmetric matrices, i.e., $P \succeq 0$, $Z \succeq 0$, and $P^T = P$, $Z^T = Z$. If $\text{trace}(P \cdot Z) = 0$, does it necessarily follow that $P \cdot Z = 0$? Provide a justification for your answer.

Answer: Yes


Question: Let $z_1, z_2, z_3$ be complex numbers such that $|z_1| = |z_2| = |z_3| = |z_1 + z_2 + z_3| = 2$ and $|z_1 - z_2| = |z_1 - z_3|$ with $z_2 \neq z_3$. Find the value of $|z_1 + z_2||z_1 + z_3|$. \(\text{Express your answer as a single number.}\)

Answer: 8


Question:
Find the smallest natural number \( x \) such that:
\[
\begin{align*}
x &\equiv 1 \pmod{2} \\
x &\equiv 2 \pmod{3} \\
x &\equiv 3 \pmod{4} \\
x &\equiv 4 \pmod{5} \\
x &\equiv 5 \pmod{6} \\
x &\equiv 6 \pmod{7} \\
x &\equiv 7 \pmod{8} \\
x &\equiv 8 \pmod{9} \\
x &\equiv 9 \pmod{10} \\
x &\equiv 10 \pmod{11} \\
x &\equiv 11 \pmod{12} \\
x &\equiv 0 \pmod{13}
\end{align*}
\]

Answer: 277199


Question: Calculate the fundamental group of the space $S^3 - S^1$. Provide your answer as a well-defined mathematical group.

Answer: \mathbb{Z}


Question: Let \( a, b \in \mathbb{N} \) such that \( a \geq 2 \), \( a \nmid b \), and \( a^n - 1 \mid b^n - 1 \) for all \( n \in \mathbb{N} \). Prove that \( b = 1 \).

Answer: 1


Question: Determine the condition on \( \alpha \) such that the function \( f(x) = |x|^\alpha \) with \( f(0) = 0 \) is weakly differentiable on the unit ball \( A = B_1(0) \subset \mathbb{R}^n \). Assume \( \alpha \neq 0 \) and \( n \geq 2 \).

Answer: \alpha > 1 - n


Question: Given $n > 0$, let $S$ be a set of positive integers less than or equal to $2n$ such that if $a$ and $b$ are in $S$ and $a \neq b$, then $a \nmid b$. What is the maximum number of integers that $S$ can contain?

Answer: n


Question: Evaluate the integral \( \int_0^4 (g \circ f \circ g)(x) \, \mathrm{d}x \), where \( f(x) = \sqrt[3]{x + \sqrt{x^2 + \frac{1}{27}}} + \sqrt[3]{x - \sqrt{x^2 + \frac{1}{27}}} \) and \( g(x) = x^3 + x + 1 \).

Answer: 156


Question: Find the area of the subset \( S \) of \( \mathbb{R}^3 \) defined by the conditions: \( x^2 + y^2 + z^2 \leq 1 \) and \( \det A = 0 \), where \( A = \begin{bmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{bmatrix} \).

Answer: 3\pi


Question: Determine the structure of a Sylow $2$-subgroup of the symmetric group $S_4$.

Answer: D_4


Question: Find the largest possible value of the product $pqr$, where $p$, $q$, and $r$ are prime numbers satisfying the conditions $p+q<111$ and $\frac{p+q}{r}=p-q+r$. What is the largest possible value of $pqr$?

Answer: 2014


Question: Evaluate the indefinite integral \( \int\sqrt{\dfrac{x}{a^3-x^3}}\,dx \) and express it in the form \( \dfrac{d}{b}\sin^{-1} \left(\dfrac{x^{3/2}}{a^{3/2}}\right) + C \), where \( b \) and \( d \) are relatively prime integers. Find the value of \( b + d \).

Answer: 5


Question:
In a village with $N$ people, each person owns one dog. A certain number of dogs, $K$, have contracted rabies. Every morning, villagers take their dogs for a walk and can identify which dogs, except their own, have rabies. They agree to shoot their own dog at night if they can conclude it has rabies. The disease is not contagious, and $K$ is constant with $K \geq 1$.

On the first night, no dogs are shot. On the second night, no dogs are shot. On the third night, no dogs are shot. On the fourth night, several gunshots are heard.

How many dogs were killed on the fourth night?

Answer: 4


Question:
Let \( Q \) be defined on the space of all polynomials with degree at most 2 by:
\[ Q(P(t)) = \int_{-1}^1 (p(t))^2 \, dt - \int_{-1}^1 (p'(t))^2 \, dt \]
Show that \( Q \) is a quadratic form and compute its signature.

Answer: (1, 2)


Question: Let $K=\mathbb{Q}(\zeta_n)$ be a cyclotomic field, and let $\mathfrak{p}$ be a prime ideal in the ring of integers $\mathcal{O}_K$. Suppose $\mathfrak{p} = (p,f_i(\zeta_n))$, where $\mathfrak{p}\cap\mathbb{Z}=p$ and $f_i(x)$ is an irreducible polynomial in the factorization of $\Phi_n(x)\bmod p$, with $\Phi_n(x)$ being the $n$th cyclotomic polynomial. If $f_j(x)$, $i\ne j$, is another polynomial in the factorization of $\Phi_n(x)\bmod p$, is it true that $N_{K/\mathbb{Q}}(f_i(\zeta_n)) = N_{K/\mathbb{Q}}(f_j(\zeta_n))$?

Answer: No


Question: Compute the second moment of a chi-squared distribution with \( n \) degrees of freedom. The chi-squared distribution is the sum of \( n \) independent squared standard normal variables. Provide the correct expression for the second moment.

Answer: n(n + 2)


Question: Determine the tangent space of the manifold \( M = \{(x,y,z):z=\sqrt{y-x}\} \) at the point \((x,y,z) = (2,6,*)\).

Answer: x - y + 4z = 4


Question: Let \( f(x) \) be a real-valued function satisfying the functional equation \( f(x+y) = f(x) + f(y) - xy - 1 \) for all real numbers \( x \) and \( y \), with the condition \( f(1) = 1 \). Determine the number of natural numbers \( n \) for which \( f(n) = n \).

Answer: 1


Question: Determine the Galois group of the polynomial \( f(x) = x^5 - 6x + 3 \) over \( \mathbb{Q} \).

Answer: S_5


Question: Find the inverse of $1+2\alpha$ in the field $\mathbb{F}_9$, where $\mathbb{F}_9 = \frac{\mathbb{F}_3[x]}{(x^2+1)}$ and $\alpha = \bar{x}$. Compute $(1+2 \alpha)^{-1}$.

Answer: 2 + 2\alpha


Question: Consider a finite state space \(\mathbb{X}\) of an irreducible Markov chain \(\{X_n\}\). Let \(T_x=\inf\{k\geq 0\mid X_k=x\}\) be the number of steps until \(\{X_n\}\) reaches state \(x\in \mathbb{X}\). Determine whether the following statement is true or false: \(\mathbb{E}[T_x\mid X_0=x_0]<+\infty\) for all \(x_0\in \mathbb{X}.\)

Answer: True


Question: Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q \neq p$. Must there exist an intermediate extension $L$ (with $\mathbb{Q}_p \subset L \subset K$) of degree $q$ over $\mathbb{Q}_p$?

Answer: Yes


Question: Let $S$ be a set and $T$ be a family of subsets of $S$. Consider a function $f: S \to T$. Suppose $S_1 \in T$ with $S_1 \neq \emptyset$ and there exists $s_1 \in S_1$ such that $f(s_1) \neq \emptyset$. Define $S_2 = f(s_1)$. Assume that for every set $S_i$ obtained in this manner, there exists $s_i \in S_i$ such that $f(s_i) \neq \emptyset$. Does there exist a function $F: \mathbb{N} \to T$ such that $F(n) \neq \emptyset$ for all $n \in \mathbb{N}$?

Answer: Yes


Question: Determine whether the following statement is true or false: In the ordinary least squares (OLS) model $y = a + bx + u$, the heteroskedasticity-robust standard errors are consistent because $\hat{u}_i^2$ (the squared OLS residual) is a consistent estimator of $E(u_i^2|x_i)$ for each $i$. Provide a justification for your answer.

Answer: False


Question: Given a function $f(x)$ satisfying the equation \( f(x) + 2f\left(\frac{1}{1-x}\right) = x \), find the value of $f(2)$. Provide your answer as a single number.

Answer: \dfrac{2}{3}


Question: Let $u \neq 0$ and $v \neq 0$ be two vectors in a vector space with an inner product such that $\| u \| = \| v \| = \| u - v \|$. Determine the angle $\theta$ between $u$ and $v$.

Answer: \dfrac{\pi}{3}


Question:
Evaluate the limit using a Taylor series centered at 0:
\[ \lim_{x\to \infty} 4x^3 \left(e^{-\frac{2}{x^3}} - 1\right) \]

Answer: -8


Question: Determine the degree of the field extension \([\mathbb{Q}(a^2 + 2) : \mathbb{Q}]\) where \(a\) satisfies the polynomial equation \(\alpha^6 - 3 \alpha^3 - 6 = 0\). Use the tower law to find your answer.

Answer: 6


Question: Let $f(z)$ be an analytic function in the domain $D = \{z \in \Bbb C : |z| < 1\}$, with the condition that $f(z) = 1$ for $Im(z) = 0$ and $-\frac{1}{2} \leq Re(z) \leq \frac{1}{2}$. Determine the value of $f\left(\frac{1}{2}+i\frac{1}{2}\right)$. \(\text{Provide your answer in a box: } \boxed{\text{your answer}}\)

Answer: 1


Question: Evaluate the integral \( \left \langle x \right \rangle = \frac{2}{a}\int_{0}^{a}x\sin^{2}\left ( \frac{m \pi x}{a} \right )\,dx \) and verify that the solution is \( \frac{a}{2} \).

Answer: \dfrac{a}{2}


Question:
Determine the value of \( \alpha \) that makes the following piecewise function continuous at \((0,0)\):
\[
f(x,y)=
\begin{cases}
 \arctan\left(\dfrac{|x|+|y|}{x^2+y^2}\right)&\text{if}\, (x,y)\ne(0,0)\\
 \alpha&\text{if}\, (x,y)=(0,0)
\end{cases}
\]

Answer: \dfrac{\pi}{2}


Question: Find the value of \( \lambda \) such that the solution of the differential equation \( \frac{d^3y}{dx^3}-8\frac{d^2y}{dx^2}=0 \) satisfies the initial conditions \( y(0)=\frac{1}{8} \), \( y'(0)=0 \), and \( y''(0)=1 \), and is given by \( y =\frac{e^{8x}-8x+7}{\lambda} \).

Answer: 64


Question:
Consider the polynomial \(P(z)=\sum_{k=0}^n a_k z^k\) of degree \(n\), and let \(z_k\) (for \(1 \leq k \leq n\)) be the \(n\)th roots of \(-1\). Evaluate the truth of the following modified inequality for \(\theta = 0\):
\[|P'(e^{i\theta})| \leq \frac{4}{n} \sum_{k=1}^n \left|\frac{z_k}{(z_k-1)^2}\right| \max_{|z|=1}|P(z)|.\] Is this inequality true?

Answer: True


Question: Determine whether the following statement is true or false: If random variables $X_n \to X$ almost surely and $|X_n| \leq Y$ with $Y \in L^1$, then $X_n \to X$ in $L^1$. Provide a proof or counterexample.

Answer: True


Question: Evaluate the integral \( \int_{0}^{\pi/2} (\cos^{5}x)\sin (7x)\,dx \).

Answer: \dfrac{1}{6}


Question: Solve the equation $2^x + x = 11$ for $x$ without using graphing methods.

Answer: 3


Question: Does every connected graph $G$ have a spanning tree $T$ such that the domination number of $G$ equals the domination number of $T$?

Answer: Yes


Question: Find the smallest positive integer \( n \geq 2 \) such that the number of friendly binary sequences \( F_n \) with \( n \) terms is greater than 100.

Answer: 9


Question: If a sequence of polynomials converges uniformly on \( \mathbb{R} \) to a function \( f \), is \( f \) necessarily a polynomial?

Answer: Yes


Question:
Let \( f: \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = 8x^3 + 3x \). Given that \( f \) is continuous, strictly increasing, and surjective, compute the following limit:
\[ \lim_{x \to \infty} \frac{f^{-1}(8x) - f^{-1}(x)}{x^{1/3}}. \]

Answer: \dfrac{1}{2}


Question: Are there perfect extensions \( K \) of \( \mathbb{F}_p \) with \( \text{trdeg}_{\mathbb{F}_p} K > 0 \) that are not algebraically closed fields?

Answer: Yes


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function. Does there exist a sequence of polynomials that converges pointwise to \( f \) on \( \mathbb{R} \)?

Answer: Yes


Question: Provide an example of a finite non-abelian group $G$ such that the intersection of every non-trivial subgroup $H$ is another non-trivial subgroup $H_o$. Here, a trivial subgroup is defined as a subgroup consisting only of the identity element, $\langle e \rangle$. What is such a group?

Answer: Q_8


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

Answer: 1


Question: Determine whether there exists a real or complex number $x$ such that $\arcsin^2 x + \arccos^2 x = 1$. Assume $\arcsin$ and $\arccos$ are the compositional inverses of $\sin$ and $\cos$, respectively.

Answer: Yes


Question: Evaluate the integral \( \int_{|z|=2} ze^{\frac{3}{z}} \, dz \) using the Residue Theorem.

Answer: 9\pi i


Question: Determine the nature of the roots of the equation \( af(x) = (x^2+2)(a-1) \), where \( f(x) \equiv x^2 +4x + a +3 \) and the constant \( a \) is such that \( f(x) \) is never negative. Deduce the value of \( a \) for which this equation has equal roots.

Answer: 1


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

Answer: 2\pi \ln 2


Question: Evaluate the limit: \[ \lim \limits_{n \to \infty}\left( 1 + \sqrt{2} + \sqrt[3]{3} + \dots + \sqrt[n]{n} \right) \ln\left(\frac{2n+1}{n}\right) \]

Answer: \infty


Question: Given that $X$ and $Y$ have a joint normal distribution with unit variance, zero mean, and a correlation of $0.5$, calculate the probability $P(X > 2Y \mid X > 0)$. Provide your answer in terms of known functions or constants.

Answer: \dfrac{1}{2}


Question: Let $p$, $q$, and $r$ be prime numbers such that their product is 19 times their sum. Find the value of $p^2 + q^2 + r^2$.

Answer: 491


Question: Approximate the value of the integral \( \int_0^\infty \left(x^x \operatorname{mod} e^x\right)\frac{dx}{x^x} \).

Answer: 3


Question: Let $g^{ij}$ be a symmetric $(2,0)$ tensor and $F_{ij}$ be an antisymmetric $(0,2)$ tensor. Determine the value of the product $g^{ij}F_{ij}$. Is it always zero, always has the same sign, or does it depend on the specific tensors $g$ and $F$?

Answer: 0


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

Answer: 5 \ln 5


Question: Given a natural number \( p \geq 1 \), let \( C_0^p(\mathbb{R}) \) be the collection of all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that their \( i \)-th derivatives are continuous for all \( 0 \leq i \leq p \) and \( f \) vanishes at infinity. The support of \( f \) is defined as \( \text{supp}(f) = \overline{\{x \in \mathbb{R}: f(x) \neq 0 \}} \). Given any \( x_0 \in \mathbb{R} \) and any open set \( I \) containing \( x_0 \), does there exist a function \( f \in C_0^p(\mathbb{R}) \) such that \( \text{supp}(f) \subseteq I \), \( f(x_0) = 1 \), and \( \sup_{x \in \mathbb{R}} |f(x)| \leq 1 \)?

Answer: Yes


Question: Given a polynomial \(P(x)\) of degree \(n\) with algebraic coefficients, where each coefficient has a minimal polynomial of degree \(\le D\) over \(\mathbb{Q}\), determine an upper bound on the degree of the minimal polynomials over \(\mathbb{Q}\) for the roots of \(P(x)\).

Answer: n D^{n+1}


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

Answer: 2


Question: Determine whether the conjecture is true: The equivalence class of every nonempty proper subset of \(\mathbb{R}^2\) under the relation defined by translation has the cardinality of the continuum.

Answer: True


Question: Determine the type of singularity at the point $z = 0$ for the function $f(z) = \frac{1}{z} + e^{\frac{1}{z}}$. Provide your answer as either removable, pole, or essential.

Answer: essential


Question:
Solve the partial differential equation:
\[
y^2\frac{\partial ^2 u}{\partial x \, \partial y} + \frac{\partial ^2 u}{\partial y^2} - \frac{2}{y}\frac{\partial u}{\partial y} = 0
\]
subject to the boundary conditions:
\[
u(x,1) = 1 - x \quad \text{and} \quad \left. \frac{\partial u}{\partial y} \right|_{y = 1} = 3.\]

Answer: u(x, y) = y^3 - x


Question: Verify the equality \((x + \sqrt{2}j)(x - \sqrt{2}ij) = (x^2 + 2x + 2)\), where \(j = \sqrt{i}\).

Answer: x^2 + 2x + 2


Question: Evaluate the contour integral \( \int_{\gamma}\frac{z+1}{z(4z^2-1)^2}dz \) where \( \gamma(t) = i + \frac{1}{2}e^{it} \) for \( 0 \leq t \leq 2\pi \). The contour \( \gamma \) is the positively oriented circle about \( i \) with radius \( \frac{1}{2} \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\to +\infty}\left(\frac{f(a+\frac{1}{n})}{f(a)}\right)^{\frac{1}{n}} \] where \( f:\mathbb{R}\to \mathbb{R} \) is differentiable at \( a\in \mathbb{R} \) and \( f(a)>0 \).

Answer: 1


Question: Calculate the surface integral \( \int \mathbf{E} \cdot d\mathbf{S} \) of the vector field \( \mathbf{E} = (x+y^2)\mathbf{i} + (y^3+z^3)\mathbf{j} + (x+z^4)\mathbf{k} \) over the entire surface of a cube defined by the region \( 0 < x < 2, -1 < y < 1, 0 < z < 2 \). The outward normal vector \( d\mathbf{S} \) points outwards from the domain of integration. Choose the correct value of the integral from the following options: \( 0, 16, 72, 80, 32 \).

Answer: 80


Question: Calculate the length of the curve given by the parametric equation \( \gamma(t) = \left( \frac{t^2}{4}, \frac{t^3}{3}, \frac{t^4}{4} \right) \).

Answer: \dfrac{1}{2}


Question: Let \(K\) be the splitting field of \(x^3-1\) over \(F_{11}\). Determine the number of roots of the polynomial \((x^2-3)(x^3-3)\) in \(K\).

Answer: 5


Question: Determine whether the following statement is true: For all $a, b \in \mathbb{F}_9$, the product $a \cdot b$ is a square if and only if $a \cdot \overline{b}$ is a square.

Answer: True


Question: Determine the region of convergence for the sequence of functions \( f_n(x) = \frac{1}{n^2} \sum_{i=1}^n i^x \) where \( x \in \mathbb{R} \), such that \( f_n(x) \to 0 \) as \( n \to \infty \).

Answer: (-\infty, 1)


Question: Determine whether there exists a positive constant \( c \in \mathbb{R} \) such that for any nonzero vectors \( a, b \in \mathbb{R}^n \), the inequality \[ \frac{1}{2}\|a-b\|^2 - \frac{1}{\pi}\sqrt{\|a\|^2\|b\|^2 - \langle a,b\rangle^2} + \langle a,b\rangle\frac{\cos^{-1}(\rho)}{\pi} \geq c\|a-b\|^2 \] holds, where \( \rho = \frac{\langle a,b\rangle}{\|a\|\,\|b\|} \).

Answer: \dfrac{1}{4}


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

Answer: -\dfrac{1}{4}


Question: In triangle ABC, the lengths of the medians from vertices A, B, and C are 13, 14, and 15, respectively. Calculate the area of triangle ABC.

Answer: 112


Question: Determine whether there exists a polynomial \( q_2(x) \) such that \((x+1)f(x/2) = (x^{2}-x-2)q_{2}(x) + (-5x-5)\), given that \(f(x) = (x^{2}-1)q_{1}(x) + (2x-7)\) for some polynomial \(q_{1}(x)\).

Answer: Yes


Question: How many ways are there to distribute 20 different presents to 4 different children such that no child receives exactly 6 presents?

Answer: 523708416736


Question: In a metric space with a bounded metric and the Heine-Borel property, is every closed set compact?

Answer: Yes


Question: Find an expression for the coefficients of the polynomial given by the falling factorial $x^{\underline n} = x (x-1) \dots (x-n+1)$, which is a polynomial of degree $n$.

Answer: (-1)^{n - k} \left[{n \atop k}\right]


Question: Evaluate the integral \( \int_{C(0;2)} \frac{z^3}{z^2+i} \, dz \), where \( C \) is the circle centered at the origin with radius 2.

Answer: 2\pi


Question: Evaluate the integral \( \int_{1/3}^{3} \frac{\sin(\frac{1}{x} - x)}{x} \, dx \).

Answer: 0


Question: Find the coefficient $a_7$ in the expansion of the product of the generating functions \( \frac{2}{(1-x)^2} \) and \( \frac{x}{1-x-x^2} \).

Answer: 158


Question: Determine all values of \( \lambda \in \mathbb{R} \) such that \((1,0)\) is a regular point of the function \( f_\lambda = (x^2 + y^2 - z^2, x + y + z\lambda) \).

Answer: \mathbb{R}


Question: Let $K$ be a number field with $[K:\mathbf{Q}] = n$. Determine if the number of embeddings of $K$ into the algebraic closure of the rational numbers, $\overline{\mathbf{Q}}$, is equal to $n$. Provide a justification for your answer.

Answer: n


Question: Suppose a continuous map \( f: S^1 \to S^1 \) satisfies \(|f^{-1}(z)| = n\) for all \( z \in S^1 \). Is it true that \( f \) is a covering map of degree \( n \)? Justify your answer.

Answer: Yes


Question: Let $a$ be a positive real number such that $a^{a^{a^{16}}} = 16$. Determine the value of $a^{a^{a^{12}}}$. \( \boxed{?} \)

Answer: 2


Question: Determine whether the statement "A given non-terminal A in a given context-free grammar (CFG) is ever used in the generation of a word" is decidable or undecidable.

Answer: Decidable


Question: Determine whether the function \( f(x,y) = -\frac{2xy}{(\frac{3}{2}\sqrt{|y|}+1+x^2)} \) satisfies the local Lipschitz condition \( |f(x,y_2)-f(x,y_1)| \leq M|y_2-y_1| \) for \( x,y \in \mathbb{R} \).

Answer: Yes


Question: Can any 3x3 rotation matrix be represented as a linear combination of fewer than 9 constant basis matrices, such that \( R = \sum_{i = 1}^{n}\lambda_i B_i \) where \( n < 9 \), \( R \) is any 3x3 rotation matrix, \( B_i \) are constant basis matrices, and \( \lambda_i \) are scalar weights specific to each \( R \)?

Answer: No


Question: In the ring of infinitesimals, which is the quotient ring \(\mathbb{R}[x] / x^2\mathbb{R}[x]\), find the value of \((3+x)(3-x)\).

Answer: 9


Question:
Determine whether there exists a subspace $W$ of the vector space $(\mathbf{Z}/2\mathbf{Z})^{64}$ such that:
- The vector $e_1$ is in $W$.
- The dimension of $W$ is 58.
- All elements with exactly two nonzero coefficients are outside of $W$.

Provide a justification for your answer.

Answer: Yes


Question: Construct all semidirect products of the cyclic group $C_p$ by $C_p$, where $C_p$ is the cyclic group of prime order $p$. Determine the structure of these semidirect products.

Answer: C_p \times C_p


Question: Calculate the value of the complex number expression \( \frac{(1+i)^{10}}{(1-i\sqrt{3})^9} \).

Answer: -\dfrac{i}{16}


Question: Consider two Gaussian distributions, where one has a variance of zero, effectively making it a unit impulse function. Determine the Overlapping Coefficient between these two distributions.

Answer: 0


Question: Two balls, A and B, are dropped simultaneously from the same height. Ball A bounces back to 90% of its previous height after each bounce, while Ball B bounces back to 80% of its previous height. How many times will the two balls cross each other during their bounces?

Answer: 2


Question: Given the sequence defined by \(a_1 = 2\) and \(a_{n+1} = a_n^2 + a_n\), calculate the sum \(\sum_{n=1}^{\infty}{\frac{a_n}{a_{n+1}}}\).

Answer: \dfrac{1}{2}


Question: Determine the stabilizer of the given 7-dimensional cross-product in $GL(7,\mathbb{R})$. Is it the compact $G_2$ or some other real form of $G_2^{\mathbb{C}}$?

Answer: G_2


Question: Find the inverse function of the polynomial \( f(x) = x^5 + x^3 + x \), if it exists.

Answer: f^{-1}(x)


Question: Calculate the expected value of the maximum result when two six-sided dice are thrown.

Answer: \dfrac{161}{36}


Question: Given a chain of fields \( F \leq E \leq K \) where the characteristic of \( F \) is neither 2 nor 3, and \( E = F(x) \) with \([E : F] = 2\), suppose \( z \in K \) and \( a \in E \) such that the minimal polynomial of \( z \) over \( E \) is \( X^3 - a \). Can we find \( y \in K \) and \( b \in E \setminus F \) such that the minimal polynomial of \( y \) over \( E \) is \( X^3 - b \) and \( K = E(y) \)?

Answer: Yes


Question: Let $K/F$ be an infinite extension and $G = \mathrm{Aut}(K/F)$. Let $H$ be a subgroup of $G$ with finite index and $K^H$ be the fixed field of $H$. Is it true that $[K^H:F]= (G:H)$?

Answer: No


Question:
Suppose that \(X_1, X_2, \ldots, X_n\) are independent observations from a uniform distribution on \([0, \theta]\) where \(\theta\) is a natural number. Determine the maximum likelihood estimator of \(\theta\):

(A) It may not exist in some cases.
(B) It is \(\lfloor X_n \rfloor\).
(C) It is \(\lceil X_n \rceil\).
(D) It is \(\lfloor X_n \rfloor\) if \((X_n - \lfloor X_n \rfloor) \leq \frac{1}{2}\), otherwise it is \(\lceil X_n \rceil\).

Answer: C


Question: Evaluate the integral \( \int_0^{2\pi}\frac{1}{1+8\cos^2\theta}d\theta \). Use the hint: \( \cos\theta = \frac{z + z^{-1}}{2} \) with \(|z|=1\).

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


Question: Find the minimum integer \( n \) such that \[ \prod_{k=1}^n \frac{\arcsin\left( \frac{9k+2}{\sqrt{27k^3+54k^2+36k+8}}\right)}{\arctan \left( \frac{1}{\sqrt{3k+1}} \right)} > 2000. \]

Answer: 7


Question: Let \( X \) be an affine scheme with structure sheaf \( \mathcal{O}_{X} \), and let \( M \) be a finitely generated \( \Gamma(X,\mathcal{O}_{X}) \)-module with generators \( s_{1}, \ldots, s_{n} \). Consider the sheaf \( \widetilde{M} \) obtained from the tilde construction applied to \( M \). Is the sheaf \( \widetilde{M} \) finitely generated in the sense that for every open set \( U \subset X \), the sections \( s_{1}\rvert_{U}, \ldots, s_{n}\rvert_{U} \) generate \( \widetilde{M}(U) \) as a \( \Gamma(U,\mathcal{O}_{X}) \)-module?

Answer: Yes


Question: Solve the differential equation \( x^2y'' - xy' + y = 0 \) given that \( y_1 = x \) using the method of reduction of order, where \( y_2 = v(x)x \). Find the function \( v(x) \).

Answer: \ln x


Question: Find all values of $b > 5$ such that the expression \( x_n = \frac{b^{2n} + b^{n+1} + 3b - 5}{b-1} \) is a perfect square for all sufficiently large integers $n$.

Answer: 10


Question: Let \( \{a_i\} \) be real numbers and \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x \), where \( n \) is fixed. Determine if there exists a constant \( C \), independent of \( a_i \) and \( N \), such that \[ \left| \int_0^N \frac{\sin(P(x))}{x} \, dx \right| \le C. \]

Answer: C


Question: Let $X$ be a set, $\mathcal{F}$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given random variables $f,g\colon X\rightarrow\mathbb{R}$ such that \(\int_\mathbb{R} h \, d{\mu_f} = \int_{\mathbb{R}} h \, d\mu_g\) for any Schwartz function $h$, is it necessarily true that $\mu_f = \mu_g$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: Given two sequences $a_n$ and $b_n$, such that $\lim_{n\to\infty}b_n=0$ and $\lim_{n\to\infty}\frac{a_n}{b_n}$ exists, then $\lim_{n\to\infty}a_n=0.$

Answer: True


Question: Find the analytic function $f(z)$ in terms of $z$ for the given real-valued function $u(x,y) = e^y \cos(x)$. Express your answer without involving the conjugate $\overline{z}$. Use the Cauchy-Riemann equations and appropriate substitutions to determine $f(z)$. Provide your final answer in terms of $z$ only.

Answer: e^{-iz}


Question: Calculate the degree of the field extension \([F_{p^2}(t)(t^{1/p}):F_p(t)]\).

Answer: 2p


Question: Calculate the limit: $$\lim_{x \to \infty}(1+e^{-x})^x$$

Answer: 1


Question: Prove using the epsilon-delta definition that \( \lim_{x \to 4} (1-\sqrt{x}) = -1 \).

Answer: -1


Question: Find the residue of the function \( f(z) = \frac{1}{z^2 \sin z} \) at the point \( z_0 = 0 \).

Answer: \dfrac{1}{6}


Question: Find the value of \( \left\lfloor \sum_{r=1}^{80} \int_0^1 x^{\sqrt{r}-1} \, dx \right\rfloor \).

Answer: 16


Question: Can the fundamental group of a quasi-affine variety over \(\mathbb{C}\) be a torsion group?

Answer: Yes


Question: In the given pattern of numbers, each row \( n \) contains the number \( n \) multiple times. Determine \( a(n) \), the number of times the number \( n \) appears in the \( n \)-th row of the pattern.

Answer: n


Question: Given an infinite cardinal \(\kappa\), determine if there exists a graph on \(\kappa\) vertices that contains \(2^\kappa\) pairwise non-isomorphic induced subgraphs.

Answer: Yes


Question: Suppose $f(t)$ is a real-valued function on $\mathbb{R}^n$ that attains its minimum at a unique point $x^*$ in the closure of a set $E$. If $x^* \notin E$, does $f$ not attain its minimum value in $E$? Answer true or false.

Answer: A


Question: Solve the equation \(2^x - 3^{x-1}=-(x+2)^2\) for \(x\).

Answer: 5


Question: Find the equation of the common tangent to the curves \( y^2 = 8x \) and \( xy = -1 \).

Answer: y = x + 2


Question: Does there exist a nontrivial ultrafilter $a$ such that there is no sequence of sets $K_0, K_1, K_2, \ldots \in a$ for which $K_0 \cap K_1 \cap K_2 \cap \ldots = \emptyset$?

Answer: No


Question: Determine the number of homomorphisms from the alternating group $A_4$ to the cyclic group $\mathbb{Z}_{12}$.

Answer: 3


Question: Determine the number of integer solutions for \( n \) such that \( 0 \leq n < 23^2 \) and \( n^5 + 2n^4 + n^3 - 3n + 2 \equiv 0 \pmod{23^2} \).

Answer: 1


Question: Given that \(x\) and \(y\) are non-zero numbers satisfying \(x^2 + xy + y^2 = 0\), find the value of \[\left(\frac{x}{x+y} \right)^{2007} + \left(\frac{y}{x+y} \right)^{2007}.\] Choose from the following options: \(\text{(A)}\ 2, \text{(B)}\ 1, \text{(C)}\ 0, \text{(D)}\ -1, \text{(E)}\ -2.\)

Answer: E


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

Answer: \dfrac{1}{4}


Question: Determine the minimum number of values needed to describe the orientation of a 3-dimensional rigid body in n-dimensional space.

Answer: 3n - 6


Question: Determine all possible orders of elements in the alternating group $A_5$. Provide your answer as a set of integers.

Answer: \{1, 2, 3, 5\}


Question: In a separable Hilbert space $H$ with inner product $(\cdot, \cdot)$, does there exist a countable family $\{ w_n \} \subset H$ such that for any vector $u \in H$, the norm $||u||$ can be expressed as $||u|| = \sup_{n \in \mathbb{N}} (u, w_n)$?

Answer: Yes


Question: Given the limit \(\lim_{x\to 0}\frac{ax^2+\sin bx+\sin cx +\sin dx}{3x^2+5x^4+7x^6}=8\), where \(a, b, c,\) and \(d\) are real constants, find the value of \(a + b + c + d\).

Answer: 24


Question: Determine whether the equation \(x^2 = 2 = -1\) has solutions in any extension field of \(\Bbb F_3\).

Answer: Yes


Question: Determine the values of \(a\) for which the system \(X' = AX\), where \(A = \begin{pmatrix} a & 1 \\ 2a & 2 \end{pmatrix}\), undergoes a bifurcation.

Answer: -2


Question: Let \( p \) be a prime divisor of the \( n \)-th Fermat number \( F_n = 2^{2^n} + 1 \). If \( k \) is the multiplicative order of 2 modulo \( p \), prove that \( k \mid (p-1) \).

Answer: k \mid (p-1)


Question: Describe the induced group homomorphism \( f_{*} : \pi_1(S^1, p) \rightarrow \pi_1(S^1, f(p)) \) for the map \( f: S^1 \rightarrow S^1 \) given by \( z \mapsto z^2 \).

Answer: 2


Question: Determine whether the following statement is true or false: For every rational function \( P \in \mathbb{C}(x, y) = \{\frac{N}{D} : (N, D) \in \mathbb{C}[x, y]\} \), if \( P(\mathbb{Z}, \mathbb{Z}) = \{P(n, m) : (n, m) \in \mathbb{Z}^2\} \subset \mathbb{Z} \), then \( P \in \mathbb{Q}[x, y] \).

Answer: True


Question: Calculate the relative homology group $H_n(M, M - \{x\})$, where $M$ is a closed manifold and $x$ is a point in $M$. Provide your answer in terms of known homology groups or other topological invariants.

Answer: \mathbb{Z}


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

Answer: \dfrac{4}{e}


Question:
Let $c_1 = -i$ and $c_2 = 3$. Let $z_0$ be an arbitrary complex number. We rotate $z_0$ around $c_1$ by $\pi/4$ counter-clockwise to get $z_1$. We then rotate $z_1$ around $c_2$ by $\pi/4$ counter-clockwise to get $z_2$.

There exists a complex number $c$ such that we can get $z_2$ from $z_0$ by rotating around $c$ by $\pi/2$ counter-clockwise. Find the sum of the real and imaginary parts of $c$. \( \boxed{} \)

Answer: 2 - \sqrt{2}


Question: Given a stochastic process $B(t,x)$ such that for every fixed $x$, $\{B(t,x)\}_t$ is a Brownian motion, and a correlation $\rho_{xy}$ defined by \(dB(t,x)dB(t,y)=\rho_{xy} dt\), find the covariance $\text{cov}(B(t,x), B(t,y))$. Assume that the correlation structure is given by \(\text{corr}(dB(t,x),dB(t,y))=\rho_{xy}\).

Answer: \rho_{xy} t


Question: Find the angle at which the parametric curve \( \gamma(t) = \begin{bmatrix} t^2-1 \\ t(t^2-1) \end{bmatrix} \) intersects itself at the origin \((0,0)\). Use the formula for the angle between two parametric curves, where \( \dot{\gamma}(t) = \begin{bmatrix} 2t \\ 3t^2 -1 \end{bmatrix} \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit: $$ \lim_{x\to 0} \frac{\sqrt{x^3+2x+1}-\sqrt{x^2-3x+1}}{\sqrt{4x^2-3x+1} - \sqrt{2x^3+6x^2+5x+1}} $$

Answer: -\dfrac{5}{8}


Question: Let $C \subset \mathbb{P}_\mathbb{C}^2$ be a projective curve, and $p \in C$ a point of multiplicity $m$. If $\mathcal{O}_p(C)$ is the local ring of $C$ at $p$, and $\mathfrak{m} \subset \mathcal{O}_p(C)$ is its maximal ideal, find $\text{dim}_\mathbb{C} \mathfrak{m}^k/\mathfrak{m}^{k+1}$.

Answer: \min(k + 1, m)


Question: Let $a, b \in \Bbb C^n$ and $A \in \Bbb C^{n \times n}$. Given that $b^* \cdot a = 1$ and $r = b^* \cdot A \cdot a$, determine if the following statement is true: $r \cdot b^* = b^* A$.

Answer: False


Question:
Show that the function \( f(x) = \sum_{i=1}^n \prod_{j \ne i} \frac{x-a_j}{a_i-a_j} = 1 \) for \( n > 1 \) without simplifying the expression. Consider the following form:
\[
\begin{align}
\frac{(x − a_2)(x − a_3)\cdots(x − a_n)}{(a_1 − a_2)(a_1 − a_3)\cdots(a_1 − a_n)}  &+  \frac{(x − a_1)(x − a_3)\cdots(x − a_n)}{(a_2 − a_1)(a_2 − a_3)\cdots(a_2 − a_n)} \\
&+ \cdots \\
&+\frac{(x − a_1)(x − a_2)\cdots(x − a_{n-1})}{(a_n − a_2)(a_n − a_3)\cdots(a_n − a_{n-1})} \\
&= 1
\end{align}
\]

Answer: 1


Question: Let $f:[1-\epsilon,1] \to \mathbb{R}$ be a strictly decreasing, continuous, and strictly convex function with $f_-'(1)=0$, where $0<\epsilon<1$. Define $g:[\log(1-\epsilon),0] \to \mathbb{R}$ by $g(x)=f(e^x)$. Is $g$ convex on some half-neighbourhood of $0$?

Answer: Yes


Question:
Given integers \(n, k \ge 1\) and sequences \(a_i, c_i \ge 0\) for \(i = 1, 2, \ldots, n\), with the conditions:

1. \(a_1 \ge a_2 \ge \cdots \ge a_n\) and \(a_1 + a_2 + \cdots + a_n = 1\).
2. For each \(m = 1, 2, \ldots, n\), \(c_1 + c_2 + \cdots + c_m \le m^k\).

Find the maximum value of \(\sum_{i=1}^{n} c_i a_i^k\).

Answer: 1


Question: Find a polynomial \( f \in \mathbb{Q}[X] \) such that \((f) = (X^2-1, X^3+1)\).

Answer: X + 1


Question: Compute \(4^{22222} \mod 121\).

Answer: 16


Question: Evaluate the surface integral \( \iint z^2 \, dS \) over the unit sphere centered at the origin.

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


Question: Let \( G = S_5 \) be the symmetric group on 5 elements. Let \( H = \langle (1,2,3) \rangle \) be a subgroup of \( G \). Determine the number of subgroups of \( G \) that are conjugate to \( H \).

Answer: 10


Question: Determine whether there exists an infinite non-nilpotent group in which every maximal subgroup is a normal subgroup.

Answer: Yes


Question: Let \( f(x) = x^3 + 4x^2 + 6x \) and \( g(x) \) be its inverse. Find the value of \( g'(-4) \).

Answer: \dfrac{1}{2}


Question: Determine whether the limit \( e^{-x}\sum_{n=0}^N \frac{(-1)^n}{n!}x^n \to e^{-2x} \) as \( N\to\infty \) is uniform on \([0,+\infty)\).

Answer: Yes


Question: Find the limit: \[ \lim_{x \to \infty}\left(\sum_{n=1}^{\infty}\left(\frac{x}{n}\right)^{n}\right)^{\frac{1}{x}}. \]

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


Question: Evaluate the limit \( \lim_{x\to\infty}\frac{\ln x}{x}\int_2^x \frac{1}{\ln t}\,dt \) using the knowledge of Riemann Integrals and L'Hospital's Rule.

Answer: 1


Question: Evaluate the double integral \( \iint_{R} xy^2 \, dA \), where \( R \) is the region in the first quadrant bounded by the curves \( x = y^2 \) and \( y = x^2 \).

Answer: \dfrac{3}{56}


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \) and \( f(1) = 1 \), with \( |f'(x)| < 2 \) for all \( x \in \mathbb{R} \). If \( a \) and \( b \) are real numbers such that the set of possible values of \( \int_0^1 f(x) \, dx \) is the open interval \((a, b)\), find the value of \( b-a \).

Answer: \dfrac{3}{4}


Question:
Consider the sequence of functions defined by
\[
f_n(x) =
\begin{cases}
1-nx, & x \in [0,1/n],\\
0, & x \in [1/n,1]
\end{cases}
\]
Determine which of the following statements is correct:

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

Answer: 4


Question: Evaluate the double integral \( \iint_D x \, dx \, dy \) where \( D \) is a triangle with vertices \((0,2), (2,0), (3,3)\) using Green's theorem.

Answer: \dfrac{20}{3}


Question: Evaluate the limit: \[ \lim_{n\to \infty } \frac {1^3+4^3+7^3+...+(3n-2)^3}{[1+4+7+...+(3n-2)]^2} \]

Answer: 3


Question: Let $f : A \to B$ be a ring homomorphism. In the left $A$-module structure on $B$ induced by $f$, determine $\operatorname{Ann}(B)$. Assume $B$ is unital.

Answer: \ker f


Question: Determine the local order of convergence of the iteration procedure given by \( x_{n+1} = x_{n} - p \cdot \frac{f(x_{n})}{f'(x_{n})} \) for a function \( f \in C^{p+1}(\mathbb{R}) \) with a \( p \)-fold root at \( x^* \), where \( p \in \mathbb{N} \).

Answer: 2


Question: Given that \(2^{100} \equiv 1 \pmod{125}\) and \(2^{100}\) is divisible by \(8\), deduce that \(2^{100} \equiv 376 \pmod{1000}\).

Answer: 376


Question: Calculate the volume of the solid whose base is the region bounded by the curve \(3x^2 + y^2 = 6\). The cross-sections perpendicular to the \(x\)-axis are equilateral triangles with one leg on the base. Express your answer in terms of \(\sqrt{6}\).

Answer: 8\sqrt{6}


Question: Find all functions \( f: \mathbb{N} \rightarrow \mathbb{R} \) such that for a given value \( n \in \mathbb{N} \), the following identity holds for all \( m, k \in \mathbb{N} \) with \( mk > n \): \[ f(m+k) = f(mk-n) \]

Answer: f(x) = c


Question: Calculate the limit: $$\lim _{x\to \infty }\frac{x^2\cdot \left(1+\dfrac{1}{\ln^2x}\right)^{1+2\ln^2x}-x^2+1}{x-2+x^2}$$ without using L'Hôpital's rule.

Answer: e^2 - 1


Question: Find the radius of convergence of the series \( \sum_{k=1}^{\infty} \frac{(2k)(2k-2)\cdots 4\cdot 2}{(2k-1)(2k-3)\cdots 3\cdot 1} z^k \).

Answer: 1


Question: Given a matrix equation \(\mathbf{H}\mathbf{\Sigma}+\mathbf{\Sigma}\mathbf{H}=\mathbf{0}\), where \(\mathbf{\Sigma}\) is a real, positive, diagonal matrix and \(\mathbf{H}\) is a complex-valued Hermitian matrix, determine if \(\mathbf{H} = \mathbf{0}\) is the only solution.

Answer: \mathbf{H} = \mathbf{0}


Question: Suppose that \( \ell_\phi \) is a reflexive Orlicz sequence space such that its dual space \( \ell_\phi^* \) is isomorphic to \( \ell_\phi \). Is \( \ell_\phi \) isomorphic to \( \ell_2 \)?

Answer: Yes


Question: Evaluate the triple integral \( \int \int \int z \, dx \, dy \, dz \) over the region \( M \), where \( M = \{ (x, y, z) \mid x \ge 0, y \ge 0, z \ge \sqrt{x^2 + y^2}, x^2 + y^2 + z^2 \le 2x \} \).

Answer: \dfrac{\pi}{64}


Question: Determine which function grows faster as \( n \) becomes larger: \( f(n) = 2^{2^{2^n}} \) or \( g(n) = 100^{100^n} \).

Answer: f(n)


Question: Determine the number of non-conjugate $6 \times 6$ complex matrices with the characteristic polynomial $(x-5)^6=0$. Consider matrices that are not diagonalizable to the diagonal matrix $D$ with all diagonal entries equal to 5, but still have the same characteristic polynomial.

Answer: 10


Question: Evaluate the limit: \[ \lim_{(x,y)\rightarrow(0,0)} xy \log\left|y\right| \]

Answer: 0


Question: Let \( \Sigma \) be a connected surface, which may or may not be compact and may have a boundary. Determine whether it is true that \( \Sigma \) is homeomorphic to the sphere if it has an Euler characteristic \( \chi(\Sigma) \geq 2 \).

Answer: True


Question: Evaluate the limit: \[ \lim_{x \to 0} \left(x^2 \left(1 + 2 + 3 + \cdots + \left\lfloor \frac{1}{|x|} \right\rfloor \right)\right) \] where \([a]\) is the largest integer not greater than \(a\) and \(|x|\) is the absolute value of \(x\). Determine if the limit is undefined.

Answer: \dfrac{1}{2}


Question: Find the smallest integer value of \(m > 2\) such that \(m^3 - 3m^2 + 2m\) is divisible by both 79 and 83.

Answer: 1660


Question: A polynomial \( p(x) \) is called self-centered if it has integer coefficients and satisfies \( p(100) = 100 \). If \( p(x) \) is a self-centered polynomial, determine the maximum number of integer solutions \( k \) to the equation \( p(k) = k^3 \).

Answer: 10


Question: Determine the number of ordered pairs \((a, b)\) such that the equation \(a(\cos x - 1) + b^2 = \cos(ax + b^2) - 1\) holds true for all real numbers \(x\).

Answer: 2


Question:
Find all values of \( p > 1 \) such that the series \( \sum_{n=1}^{\infty} f_n \) converges in \( L^p(\mathbb{R}) \), where:
\[
f_n(x) = \frac{1}{1+n^2\sqrt{x}}\chi_{[\exp(2n),2\exp(2n+2)]}(x)
\]

Answer: p \geq 2


Question: Find a quartic polynomial in \( \mathbb{F}_2[x] \) whose roots are the four elements of the field \( \mathbb{F}_2[x]/(x^2+x+1) \).

Answer: x^4 + x


Question: Suppose \( \Phi: X^* \to \mathbb{C} \) is a linear map that is continuous with respect to the weak* topology on \( X^* \). Is it necessarily true that \( \Phi = \Phi_x \) for some \( x \in X \)? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: "The order of the wreath product of $\mathbf{Z}_{p^n}$ and $\mathbf{Z}_{p^m}$ is $p^{m+n}$."

Answer: B


Question:
Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) satisfying the equation:
\[ f(x) + f\left(\frac{x}{2}\right) = \frac{x}{2} \]
for all \( x \in \mathbb{R}^+ \).

Answer: \dfrac{x}{3}


Question: Given a function \( F(x,p):\mathbb{R}^n\times\mathbb{R}^n\to \mathbb{R} \), where for each \( p \in \mathbb{R}^n \), \( F(\cdot,p) \) is continuous, and there exists \( L>0 \) such that \( |F(x,p_1)-F(x,p_2)|\leq L|p_1-p_2| \) for all \( x,p_1,p_2 \in \mathbb{R}^n \). Is \( F \) continuous on \( \mathbb{R}^n\times\mathbb{R}^n \)?

Answer: Yes


Question: Consider the fixed-point equation \( x = (1-x)^{1-\frac{2}{a+1}} - 1 \), where \( x \in [0,1] \) and \( a \in [0,1] \). If the solution of \( x \) in terms of \( a \) is denoted as \( x(a) \), compute \( \lim_{a \rightarrow 1} x(a) \).

Answer: 0


Question: Find the infimum of the sequence \( y_n = \sum_{k=n}^{\infty} \frac{1}{k} \) for \( n \in \mathbb{N} \).

Answer: \infty


Question: Find a polynomial $f(x)$ such that the ring $\mathbb{Q}[x]/(f)$ is isomorphic to the set $R = \{ a + b\sqrt{2} : a,b \in \mathbb{Q} \}$. What is the polynomial $f(x)$?

Answer: x^2 - 2


Question: Determine the dimension of $H^0(T, \mathcal{L})$, where $T$ is a complex torus of dimension 2 and $\mathcal{L}$ is a line bundle that gives $T$ an $(m,n)$-polarization.

Answer: mn


Question:
Determine if the zero function is the only function that satisfies the following conditions for a function $f: \mathbb{R}^+ \to \mathbb{R}^+$:
1) $f$ is continuously differentiable on $(0, \infty)$.
2) $f(0) = 0$.
3) Both $f$ and its derivative $f'$ are non-decreasing.
4) There exists a positive constant $c < 1$ such that for all $x > 0$, $xf'(x) \leq cf(x)$.

Answer: Yes


Question: Determine the number of distinct cycle subgraphs of the complete bipartite graph $K_{3,3}$.

Answer: 15


Question: Let $f: \mathbf{D} \to \mathbf{C}$ be a holomorphic function on the unit disc such that $f(0) \neq 0$. Suppose $|f|$ is bounded from below by some real number $C > 0$ on some annulus contained in $\mathbf{D}$. Does it follow that $|f|$ is bounded from below on $\mathbf{D}$ by some positive real number $C'$?

Answer: No


Question: If an operator \( A \) has an inverse, is it true that \((A^{-1})^{-1} = A\)?

Answer: A


Question: Solve the recurrence relation \( t(n) = (2^n)(t(n/2))^2 \) with the initial condition \( t(1) = 1 \).

Answer: n^n


Question: Given the group \( G = C_{25} \times C_{45} \times C_{48} \times C_{150} \), where \( C_n \) denotes a cyclic group of order \( n \), determine the number of subgroups of order 5 in \( G \).

Answer: 31


Question: Calculate the circulation of the vector field \( F = (2x - 4y)i + (x + 3y)j \) around the circle \( C \) of radius 10 centered at the origin in the \( xy \)-plane, oriented clockwise as viewed from the positive \( z \)-axis, using Stokes' Theorem.

Answer: -500\pi


Question: Calculate the value of the infinite series: \[ \sum_{n=2}^{\infty}\frac{3n-5}{n(n^2-1)} \]

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 2} \frac{\sqrt[3]{x^{2}+4}-\sqrt{x+2}}{x-2} \] without using L'Hôpital's rule.

Answer: \dfrac{1}{12}


Question: Five students and their five parents want to stand in a line. For every arrangement of these 10 people, define 'adj' as the number of positions where a student is standing next to a parent (not necessarily their own parent). Calculate the average value of 'adj' over all possible permutations of these 10 people.

Answer: 5


Question: Calculate the integral \( \int_{-\pi}^\pi s(x) \, dx \), where \( s(x) = \sum_{k=1}^\infty a_k \cos(kx) \) and \( \sum_{k=1}^\infty a_k \) is an absolutely convergent series.

Answer: 0


Question: Determine the number of group homomorphisms from \(\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z\) to \(S_3\).

Answer: 10


Question: Let $X$ be a proper scheme defined over an algebraically closed field of characteristic $p > 0$. Let $F : X \rightarrow X$ be the absolute Frobenius morphism. Determine the dimension of $H^0(X, F_*\mathcal{O}_X)$. \( \text{Provide your answer as a single integer.} \)

Answer: 1


Question: Calculate the minimum value of \(\left|\frac{a^2 - bc}{b - c}\right| + \left|\frac{b^2 - ca}{c - a}\right| + \left|\frac{c^2 - ab}{a - b}\right|\) given that \(a, b, c\) are positive numbers such that \(abc = 1\).

Answer: 6


Question: Dan, Dominic, and Doug are waiting for calls from their girlfriends. Their waiting times (in hours) are independent exponential random variables with parameters 2.1, 3.7, and 5.5, respectively. What is the probability that the first call will arrive within the next 30 minutes?

Answer: 0.9965


Question: A square is inscribed in a circle with a unit radius. The area of the square is \(\frac{\pi}{12}\). Determine the measure of angle \(CAB\) in degrees.

Answer: 45


Question: If \( \sqrt{2 + (\sqrt{3} + \sqrt{5})} \) is a root of a polynomial of eighth degree, find the sum of the absolute values of the coefficients of this polynomial.

Answer: 93


Question: Prove that if \(n, k \in \mathbb{Z}^+\) and \(n+k\) divides \(n^2\), then \(k > \sqrt{n}\).

Answer: k > \sqrt{n}


Question: Calculate the limit: \[ \lim _{ x\to0^+}{x\int _{x}^{1}\frac{\cos(t)}{t^2}\,dt} \]

Answer: 1


Question: Find the expected value of a random variable with the probability distribution given by \( P(X = k) = \frac{1}{k^2-k} \) for \( k \in \{2, 3, 4, \ldots\} \).

Answer: \infty


Question: Given the differential equation $ty''(t)+2y'(t)+ty=0$ for $t>0$, with initial conditions $y(0+)=1$ and $y'(0+)=0$, find the value of $Y(1)$, where $Y(s)$ is the Laplace transform of $y(t)$. Assume that $Y(s) = -\tan^{-1}(s) + C$ and determine the constant $C$ to find $Y(1)$. Provide your answer as a single value.

Answer: \dfrac{\pi}{4}


Question:
Calculate the area of the loop in the Folium of Descartes, given by the parametric equations:

\[ x = \frac{3at}{1+t^3}, \quad y = \frac{3at^2}{1+t^3} \]

Use the formula for the area enclosed by a closed curve:

\[ \int_{t_{1}}^{t_2}\left[x(t)y'(t)-y(t)x'(t)\right]dt \]

Find the values of \( t_1 \) and \( t_2 \) where the curve intersects itself by solving the system:

\[ \frac{3at_1}{1+t_1^3} = \frac{3at_2}{1+t_2^3} \]
\[ \frac{3at_1^2}{1+t_1^3} = \frac{3at_2^2}{1+t_2^3} \]

Then, compute the area of the loop.

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


Question:
Consider the set \(A\) and the function \(f(x, y)\) defined as follows:
\[A = \{(x, y)\in\mathbb{R}^2; x > 0, y < e^{-1/x}\}\]
\[f(x, y) = \begin{cases} 2x & (x, y) \in A \\\\ y & (x, y) \not\in A \end{cases}\]
Determine the points on the boundary \(\partial A\) where the function \(f(x, y)\) is continuous.

Answer: (0, 0)


Question: Evaluate the double sum: \[ \sum_{k=2}^{\infty}\sum_{n=2}^{\infty}\frac{1}{k^n} \]

Answer: 1


Question: Let $X$ be a projective variety over an algebraically closed field $k$, and let $L_1, \dots, L_n$ be line bundles on $X$. Is there a very ample line bundle $M$ on $X$ such that $M \otimes L_i$ is very ample for each $i$ from 1 to $n$?

Answer: Yes


Question: Find the number of horizontal tangents to the function \( f(x) = xe^{-x^2-x^{-2}} \) for \( x \neq 0 \) and \( f(0) = 0 \).

Answer: 3


Question: Is there a norm space $X$ such that there exists a linear continuous functional $f$ with $\|f\|=1$, but there is no $x \in X$ such that $\|x\|=1$ and $f(x)=1$?

Answer: Yes


Question: Given that $X, Y, Z \sim N(0,1)$, $\text{corr}(X,Y) = \rho$, and $Y$ and $Z$ are independent, determine $\text{corr}(X,Z)$.

Answer: 0


Question: Find the volume of the solid obtained by rotating the region bounded by the curves \(xy = 1\), \(y = 0\), \(x = 1\), and \(x = 2\) about the line \(x = -1\).

Answer: 2\pi(1 + \ln 2)


Question: In an acute triangle \( \triangle ABC \), the bisector of angle \( C \) intersects side \( AB \) at point \( D \) and the circumcircle of the triangle at point \( E \). Given that \( \frac{CE}{DE} = \frac{(a+b)^2}{kc^2} \), find the value of \( k \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{k \to 0^{+}} \frac{(k+1)A-3kA+2A^3}{(k-A^2)^{\frac{3}{2}}} \] where \( A = -\sqrt{2} \frac{\Gamma(\frac{k+1}{2})}{\Gamma(\frac{k}{2})} \) and \( \Gamma \) is the Gamma function.

Answer: -\infty


Question: Determine whether the function \( f(x) = x + A e^{Bx} \), where \( A \) and \( B \) are known constants, is invertible.

Answer: AB \geq 0


Question:
Determine the weight function $w(x)$ such that the eigenfunctions corresponding to the distinct eigenvalues of the Sturm-Liouville problem \( y'' - 3y' = \lambda y \) are orthonormal. Choose the correct option:

(A) \( e^{-3x} \)
(B) \( e^{-2x} \)
(C) \( e^{2x} \)
(D) \( e^{3x} \)

Answer: A


Question: Determine the isomorphism class of the quotient group $D_{16} / Z(D_{16})$, where $D_{16}$ is the dihedral group of order 16.

Answer: D_4


Question: Find the points on the graph of the function \( g: \mathbb{N} \rightarrow \mathbb{R} \) for which both coordinates are natural numbers, where \( g(n) = \log_2(1 + f(n)) \). The function \( f: \mathbb{N} \rightarrow \mathbb{R} \) is defined by \( f(2) = \frac{3}{2} \) and \( 2f(n+1) = f(n) + n + 1 \) for all \( n \). Determine these points.

Answer: (1, 1)


Question: Determine the fundamental group of the space $X$, where $X$ is the unit square with its corners identified.

Answer: \mathbb{Z} \ast \mathbb{Z} \ast \mathbb{Z}


Question: Determine the 3-manifold obtained by performing surgery on the link $L_n = \{p_1, \dots, p_n\} \times S^1$ in $S^2 \times S^1$, where $p_1, \dots, p_n$ are points in $S^2$. Is the resulting manifold diffeomorphic to a known space such as $S^3$, $S^1 \times S^1 \times S^1$, or a lens space?

Answer: S^3


Question: Determine the degree of the continuous map \( \mathbb{S}^n \to \mathbb{S}^n \) induced by multiplication with an orthogonal matrix \( A \).

Answer: \det A


Question: Determine the number of homomorphisms from the group \( G = \langle x,y \mid x^{-1}yx = y^6 \rangle \) to the symmetric group \( S_5 \).

Answer: 240


Question: Calculate the value of the homology group $H_6(RP^3 \times RP^3; \mathbb{Z}_2)$ using the Kunneth formula. Verify whether the result is $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ or $\mathbb{Z}_2$.

Answer: \mathbb{Z}_2


Question: Let $f$ and $g$ be entire functions of $n$ complex variables such that for all $z \in \mathbb{C}^n$, $|f(z)| = |g(z)|$. Does this imply that $g = cf$ for some constant $c \in \mathbb{C}$ with $|c| = 1$?

Answer: Yes


Question: Given that \(x + y + z = 0\), \(x^3 + y^3 + z^3 = 3\), and \(x^4 + y^4 + z^4 = 15\), find the value of \(x^2 + y^2 + z^2\).

Answer: \sqrt{30}


Question: Given that the range of the function \( f(x) = \frac{x^2 + ax + b}{x^2 + 2x + 3} \) is \([-5, 4]\) and \(a, b \in \mathbb{N}\), find the value of \(a^2 + b^2\).

Answer: 277


Question: If $X$ and $Y$ are continuous random variables uniformly distributed over $[0,1]$, find the expected value $E(X^Y)$. Express your answer in terms of known constants or functions.

Answer: \ln 2


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

Answer: e


Question: Evaluate the integral \( \int_{-\pi/2}^{\pi/2} \frac{\cos^3 x}{e^x+1} \, dx \).

Answer: \dfrac{2}{3}


Question: Is it true that a subspace Y of a Banach space X, considered as a normed space, does not need to be complete?

Answer: Yes


Question: Is there a discrete non-archimedean valued field \( \mathbb{K} \) with an infinite residue field?

Answer: Yes


Question: Is the set $B_n$ of integer matrices $A \in \text{Mat}(n \times n, \mathbb{Z})$ such that the sequence $(A^k)_{k \in \mathbb{N}}$ is eventually periodic computable for every $n \in \mathbb{N}$?

Answer: Yes


Question: Evaluate the summation: \[ 2^{x-1}\sum_{y=x}^{\infty} \frac{y}{2^y} \] for \( y \ge x \ge 1 \).

Answer: x + 1


Question: Find the value of $A + B + C$ such that \[ S = \sum_{n=1}^{\infty} \frac{1}{n^2 \binom{2n}{n}} = \frac{A}{B} \zeta(C). \]

Answer: 6


Question:
Calculate the sum of the series given by:
\[ 2\sum_{i=1}^{48}\left(\frac{i(i+1)}{2}\cdot\sum_{n=1}^{i}\frac{n(n+1)}{2}\right) \]

Answer: 398273960


Question: Find the norm of the Volterra integral operator \( A: (C[0,1],\mathbb{R}) \rightarrow (C[0,1],\mathbb{R}) \) defined by \( (A\phi)(x) = \int_0^x \phi(y) \, dy \).

Answer: 1


Question: Does there exist a sequence of real numbers \((a_n)_{n\in\mathbb{N}}\) such that for any real number \(L\), there is a subsequence \(a_{k_n}\) with \(\lim_{n\to\infty} a_{k_n} = L\)?

Answer: Yes


Question: Find the expectation \( E[e^{iu(X_{4/5}-\frac{1}{2} X_{3/5})}|X_{3/5}] \) where \( X_t \) is a Brownian bridge.

Answer: e^{-u^2/20}


Question: Let $R$ be a commutative ring and $M$, $N$ be $R$-modules such that $M$ and $M \otimes_R N$ are finitely generated projective modules. Suppose that $\text{rk}_R(M) = \text{rk}_R(M \otimes_R N)$. Is $N$ necessarily a finitely generated projective module?

Answer: No


Question: Find the length of the short diagonal of a dodecahedron with side length $a$. The dodecahedron has $10$ long diagonals, $30$ medium diagonals, and $60$ short diagonals. The length of a long diagonal is $a\sqrt{1+\phi^{4}}$ and the length of a medium diagonal is $a\phi^{2}$. What is the length of a short diagonal?

Answer: a\phi


Question: Find the remainder when the expression \( (257^{33} + 46)^{26} \) is divided by 50.

Answer: 29


Question: Determine the range of values for the constant term \(a\) such that the polynomial \(3x^{4}-8x^{3}-6x^{2}+24x+a=0\) has four distinct real roots.

Answer: (-13, -8)


Question: Verify whether the integral of a function over the circle |z| = 2 is zero, given that the point Z = 4 is outside the circle.

Answer: 0


Question: Determine the complexity class for subsumption in the description logic \(\mathcal{AL}(\circ, ^{-})\). Is it in \(\mathrm{P}\) or does it belong to a different complexity class?

Answer: \mathrm{P}


Question: Determine whether the following statement is true or false: If $X$ and $Y$ are Banach spaces and $X \subset Y$, then $X^* \subset Y^*$. Justify your answer.

Answer: False


Question: Find the point in 2D space that minimizes the sum of the squared distances to the following two lines: the line passing through points \( (1, 3) \) and \( (2, 2) \), and the line passing through points \( (1, 1) \) and \( (2, 2) \).

Answer: (2, 2)


Question: Let \( X \sim N(2,4) \). Calculate the expectation \( \mathbb{E}(e^X) \).

Answer: e^4


Question: Given a Voronoi diagram created in \(\mathcal{O}(n)\), is it possible to find the closest pair of points in \(\mathcal{O}(n)\) time?

Answer: Yes


Question: Does there exist a vector space $V$ and a bilinear form $w$ on $V \oplus V$ such that $w$ is not identically zero but $w(x, x) = 0$ for every $x \in V$?

Answer: Yes


Question: Does there exist an ordered field \( \mathbb{F} \neq \mathbb{R} \) such that \( \mathbb{R} \) can be embedded into \( \mathbb{F} \)?

Answer: Yes


Question: Let $X$ be a Banach space and $X^*$ its dual space. Suppose $u \in X$ satisfies $\langle f, u \rangle_{X^* \times X} = 0$ for all $f \in X^*$. Is it necessarily true that $\|u\|_X = 0$?

Answer: 0


Question: Determine whether the integral \( \int_1^{\infty} \frac{e^{-\sqrt{x}}}{\sqrt{x}}\,\mathrm{d}x \) is convergent or divergent. If it is convergent, evaluate the integral.

Answer: \dfrac{2}{e}


Question: Find the derivative of $Ad(\exp(ty))(tH)$ with respect to $t$ at $t=0$, where $G$ is a Lie group, $Ad: G \rightarrow \text{End}(\mathfrak{g})$ is the adjoint representation, and $y, H \in \mathfrak{g}$. Assume that $ad$ is the derivative of $Ad$ at $t=0$. What is the value of this derivative?

Answer: H


Question:
Given that the graph of the function $f(x)$ is defined over a certain interval and each side of the quadrilateral in the figure is equal to $a$, suppose that the product of the integers satisfying the inequality $(f\circ f)(x)\times f'(x)<0$ is less than the largest element in the image set of the function $f(x)$. What is the value of the integral \( \int_{3a}^{5a}f(x)\mathrm {d}x \)?

A) 4 \quad B) 5 \quad C) 6 \quad D) 7 \quad E) 8

Answer: C


Question: Find the maximum value of the function \( F(x_1, x_2, \ldots, x_N) = - \sum_{i=1}^N x_i \log(x_i) \) subject to the constraint \( \sum_{i=1}^N x_i = 1 \) and \( x_i > 0 \) for all \( i \).

Answer: \log N


Question: Let $H$ be a complex infinite-dimensional Hilbert space, and let $T$ be a continuous linear functional on $H$. Determine whether there exists $v_0 \in H$ such that $\lVert T \rVert = \frac{\lvert T(v_0)\rvert}{\lVert v_0 \rVert}$.

Answer: Yes


Question: Determine whether the inequality \(1+\sqrt[3]{e^{2a}}\sqrt[5]{e^{b}}\sqrt[15]{e^{2c}} \leq \sqrt[3]{(1+e^{a})^2}\sqrt[5]{1+e^{b}}\sqrt[15]{(1+e^{c})^2}\) holds for all real numbers \(a, b, c\).

Answer: Yes


Question: Determine the VC dimension of a Support Vector Machine (SVM) using the polynomial kernel \(k(x,x')=(1+\langle x,x'\rangle_{\mathbb{R}^2})^2\) for binary classification in \(\mathbb{R}^2\).

Answer: 6


Question:
Solve the inequality for $z$:
$$(z+1) \left(1-e^x\right)-e^y \geq 0$$
subject to the conditions $-\infty <x\leq \log \left(\frac{1}{z+1}\right)$ and $-\infty <y\leq 0$.

Answer: [1, \infty)


Question: In the tensor product \( \mathbb{Z}/n\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z}/m\mathbb{Z} \), where \( m,n \neq \pm 1 \), does \( \bar{a} \otimes 1 = 0 \) imply that \( \bar{a} = 0 \)? Justify your answer.

Answer: No


Question: Determine the number of ways to partition a set of \(2n\) items into \(n\) pairs. Express your answer in terms of double factorials.

Answer: (2n-1)!!


Question: For an integer $x$, compute $y = x^2$. If $y \geq N$, set $y = y \mod N$. Assign the value of $y$ to $x$, and repeat the process until a duplicate $y$ is encountered. Determine the number of distinct values of $y$ encountered before a duplicate appears. For example, with $x = 26$ and $N = 1000$, the sequence of $y$ values is 676, 976, 576, 776, 176, and then 976 again, resulting in 5 distinct values. Can you find the number of distinct $y$ values, $Y$, without performing the entire process?

Answer: 5


Question: Determine whether the following statement is true or false: For any integer \( n \geq 2 \), there exists an \( n \times n \) real matrix \( A \) such that the set \( \{A^p \mid p \geq 1\} \) spans the real vector space \( M_n(\mathbb{R}) \).

Answer: False


Question: Find the function \( u(x, y) \) that satisfies the partial differential equation \( \frac{\partial^2 u}{\partial x^2} = 6xy \) with the boundary conditions \( u(0, y) = y \) and \( \frac{\partial u}{\partial x}(1, y) = 0 \).

Answer: y(x^3 - 3x + 1)


Question: Find the limit of the sequence \(x_n\) that satisfies the equation \(x_{n}^{n} + x_{n}^{2} + 2x_{n} - 1 = 0\) and converges to some \(l \in [0,1]\) for all \(n \geq 3\).

Answer: \sqrt{2} - 1


Question: Determine the multiple eigenvalue of the matrix \( A = \begin{bmatrix} 2 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 3 & 5 & -2 & -2 \\ -2 & 3 & 3 & 5 \\ \end{bmatrix} \).

Answer: -1


Question:
Determine whether the symmetrization of a function $f \in L^2(\mathbb{R})$ is equal to the orthogonal projection of $f$ onto the space of square integrable symmetric functions $\widehat{L}^2(\mathbb{R}^n)$. Consider the symmetrization defined by:
\[ \tilde{f}(x_1, \ldots, x_n) = \frac{1}{n!} \sum_{\sigma \in \mathcal{P}_n} f(x_{\sigma(1)}, \ldots, x_{\sigma(n)}). \]
Is $\tilde{f}$ the orthogonal projection of $f$ onto $L^2(\mathbb{R})^{\widehat{\otimes} n} \cong \widehat{L}^2(\mathbb{R}^n)$?

Answer: Yes


Question: Determine the number of positive integers \( n = 2^a 3^b \) such that \( n^6 \) does not divide \( 6^n \).

Answer: 6


Question: Evaluate the limit: $$\lim_{x\to 0^+} \frac{\ln(x+ \sqrt{x^2+1})}{\ln{(\cos{x})}}$$

Answer: -\infty


Question: Calculate the flux of the vector field \( \boldsymbol{F}(x,y,z) = \nabla \phi + \nabla \times \boldsymbol{G} \) out of the cylindrical surface of the region \( D = \{(x,y,z) : x^2 + y^2 \leq 1, 0 \leq z \leq 1\} \), where \( \phi(x,y,z) = x^2 - y^2 + \frac{3}{2\pi}z^2 \) and \( \boldsymbol{G}(x,y,z) = \frac{3}{\pi}zy\boldsymbol{i} + \arctan(x^2)\boldsymbol{j} + x^5 y \sqrt{z^2 +1}\boldsymbol{k} \). Use the divergence theorem and the identity \( \nabla \cdot(\nabla \times \boldsymbol{F}) = 0 \).

Answer: 3


Question: Compute the volume of the set $T_n = \{(x_1, x_2, \ldots, x_n) \mid x_i \ge 0, x_1 + x_2 + \cdots + x_n \le 1\}$ for any positive integer $n$.

Answer: \dfrac{1}{n!}


Question: Consider the set of real bounded sequences \(S_b:=\{(x_n)_n\in\mathbb{R}^{\mathbb{N}} \mid \sup_{n}|x_n|<\infty \}\) and an operator \(A=(a_{ij})_{i,j\in\mathbb{N}}\) such that for each fixed \(i\in\mathbb{N}\), there exists \(n_i\in\mathbb{N}\) with \(\sum_{j\in\mathbb{N}} a_{ij} = \sum_{k=1}^{n_i} a_{ij_k} = 0\). This means each row of \(A\) has only finitely many non-zero elements that sum to zero. Is the set \(S_b\) invariant under \(A\)? Provide a yes or no answer.

Answer: No


Question:
Let $X \subset \mathbb{R}$ and let $f, g: X \to X$ be continuous functions such that $f(X) \cap g(X) = \emptyset$ and $f(X) \cup g(X) = X$. Which one of the following sets cannot be equal to $X$?

A) $[0,1]$
B) $(0,1)$
C) $[0,1)$
D) $\mathbb{R}$

Answer: A


Question: Let $X_1, \ldots, X_n$ and $Y_1, \ldots, Y_m$ be mutually independent random variables. Each $X_i$ follows a normal distribution $N(\mu, \sigma_1^2)$, and each $Y_j$ follows a normal distribution $N(\mu, \sigma_2^2)$, where $\mu \in \mathbb{R}$, $\sigma_1^2 > 0$, and $\sigma_2^2 > 0$. Define $\tilde{X} = \frac{1}{n}(X_1 + X_2 + \ldots + X_n)$ and $\tilde{Y} = \frac{1}{m}(Y_1 + Y_2 + \ldots + Y_m)$. Find the probability $P(\tilde{X} \leq \tilde{Y})$. Assume $Z = \tilde{X} - \tilde{Y}$ is normally distributed and calculate $P(Z \leq 0)$. What is the value of this probability?

Answer: \dfrac{1}{2}


Question: Determine whether any conformal map \( f: D(0,1) \rightarrow D(0,1) \) with a non-zero derivative is necessarily a homography (Möbius transformation).

Answer: Yes


Question: Find the volume of the solid bounded by the surfaces $z = y^2$ and $z = 2 - x^2$. Provide the setup for the integral and calculate the volume.

Answer: 2\pi


Question: Solve the inequality: \( \tan^{-1}x > \cot^{-1}x \).

Answer: (1, \infty)


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy} - 1}{y^3}dy}{\ln(1+x)}. \]

Answer: -1


Question: Determine whether the series \( \sum_{n=0}^\infty \frac{n^3}{n!} \) is convergent. If it is convergent, find its sum.

Answer: 5e


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

Answer: \dfrac{4}{e}


Question: Compute the limit \( \lim\limits_{h\to0^+}{h\sum_{n=1}^\infty f(nh)} \) as an improper integral, where \( f(x)=\frac {\ln(x)}{1+x^2} \).

Answer: 0


Question: Find the minimum value of \( \sec A + \sec B + \sec C \) for an acute-angled triangle \( ABC \).

Answer: 6


Question: Compute the limit \( \lim\limits_{x\to 0}\frac{e^{x^2}-\cos{(2x)}-3x^2}{x^2\sin{x^2}} \) using Taylor series expansions.

Answer: -\dfrac{1}{6}


Question: Determine whether the element $(\bar{1}, \bar{1})$ is a generator of the group $\mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$, given that $m$ and $n$ are coprime and the group is isomorphic to $\mathbb{Z}/mn\mathbb{Z}$.

Answer: Yes


Question: Determine the values of $p$ for which the series \( \sum\limits_{n=3}^\infty\frac{1}{n\cdot\ln(n)\cdot\ln(\ln(n))^p} \) converges.

Answer: p > 1


Question: Determine the number of zeros of the polynomial \( z^{113} - 180z^{23} + 115z^{7} - 32z^2 - 3z - 10 \) within the annulus defined by \( 1 \leq |z| \leq 2 \).

Answer: 90


Question: Let $Z_1, \ldots, Z_n$ be independent and identically distributed standard normal random variables. Determine the constants $c_1, \ldots, c_n$ (not all zero) such that the random variable \( M = c_1Z_1 + \ldots + c_nZ_n \) is independent of \( Z_i - M \) for each \( i = 1, \ldots, n \).

Answer: \dfrac{1}{n}


Question:
Find the set of discontinuities of the function:
\[ f(x) = \lim_{n \to \infty} \left( \lim_{t \to 0} \frac{(\sin(n!\pi x))^2}{(\sin(n!\pi x))^2 + t^2} \right) \]

Answer: \mathbb{R}


Question:
Compute the reduced homology group \( H_1(\mathbb{R}^n, x) \) for \( x \in \mathbb{R}^n \) and \( n > 1 \). Use the exact sequence:
\[ H_1(\mathbb{R}^n) \to H_1(\mathbb{R}^n, x) \to \widetilde{H}_0(x) \to \widetilde{H}_0(\mathbb{R}^n) \]
Given that \( \widetilde{H}_p(\mathbb{R}^n) \cong 0 \) for all \( p \), determine if \( \widetilde{H}_0(x) \cong 0 \) and verify the computation of \( H_1(\mathbb{R}^n, x) \).

Answer: 0


Question: Evaluate the integral \( \int_c \frac{z^2}{z^3-2} \, dz \) where \( c: |z| = 3 \). Determine if the value of the integral is \( 0 \), \( -\pi i \), \( \pi i \), or \( 2\pi i \).

Answer: 2\pi i


Question:
Determine whether there exists a constant \( C \), independent of the real numbers \( a, b, \) and \( N \), such that the following inequality holds:
\[
\left| \int_{-N}^N \frac{e^{i(ax^2+bx)}-1}{x} \, dx \right| \le C.
\]

Answer: C


Question: Use Stokes' Theorem to evaluate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \), where \( \mathbf{F} = (-y^2, x, z^2) \) and \( C \) is the curve of intersection of the plane \( y+z=2 \) and the cylinder \( x^2+y^2=1 \). The curve \( C \) should be oriented anticlockwise when viewed from above.

Answer: \pi


Question: Find the greatest common divisor of the set \( S = \{ n^{13} - n \mid n \in \mathbb{Z} \} \).

Answer: 2730


Question: For what value of \( k > 1 \) is the integral \( \int_k^{k^2} \frac{1}{x} \log\frac{x-1}{32} \, \mathrm{d}x \) minimal?

Answer: 3


Question: For a point $P = (x, y)$ in the plane, define $f(P) = ax + by$, where $a$ and $b$ are given real numbers. Given that $f(A) = f(B) = 10$ for points $A$ and $B$, and $C$ is a point not on the line joining $A$ and $B$, let $C_0$ be the reflection of $C$ with respect to this line. If $f(C) = 15$, determine the value of $f(C_0)$. Provide your answer as a single number.

Answer: 5


Question: Calculate the volume of Gabriel's horn, which is formed by revolving the curve \( y = \frac{1}{x} \) for \( x \in [1, \infty) \) about the \( x \)-axis.

Answer: \pi


Question: Evaluate the limit: $$\lim_{x\to 14} \left(\left(x^2-30\cdot x+225\right)^\frac{1}{x^2-29\cdot x +210}\right)$$

Answer: e^2


Question: Find the smallest integer $n$ such that the cyclic group $C_{12}$ is a subgroup of the symmetric group $S_n$. Provide your answer as a single integer.

Answer: 7


Question: Evaluate the contour integral \( \oint_{\gamma}(2z-3\bar{z}+1)\,dz \) where \( \gamma \) is the ellipse \( \frac{x^{2}}{4}+\frac{y^{2}}{9}=1 \) traversed clockwise.

Answer: 36\pi i


Question: Given that $y(x)$ is a solution of the differential equation \[ x^2 y^{\prime \prime}+x y^{\prime}-4 y=x^2 \] on the interval $(0, \infty)$ such that $\lim _{x \rightarrow 0^{+}} y(x)$ exists and $y(1)=1$, find the value of $y^{\prime}(1)$. \[ y(x) = c_1 x^2 + c_2 \frac{1}{x^2} + \frac{\log x}{4} x^2 \]

Answer: \dfrac{9}{4}


Question:
Find the closed form of the sequence defined by the recurrence relation:
\[ a_{n+2} = 2a_{n+1} - a_n + 2^n + 2 \]
with initial conditions \( a_1 = 1 \) and \( a_2 = 4 \).

Answer: 2^n + n^2 - 2n


Question: Find the quadratic factor \( f(x) = x^2 + bx + c \), where \( b, c \in \mathbb{R} \), that is common to both quartic polynomials \( x^4 + 6x^2 + 25 \) and \( 3x^4 + 4x^2 + 28x + 5 \). Determine \( f(x) \).

Answer: x^2 - 2x + 5


Question: Evaluate the limit: \[ \lim_{x\to 0^+}\frac{n \left((1-x)^n+x-1\right)}{(1-x) \left((1-x)^n-(-x)^n-1\right)} \] for \( n > 2 \).

Answer: n - 1


Question: Evaluate the integral \( \int_{0}^{1} \left( \left[ \frac{2}{x} \right] - 2 \left[ \frac{1}{x} \right] \right) dx \), where \([x]\) denotes the greatest integer less than or equal to \(x\).

Answer: 2 \ln 2 - 1


Question: Evaluate the limit \( \lim_{\epsilon\to 0^+} \frac{3}{4\pi\epsilon^3}\oint_{S_\epsilon} F \cdot\hat{N}dS \) for the vector field \( F(x,y,z) = (yx^2, xyz, x^2y^2) \), where \( S_\epsilon \) is the sphere centered at \((0,0,0)\) with radius \( \epsilon \), and \( \hat{N} \) is the outward unit normal vector to \( S_\epsilon \).

Answer: 0


Question: Calculate \(\mathbb{E}X_{t}^{2}\) using Itō isometry, where \(X_{t}= \int_{0}^{t}B_{s}{\boldsymbol d}B_{s}\). Recall that \(\int_{0}^{t}B_{s}{\boldsymbol d}B_{s}= \frac{1}{2}B_{t}^{2}- \frac{t}{2}\) and \(\mathbb{E}\left [ B_{t}^{2} \right ]= t\).

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


Question: Find the volume of the region determined by the inequalities \(x>0, y>0, z<8\) and \(z>x^2+y^2\) using a triple integral. Use the limits: \(z\) varies from \(x^2+y^2\) to \(8\), \(x\) varies from \(0\) to \(\sqrt{8-y^2}\), and \(y\) varies from \(0\) to \(2\sqrt{2}\). Are these limits correct?

Answer: 8\pi


Question: A circle with radius 5 is centered at the origin (0, 0). From the points A(6, 8) and B(-4, -6), tangents are drawn to the circle. Let E and F be the points where the tangents from A touch the circle, and G and H be the points where the tangents from B touch the circle. Find the arc length along the circle between points E and F.

Answer: \dfrac{10\pi}{3}


Question: Given the functional equation \( f(x) + f(2x+1) = 6x + 10 \), find the value of \( f(4) \).

Answer: 12


Question: Find the residue of \( \frac{z^2}{(z-1)(z-2)(z-3)} \) at \( \infty \).

Answer: -1


Question: Evaluate the limit of the improper integral: \[ \lim_{x\rightarrow 0 } \dfrac{1}{x} \int_{x}^{2x} e^{-t^2}\,\mathrm dt \]

Answer: 1


Question: Let $A$ and $B$ be $n \times n$ Hermitian matrices with eigenvalues $\{\lambda_{A1}, \ldots, \lambda_{An}\}$ and $\{\lambda_{B1}, \ldots, \lambda_{Bn}\}$, respectively. Given that $A + B = \Lambda$, where $\Lambda$ is a diagonal matrix with diagonal elements $\{\lambda_1, \ldots, \lambda_n\}$ and $\lambda_{Bn} = 0$, determine the eigenvalue $\lambda_{An}$.

Answer: \lambda_n


Question:
Consider the quadratic equation \(x^2 + 2x = k + \int_0^1 |t + k| \, dt\). Determine the nature of the roots of this equation. Choose the correct option(s):

(A) Roots are Real
(B) Roots are Imaginary
(C) Roots are Distinct
(D) Roots are Complex Numbers

Answer: AC


Question: Given the function \( f(x) = \frac{1}{16}(e^{\arctan(\frac{x}{7})} + \frac{x}{7}) \), where \( f \) is one-to-one and its inverse \( f^{-1} \) is differentiable on \( \mathbb{R} \), and knowing that \( f(0) = \frac{1}{16} \), find the value of \((f^{-1})'(1/16)\).

Answer: 56


Question: If a matrix $A \in \mathbb{C}^{n \times n}$ satisfies $A^k \to 0$ as $k \to \infty$, does this imply that the spectral radius of $A$ is less than 1?

Answer: Yes


Question: Two people, A and B, arrive at random times within a 10-minute interval [0, 10], with A arriving before B. Assuming a uniform distribution, calculate the probability that B arrives in the last 5 minutes of the interval, i.e., within the time frame [5, 10].

Answer: \dfrac{3}{4}


Question: Let $K$ be a random variable with mean $\mathbf{E}[K]=e$ and variance $\text{var}(K)=v$. Let $X_1, X_2, \ldots$ be a sequence of independent identically distributed random variables, all independent of $K$, with mean $1$ and variance $1$. Let $X=\sum_{i=1}^K X_i$. Calculate $\text{var}(X)$. Provide your answer in terms of $e$ and $v$. \( \boxed{\text{var}(X) = e + v} \)

Answer: e + v