Question: Calculate the derivative of the energy function \( \psi(B) = \|B - J\|_F^2 \) with respect to the matrix \( B \), where \( J \) is a constant matrix.

Answer: 2(B - J)


Question: Evaluate the integral \( \int_{[0,1]} f \, d\mu \) using the Lebesgue integral, where \( d\mu \) is the Borel measure and \( f \) is defined as follows: \[ f(x) = \begin{cases} x & \text{if } x \in C, \\ 0 & \text{if } x \in [0,1] \setminus C, \end{cases} \] with \( C \) being the Cantor set. Given that \( \mu(C) = 0 \), determine the value of the integral.

Answer: 0


Question: Is it always possible to draw the edges of a planar graph without crossings, given arbitrary vertex positions in the plane?

Answer: Yes


Question: Determine the value of $k$ such that the determinant of the matrix \[\begin{vmatrix}(b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2\end{vmatrix}\] equals $k(abc)(a+b+c)^3$. Provide your answer as a single integer or fraction.

Answer: 2


Question: Are two matrices with identical characteristic and minimal polynomials but different ranks similar?

Answer: No


Question: Evaluate the integral \[ I=\frac{1}{2\pi i}\int_{\vert z \vert =R}(z+1)\cos\left(\frac{1}{z-1}\right)dz \] where \( R \geq 2 \).

Answer: -\dfrac{1}{2}


Question: Suppose $P$ is a poset and let $G$ be $P$-generic over $V$. If $M$ is a model of ZFC such that $V \subset M \subset V[G]$, does there exist a poset $Q \subset P$ such that $M = V[H]$ for some $H$ which is $Q$-generic over $V$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to \infty} x\left(e^{\frac{1}{x}}-1\right)^x \] without using L'Hopital's rule or Taylor expansion.

Answer: 0


Question: Is a bounded function \(f:[a,b] \rightarrow \mathbb{R}\) Riemann integrable if there exists a sequence of partitions \((P_{n})_{n}\) such that \(\lim_{n\rightarrow \infty} L(f,P_{n}) = \lim_{n \rightarrow \infty} U(f,P_{n}) = L\)?

Answer: Yes


Question: Determine whether the canonical morphism $X \to \mathrm{Spec}\;\Gamma(X,\mathcal{O}_X)$ is an isomorphism when $X$ is an affine scheme.

Answer: Yes


Question: Determine if the function \( f(x) = x^2 \) is Riemann-Stieltjes integrable with respect to \( g(x) = 3x \) on the interval \([0,1]\), given that \( g(1) = 4 \). If it is integrable, find the value of the integral.

Answer: 2


Question: Find the smallest positive integer $x$ that satisfies the following system of congruences: \[ 77x \equiv 1 \pmod{144} \] \[ 67x \equiv 3 \pmod{77} \]

Answer: 4997


Question: Evaluate the limit: $$ \lim_{x\to\infty}\left(\int_0^x e^{t^2}\,dt\right)^{\frac{1}{x^2}} $$

Answer: e


Question: Calculate the derivative of \( \log(\det((X+X^T)/2)) \) with respect to the matrix \( X \).

Answer: \left( \frac{X + X^T}{2} \right)^{-1}


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac{\sum^{n}_{k=1}(n-k+1)^d}{n^{d+1}}$$

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


Question: In a truth table with $n^2$ entries, an algorithm takes two rows, $a$ and $b$, and performs $s$ logical operations, resulting in Boolean values. If each iteration returns $c$ TRUE values, where $c > b - a + 1$, determine the number of additional operations required before selecting a new pair of rows.

Answer: c - (b - a + 1)


Question: Determine whether the statement is true or false: If \(T^2 = T\), then \(T\) must be the identity operator.

Answer: B


Question: Evaluate the limit: $$\lim\limits_{n\to\infty}n\left(n\ln{n}+\ln{\sqrt{2}}-n-\ln\left(\frac{(2n)!}{2^n n!}\right)\right)$$

Answer: -\infty


Question: Given two complex matrices $A$ and $B$, both symmetric and positive semi-definite, with $B$ having full rank, find a condition equivalent to $\operatorname{tr}(A^H B) = 0$. Provide your answer as a mathematical statement or condition.

Answer: A = 0


Question: Evaluate the limit \( \lim_{n\to \infty}\int_{0}^{1}nx^nf(x)\,dx \) for a continuous function \( f \) defined on the interval \([0,1]\).

Answer: f(1)


Question: Solve the equation \( \log_3(3-2\cdot3^{x+1})=2+2x \).

Answer: -1


Question: Determine the conditions on the fixed point (a, b) such that the mapping from the group of all polynomials with real coefficients under addition to itself, which assigns to each polynomial its antiderivative passing through (a, b), is a homomorphism.

Answer: (a, 0)


Question: Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed random variables, where $X_i > 0$ and $E[X_i] = \mu$, $\text{Var}(X_i) = \sigma^2$. Calculate $E\left(\frac{X_1 + X_2 + \cdots + X_n}{X_1 + X_2 + \cdots + X_m}\right)$, given that $m > n$. Provide your answer in terms of $\mu$, $\sigma^2$, $n$, and $m$. \(\text{Assume } m > n.\)

Answer: \dfrac{n}{m}


Question: Find all values of $a$ such that the equation $\cos(ax) = a \cdot \cos(x)$ holds true for every $x \in \mathbb{R}$. It is given that $a = 1$ is a solution. Are there any other values of $a$ that satisfy the equation?

Answer: 1


Question: Let $a$, $b$, and $c$ be positive real numbers such that \(\frac{a^3 + 2a}{a^2 + 1} + \frac{b^3 + 2b}{b^2 + 1} + \frac{c^3 + 2c}{c^2 + 1} = \frac{9}{2}.\) Determine if it is true that \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 3.\)

Answer: True


Question: Determine the supremum of the ratio $\frac{m}{n}$, where $m$ is the sum of the divisors of a positive integer $n$. Provide your answer as a specific value or expression.

Answer: \infty


Question: Solve the differential equation: $$\frac{yy''-y'^2}{y^2}+\frac{y'}{xy}=0$$.

Answer: y = C_1 x^{C_2}


Question: Evaluate the limit: $$\lim _{x \to 0} |x|^{\frac{1}{x^2}}$$ without using series expansions or L'Hôpital's rule.

Answer: 0


Question: Given that \(\frac{\sum\limits_{i=1}^n x_i}{n}\rightarrow 0\), does it imply that \(\frac{\sum\limits_{i=1}^n x_i^2}{n}\rightarrow 0\) if \(x_i\geq 0\)?

Answer: No


Question: Solve the differential equation using Laplace Transform: \[ y''(t) + 2y'(t) + 5y(t) = xf(t), \ y(0) = 1, y'(0) = 1 \] where \( x \) is a constant. Once the solution is found, evaluate \( \lim_{t \to \infty} y(t) \).

Answer: \dfrac{x}{5}


Question: Determine a sequence of real numbers $(\alpha_n)_n$ such that the process $(Y_n)_{n \geq 0}$, defined by $Y_n := S_n^3 + \alpha_n \cdot S_n$, is a martingale with respect to the natural filtration $(F_n)_{n \geq 0}$ of a simple symmetric random walk $(S_n)$ on $\mathbb{Z}$. Provide the sequence $(\alpha_n)_n$.

Answer: \alpha_n = -3n


Question: Calculate the value of $i^i$.

Answer: e^{-\pi/2}


Question: Find an integer $k$ such that $7^k \equiv 2^{12} \pmod{289}$, given that $7$ is a primitive root modulo $289$. \(\text{Express } k \text{ as a single integer.}\)

Answer: 120


Question: Find the possible minimal polynomials of a matrix $B$ such that $B^3 = I$ and $B$ has only $-1$ as an eigenvalue.

Answer: x + 1


Question: Let $a,b,c,d,e,f$ be real numbers such that the polynomial \( p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f \) factorizes into eight linear factors \(x-x_i\), with \(x_i>0\) for \(i=1,2,\ldots,8\). Determine all possible values of \(f\).

Answer: \dfrac{1}{256}


Question: Find a group $G$ and an element $a \in G$ such that $|a| = n$ for some positive integer $n$, and the centralizers of $a$ and $a^k$ are different for some positive integer $k < n$. Provide an example of such a group and element.

Answer: D_4


Question: Determine if the set \( L = \{ amk + bmh \mid k,h,m \in \mathbb{Z} \} \) is a cyclic group. If it is cyclic, identify its generator.

Answer: \gcd(a, b)


Question: Evaluate the improper integral: $$\int_{0}^{1}\ln(e^{1/x}+1)\ dx$$

Answer: \infty


Question: Find the sum of all integer values of $m$ that satisfy the equation $2^{m^2-8m+7}=3^{m^2-5m-14}$.

Answer: 7


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

Answer: \dfrac{1}{2}


Question:
Evaluate the following limit:

$$\lim_{n \to \infty} \sum_{a=2}^\infty \frac{\left(\int_0^1(1-x^a)^n dx\right)^{\frac{1}{n}}}{a!}$$

Answer: e - 2


Question: Find the norm of the linear operator $T$ on the Hilbert space $\ell^2$, given that for any $x \in \ell^2$, $$\|Tx\| = \sqrt{\sum_{i=1}^{\infty} \frac{x_i ^2}{i^2}}.$$

Answer: 1


Question: Calculate the limit: $$\lim_{n \to \infty }n\int_{0}^{\pi}\left \{ x \right \}^{n}dx$$

Answer: 3


Question: For a real matrix \( A \), is it possible to find a singular value decomposition (SVD) such that the matrices \( U \) and \( V \) are both orthogonal and real?

Answer: Yes


Question: Find the remainder when the polynomial $p(x)$, which is the remainder of $x^{2019} - 1$ divided by $x^6 + 1$, is divided by $x - 3$. Express your answer as a single number.

Answer: 26


Question: Evaluate the limit: $$\lim_{n \to \infty} \frac{n^n + n!}{2^n + (n + 1)^n}$$

Answer: \dfrac{1}{e}


Question: Let $f: [0, +\infty) \to \mathbb{R}$ be a continuous function. If the integral $\int_{0}^{+\infty}f(x)dx$ converges, does it imply that the series $\sum_{n\geq1}f(n + x)$ converges uniformly for $x \in [0, 1]$? Answer yes or no and justify your answer.

Answer: No


Question: Find the limit of the sequence \((a_n)_{n=0}^{\infty}\) defined by \(a_0=2\) and \(a_{n+1}=a_n+\left(1+\frac{a_n}{n}\right)^{1/2}\) for all \(n \ge 0\).

Answer: \infty


Question: Determine all values of $a$ such that the range of the function \( y=\frac{\sqrt{a}-2\cos x+1}{\sin^2x+a+2\sqrt{a}+1} \) includes the interval \([2, 3]\).

Answer: 0


Question: Calculate the length of the longest rod that can fit inside the given figure and rotate around a corner.

Answer: 2\sqrt{2}


Question: Evaluate the limit \( \lim_{n\to\infty}\int_{-\pi/4}^{\pi/4}\frac{n\cos(x)}{n^2x^2+1}\,dx \).

Answer: \pi


Question: Calculate the limit \( \displaystyle \lim_{x \to0^+} (\sin x)^{\large {\frac1 {\ln x}}} \) without using l'Hopital's rule, integration, or Taylor's theorem.

Answer: e


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

Answer: 3


Question: If $x$ and $y$ are two complex numbers such that the polynomial ring $\mathbb{Q}[x,y]$ forms a field, are $x$ and $y$ necessarily algebraic over $\mathbb{Q}$? Provide a justification for your answer.

Answer: Yes


Question: Determine the minimum number of steps required for a random walker, who moves with a 50% probability in either direction, to have at least a 99% probability of being at least 10 meters away from the starting point.

Answer: 640000


Question: Evaluate the limit \( \lim_{h \to 0} \frac{1}{h} \int_{t-h}^{t+h} f(s) \, ds \) for a continuous function \( f: \mathbb{R} \rightarrow \mathbb{R} \) and any \( t \in \mathbb{R} \). How does this limit relate to the antiderivative of \( f \)?

Answer: 2f(t)


Question: Determine whether a non-Lebesgue-measurable set $A \subset \mathbb{R}$, such that for all real numbers $a < b$, $A \cap [a, b]$ is also non-Lebesgue-measurable, is necessarily dense in $\mathbb{R}$. Provide a justification for your answer.

Answer: Yes


Question: Find the value of the limit \( \lim\limits_{x \to 0}\frac{2x^3+x^2+x\sin(x)} {(\exp(x)-1)^2} \) without directly applying L'Hôpital's rule.

Answer: 2


Question: Determine the degree of the field extension \( \mathbb{Q}(\sqrt[3]{2}, i, \sqrt{2}) \) over \( \mathbb{Q} \).

Answer: 12


Question: Find the smallest integer value of \( n \) such that the remainder \( R_{n}f(x) \) of the Taylor polynomial of degree \( n \) for \( \sin(x) \) around \( a=0 \) satisfies \( R_{n}f(x) \leq 0.01 \) for all \( x \in [0, 1/3] \).

Answer: 1


Question: In the expansion of \((2x+3y-4z+w)^9\), determine the coefficient of the term \(x^3y^2z^3\).

Answer: -23224320


Question: Find the value of the limit \(\lim_{n \to \infty} \frac{\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor + \ldots + \lfloor nx \rfloor}{n^2}\), where \(\lfloor . \rfloor\) represents the greatest integer function.

Answer: \dfrac{x}{2}


Question: In triangle $\Delta ABC$, the circumradius $R$ is 1. Given that the sum of the side lengths satisfies $AB + BC + AC \leq 2 + \sqrt{3}$, determine the maximum value of the product $P = AB \cdot BC \cdot AC$. Provide your answer as a single number.

Answer: \sqrt{3}


Question: Determine the total number of possible game situations in Connect Four on a $7 \times 6$ grid, considering that discs must be stacked on top of each other.

Answer: 127^7


Question: Evaluate the integral: $$\int_{-2}^2\frac{\sqrt{2-x}-3\sqrt{2+x}}{\sqrt{4-x^2}}\,dx$$

Answer: -8


Question: In an ergodic Markov chain with a finite state-space, is the long-term proportion of time spent in state $i$ equal to the probability of state $i$ in the stationary distribution?

Answer: Yes


Question:
Find the derivative of the function at $x = 0$ given by:
\[ f(x) = \begin{cases} e^{-\frac{1}{x^2}} \sin \frac{1}{x}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \]

Answer: 0


Question: Evaluate the limit of the integral: \[ \lim_{n\to\infty}\int_0^\infty\frac{n\cos^2(x/n)}{n+x^4}dx. \] Determine whether the integral converges to a finite value or diverges to infinity.

Answer: \infty


Question: Find the residue of the function \( z\cos\left(\frac{1}{z}\right) \) at \( z = 0 \).

Answer: -\dfrac{1}{2}


Question: Evaluate the nested integral: $$\int_0^{\int_0^{\int_{0}^{\int_0^{.^{.^{.^{.^.}}}}} 2x\,dx} 2x\,dx} 2x\,dx$$

Answer: 1


Question: Verify that the matrix $A = H - \frac{1}{n}J$ is idempotent, where $H = X(X'X)^{-1}X'$ is the hat matrix for multilinear regression and $J$ is the matrix of all 1's. Assume that $X$ includes a column of ones.

Answer: A


Question: Is it true that for a compact operator $A$ on a complex Hilbert space $H$, the inequality $A \leq |A|$ holds, meaning $\langle x, Ax \rangle \leq \langle x, |A|x \rangle$ for all $x \in H$?

Answer: No


Question: What is the probability that a random graph with $n$ vertices has no vertices with degree greater than 5, for sufficiently large $n$?

Answer: 0


Question: Find the derivative of the inverse function of \( f(x) = x^2 + x + e^x \) at \( x=1 \), if it exists.

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{0}^{2\pi}\frac{(x-\pi)|\sin (x-\pi)|}{1 + |\cos (x-\pi)|} dx \) using the properties of definite integrals.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0} {x\sqrt{y^2-(y-x)^2}\over (\sqrt{8xy-4x^2}-\sqrt{8xy})^3}$$

Answer: -\infty


Question: Is the set \( D:=\{ f \in C[0,1] \mid f \text{ is differentiable} \} \) dense in \((C[0,1],||.||_{\infty})\)?

Answer: Yes


Question: Find the derivative of the function defined by the series \( \sum_{n\geq 1}\frac{e^{nx}}{n!} \) and evaluate it at \( x=0 \).

Answer: e


Question: Evaluate the limit: $$ \lim_{x\to 1} \left(\frac{x+1}{x-1}\right)^{\sqrt{x^2-1}} $$

Answer: 1


Question: Suppose $f(x) = x^{2k} + a_{2k-2} x^{2k-2} + \dots + a_2 x^2 + a_0 \in \mathbb{R}[x]$ with $k \ge 2$. Determine the number of distinct real roots that $f(x)$ can have.

Answer: 2k


Question: Determine whether the following statement is true or false: If $f:\mathbb{R}^n\to\mathbb{R}^n$ is a continuously differentiable function with compact support, then the integral of the divergence of $f$ over $\mathbb{R}^n$ is 0.

Answer: True


Question: Given the equation \((x-1)g(y) + yg(-y) = y^3\), determine whether the function \(g(y)\) is odd or even, and find an expression for \(g(y)\).

Answer: -y^3


Question: Evaluate the limit: $$\lim_{x \to 0^+} \left(\ln\left(\cot(x)\right)\right)^{\tan(x)}$$

Answer: 1


Question: Evaluate the sum \( \sum_{n=1}^{32}(-1)^n \left\lfloor \dfrac{32}{n} \right\rfloor \varphi(n) \).

Answer: -156


Question: Determine whether there exist matrices of the form $M^k$, where $k$ is a positive integer greater than 2, that have an LU decomposition.

Answer: Yes


Question: Determine the time complexity of an algorithm that uses Stirling's approximation to compute the hyperfactorial function for large input values $m$.

Answer: O(1)


Question: Find the limit of the sequence \((S_n)\), where \(S_n = \sum_{i=1}^{3n-1} \frac{1}{\sqrt{n^2+i}}\).

Answer: 3


Question: Calculate \(2023^{20^{100800}} \mod 75\).

Answer: 1


Question: Evaluate the limit: $$\lim_{x\to\frac{\pi}{2}}\sqrt{\frac{\tan x-\sin\left(\tan^{-1}\left( \tan x\right)\right)}{\tan x+\cos^2(\tan x)}}$$

Answer: 1


Question: Given that $A$ and $B$ are matrices satisfying $AB^2 = BA$ and $A^4 = I$, determine the value of $B^{16}$.

Answer: B


Question: In a flock of pigeons, each pigeon dislikes exactly $k$ other pigeons. Determine the minimum number of groups required to arrange the pigeons such that no pigeon is in a group with any pigeon it dislikes.

Answer: k + 1


Question: Given the joint mass function of two random variables $X$ and $Y$: $$f(x,y)=\frac{C}{(x+y-1)(x+y)(x+y+1)}\quad \text{for} \quad x,y=1,2,3,\ldots$$ find the marginal mass function of $X$ and calculate the value of the constant $C$. Provide your answer for $C$ in a boxed form.

Answer: 2


Question: Evaluate the integral \( \int_{-\infty}^\infty f(\alpha)\delta(\beta-\alpha) \, d\alpha \).

Answer: f(\beta)


Question: Evaluate the integral \[ I = \frac{1}{\pi^2} \int_{0}^{\infty} \frac{(\ln{x})^2}{\sqrt{x}(1-x)^2} \, \mathrm{d}x \] using suitable substitutions and techniques.

Answer: 2


Question: Determine whether the vector field $F(x,y) = y^2e^{xy}\vec{i} + (1+xy)e^{xy}\vec{j}$ is conservative. If it is, find a function $f(x,y)$ such that $F = \nabla f$. Provide your answer as the function $f(x,y)$.

Answer: y e^{xy}


Question: Given a real matrix $M$ with $m$ rows and $n$ columns, where $n > m$, and $M$ has full row rank, is it always possible to augment the matrix with $n-m$ unique standard basis vectors such that the resulting $n \times n$ matrix has full rank? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the space $M_2(\Gamma_0(N))$ has positive dimensions for all integers $N > 1$. Provide a true or false answer.

Answer: True


Question: Determine the cardinality of the union of an uncountable collection of disjoint sets, each having the cardinality of the real numbers.

Answer: \mathfrak{c}


Question: Evaluate the limit: $$\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \ln \sin x } { \cos ^ { 2 } x }$$ without using L'Hopital's theorem.

Answer: -\dfrac{1}{2}


Question: Find the minimum value of the function $f(x, y, z) = x^2 + y^2 + z^2$ subject to the constraints $x^2 + 2y^2 = 1$ and $xy + xz = 2$.

Answer: \dfrac{9}{2}


Question: Evaluate the line integral \(\int_C \mathbf{F} \cdot \, \mathrm{d} \mathbf{r}\) where \(\mathbf{F} = (y^2 - xz) \mathbf{i} + (x^2 + yz) \mathbf{j} + (z^2 + xy) \mathbf{k}\) and \(C\) is the ellipse \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) oriented counterclockwise.

Answer: 0


Question: Given that $a + b + c = 1$, find the minimum value of the expression $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}$.

Answer: 27


Question: Evaluate the integral: $$ \int_0^{\pi/2} \log\left(\frac{1 + a\cos\left(x\right)}{1 - a\cos\left(x\right)}\right)\, \frac{1}{\cos\left(x\right)}\,\mathrm{d}x\,, \qquad\left\vert\,a\,\right\vert \le 1$$

Answer: \pi \arcsin a


Question: Determine the function $f(M)$ that represents the total number of vertices visible after $M$ steps on an N-dimensional hypercube, starting from the origin $(0,0,0,...,0)$. Additionally, find the minimum number of steps $m$ such that $f(m) = 2^N$, indicating all vertices are visible.

Answer: N


Question: Evaluate \( \lim_{x\to 2} \int^{f(x)}_6 \frac{4t^3}{x-2}\,dt \) given that \( f: \mathbb{R} \to \mathbb{R} \) is a differentiable function with \( f(2) = 6 \) and \( f'(2) = \frac{1}{48} \).

Answer: 18


Question: Determine whether a morphism between integral curves induces a map between the corresponding function fields.

Answer: Yes


Question: Find all continuous functions \( g: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the condition: for all real numbers \( x \) and natural numbers \( n \), \[ g(2^n x) + (-1)^n g(2^{n+1} x) = 0. \]

Answer: 0


Question:
Consider the function \(f(z)=\frac{\sin\left(\frac{\pi z}{2}\right)}{\sin(\pi z)}.\) At which values of \(z\) does \(f\) have poles? Choose the correct option:

A) All integers
B) All even integers
C) All odd integers
D) All integers of the form \(4k+1\), where \(k \in \mathbb{Z}\)

Answer: C


Question: Find the minimum value of the expression \( \frac{(a^2+b^2+c^2)^3}{(a+b+c)^3|(a-b)(b-c)(c-a)|} \) where \(a, b, c \ge 0\) are distinct real numbers.

Answer: \dfrac{125}{54}


Question:
Let $a,b,c \ge 0$ such that $a^2 + b^2 + c^2 + abc = 4$. Find the minimum value of the expression:
\[ Q = \sqrt{\frac{2b+ca}{3}} + \sqrt{\frac{2c+ab}{3}} + \sqrt{\frac{2a+bc}{3}} - \frac{3(a+b+c+abc)}{2}. \]

Answer: -3


Question: Determine the minimum degree of an irreducible polynomial over a field whose Galois group is isomorphic to the quaternion group of order 16, $Q_{16}$. Provide your answer as a single integer.

Answer: 8


Question: In a finite-dimensional vector space $V$ of dimension $n$, consider the space of all alternating $k$-fold multilinear maps, denoted by $\Lambda^k(V)$. Given that $\dim \Lambda^n(V) = 1$, determine the geometric significance of the scalar $\alpha$ in the equation $T(v_1, \ldots, v_n) = \alpha \cdot \det(v_1, \ldots, v_n)$, where $T$ is an $n$-fold alternating multilinear map defined by $\varphi : V \to \mathbb{R}$ as $T(v_1, \ldots, v_n) := \sum_{\sigma \in S_n} \text{sgn}(\sigma) \varphi(v_{\sigma(1)}) \cdots \varphi(v_{\sigma(n)})$. What does $\alpha$ represent geometrically?

Answer: 0


Question: Evaluate the line integral \( \oint_{C_1} \left(\frac{y\hat{i} - x\hat{j}}{x^2 + y^2}\right) \cdot d\vec{r} \), where \( C_1 \) is a circle of radius 3 centered at the origin in \( \mathbb{R}^2 \). Determine the result without relying solely on parametrization.

Answer: -2\pi


Question: Consider the initial value problem for the heat equation: \(u_{t} = u_{xx}, \quad x\in \mathbb{R}, \quad t > 0\) with the initial condition \(u(x,0) = \begin{cases} 1, & x\in[0,1] \\ 0, & \text{otherwise} \end{cases}\). Determine \(\lim_{t\to 0^+} u(1,t)\).

Answer: \dfrac{1}{2}


Question:
Evaluate the limit \( \lim_{k\to\infty} h_{2k}^2 \) for the recurrence relations given by:
\[
\begin{align*}
h_{2k+1} &= h_{2k} + h_{2k-1} \\
h_{2k} &= \frac{1}{\frac{1}{h_{2k-1}} + \frac{1}{h_{2k-2}}}
\end{align*}
\]
where \( h_0 = h_1 = 1 \).

Answer: 0


Question: Calculate \(314^{289} \mod 367\) using the Chinese remainder theorem.

Answer: 171


Question: In an algebraic poset, must every element be a compact element?

Answer: No


Question: Does a convex closed curve in the plane necessarily need to be simple, meaning it cannot intersect itself except at its endpoints? Answer yes or no.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{N \to \infty} \frac{\sum\limits_{k=1}^{d} {\varphi(N) \choose k} {d-1 \choose k-1}}{(\varphi(N)+d)^d} \] where \( \varphi(N) \) is a function that tends to infinity as \( N \) increases.

Answer: \dfrac{1}{d!}


Question: Determine whether the sequence \( v_n = \int_0^\pi x \cdot \cos(nx) \,dx \), where \( n \in \mathbb{N} \), converges.

Answer: 0


Question: Compute \(123^{456} \pmod{567}\) using the Chinese Remainder Theorem.

Answer: 162


Question: Evaluate the sum of the series: \( \sum_{n=1}^\infty \frac{n}{\sqrt{n^3+2}} \)

Answer: \infty


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

Answer: e^3


Question: Does there exist a function that has only countable noncontinuous points?

Answer: Yes


Question: For two independent Brownian motions \( W_t \) and \( \tilde{W}_t \), and a constant \( \rho \) such that \( -1 \leq \rho \leq 1 \), define \( X_t = \rho W_t + \sqrt{1-\rho^2} \tilde{W}_t \). Find the conditions on \( \rho \) for which \( X_t \) is a martingale.

Answer: -1 \leq \rho \leq 1


Question: Evaluate the Riemann-Stieltjes integral \( \int_0^2 x^3 \, dg(x) \), where \( g(x) = \begin{cases} 0, & x < 1 \\ 2, & x \ge 1 \end{cases} \).

Answer: 2


Question: Calculate the limit \( \lim_{n\to\infty} n^2 \int_{0}^{1} \frac{x\sin{x}}{1+(nx)^3} \, \mathrm{d}x \) using the dominated convergence theorem. Define the sequence of functions \( f_n(x) = \frac{n^2 x \sin{x}}{1+(nx)^3} \) and confirm that \( f_n \) is measurable. Determine the limit of \( f_n(x) \) as \( n \) approaches infinity.

Answer: 0


Question: Evaluate the integral \( \int_0^{\infty} e^{-y} y^3 \, dy \) using integration by parts and the Gamma function.

Answer: 6


Question: Let $A$ be a $5 \times 6$ matrix with real entries and rank 4. Consider the block matrix $D = \begin{pmatrix} I_5 & A \\ A^T & 0 \end{pmatrix}$. Determine the rank of the matrix $D$. \( \boxed{} \)

Answer: 9


Question: Determine the dimension of the vector space $W$ defined as: \[W=\{\psi :\mathbb{R}^{7}\to\mathbb{R}^5\text{ linear}: V\subseteq\ker\psi\},\] where $V\subseteq\mathbb{R}^{7}$ is a subspace of dimension $3$. Provide your answer as a single integer.

Answer: 20


Question: Given that $f(x) = \int x(x^2-a)^2 \, dx$ and $f(a) = 7$, find the value of $f(-a)$. \( \boxed{?} \)

Answer: 7


Question: If $X$ is a random variable with a given distribution and $Y=f(X)$ where $f$ is a deterministic function, find the mutual information $I(X;Y)$.

Answer: H(Y)


Question: Determine the radius of the circle that is tangent to the parabola \(y^2 = 8x\) at the point \((2, 4)\) and passes through the focus of the parabola.

Answer: 2\sqrt{2}


Question: Evaluate the surface integral \(\int_S (\nabla \times \vec{F}) \cdot d\vec{S} \) where \(\vec{F}(x, y, z) = (z, x, y)\) and \(S\) is the portion of the surface \(x^2 + y^2 + z = 1\) lying above the xy-plane, oriented upward.

Answer: \pi


Question: Determine the value of the polynomial \( f(x) \) of degree 5 with a leading coefficient of 5, given that it satisfies the conditions \( f(1)=1 \), \( f(2)=3 \), \( f(3)=5 \), \( f(4)=7 \), and \( f(5)=9 \). Find \( f(6) \).

Answer: 611


Question:
Given the following conditions:
1) \(\int_{0}^{\pi} [f(x) +f''(x)]\sin(x)\,dx = 2\)
2) \(f(\pi) = 1\)
3) \(f''(x)\) is continuous

Calculate the value of \(f(0)\).

Answer: 1


Question: Determine whether the function \( g(x) \) defined as \( g(x) = \arccos(1-x^2) \) for \( x \ge 0 \) and \( g(x) = -\arccos(1-x^2) \) for \( x < 0 \) can be expanded into a Maclaurin series.

Answer: Yes


Question: Consider the Ordinary Differential Equation (ODE) in $\mathbb{R}^n$ given by $x' = f(x, t)$, where $t \in \mathbb{R}$. Suppose $\gamma(t)$ is a solution to this ODE with the initial conditions $\gamma(0) = 0$ and $\gamma(1) = x_0$. Given a neighborhood $U_{x_0}$ around $x_0$, determine if there exists an $\epsilon > 0$ such that if $g(x, t)$ satisfies $|g(x, t) - f(x, t)| < \epsilon$ for all $x$ and $t$, then the solution $\eta(t)$ of the perturbed ODE $x' = g(x, t)$ with $\eta(0) = 0$ will result in $\eta(1) \in U_{x_0}$. Answer with 'True' or 'False'.

Answer: True


Question: Determine the cardinality of the set \( A = \bigcup_{n=1}^{\infty} \mathcal{P}(A_n) \), where \( A_1 = \{ 5 \} \) and \( A_{n+1} = \{ A_{n} \} \) for each \( n \in \mathbb{N} \).

Answer: \aleph_0


Question:
Given a symmetric, doubling, and strictly monotone increasing function $F: [0,1] \to [0,1]$ with the following properties:
1. $F(1) = 1$, $F(\frac{1}{2}) = \frac{1}{2}$, $F(0) = 0$
2. For all $n \in \mathbb{Z}$, if $(2^n \cdot x) \in [0,1]$, then $F(\frac{1}{2^n} \cdot x) = \frac{1}{2^n} \cdot F(x)$
3. Symmetry: $F(1-x) + F(x) = 1$ for all $x \in [0,1]$
4. $F$ is strictly increasing: $x>y \Rightarrow F(x)>F(y)$, $x=y \Rightarrow F(x)=F(y)$, $x<y \Rightarrow F(x)<F(y)$

Determine if $F(x) = x$ for all $x \in [0,1]$. Provide a justification for your answer.

Answer: F(x) = x


Question: Calculate the line integral \(\int_\gamma \mathbf{f} \cdot d\mathbf{x}\) for the vector field \(\mathbf{f}(x_1, x_2, x_3) = (x_1^2 + 5x_2 + 3x_2x_3, 5x_1 + 3x_1x_3 - 2, 3x_1x_2 - 4x_3)\), where \(\gamma\) is the helix with radius 1 and pitch 1 from \((0, 1, 0)\) to \((0, 1, 2\pi)\). Use the parametrization \(\gamma(t) = (\cos{t}, \sin{t}, t)\) for \(t\) in the interval \([0, 2\pi]\).

Answer: -8\pi^2


Question: Let $H$ be a Hilbert space with an orthonormal basis $\{e_k\}$. Consider an operator $A: D(A) \subset H \to H$ that generates a strongly continuous semigroup $e^{At}$, where $A$ satisfies $Ae_k = -\lambda_k e_k$ with eigenvalues $\lambda_k > 0$ such that $\inf_k \lambda_k = \lambda > 0$. According to the Hille-Yosida theorem, $|e^{At}| \leq M e^{-\lambda t}$ for some $M > 0$. Determine if it is possible for $M = 1$ such that $e^{At}$ forms a semigroup of contractions, i.e., $|e^{At}| \leq e^{-\lambda t}$ for all $t \geq 0$.

Answer: Yes


Question: Given a family of $R$-modules $\{ P_{i}\} _{i\in I}$, if the direct sum $\oplus_{i\in I}P_{i}$ is a projective module, does it imply that each module $P_{i}$ is projective for all $i\in I$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Determine the values of $m$ for which the integral \( \int_0^{\pi/2} \frac{1-\cos(x)}{x^m} \, dx \) converges.

Answer: (-\infty, 3)


Question: Given a closed subspace $W$ of $L^2[-1,1]$, where $W^{\perp} = \text{span}(x^3)$, and $Qk$ is the orthogonal projection of $k(x) = x^4$ onto $W$, find the expression for $(x^4 - Qk)(x)$ for $x \in [-1,1]$. \(Qk\) is the orthogonal projection of $x^4$ onto $W$, and $(x^4 - Qk)(x)$ is the component of $x^4$ orthogonal to $W$. Determine this expression.

Answer: 0


Question: Find the coefficient of \(x^{30}\) in the expansion of the generating function \(\frac{(1-x^{10})^4}{(1-x)^4}\) using the Negative Binomial Series.

Answer: 84


Question: In triangle $LMN$, the area is 72 square units. The medians of triangle $LMN$ are $LD$, $ME$, and $NF$. Triangle $PQR$ is constructed such that $PQ = LD$, $QR = ME$, and $PR = NF$. Find the area of triangle $PQR$. \( \boxed{54} \)

Answer: 54


Question: What is the common name for the set of formulas \(\Gamma = \left\{ \phi \wedge \psi \mid \phi \in \Sigma_n, \psi \in \Pi_n \right\}\) in the context of the Levy hierarchy?

Answer: \Delta_{n+1}


Question: Determine the values of \(\alpha \in \mathbb{R}\) for which the integral \(\int_{0}^{\infty} \frac{\sin u}{(u+1)^\alpha} \, du\) converges.

Answer: \alpha > 0


Question: Determine the cardinality of \( \bigcup_{\alpha \in I} E_{\alpha} \), where \( I \) is an uncountable directed set and \( T = \{E_{\alpha}:\alpha \in I\} \) is a collection of countably infinite sets such that \( E_{\alpha} \subseteq E_{\beta} \) whenever \( \alpha \leq \beta \), and \( T \) is totally ordered with respect to set inclusion.

Answer: \aleph_1


Question: In a Routley model, is it possible for there to be a world \( w \) such that both \( A \) and \( \neg A \) are true at \( w \)?

Answer: Yes


Question:
Find the value of \(\cot\left(\frac{5\pi}{12}\right)\) using the identities:

$$\cot(2x)=\frac{1}{2}\left(\cot(x)-\tan(x)\right)$$

$$\cot(\theta-\phi)=\frac{\cot\theta \cot\phi+1}{\cot\phi-\cot\theta}$$

Answer: 2 - \sqrt{3}


Question: Evaluate the integral: $$\int_{0}^{1} (x-1)\sqrt{1-x} \, dx$$

Answer: -\dfrac{2}{5}


Question: Assume $A, B \subsetneq \mathbb{R}^{n}$ ($n\ge 2$ arbitrary) are bounded connected sets with disjoint connected boundaries and $A\cap B\neq \emptyset$. Determine whether it is necessarily true that either $A \subsetneq B$ or $B \subsetneq A$.

Answer: Yes


Question: If a map \(f:\mathbb{C}_\infty \to \mathbb{C}_\infty\) preserves all cross ratios, is it necessarily a Möbius map? Provide a justification for your answer.

Answer: Yes


Question: Is a complex matrix $A$ of size $n$ diagonalizable if there exists a positive integer $k$ such that $A^k = I_n$?

Answer: Yes


Question: In triangle $ABC$, the inradius $r$ is 2, the circumradius $R$ is 4, and the semiperimeter $s$ is 6. Find the value of $ab + bc + ca$. Express your answer as a single number.

Answer: 72


Question:
Find the value of $x_0$ for the system of equations:
$$(3x)^{\ln{3}}=(2y)^{\ln{2}}$$
$$2^{\ln{x}}=3^{\ln{y}}$$

Answer: \dfrac{1}{3}


Question: Given the conditional density function \(f_{X\mid Y}(x\mid y) = \dfrac{3(x^2+y^2)}{3y^2+1}\) for \((x,y)\in(0;1)^2\), calculate \(\mathsf{P}(\tfrac{1}{4} < X < \tfrac{1}{2} \mid Y=\tfrac{3}{4})\).

Answer: \dfrac{17}{86}


Question: Determine the dimension of the subspace $V$ of $M_n(\mathbb{R})$, where $V$ consists of all $n \times n$ matrices such that the sum of the entries in each row and each column is zero.

Answer: (n-1)^2


Question: Find the limit of the sequence defined by \( a_n = \frac{1}{n} \sum_{k=1}^{n} |\cos(k)| \).

Answer: \dfrac{2}{\pi}


Question: Evaluate the improper integral \( \int_0^\infty \frac{x}{x^4 + 9} \, dx \).

Answer: \dfrac{\pi}{12}


Question: Calculate the volume of the region bounded by the surfaces $x^2 + z = 1$ and $y^2 + z = 1$ in the first octant.

Answer: \dfrac{1}{2}


Question: Determine the asymptotic bound of the recurrence relation \(T(n) = T(n/3) + T(n/2) + n^3\) using the guess and verify method.

Answer: \Theta(n^3)


Question: Determine whether every Riemann integrable function on a closed interval $[a, b]$ has discontinuities with zero Jordan measure.

Answer: No


Question: Determine the maximum value of \(|a| + |b| + |c| + |d|\) given that the polynomial \(p(x) = ax^3 + bx^2 + cx + d\) satisfies \(|p(x)| \leq 1\) for all \(|x| \leq 1\).

Answer: 7


Question: Let \(g: [0, \infty) \rightarrow \mathbb{R}\) be a continuous function. Define \[ G(s) = \int_{0}^{\frac{1}{s}} g(sx)\sin(s^2x) \, dx. \] Calculate \(\lim_{s \rightarrow \infty} sG(s)\).

Answer: 0


Question: Let $\{Y_n\}_{n \in \mathbb{N_0}}$ be a sequence of random variables with $\mathbb{P}(Y_0=0)=1$. Given that $Y_n$ is conditionally uniformly distributed on the interval $(Y_{n-1}, Y_{n-1}+1)$ given $\{Y_1=y_1,...,Y_{n-1}=y_{n-1}\}$ for all $n \in \mathbb{N}$, find the expected value $\mathbb{E}(Y_n)$.

Answer: \dfrac{n}{2}


Question:
Determine the radius of convergence for the power series:

$$ \sum_{n=1}^\infty \frac{x^{n^2}}{2^{n-1} n^n} $$

Answer: 1


Question: Given a sequence of independent and identically distributed (i.i.d.) uniform random variables $X_1, X_2, \dots$ on the interval $(0,1)$, let $N$ be the minimum value of $n \geq 2$ for which $X_n < X_{n-1}$. Calculate the expected value of $N$, denoted as $E(N)$. \( \text{Express your answer in terms of } e. \)

Answer: e


Question: Is a matrix diagonalizable if its minimal polynomial is a product of distinct linear factors?

Answer: Yes


Question:
Evaluate the second derivative of the integral at $t=0$:

$$\frac{d^2}{dt^2}\left(\left.\int_0^1 e^{tf(x)} \, dx\right)\right|_{t=0}$$

where $f(x)$ is defined as follows: Fix $n$. If $x \in (0,1]$ can be written in binary in non-terminating fashion as $0.a_1a_2a_3\ldots a_n\ldots$, then $f(x)$ is the number of $1$'s among $\{a_1,\ldots,a_n\}$ minus the number of $0$'s among the same set.

Answer: n


Question: Consider a sequence $\{f_n\}$ of measurable real-valued functions defined on the interval $[0, 1]$, which is uniformly bounded. Does there exist a subsequence $\{f_{n_j}\}$ such that the integral of $f_{n_j}$ over any Borel subset $A$ of $[0, 1]$ converges?

Answer: Yes


Question: Let $H$ be a separable Hilbert space, and consider a densely defined self-adjoint operator $T:{\frak Dom}(T) \to H$. If $\mathrm{spec}(T)$ denotes the spectrum of $T$, and $f:\mathrm{spec}(T) \to \mathbb{C}$ is a bounded continuous function, we can use the function calculus to define a bounded operator $f(T):H \to H$. Suppose $T$ is diagonalizable with an orthonormal basis $\{b_k\}_{k \in \mathbb{N}_0}$ satisfying $T(b_k) = c_k b_k$ for real numbers $c_k$ and all $k$. Is $f(T)$ the unique bounded operator on $H$ that acts on the basis elements as $f(T)(b_k) = f(c_k)b_k$ for all $k$?

Answer: Yes


Question: Determine whether the reciprocal function, \( f(x) = \frac{1}{x} \), transforms a set of measure zero in the interval \((0, 1)\) to another set of measure zero in its image.

Answer: Yes


Question: Determine whether a group can have two distinct characteristic Sylow-$p$-subgroups. Answer true or false.

Answer: False


Question: Find a number $a$, with $a < 2000$, that is not divisible by 7 and satisfies the congruence $a \equiv 7^{3003} \pmod{m}$, given that $\phi(m) = 1000$. Use Euler's Theorem, which implies $7^{1000} \equiv 1 \pmod{m}$, to solve this without a calculator.

Answer: 1454


Question: A differentiable function $g$ on the interval $[0,1]$ satisfies $g(g(x))=x$ and $g(1)=0$. Calculate the value of the integral \( \int_{0}^{1} (g(x)-x)^{2017} \, dx \).

Answer: 0


Question:
Given that $x$, $y$, and $z$ satisfy the following equations:

$$x+y+z=1$$

$$x^2+y^2+z^2=2$$

$$x^3+y^3+z^3=3$$

Find the value of $x^4+y^4+z^4$. \(\boxed{\ }\)

Answer: \dfrac{25}{6}


Question: Suppose $N_t$ is a Poisson process and $X_t$ is a continuous process of finite variation. Determine whether the quadratic covariation $\langle X, N \rangle_t$ is equal to zero.

Answer: 0


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

Answer: -\dfrac{1}{4}


Question: Evaluate the integral \(\int_{-\infty}^{+\infty} \frac{b}{1+2ax+(cx)^2} \cdot \frac{dx}{1-2ax+(cx)^2},\) given that \(a^2+b^2=c^2\).

Answer: \dfrac{\pi}{2}


Question: Find the directional derivative of the function \( f(X) = AX \) in the direction of the matrix \( U \), where \( U, A \in \mathbb{R}^{n \times n} \).

Answer: AU


Question: Determine whether the limit \( \lim_{n\to \infty }\sum_{i=0}^{m_n-1}o(\Delta x_i^n)=0 \) holds for a subdivision \( \{x_i^n\}_{i=0}^{m_n-1} \) of \([0,1]\) such that \( \max_{i=1,...,m_n-1}\Delta x_i^n\to 0 \).

Answer: 0


Question: Evaluate the integral \( \int_{Q'} \frac{xy}{x^2+y^2} \, dx \, dy \) where \( Q' = \{(x,y) \mid x^2+y^2 < b^2\} \) with \( b < 1 \).

Answer: 0


Question: Given a matrix $A$ that is not a scalar multiple of the identity matrix, determine whether there always exists a matrix $B$ such that the $(1,1)$ element of $BAB^{-1}$ is $0$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the infinite sum: $$ \sum_{n=1}^{\infty} \arctan\left(\frac{2}{1+4n^{2}-4n}\right). $$

Answer: \dfrac{\pi}{2}


Question: Given the differential equation \(y^{(4)}+a_3y^{(3)}+a_2y''+a_1y'+a_0y=0\), where \(3te^{-t}+e^{-4t}\sin(t)\) is a solution, determine the value of \(a_0\).

Answer: 17


Question: Let $V$ be a finite-dimensional vector space over a field $F$ with dimension $n$, and consider a basis $\{\alpha_1, \alpha_2, \ldots, \alpha_n\}$ for $V$. Given $r$ subspaces $\{W_1, W_2, \ldots, W_r\}$ of $V$ and their prescribed images $\{U_1, U_2, \ldots, U_r\}$ (also subspaces), determine the minimum number of subspaces whose images are sufficient to guarantee a unique vector space isomorphism $\varphi: V \rightarrow V$. Assume the conjecture is that this number is $n$, the dimension of $V$. What is the minimum number?

Answer: n


Question: Consider a meromorphic function $g$ in a region containing the unit disk. Suppose $g$ has a simple pole at $z=2$ and no other poles in the region. If the Maclaurin series of $g$ is $\sum_{n=0}^{\infty}b_nz^n$, determine the value of $\lim_{n\to\infty}b_n$. \(\text{Express your answer as a number or state if it does not exist.}\)

Answer: 0


Question: Evaluate the integral: \[ u = \int_0^\infty \frac{dx}{x^4 + 7x^2 + 1} \]

Answer: \dfrac{\pi}{6}


Question: Given a positive real sequence $\{x_n\}$, if $\liminf_{n \to \infty} x_n < x$ for some real number $x$, does there exist a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $x_{n_k} \leq x$ for all $k \geq 1$? Provide a proof or counterexample.

Answer: Yes


Question: Under ideal conditions of uniform gravitational acceleration and no friction, what shape would the top ends of a set of straight ramps form if they were arranged such that they all ended at the same base point and took an equal amount of time for a point-like body to slide down from the top to the base?

Answer: a circle


Question: Consider the metric space $(X, d)$ where $X = \{ (x_1,x_2,\cdots): x_i \in \mathbb{R} \text{ and only finitely many } x_i\text{'s are non-zero}\}$ and the metric $d: X \times X \to \mathbb{R}$ is defined by \[d(x,y) = \sup_{i \in \mathbb{N}}|x_i-y_i|\] for $x=(x_1,x_2,\cdots)$ and $y=(y_1,y_2,\cdots)$. Determine which of the following statements is true: \(P\): $(X,d)$ is a complete metric space. \(Q\): The set $\{x\in X: d(\bar{0},x) \leq 1\}$ is compact, where $\bar{0}$ denotes the zero element of $X$. Choose the correct option: (a) both $P$ and $Q$ (b) $P$ only (c) $Q$ only (d) neither $P$ nor $Q$. \[\boxed{}\]

Answer: D


Question: Evaluate the surface integral \(\iint_S F \cdot n \, dS\), where \(F(x,y,z) = (x,y,z)\) and \(S\) is the surface defined by \(x^2 - y^2 + z^2 = 0\) with \(y \in [0, 1]\). Here, \(n\) is the unit normal vector to the surface.

Answer: 0


Question: Evaluate \( \bigtriangleup ^{6} p(x) \big| _{x=0} \), where \( p(x) = \sum_{k=0}^{10} k^{2} x^{ \underline{k} } \).

Answer: 25920


Question: Given the sum \( \sum_{j=1}^M c_M(j) \), where \( \lim_{M\to \infty}c_M(j)=0 \) and there exists a sequence \( d_j \ge 0 \) with \( \sum_{j=1}^\infty d_j < \infty \) such that \( |c_M(j)| \le d_j \) for all \( j \) and \( M \), determine if the sum converges to zero as \( M\to \infty \).

Answer: 0


Question: Evaluate the limit: \[ \lim _{x\to 0^+}\left(\left(\left(1+x\right)^x-1\right)^x\right) \] as \( x \) approaches 0 from the positive numbers.

Answer: 1


Question: Determine whether there exists an arc $\overset{\frown}{AB}$ in the unit circle $\mathbb{S}^{1}$ that connects two disjoint compact sets $F_{1}$ and $F_{2}$, such that $A \in F_{1}$, $B \in F_{2}$, and the open arc $\overset{\frown}{AB}$ contains no points from $F_{1} \cup F_{2}$. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Given a finite abelian group $G$ with order $|G| = p_1^{n_1} \cdot \ldots \cdot p_m^{n_m}$, where $p_1, \ldots, p_m$ are distinct primes, determine if $G$ necessarily has a subgroup of order $p_1^{n_1}$ without using Sylow theorems. Consider the structure theorem for finitely generated abelian groups.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty} \left(n \cdot \sqrt[n]{\sin\left(\frac{1}{1}\right) \cdot \sin\left(\frac{1}{2}\right) \cdot \ldots \cdot \sin\left(\frac{1}{n}\right)}\right)$$ without using continuity, Taylor series expansion, or L'Hôpital's rule.

Answer: e


Question: Given that $G(s)$ is the Mellin transform of $g(x)$ which converges for $\Re(s)>\beta>0$, and $K(s)$ is a meromorphic function on $\Re(s)>0$ with simple poles at $\Re(s)=\beta$. If $G(s)=K(s)$ for $\Re(s)>\beta$, does the meromorphic continuation of $G(s)$ have poles on $\Re(s)=\beta$?

Answer: Yes


Question: Find the value of \( \alpha \) such that the field extension \( \mathbb{Q}(i,\sqrt[3]{2}) \) is equal to \( \mathbb{Q}(\alpha) \), where \( i \) is the imaginary unit and \( \sqrt[3]{2} \) is the cube root of 2.

Answer: i + \sqrt[3]{2}


Question: Calculate \( \lim_{x \to 0^+} x^{x^{x-1}} \).

Answer: 0


Question: Evaluate the improper integral: $$ \int_a^b \frac{dt}{\sqrt{(b-t)(t-a)}} $$

Answer: \pi


Question: Compute the integral \( \iint_S \vec{F} \cdot \vec{n} \ dS \) using Gauss's divergence theorem, where \( \vec{n} \) is the outward normal, \( S \) is the exterior surface of the cylinder defined by \( x^2 + y^2 \leq 1 \) for \( 0 \leq z \leq 1 \), and \( \vec{F} (x, y, z) = (x^2y, z - xy^2, z^2) \).

Answer: \pi


Question:
Given distinct prime numbers $p$ and $q$, find the number of positive integer solutions to the equation:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{pq} \]

Answer: 9


Question: Verify the claim that for an even integer $n$, the number of $(n-1)$-cycles in the symmetric group $S_n$ is given by $\frac{n!}{n-1} = n(n-2)!$. Provide a detailed explanation or proof.

Answer: n(n-2)!


Question: Evaluate the triple product: $$\left ( \frac{\partial y}{\partial x}\right )_z \left ( \frac{\partial z}{\partial y}\right )_x\left ( \frac{\partial x}{\partial z}\right )_y$$ given that $x$, $y$, and $z$ are implicit functions of each other.

Answer: -1


Question: Calculate the limit: $$\lim_{n \to \infty } \frac{\sum_{k=1}^{n} \left(\frac{1}{k}\right)^{1/k}}{n}$$

Answer: 1


Question: Find all differentiable functions \( f \) such that for all \( x, y \in \mathbb{R} \), the equation \( f(x^4 + y) = x^3 f(x) + f(y) \) holds.

Answer: f(x) = kx


Question: Determine whether the set of all $2 \times 2$ Hermitian matrices forms a 4-dimensional real vector space that is isomorphic to $\mathbb{R}^4$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether every element in the set $G = \mathbb{R}$ has an inverse under the operation $*$ defined by $x * y = x + y + x^3 y^3$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the linear mapping \(T: \ell^{1}(\mathbb{R})\rightarrow\mathbb{R}\) defined by \[T({x_n})=\sum_{n=1}^{\infty}{x_n}/(n^2)\] is bounded. If it is bounded, find its operator norm.

Answer: 1


Question: If an isomorphism maps a subgroup $H$ to a subgroup isomorphic to $H$, does this imply that $H$ is a normal subgroup?

Answer: No


Question: Is there a field that is isomorphic to \(\mathbb{Q}\)? If so, provide an example or construct such a field.

Answer: \mathbb{Q}


Question: Determine whether there exists a convex program with a zero duality gap that does not satisfy Slater's condition.

Answer: Yes


Question: Evaluate the limit: \( \lim_{x \to \infty} x^2\ln\left(x\cot^{-1}(x)\right) \).

Answer: -\dfrac{1}{3}


Question: Evaluate the limit: $$\lim_{(x,y,z)\to (0,0,0)} \frac{(\arctan(x)-x)(1-\cos(y))\sin(z)}{(x^2+y^2+z^2)^2}$$

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^\infty g(u-t) \, dt \) where \( g(u)= \begin{cases} 1& \text{if}\quad -1\leq u\leq0\\ 0 & \text{otherwise}. \end{cases} \)

Answer: 1


Question: Determine whether the following statement is true or false: If $A$ is a dense subset of a metric space $X$ and $B$ is an open subset of $X$, then every point in $B$ is a limit point of $A \cap B$. Provide a justification for your answer.

Answer: B


Question: Determine the Lebesgue measure of the set $A$ of all real numbers in the closed interval $[0,1]$ whose decimal representation contains an infinite number of sevens.

Answer: 1


Question: Consider the unit circle $S^1$ in the Euclidean plane $\Bbb{R}^2$ and divide it into pairs of antipodal points. The resulting identification space is denoted as $P$. Determine a space that is homeomorphic to $P$ and provide a homeomorphism or an explanation for your answer.

Answer: S^1


Question: Given that \(\phi=\frac{1+\sqrt{5}}{2}\), let \[ n=\frac{1}{1}+\frac{1}{1+\phi}+\frac{1}{1+\phi+\phi^2}+\frac{1}{1+\phi+\phi^2+\phi^3}+\dots \] Find the value of \(\lfloor2n\rfloor+\lceil2n\rceil\).

Answer: 7


Question: Determine the maximum possible value of \( \lvert x-y \rvert \) for non-negative integers \( x \) and \( y \) that satisfy the equation \( 2^{x} + 2^{y} = x^{2} + y^{2} \).

Answer: 3


Question: Compute the integral \( \int_C \frac{z}{1-e^z}dz \), where \( C \) is the union of two circles of radius 1 and radius 2 centered at 0, oriented in opposite directions.

Answer: 0


Question: Given the ideal \( I = (2, 1 + \sqrt{-5}) \subset \mathbb{Z}[\sqrt{-5}] \) generated by 2 and \( 1 + \sqrt{-5} \), explicitly find the set \( II = \{xy \mid x, y \in I\} \).

Answer: (2)


Question: Find the value of \( \sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{14-6\sqrt{5}}}} \).

Answer: 2


Question: Determine whether the inequality \(2\mathbb{E}[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)\) holds for any real-valued random variable \(X\).

Answer: No


Question: Determine the number of real roots of the equation \(2\cos\left(\frac{x^2+x}{6}\right) = 2^x + 2^{-x}\).

Answer: 1


Question: Evaluate the limit: $$\lim_{n \to +\infty}{n \left(\arctan\frac{1}{\sqrt n}\right)^{2n}}$$

Answer: 0


Question: Determine the number of elements in the quotient ring \( \mathbb{Z}[x]/(x^2+5,2) \).

Answer: 4


Question: Is it true that for $N=p=1$, we have $W^{1,p}(I) \subset C^0(I)$, but this is not true for $N=p=2,3,4,...$?

Answer: True


Question: Evaluate the integral \( \int \int_R x \, dx \, dy \) where \( R \) is the region defined in polar coordinates by \( 0 \leq r \leq \cos \theta \) and \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \).

Answer: \dfrac{\pi}{8}


Question: Find a nonzero solution \((x, y, z)\) in the finite field \(\mathbb{F}_7\) for the equation \(2x^2 + y^2 + 2yz + 2z^2 = 0\).

Answer: (1, 1, 1)


Question: Consider the set \( Y:=\{x\in[0,1]|(0.x_{1}x_{2}...),x_{i}\in\{0,1\},x_{2j+1}=1 \text{ for some } j\in\mathbb{N}\} \). Determine the measure of \( Y \).

Answer: 1


Question: A line $L_1$ with slope $9$ is parallel to line $L_2$. Another line $L_3$ with slope $-\frac{1}{25}$ is parallel to line $L_4$. All these lines are tangent to the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$. Calculate the area of the parallelogram formed by these lines.

Answer: 60


Question: What is the order of the automorphism group $Aut(P)$ for a group $P$ of prime order?

Answer: p-1


Question:
Determine whether the function \( g: \mathbb{R}^3 \to \mathbb{R} \) defined by
\[
g(\mathbf{y}) = \dfrac{y_1y_2y_3}{y_1^2+y_2^4+y_3^2} \quad \text{if } \mathbf{y} \neq \mathbf{0}
\]
\[g(\mathbf{0}) = 0\]
has a limit of 0 as \( \mathbf{y} \to \mathbf{0} \).

Answer: 0


Question: Determine whether there exists a non-continuous function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(f(f(x))) = x \) for all \( x \in \mathbb{R} \).

Answer: Yes


Question: In \(\Delta ABC\), it is given that \(AB' = 2AB\), \(CC' = 2BC\), and \(AA' = 3CA\). If the area of \(\Delta ABC\) is 1, find the area of \(\Delta A'B'C'\).

Answer: 18


Question: Determine whether the sequence of functions $f_n(x) = \cos\left(\frac{x}{n}\right)$ converges to the constant function $g(x) = 1$ in measure on the interval $X = [0, \pi]$ with respect to the Lebesgue measure $\mu$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the series \( \sum_{n=1}^{\infty}\frac{1}{n^k+1} \) converges to \( \frac{1}{2} \) as \( k \) approaches infinity.

Answer: \frac{1}{2}


Question: Determine whether there exists a divergent sequence \(a_n\) such that the sequence \(b_n = \frac{\ln(-2a_n - 1)}{a_n + 1}\) converges, given that the function \(f(x) = \frac{\ln(-2x - 1)}{x + 1}\) is strictly decreasing.

Answer: Yes


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

Answer: \dfrac{\pi}{2}


Question: Given three random variables $X$, $Y$, and $Z$, if $X$ is independent of $Z$ ($X\perp Z$) and $Y$ is independent of $Z$ ($Y\perp Z$), does the independence of $X$ and $Y$ ($X\perp Y$) necessarily imply their conditional independence given $Z$ ($X\perp Y | Z$)? Answer 'yes' or 'no' and justify your answer.

Answer: no


Question: Determine whether the conjecture that the order of 3 in the multiplicative group of \( \mathbb{Z}_{10^k} \) divides \( 10^k \) for all \( k \geq 2 \) is true.

Answer: True


Question: Evaluate the expression \( S = \frac{A^3-B^3}{B^3}+\frac{A^4+B^4}{B^4} \) where \( A=\int_0^1(1-x^{2023})^{\frac{1}{2022}}dx \) and \( B=\int_0^1(1-x^{2022})^{\frac{1}{2023}}dx \).

Answer: 2


Question: Determine the number of ways to completely cover a $2 \times 16$ rectangle using $2 \times 2$, $2 \times 3$, and $2 \times 4$ rectangles.

Answer: 165


Question: Is the localization of a commutative ring $R$ by a left and right Ore set $S$, denoted as $S^{-1}R$, always a flat $R$-module?

Answer: Yes


Question: Let $X_1, X_2, \ldots, X_n$ be i.i.d. random variables with a continuous probability distribution. Calculate the probability that $X_1$ is the second largest value among $X_1, X_2, \ldots, X_n$. Express your answer in terms of $n$.

Answer: \dfrac{1}{n}


Question: In a non-cyclic group of 9 elements, does every element (except the identity) have an order of 3?

Answer: Yes


Question: Compute the flux of the vector field \( F = \langle x, y, 1 \rangle \) through the surface of the paraboloid \( z = x^2 + y^2 \) bounded by the plane \( z = 1 \) using the Divergence Theorem.

Answer: 0


Question: Given a set of linearly independent vectors \( \{ v_1, v_2, \ldots, v_k \} \) where \( \Vert v_i \Vert \leq 1 \) for all \( 1 \leq i \leq k \), and real numbers \( \alpha_i \) such that \( \Vert \sum_{i=1}^k \alpha_i v_i \Vert = 1 \), determine if there exists a constant \( M > 0 \) such that \( |\alpha_i| \leq M \) for all \( 1 \leq i \leq k \).

Answer: Yes


Question:
Let $g(x)$ be a continuous function from $[-1,1]$ to $[0,1]$ with a fixed point $p$. Suppose $f(x)$ is a continuous function from $[-1,1]$ to $\mathbb{R}$ satisfying the following properties:

1. $f(x) = g(x) \cdot f(g(x))$
2. $\lim\limits_{x\to1^{-}}\dfrac{f(x)}{\sqrt{1-x}}$ exists and is finite.

If $f(0) = 0$, determine the possible values of $f(p)$. \( \boxed{} \)

Answer: 0


Question: Find the limit of the sequence \(a_n=\int_{2n}^{4n} \frac{1}{x}e^\frac{1}{2x} \,dx\) as \(n\) approaches infinity.

Answer: \ln 2


Question: Determine whether a group of order 128 is solvable.

Answer: Yes


Question:
Let $V=C^1[0,1]$, $X=(C[0,1],\|\cdot\|)$ where the norm on $X$ is the sup norm, and $Y=(C[0,1],\|\cdot\|_2)$ where $\|f\|_2=\sqrt{\int_0^1 |f(x)|^2 \, dx}$. Determine which of the following statements is true:

(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 neither dense in $X$ nor dense in $Y$.

Choose the correct option.

Answer: C


Question: In an $n \times n$ idempotent matrix, what is the minimum number of entries that need to be known to determine the rest of the matrix?

Answer: 2n - 1


Question: Calculate the length of the curve defined by the parametric equations \(x(t) = t\sin(2t)\), \(y(t) = t\cos(2t)\), and \(z(t) = \frac{4}{3}t\sqrt{t}\) from \(t = 0\) to \(t = 2\pi\).

Answer: 4\pi^2 + 2\pi


Question: Consider the linear space $C[0, 1]$ of all continuous functions on the interval $[0, 1]$, equipped with the norm $\|f\| = \max_{0\leq x\leq 1} |f(x)|$. Define an operator $A$ by $$Af(x) = \int^{1}_0 K(x, y)f(y) \, dy,$$ where $K(x, y) = x$. Determine the norm of the operator $\|A\|$.

Answer: 1


Question: Evaluate the limit \( \lim_{x \to +\infty} \int_{0}^{x}|\sin(t)|dt \). Is the reasoning that \( \lim_{x \to +\infty} \int_{0}^{x}|\sin(t)|dt = \sum_{k=1}^{+\infty}2 = \lim_{n \to +\infty}2n = +\infty \) correct? Provide a detailed analysis.

Answer: +\infty


Question: Can a process with independent increments and continuous variance have a discontinuous expected value function?

Answer: Yes


Question: Evaluate the definite integral \( \int_0^{\pi/2} \frac{\cos x}{\sqrt{1-\sin x}} \, dx \).

Answer: 2


Question: What is the maximum number of digits a number can have if it is equal to the sum of the factorials of its digits?

Answer: 5


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \int_{0}^{\pi/6} \frac{\cos^2\left(\frac{nx}{3}\right)}{n\,\cos^2\left(\frac{x}{3}\right)} \, dx \]

Answer: 0


Question: Determine if there exists a sequence of polynomials that converges uniformly to \( g(x) = \frac{\sin(1/x)}{1/x} \) on the interval \((0, 1]\).

Answer: Yes


Question:
Given the system of equations:
\[ x^2 - y^2 - u^3 + v^2 + 4 = 0 \]
\[ 2xy + y^2 - 2u^2 + 3v^4 + 8 = 0 \]
find \( \frac{\partial u}{\partial x} \) at the point \((x,y) = (2,-1)\).

Answer: \dfrac{13}{32}


Question: Determine all positive integers $n$ such that $3^n + 5^n$ is a perfect cube.

Answer: 1


Question: Determine the minimum value of \((f^{-1})'f(x)\) for an invertible increasing function \(f(x)\) that satisfies \(f''(x)+2f'(x)\geq 2\) and \(f'(0)=2\).

Answer: \dfrac{1}{2}


Question: Find the coefficient of \(x^{1998}\) in the expansion of \(\prod_{n=1}^{1996}(1+nx^{3^n})\).

Answer: 0


Question: Given that the $n$th moment of a random variable $X$ is $E[X^n] = 3n$ for all positive integers $n$, find the expected value $E[e^X]$. Express your answer in a closed form.

Answer: 1 + 3e


Question: Find the area enclosed by the curve \(x^4 + y^4 = 4xy\) in the first quadrant.

Answer: \dfrac{\pi}{2}


Question: Determine whether every diagonalizable matrix can be diagonalized using an orthogonal basis of eigenvectors, even over the complex numbers. Answer with 'Yes' or 'No'.

Answer: No


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

Answer: 0


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty}\left[\prod_{k=1}^{n}\left(1+\frac{(k+1)^{n}}{n^{n+1}}\right)\right]$$

Answer: 1


Question: Suppose U and V are subsets of a topological space X, and U and V are homeomorphic with the subspace topology. If U is open in X, does it follow that V is open in X? Provide a justification for your answer.

Answer: No


Question: If $f(n)$ and $g(n)$ are monotonically increasing functions and $f(n) = O(g(n))$, does it follow that $(f(n))^{\log(n)} = O((g(n))^{\log(n)})$? Answer yes or no.

Answer: No


Question: Determine the initial conditions for the differential equation \( y' = y(y-4)(y+4)(y-6) \) such that \( \lim_{t \rightarrow \infty} y(t) = 4 \).

Answer: (0, 6)


Question: A graph has 20 vertices. If two vertices have the same degree, they are not connected by an edge. What is the maximum possible number of edges in such a graph?

Answer: 155


Question: Determine whether there exists a function $g$ such that there are infinitely many points where both $g(x) = 1$ and $g'(x) = 1$.

Answer: Yes


Question: Suppose $g$ is the inverse function of a differentiable function $f$, and $G(x) = \frac{-4}{g^2(x)}$. Given that $f(5) = 3$ and $f'(5) = \frac{1}{125}$, find $G'(3)$. \( \boxed{} \)

Answer: 8


Question: Evaluate the integral $$ \int_{0}^{\infty}{\cos\left(x\right)\cos\left(2x\right)\cos\left(3x\right)\ldots \cos\left(nx\right)\cos\left(n^{2}x\right) \over x^{n + 1}}\,\mathrm{d}x ?$$

Answer: 0


Question: Determine whether the following statement is true or false: If $A$ is relatively closed in $X \subset \mathbb{R}^{n}$, then $\overline{A} \cap X = A$, where $\overline{\cdot}$ denotes the closure operator in $\mathbb{R}^{n}$, and $X$ is equipped with the subspace topology inherited from $\mathbb{R}^{n}$. Provide a proof or counterexample.

Answer: True


Question: Given two random variables $X$ and $Y$ with joint distribution function $f_{X,Y}(x,y) = \frac{8xy}{3}$ for $0 < x < 1$ and $x < y < 2x$, determine the covariance of $X$ and $Y$. Consider how the inequality constraint $x < y < 2x$ should be applied when integrating over $Y$.

Answer: \dfrac{28}{675}


Question: Determine if the equation \(x^3 y^2 + z^2 - xz \ln(x) = 5\) defines an implicit function \(z = \psi(x, y)\) around the point \((1, 1, 1)\). If it does, find the equation of the tangent plane to the surface defined by \(x^2 \psi(x, y) = 9\) at the point \((3, 1, 1)\).

Answer: 2x + 3z = 9


Question: Determine the number of eigenvalues in the interval (0, 1) for the matrix \( A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix} \) by examining the signatures of the quadratic forms associated with the matrices \( A - 0I \) and \( A - 1I \).

Answer: 0


Question: Determine the size of the symmetry group of a regular n-gon using the orbit-stabilizer theorem.

Answer: 2n


Question: Consider a continuous function \(f(x):[0, 1] \rightarrow \mathbb{R}\) that attains a maximum at exactly 3 points. Determine whether the function \(g(f(x)) = f^3(x) - f^2(x) + 5f(x) - 14\) also attains a maximum at exactly 3 points.

Answer: Yes


Question:
Determine which of the following sets form an ideal in the ring of all continuous real-valued functions on \(\mathbb{R}\), denoted as \(C(\mathbb{R})\):

A. The set of all \(C^{\infty}\) functions with compact support.
B. The set of all continuous functions with compact support.
C. The set of all continuous functions that vanish at infinity, i.e., functions \(f\) such that \(\lim_{x\to\infty}f(x)=0\).

Answer: B


Question: Does the disjoint union of $n$ mutually unlinked unknots, each with a framing that twists once along the knot (corresponding to $1\in\pi_1(SO(2))$), represent the unknot with framing $n\in\pi_1(SO(2))$?

Answer: No


Question: Given $x > 1$ and $y > 1$, with $\left(\log_e x \right)^2+\left(\log_e y \right)^2=\log(x^2)+\log(y^2)$, determine the maximum value of $x^{\log_e y}$.

Answer: e^4


Question: Determine the values of $a \in \mathbb{R}$ for which the function $f(x) = x^{100} - ax^{97} + 5$ has multiple local minimum values.

Answer: \emptyset


Question: Evaluate the limit: $$\lim_{x\to-7} \frac{[x]^2+15[x]+56}{\sin(x+7)\sin(x+8)}$$ where $[x]$ denotes the greatest integer less than or equal to $x$. Provide your answer in simplified form.

Answer: 0


Question: A variable plane intersects the coordinate planes to form a tetrahedron with a constant volume of \(96k^3\). Determine the locus of the center of gravity of this tetrahedron.

Answer: xyz = 9k^3


Question: Evaluate the limit: \[ \lim _{ n\rightarrow \infty }{ \frac { { 2 }^{ n }! }{ { 2 }^{ n! } } } \]

Answer: 0


Question: Evaluate the integral \( \int_C \frac{2}{z^2-1}\,dz \) where \( C \) is the circle of radius \( \frac{1}{2} \) centered at \( 1 \), positively oriented.

Answer: 2\pi i


Question: Evaluate the limit: \[ \lim_{n\to\infty} e^{-2n} \sum_{k=0}^{2n} \frac{(2n)^k}{k!} \]

Answer: \dfrac{1}{2}


Question: Calculate the double integral \( \int\int _{[1,2]\times[2,3] }x^2y^2 \, d(x,y) \) using the definition of the double integral, without applying Fubini's theorem.

Answer: \dfrac{133}{9}


Question: Find the limit as \(x\) approaches 0 for the expression: \[ \lim_{x \to 0} \left( \frac{\arcsin \lfloor (a + 2)x \rfloor}{x + a} - \cos \lfloor |ax| \rfloor \right), \] where \(\lfloor x \rfloor\) denotes the floor value of \(x\). Determine the conditions on \(a\) for the limit to exist.

Answer: -1


Question: How many ordered triples \((a, b, c)\) of natural numbers satisfy the conditions \(abc = 2^5 \times 3^7\) and \(\gcd(a, b, c) = 1\)?

Answer: 315


Question: Given that the angle bisector of the lines represented by the equation \(a(x-1)^2+2h(x-1)(y-2)+b(y-2)^2=0\) is \(x+2y-5=0\), and the ratio \(\frac{b-a}{h}\) can be expressed as \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, find the value of \(p+q\).

Answer: 5


Question: Given the joint density function \( f(x, y) = e^{-y} \) for \( 0 < x < y \) and 0 otherwise, find the covariance of the random variables \( X \) and \( Y \).

Answer: 1


Question: Determine the relationship between the coefficients of the expansion of \((1+x+x^2)^n\) and the sum of every third coefficient.

Answer: 3^{n-1}


Question: Find the number of ideals in the ring $R[X]/(X^3)$, where $R$ is a field.

Answer: 4


Question: Find the Wirtinger derivative of $z^{-1}$ with respect to $\bar{z}$, where $z$ is a complex variable.

Answer: 0


Question: Determine the values of $x$ for which the infinite series \( \sum \left[(n^3+1)^{\frac{1}{3}} - n\right] x^n \) converges.

Answer: [-1, 1]


Question: Determine whether the sequence \(a_n=\sqrt[n]{\sum_{k=1}^{n}\left(2-\frac{1}{k}\right)^k}\) converges. If it converges, find the limit.

Answer: 2


Question: Find the values of $p$ for which the series $\displaystyle\sum_{n=1}^\infty \sin(n^p)$ converges.

Answer: p < -1


Question: The graph of the function \( f(x) = x^4 + 4x^3 - 16x^2 + 6x - 5 \) has a common tangent line at \( x = p \) and \( x = q \). Find the product \( pq \).

Answer: -10


Question: Find the maximum value of the complex function \( f(z) = |z^3 + 2iz| \) over the closed unit disk \( \Delta = \{ z \in \mathbb{C} : |z| \leq 1 \} \).

Answer: 3


Question: Determine whether there exists a non-one-to-one linear transformation $T: V \to V$ such that the null space $N(T)$ is a proper subset of the range $R(T)$. Provide a justification for your answer.

Answer: Yes


Question: Find the sum of the series \(\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}\) using the telescoping series technique.

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


Question: Determine the last two digits of the number $9^{9^9}$ without using Euler's theorem.

Answer: 89


Question: Evaluate the line integral \( \int_{C}\left(-\frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy\right) \) where \( C \) is the triangle with vertices at \((5,5)\), \((-5,5)\), and \((0,-5)\), traced counterclockwise. Consider the hypotheses of any theorem you apply.

Answer: 2\pi


Question: Calculate the expected number of coin tosses needed to get three consecutive heads or tails using the conditional expectation technique.

Answer: 7


Question: Determine whether the following statement is always true: The expected value of the exponential of a random variable is less than or equal to the exponential of the expected value of the random variable.

Answer: False


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

Answer: -\ln\left(1 - \frac{1}{e}\right)


Question: Let $G: \mathbb{R}^{3} \rightarrow \mathbb{R}$ be a differentiable function such that $\lim_{|w| \to \infty} G(w) = -\infty$. Determine whether $G$ necessarily has a local maximum.

Answer: Yes


Question: Determine the order of the quotient module $M/K$ for the ring $D = \mathbb{Z}[\sqrt{-2}]$ and the free $D$-module $M = D^3$, where $K$ is the submodule generated by $(1, 1 + \sqrt{-2}, 2), (0, 1, 3), (2, 0, 1)$. Provide your answer as a number or expression.

Answer: 81


Question: Let $X$ be a Tychonoff space, and let $f_1, f_2: X \to \mathbb{R}$ be continuous functions. Suppose $A$ and $B$ are completely separated sets in $X$. Determine whether there exists a continuous function $g: X \to \mathbb{R}$ such that $g(x) = f_1(x)$ for all $x \in A$ and $g(x) = f_2(x)$ for all $x \in B$. Provide a justification for your answer.

Answer: Yes


Question: In the group of invertible matrices \( \operatorname{GL}_n \), can two distinct permutation matrices belong to the same double coset \( LpU \) for some permutation matrix \( p \)?

Answer: No


Question: Determine whether the following statement is true or false: If $X_n \to X$ in distribution and $\{X_n^2\}$ is uniformly integrable, then $X_n \to X$ in $L^2$. Provide a justification for your answer.

Answer: False


Question: Determine whether the following statement is true or false: A function $f$ is weakly differentiable if and only if there exists a sequence of smooth functions $f_n$, which converges locally to $f$ in $L^1$ and such that $f'_n$ is locally Cauchy in $L^1$. Then the weak differential is the (local) limit of $f'_n$ in $L^1$. Provide a justification for your answer.

Answer: True


Question: Find a function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(\overline{A}) \subset \overline{f(A)} \) and \( f(\overline{A}) \neq \overline{f(A)} \). Provide an example of such a function.

Answer: f(x) = \arctan(x)


Question: Let $f$ be a Lebesgue integrable function on $[0,1]$ with $\lim_{x\rightarrow 1}f(x) = c$, where $c$ is a constant. Determine the value of $\int_0^1 nx^n f(x) \, dx$. \(n \in \mathbb{N}\)

Answer: c


Question: Determine the range of the positive integer $w$ such that there exist $a, b, c$ with $\pi \le a < b < c \le 2\pi$ satisfying the equation $\sin(aw) + \sin(bw) + \sin(cw) = 3$. Provide your answer as an interval or set of values for $w$.

Answer: \{6, 7, 8, \ldots\}


Question: Let $(x_n)_{n\in \mathbb{N}}$, $(y_n)_{n\in \mathbb{N}}$, and $(z_n)_{n\in \mathbb{N}}$ be sequences of positive real numbers. Given that \( \lim \limits _{n\to \infty}\frac{x_n}{z_n}=0 \) and \( \liminf \limits _{n\to \infty}\frac{y_n}{z_n}>0 \), determine whether \( \liminf \limits _{n\to \infty}\frac{x_n}{y_n}=0 \).

Answer: 0


Question: Given a polynomial function $f(x)$ that satisfies the equation \( f(x) \cdot f\left(\frac{1}{x}\right) + 3f(x) + 3f\left(\frac{1}{x}\right) = 0 \) and the condition \( f(3) = 24 \), find the value of \( f(2) + f(-2) \).

Answer: 18


Question: Suppose that a sequence of functions \(f_n\) converges uniformly to a function \(f\) on the interval \((a, b]\). If both \(f\) and \(f_n\) are continuous for all \(n \in \mathbb{N}\), does \(f_n\) converge uniformly to \(f\) on the interval \([a, b]\)? Provide a justification for your answer.

Answer: Yes


Question: Find an approximation for the constant coefficient in the asymptotic expansion of $k\log(x)$ as $k$ tends to infinity, where $x$ is the solution to the equation $p_K(x)=\frac{1}{2}$. Provide your answer as a single value or expression.

Answer: -\dfrac{1}{3}


Question: Find the integral part of the expression \((2+\sqrt{5})^{2016}+(2-\sqrt{5})^{2016}\). Express the result as a Lucas number.

Answer: L_{6048}


Question: Evaluate the integral \( \oint_C \frac{1}{z+2} \, dz \) where \( C \) is the circle \( \left\vert z-1 \right\vert=3 \).

Answer: \pi i


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

Answer: \infty


Question: Find the number of polynomial functions $g$ of degree at least 2 that satisfy the equation $g(x^3) = (g(x))^3 = g(g(g(x)))$ for all real numbers $x$. Provide your answer as a single integer.

Answer: 0


Question: Find the inverse of the element \( \beta \) in the finite field \( GF(81) \), where \( \beta \) satisfies the equation \( \beta^4 + \beta^2 + 3 = 0 \).

Answer: 2\beta


Question: Evaluate the limit \( \lim_{x\to -\infty}\frac{g(x)}{e^{-2x}} \) given that \( g:\mathbb{R}\to\mathbb{R} \) is a differentiable function with \( g' \) continuous and satisfying \( |g'(x)-e^{-2x}|\leq 5 \) for all \( x\in\mathbb{R} \).

Answer: -\dfrac{1}{2}


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

Answer: \infty


Question: Determine the highest order for which all groups have been classified, excluding groups of order 1024 and cube-free groups of order 50000.

Answer: 2000


Question:
Determine the limit of the sequence defined by the following terms as a function of $n$ and $k$:
\[ a_1 = \text{LOG }n \]
\[ a_2 = \text{LOG }k \cdot \text{LOG }\text{LOG }n \]
\[ a_3 = \text{LOG }k \cdot \text{LOG }\text{LOG }k \cdot \text{LOG }\text{LOG }\text{LOG } n \]
\[ a_4 = \text{LOG }k \cdot \text{LOG }\text{LOG } k \cdot \text{LOG }\text{LOG }\text{LOG }k \cdot \text{LOG }\text{LOG }\text{LOG }\text{LOG }n \]
...
Express the limit in terms of big-Oh notation, if applicable.

Answer: O(1)


Question: Given an even integer $n$, let $f(n)$ be the number of ways to represent $n$ as the sum of two primes. Determine whether there exists a constant $c$ such that for all sufficiently large $n$, $f(n) \geq c \cdot \frac{n}{\log n}$. Provide a justification for your answer.

Answer: No


Question: Let $V$ be a finite-dimensional complex vector space, $a \neq 0$ an element in $V$, and $f$ a linear functional on $V$. Define the linear transformation $A: V \to V$ by $A(x) = x - f(x)a$. Find the determinant of the transformation $A$. \( \text{det}(A) = ? \)

Answer: 1 - f(a)


Question: Let $X$ be a dual Banach space, i.e., $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, which is the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-topology. Determine whether the set $B_{w^*}(X)$ of $w^*$-continuous bounded operators on $X$ is a closed subset of $B(X)$ for the weak* topology of $B(X)$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the result of the operation \((2'*3'*4'*\ldots*26') \mod 29\), where the operation \(*\) is defined as \(x*y = xy + x + y\) for the monoid \((\mathbb{N},*)\).

Answer: 14


Question: Determine the constant \( C_p \) such that for the linear operator \( T : L^{p}([0,1]) \rightarrow F \) defined by \( T(f)(x) = f(2x) C_p \) if \( x \leq \frac{1}{2} \) and \( T(f)(x) = 0 \) if \( x > \frac{1}{2} \), the equality \( \| Tf \|_{p} = \| f \|_{p} \) holds for all \( f \in L^{p}([0,1]) \).

Answer: 2^{1/p}


Question: Find the limit of the function \(f(x,y)=\frac{(xy)^3}{x^2+y^6}\) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: Evaluate the integral \( J = \int_{-\pi}^{\pi} e^{\cos\theta}\cos(\sin\theta)\ d\theta \) using a change of variable.

Answer: 2\pi


Question:
In constructing a highway across a region with hills and valleys, an engineer wants to ensure that the highway surface can be laid level using the tops of the hills as fill material for the valleys without additional dirt. Let S be a rectangular region bounded by x = a, x = b, y = c, and y = d, and let f be a continuous function on S with M and m as the maximum and minimum values of f on S, respectively. Which of the following best represents this situation?

(A) There exists a point p in S such that m ≤ f(p) ≤ M.
(B) There exists a value q of f such that M - m = q.
(C) There exists a point p in S such that ∬Sf = f(p) (area of S).
(D) ∫ab(∫cd f(x,y)) dx = ∂f/∂x |(a,b) + ∂f/∂y |(c,d)
(E) There exists a value q of f such that q/(area of S) = ∂f/∂x |(a,b) + ∂f/∂y |(c,d)

Answer: C


Question: Determine the limit \( \lim \limits_{x \to \infty} \dfrac{\lfloor P(x) \rfloor}{P(\lfloor x \rfloor)} \) for any polynomial \(P(x)\) with positive real coefficients. Provide a proof of your result.

Answer: 1


Question: Determine whether the sequence \( y_n = \frac{20}{2} \cdot \frac{21}{4} \cdot \ldots \cdot \frac{n+19}{2n} \) converges, and if so, find its limit.

Answer: 0


Question: Determine the number of real roots of the equation \(2^x = 1 + x^2\).

Answer: 3


Question: Find the radius of a fixed concentric circle that a variable chord of the hyperbola \(\frac{x^2}{4} - \frac{y^2}{8} = 1\) touches, given that the chord subtends a right angle at the center of the hyperbola.

Answer: 2\sqrt{2}


Question: Does a group of order 16 necessarily have a normal subgroup of order 8?

Answer: Yes


Question: Compute \(12^{45} \mod 47\) using an efficient method other than repeated squaring.

Answer: 4


Question: Determine whether the unit circle \( C \subset \mathbb{R}^2 \), defined as \( \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1\} \), is Jordan-negligible. Specifically, can it be covered by rectangles with a total area less than any given \( \epsilon > 0 \)?

Answer: Yes


Question: Provide an example of a non-abelian group $G$ such that $T_G = \{ x \in G \mid x^n = 1 \text{ for some } n \geq 1 \}$ is not a subgroup of $G$.

Answer: D_\infty


Question: Evaluate the limit: $$\lim _{x\to 0}\frac{\exp(x^2)-\cos x-x\sin x}{\sinh^2x}$$

Answer: \dfrac{1}{2}


Question:
Given non-negative real numbers $a, b, c$ such that $ab + bc + ca = 1$, determine the maximum value of the expression:

$$P = \frac{\sqrt{a + bc} + \sqrt{b + ca} + \sqrt{c + ab}}{\sqrt{3a + 3b + 3c + 10}}.$$

Answer: \dfrac{3}{4}


Question: Find the remainder when \((x + 1)^{100} + (x + 2)^{200}\) is divided by \(x^2 + 3x + 2\).

Answer: 1


Question: Find the limit of the sequence as \(n\) approaches infinity: \[\lim_{n\to\infty}n^2\left(\sqrt[n]{x}-\sqrt[{n + 1}]{x}\right)\] where \(x > 0\).

Answer: \ln x


Question: Is a matrix with non-positive off-diagonal entries, positive diagonal entries, and positive row sums invertible according to the Gershgorin circle theorem?

Answer: Yes


Question: Does weak-$\star$ convergence of a sequence of $L^p(\mathbb{R}^d)$ functions $(f_n)_{n\in\mathbb{N}}$ to $f$ imply that $\int_{\mathbb{R}^d}f_n g \to \int_{\mathbb{R}^d}f g$ for all $g\in L^q(\mathbb{R}^d)$, where $1/p+1/q=1$?

Answer: Yes


Question: Let $f$ be a real-valued function such that $f(x+y) = f(x)f(y)$ for all real numbers $x$ and $y$. Given that $f(3) = 3$, determine the value of $f(6)$. Express your answer as a single number.

Answer: 9


Question: Let $I$ be the ideal in $\mathbb{Q}[T]$ generated by the set $L = \{T^2 - 1, T^3 - T^2 + T - 1, T^4 - T^3 + T - 1\}$. Determine a polynomial $f \in \mathbb{Q}[T]$ such that $I = (f)$. Can $f = T - 1$ be the generator of $I$?

Answer: T - 1


Question: Is the field obtained by adjoining all the cube roots of \(-3\) to \(\mathbb{Q}\) a number field? The cube roots of \(-3\) are \(-\sqrt[3]{3}\), \(\sqrt[3]{3}e^{\frac{i\pi}{3}}\), and \(\sqrt[3]{3}e^{-\frac{i\pi}{3}}\). Determine if \(\mathbb{Q}(\sqrt[3]{3},\sqrt[3]{3}e^{\frac{i\pi}{3}}, \sqrt[3]{3}e^{-\frac{i\pi}{3}})\) is a finite extension of \(\mathbb{Q}\) with characteristic 0.

Answer: Yes


Question: Determine the value of the derivative \( \frac{d z^*}{dz} \), where \( z^* \) denotes the conjugate of the complex variable \( z \).

Answer: 0


Question: Evaluate the limit using L'Hopital's rule: $$\lim_{x\to \infty}e^{-x^2}\int_x^{x+1/(x^2)}e^{t^2}dt$$

Answer: 0


Question: Determine whether there exists a constant \( M \) such that for all integers \( c \ge M \), there is no prime number \( p \) for which \( p^c \) divides \((p-1)!+1\).

Answer: 3


Question: Determine the values of \(\alpha\) for which the improper integral \(I=\int_1^\infty \frac{x\sin{x}-\cos{x}}{x^\alpha}dx\) converges.

Answer: \alpha > 1


Question: Given a sequence $\{ x_n \} \in l_2$ that weakly converges to $x$, determine if there exists a subsequence $\{ x_{n_k} \}$ such that the Cesàro means of this subsequence converge to $x$ in the $l_2$ norm.

Answer: Yes


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

Answer: 0


Question:
Given the function \( f: \mathbb{R}\to \mathbb{R} \) satisfies:
\[\begin{cases} f(1)=1 \\ f(x+y)=f(x)+f(y)+2xy,\quad&\forall x,y \in \mathbb{R}\\ f\left(\frac{1}{x}\right)=\frac{f(x)}{x^4},&\forall x\neq0 \end{cases}\]
Find the limit:
\[ L=\lim_{x \to 0} \frac{\frac{1}{e^{2f(x)}}-\sqrt[3]{1+f(x)}}{\ln\bigl(1+f(x)\bigr)} \]

Answer: -\dfrac{7}{3}


Question: Find the maximum volume of a rectangular prism that can be inscribed in the ellipsoid defined by the equation $9x^2 + 4y^2 + 16z^2 = 144$. The volume of the prism is given by $8xyz$. What is this maximum volume?

Answer: 64\sqrt{3}


Question: Determine whether there exist uniform constants \(c_1\) and \(c_2\) such that for a set of functions \(f_i(x)\) defined on \([a,b]\) for \(i=1,\ldots,N\), the inequality \(c_1\|f_i\|_1 \le \|f_i\|_2 \le c_2\|f_i\|_1\) holds for all \(i=1,\ldots,N\), where \(\|f\|_1 = \int_a^b |f(x)| \, dx\).

Answer: Yes


Question: Determine whether the sequence \( \left( \sqrt[n]{(2n)!} \right)_{n \in \mathbb{N} } \) converges or diverges. If it converges, find its limit.

Answer: \infty


Question: Evaluate the integral: $$\int_{0}^{1} \frac{\arctan(\sqrt{x})}{(1+x)^2} \, dx$$

Answer: \dfrac{1}{4}


Question: Let $f$ be a continuous function on the interval $[0,1]$. Suppose that for every $x$ in the interval $(0,1)$, the following condition holds: \[ \int_0^x f(t) \, dt = \int_x^1 f(t) \, dt. \] Determine the function $f$.

Answer: 0


Question: Determine whether the following statement is true or false: If $\overline{A}$, the closure of a bounded subset $A$ of $\mathbb{C}^d$, is convex, then $A$ must also be convex.

Answer: B


Question: Determine whether the weak law of large numbers (WLLN) holds for the sequence of independent random variables $\xi_k$, where $P(\xi_k = \sqrt{k}) = P(\xi_k = -\sqrt{k}) = \frac{1}{2\sqrt{k}}$ and $P(\xi_k = 0) = 1 - \frac{1}{\sqrt{k}}$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the sequence \( \left( x_n \right)_{n \in \mathbb{N} } \), defined by \[ x_n = \frac{1}{\sqrt{n}} \left( \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}}+ \cdots + \frac{1}{\sqrt{n}} \right), \] converges. If it converges, find its limit.

Answer: 2


Question: Let \(A\) be a unital Banach algebra and \(a \in A\). If \(a\) is idempotent and \(a \neq 0, 1\), is the spectrum of \(a\) equal to \(\{0, 1\}\)?

Answer: \{0, 1\}


Question: Calculate the limit: $$ \lim_{h\rightarrow0}\frac{f(a+3h)-f(a-2h)}{f(a-5h)-f(a-h)} $$ where $f(x)$ is a differentiable function and $f'(a) \ne 0$ for some $a$. Provide your answer in terms of $f'(a)$.

Answer: -\dfrac{5}{4}


Question: Evaluate the triple integral \( \int \int \int_{V} xyz \, dV \), where \( V \) is the volume of the tetrahedron with vertices at \((0,0,0)\), \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\).

Answer: \dfrac{1}{720}


Question: Find the limit of the expression \( \lim_{x \to 0} \frac{\ln\cos mx}{\ln \cos x} \).

Answer: m^2


Question: Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function which is differentiable on all of \(\mathbb{R}_{0}\) except at \(0\), and suppose that \(\lim_{x \rightarrow 0}f'(x)\) exists and is finite. Can we conclude that \(f\) is differentiable at \(0\)?

Answer: Yes


Question: Evaluate the sum: \( \tan^{2}\left(\frac{\pi}{16}\right) + \tan^{2}\left(\frac{2\pi}{16}\right) + \tan^{2}\left(\frac{3\pi}{16}\right) + \cdots + \tan^{2}\left(\frac{7\pi}{16}\right) \) using trigonometric identities.

Answer: 35


Question: Determine the form of the polynomial $f(x)$ that satisfies the following conditions: \( f(1/x) = f''(x) \) for all \( x \), \( f(x) \) approaches a constant value as \( x \to \infty \), and \( f(x) \geq f(1) \) for all \( x \).

Answer: 0


Question: Determine the range of values of \(\alpha\) for which the improper integral \(\int_{0}^{\infty}\frac{\sin x}{x^{\alpha}}\,dx\) is convergent.

Answer: (0, 2)


Question: Use the divergence theorem to evaluate the surface integral \( I = \int \int_{S} F \cdot n \, dS \), where \( F = x^3 \hat{i} + y^3 \hat{j} + z^3 \hat{k} \) and \( S \) is the surface of the cylinder defined by \( x^2 + y^2 = 4 \) for \( -1 < z < 1 \). Determine the correct limits for the integral in cylindrical coordinates and compute the value of \( I \).

Answer: 56\pi


Question: Evaluate the complex integral \( \int \limits_{\beta} f(z) \, dz \), where \( \beta \) is the path consisting of the line segment from \(-i\) to \(2+5i\), followed by the line segment from \(2+5i\) to \(5i\), and \( f(x+yi) = iy + x^2 \).

Answer: -12 + 2i


Question:
Consider the Hilbert space $l_2[0,\infty)$ with the right shift operator $S_r$ and the left shift operator $S_l$ defined as follows:

$$S_r (x_0, x_1, x_2, \dots) \mapsto (0, x_0, x_1, \dots)$$
$$S_l (x_0, x_1, x_2, \dots) \mapsto (x_1, x_2, x_3, \dots)$$

The combined operator $T = S_r + S_l$ acts as:

$$T(x_0, x_1, x_2, \dots) \mapsto (x_1, x_0 + x_2, x_1 + x_3, \dots)$$

Describe the spectrum of the operator $T$. Provide your answer as a set of complex numbers.

Answer: [-2, 2]


Question: Evaluate the Riemann-Stieltjes integral: \[ \int_{0}^{2}x^2d\left\lfloor 2x\right\rfloor \]

Answer: \dfrac{15}{2}


Question: Evaluate the integral \( \int_{0}^{a}\frac{1}{1+f(x)}\,dx \) given that \( f(x)f(a-x)=1 \), \( a>0 \), and \( f(x) \) is continuous and positive on \([0,a]\). Use the fact that the integral is independent of \( f(x) \) to find the answer.

Answer: \dfrac{a}{2}


Question:
Does there exist a sequence of continuous functions \( \{f_n\} \) that converges pointwise to the function \( f \) defined by:
\[
f(x)=\begin{cases} \frac{1}{g^{-1}(x)} & x\in\mathbb{Q}\\ 0 &x\not\in\mathbb{Q} \end{cases}
\]
where \( g:\mathbb{N}\to\mathbb{Q} \) is a bijection, and \( f \) is continuous at irrational numbers and discontinuous at rationals?

Answer: Yes


Question: Given two matrices $M, M' \in M_{2 \times 2}(\mathbb{Z})$ with non-zero determinants, such that for all $\lambda, \mu \in \mathbb{Z}$ there exists $U_{\lambda, \mu} \in \mathrm{GL}_2(\mathbb{Z})$ satisfying the equation $M' \begin{pmatrix} \lambda \\ \mu \end{pmatrix} = U_{\lambda, \mu} M \begin{pmatrix} \lambda \\ \mu \end{pmatrix}$. Determine if there exists a matrix $U \in \mathrm{GL}_2(\mathbb{Z})$ such that $M' = UM$. Provide a proof or counterexample.

Answer: Yes


Question: If the order of a group \(G\) admits a prime decomposition with a factor \(p^k\), does this imply that there must exist an element in \(G\) of order \(p^k\)?

Answer: No


Question: Let $X$ be a magma and $x \in X$. If there exists a magma $Y$ and an injective homomorphism $f : X \rightarrow Y$ such that $f(x)$ has a left-inverse, then $x$ is left-cancellative. Does the converse hold? Specifically, if $x \in X$ is left-cancellative, does there exist a magma $Y$ and an injective homomorphism $f : X \rightarrow Y$ such that $f(x)$ has a left-inverse? Here, "$x$ is left-cancellative" means $\forall a,b : x*a = x*b \rightarrow a=b$, and "$x$ has a left-inverse" means $\exists y \forall a : y*(x*a)=a$. Answer with "Yes" or "No" and justify your answer.

Answer: Yes


Question: Find the limit of the function \( f(x,y) = x^2y - 3xy^3 + 2 \) as \((x,y)\) approaches \((2,-1)\).

Answer: 4


Question: Let $A \in \mathbb{R}^{3\times3}$ be a matrix with eigenvalues $\lambda_1 = -1$, $\lambda_2 = 1$, and $\lambda_3 = 5$. Determine the dimension of the subspace $V \subset \mathbb{R}^{3}$ such that $A^3 v = v$ for all $v \in V$. \( \text{Provide your answer as a single integer.} \)

Answer: 1


Question: Evaluate the limit: \[ \lim_{s\to\infty} s^n P(|Y| > s) \] using the inequality \[ x^pP(|X|>x) \leq x^p \frac{E|X|^p}{x^p} \leq E|X|^p < \infty. \]

Answer: 0


Question: Evaluate the integral: $$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} \, dx$$

Answer: \dfrac{\pi}{4}


Question: Find the maximum value of the function $f(x,y,z) = y$ subject to the constraints $y^2 + x^2 + z^2 = 3$ and $y + x + z = 1$.

Answer: \dfrac{5}{3}


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{dx}{(x^2+1)(x^2+9)} \).

Answer: \dfrac{\pi}{12}


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

Answer: 0


Question: In the group $A_5$, what is the probability that the subgroup generated by two randomly chosen elements is cyclic?

Answer: \dfrac{3}{40}


Question: In triangle $ABC,$ the angle bisectors $AD,$ $BE,$ and $CF$ intersect at the incenter $I.$ Given that $DI = 3,$ $BD = 4,$ and $BI = 5,$ find the area of triangle $ABC.$

Answer: \dfrac{384}{7}


Question: Determine all natural numbers $e$ for which there exists a natural number $f$ such that $e > f$ and a polyhedron exists with $e$ vertices and $f$ faces.

Answer: e \geq 6


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^{1} \left[x^n + (1-x)^n \right]^{1/n} \, \mathrm{d}x \]

Answer: \dfrac{3}{4}


Question: Determine the largest integer $n$ such that $1000!$ is divisible by $1001^n$, where $1001 = 7 \times 11 \times 13$. Provide your answer as a single integer $n$.

Answer: 81


Question: Determine whether a non-identity element $a$ can belong to both $P_1$ and $P_2$, where $P_1$ and $P_2$ are distinct Sylow $p$-subgroups of a group $G$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral: $$\int_{-\infty}^\infty e^{-x^2} \, dx$$.

Answer: \sqrt{\pi}


Question: Is it possible to construct a sequence of infinite, disjoint subsets of the natural numbers \(\mathbb{N}\) such that their union does not equal \(\mathbb{N}\)?

Answer: Yes


Question: After standing on the real line at the point 3, you take steps to the points 3.1, 3.14, and so on, following a Cauchy sequence of decimal approximations of \(\pi\). Each step takes \(\frac{1}{2^n}\) seconds for the \(n\)-th step. Where will you be after continuing this process for 1 second?

Answer: \pi


Question: Determine whether there exists a function $e: \mathcal{F}(X) \rightarrow (0, \infty)$ for an infinite Polish space $X$, where $\mathcal{F}(X)$ denotes the set of all non-empty finite subsets of $X$, such that for any sequence $(x_n)_{n \geq 1} \in X^\mathbb{N}$, $e(\{x_1, \ldots, x_n\})$ converges to $0$ as $n \to \infty$ if and only if the sequence $\{x_n : n \geq 1\}$ is dense in $X$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to 0^+}\int_x^{2x} \frac{\sin t}{t^2} \,dt$$

Answer: \ln 2


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

Answer: \dfrac{e}{2}


Question: Find the minimum distance between the curves $y = e^x$ and $y = \ln(x)$. Provide your answer as a numerical value.

Answer: \sqrt{2}


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

Answer: 0


Question: Determine whether it is true that \( \mathbb{P}(X=Y)=0 \) for all \( x \in \mathbb{R} \) if \( X \) and \( Y \) are independent random variables on the same probability space with \( \mathbb{P}(X=x)=\mathbb{P}(Y=x)=0 \) for all \( x \in \mathbb{R} \).

Answer: True


Question: Use the power series expansion of \( \frac{z}{e^z-1} \) to find the value of \( B_5 \).

Answer: 0


Question: Given two equivalent norms $p_{1}:V\to\mathbb{R}$ and $p_{2}:V\to\mathbb{R}$ on a finite-dimensional vector space $V$, if $p_{1}(x) \geq p_{1}(y)$ for vectors $x, y \in V$, does it necessarily follow that $p_{2}(x) \geq p_{2}(y)$? Provide a justification for your answer.

Answer: No


Question: Determine the minimal polynomial of $\sqrt{5} + \sqrt{7}$ over the field $\mathbb{Q}(\sqrt{3})$. Provide your answer as a polynomial with rational coefficients.

Answer: x^4 - 24x^2 + 4


Question: Let $R$ be a commutative ring with $1$, and let $J$ be a proper ideal of $R$ such that $R/J \cong R^n$ as $R$-modules, where $n$ is some natural number. Does this imply that $J$ is the trivial ideal?

Answer: Yes


Question: Evaluate the triple integral \( \iiint_S z \sqrt{x^2+y^2} \, dx \, dy \, dz \) over the region \( S \) bounded by \( y^2 + z^2 = 1 \), \( x = 0 \), and \( x = y \) with \( x \ge 0 \), using cylindrical coordinates.

Answer: 0


Question: Let $K$ be a field and $L$ be a field extension of $K$ obtained by adjoining a root of an irreducible polynomial of degree $n$ over $K$. If $n$ is not prime, determine whether there can be elements of $K$ which are an $n$th power in $L$, but not in $K$. Provide a justification for your answer.

Answer: Yes


Question: Compute the integral \( \int_{M=L(3;1,1,1)} X p_1(M) \), where \( M = S^5/\mathbb{Z}_3 \) is a Lens space, \( X \in H^1(M, \mathbb{Z}_3) \), and \( p_1(M) \) is the first Pontryagin class of \( M \). What is the value of this integral?

Answer: 0


Question: Evaluate the integral \( \int_2^\infty \frac{\ln(t-1)}{(t-1)^5}\,dt. \)

Answer: \dfrac{1}{16}


Question: Find all critical points of the function \( F(x,y) = 4x^2 + 3y^2 + \cos(2x^2 + y^2) - 1 \).

Answer: (0, 0)


Question: Derive the formula for the sum of the first \( n \) terms of the series: \( 1^3 + 3^3 + 5^3 + \ldots + (2n-1)^3 \).

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


Question: Let $A$ be an $m \times n$ matrix, and consider linearly independent vectors $v_1, v_2, \ldots, v_k$ in $\mathbb{R}^n$. If the null space of $A$, denoted by $\text{Ker}(A)$, consists only of the zero vector, must the vectors $Av_1, Av_2, \ldots, Av_k$ be linearly independent? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty} \sqrt[n]{\frac{n!}{\sum_{m=1}^n m^m}}$$

Answer: \dfrac{1}{e}


Question: Does the condition \( \lim_{n \to \infty} x_n = x_0 \) implying \( \lim_{n \to \infty} f(x_n) = f(x_0) \) mean that the function \( f \) is continuous at \( x_0 \)?

Answer: Yes


Question: Compute the differential \(d(x_1x_2)\) using the product rule for the stochastic differential equations \(dx_1 = a_1 dt + b_1 dw_1(t)\) and \(dx_2 = a_2 dt + b_2 dw_2(t)\), where \(w_1\) and \(w_2\) are independent Wiener processes. What is the term \((dx_1)(dx_2)\) when \(w_1\) and \(w_2\) are independent?

Answer: 0


Question: Determine whether the following statement is true or false: For every positive integer \( k \geq 2 \), there exist distinct natural numbers \( w_1, w_2, \dots, w_k \neq 1 \) such that there is a prime number \( p \neq 2 \) dividing the sum of their squares, \( \sum_{i=1}^k w_i^2 \), without dividing any of the numbers \( w_1, w_2, \dots, w_k \).

Answer: True


Question: Find the Laplace Transform of the function \( f(t) = \delta(t-2\pi)\cos(t) \), where \( \delta(t) \) is the Dirac Delta Function.

Answer: e^{-2\pi s}


Question: Given an $n \times n$ matrix $A$ and an $m \times m$ matrix $B$, determine whether there always exists an $m \times n$ matrix $C$ such that the matrix product satisfies $CA = BC$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to \infty}n\int_1^{e^2}{(\ln x)^n}dx$$

Answer: \infty


Question: Given a function $f: \mathbb{N} \to \mathbb{R}$, assume that \( A := \sup_{k \geq 0} \left| \sum_{n=0}^\infty \frac{t^n}{n!} f(k+n) \right| < \infty \) for some \( t \in (0,1) \). Can we conclude that $f$ is bounded on $\mathbb{N}$? Provide a justification for your answer.

Answer: Yes


Question: Given that $G(x)$ is a composite function of $f(x)$ and an integral involving $\sqrt{\tan(t)}$, where $f(x)$ is a differentiable even function such that $f(-x) = f(x)$, and it is found that $G'(x) = f'(x)f(\sqrt{\tan(f(x))})$. If $f'(0) = 0$, determine the value of $G'(0)$. \( \boxed{0} \)

Answer: 0


Question: Find the smallest real number \( a \) such that the inequality \( 4ax^2 + \frac{1}{x} \geq 1 \) holds for all \( x > 0 \).

Answer: \dfrac{1}{27}


Question: Determine whether the Cartier dual of an abelian étale finite group scheme $G$ is étale, given that the characteristic of the field $k$ is $p > 0$ and the dimension of $k[G]$ is not divisible by $p$.

Answer: Yes


Question: Find the sum of the infinite series: \[1+\frac{1}{2!}\cdot\frac{1}{2^2}+\frac{1\cdot3}{3!}\cdot\left(\frac{1}{2^3}\right)+\frac{1\cdot3\cdot5}{4!}\cdot\left(\frac{1}{2^4}\right)+\cdots\]

Answer: \dfrac{3}{2}


Question: Calculate the quotient group $D_{10}/\langle \sigma \rangle$ for the dihedral group $D_{10} = \{1, \sigma, \sigma^2, \sigma^3, \sigma^4, \tau, \sigma\tau, \sigma^2\tau, \sigma^3\tau, \sigma^4\tau\}$, where $\sigma = (12345)$ and $\tau = (13)(45)$. Describe the structure of the resulting quotient group.

Answer: \mathbb{Z}_2


Question: Given a fourth-degree polynomial $p(x)$ with extreme points at $x=1$ and $x=2$, and \(\lim\limits_{x \to 0} \left(1 + \frac{p(x)}{x^2}\right) = 2\), find the value of $p(2)$. Determine the coefficients of $p(x) = Ax^4 + Bx^3 + Cx^2 + Dx + E$ using the given conditions, and then calculate $p(2)$. What is the value of $p(2)$?

Answer: 0


Question: Determine whether the number of involutions in a finite abelian group uniquely determines its invariant factors. Answer with True or False.

Answer: False


Question: Evaluate the limit \( \lim_{n \to +\infty} a_n \) where \( a_n = \prod_{k=1}^n \sin(ak) \) and \( a \) is an irrational real number.

Answer: 0


Question: Determine all real values of the parameter $a$ such that the inequality \(1+\log_5(x^2+1) \geq \log_5(ax^2+4x+a)\) holds for all real numbers \(x\).

Answer: (2, 3]


Question: Consider the special linear group $SL(2,F_q)$, where $F_q$ is a finite field of order $q$. Determine whether all matrices in the upper triangular subgroup of $SL(2,F_q)$ share a common eigenvector.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to -\infty}\left(1+\tan\left(-\frac{4}{x}\right)\right)^\frac{1}{\arctan \frac{3}{x}} \] without using L'Hopital's rule.

Answer: e^{-\frac{4}{3}}


Question: Evaluate the limit \( \lim_{p\rightarrow\infty}\int_0^{\pi/2}e^{-px}\sin^2 x\,\text{d}x \) using integration by parts.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\tan x-\sin x}{\sin^3x}$$ without using series expansion or l'Hopital's rule.

Answer: \dfrac{1}{2}


Question: Suppose $a_n \to a \neq 0$ and \( \lim_{n\to\infty} \int| a_n f_n(x)+b_n g_n(x)|dx =M<+\infty \). Determine if there exists a constant $C>0$ such that \( \int| a_n f_n(x)+b_n g_n(x)|dx \leq C \) for all $n$. Provide a justification for your answer.

Answer: Yes


Question: Find the minimum value of the function \( \sqrt {a^2+c^2} + \sqrt {(b-d)^2 +(e-c)^2} + \sqrt{(f-b)^2 + g^2} \) where \(a, b, c, d, e, f, g\) are real numbers.

Answer: 0


Question: For a bounded function \( f \) on an interval \([a, b]\), does the condition \( L(f) = U(f) \) imply that \( f \) is Riemann integrable on \([a, b]\)? Provide a justification for your answer.

Answer: Yes


Question: If $a$, $b$, $c$, and $d$ are distinct integers such that the polynomial equation $(x-a)(x-b)(x-c)(x-d)=4$ has an integral root $r$, find the value of $a+b+c+d$ in terms of $r$.

Answer: 4r


Question: If an integral domain $R$ is a free $\mathbb{Z}$-module, and $I$ is a non-zero ideal of $R$, do $R$ and $I$ have the same rank as $\mathbb{Z}$-modules? Answer yes or no.

Answer: Yes


Question: Compute $D_{\alpha}f(z)$ for $f(z) = \bar{z}$, where $D_{\alpha}f(z) = \lim_{r\to 0}\frac{f(z+re^{i\alpha})-f(z)}{re^{i\alpha}}$. Then, determine $D(z) = \frac{\max_{\alpha}|D_{\alpha}f(z)|}{\min_{\alpha}|D_{\alpha}f(z)|}$.

Answer: 1


Question: Find the length of the curve defined by the equations $z^2 = 2x$ and $3y = xz$ between the points $(0,0,0)$ and $(\frac{1}{2},\frac{1}{6},1)$. Express the curve parametrically as $\vec{r}(t) = (x(t), y(t), z(t))$ and compute the arc length integral.

Answer: \dfrac{7}{6}


Question: Determine the smallest integer $N$ such that the inequality $x^N \ge \ln x$ holds for all $x > 0$. Provide a proof for your answer.

Answer: 1


Question: Find the value of $a$ such that the solution to the differential equation \(9y''-12y'+4y=0\) with initial conditions \(y(0)=a>0\) and \(y'(0)=-1\) changes from positive to negative. The solution is given by \(y=ae^{\frac{2}{3}t}+(-1-\frac {2}{3}a)te^{\frac{2}{3}t}\). Determine the value of $a$ that separates solutions that become negative from those that remain positive.

Answer: \dfrac{3}{2}


Question:
Calculate the limit as \(x\) approaches 0 for the following expression using Taylor series expansion:
\[\lim_{x \to 0} \frac{2\arcsin(1-\cos(x))-\tan^2x}{\ln(1+x^2)-\sin^2(x)}\]
Given the Taylor series expansions:
\[\ln(1+x^2)=x^2+o(x^2)\]
\[\sin(x)=x-\frac{x^3}{6}+o(x^3)\]
\[\cos(x)=1-\frac{x^2}{2}+o(x^2)\]
\[\arcsin(x)=x+\frac{x^3}{6}+o(x^3)\]
\[\tan(x)=x+\frac{x^3}{3}+o(x^3)\]

Answer: -2


Question: Determine whether the connected components of a graph can be identified by performing a series of matrix operations on its adjacency matrix.

Answer: Yes


Question: Let \(x > 1, y > 1\) and \((\ln x)^2 + (\ln y)^2 = \ln x^2 + \ln y^2\). Find the maximum value of \(x^{\ln y}\).

Answer: e^4


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

Answer: \dfrac{1}{2}


Question: Determine the radius of convergence for the power series: $$\sum_{n=1}^\infty 2^n x^{n^2}$$

Answer: 1


Question: Let $f$ be a holomorphic function on the open unit disk $\mathbb{D}$. Suppose that the power series expansion of $f$ around $0$ has coefficients $a_n$ such that $a_n \ne 0$ for all but finitely many $n$. If the limit $\lim_{n \to \infty} |a_n / a_{n+1}|$ exists and is equal to $L$, determine the radius of convergence of the power series.

Answer: L


Question: Find a subgroup of order 4 in the multiplicative group of nonzero elements of $\mathbb{Z}_{13}$, denoted as $\mathbb{Z}^*_{13}$. The subgroup should contain the identity element and ensure that every element has an inverse within the subgroup.

Answer: \{1, 5, 8, 12\}