Question: Determine the values of \(\alpha > 1\) for which the inequality \(\log^\alpha(x) \leq x\) holds as \(x\) approaches infinity.

Answer: \alpha > 1


Question: Determine the number of functions \( g \) from the set \( A = \{1, 2, 3, 4, 5, 6, 7\} \) to itself such that \( g(g(g(x))) \) is a constant function. Find the remainder when this number is divided by 1000.

Answer: 249


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

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


Question: Evaluate the limit \( \lim_{z\rightarrow 0}\frac{\ln(\sin (x+z))-\ln(\sin x)}{z} \) without using L'Hopital's rule, where \( x\in\left(0,\dfrac{\pi}{2}\right) \).

Answer: \cot x


Question: Determine whether every submultiplicative matrix norm on real matrices can be extended to a submultiplicative matrix norm on complex matrices.

Answer: Yes


Question: Is the derivative of a differentiable function on a closed domain $J$ always continuous on the domain $J$? Answer yes or no.

Answer: No


Question: Calculate the limit: $$\lim_{n\to\infty}\frac{[x] + [2x] + \dots + [nx]}{n^2}$$ where $[x]$ denotes the floor function.

Answer: \dfrac{x}{2}


Question: Calculate the number of non-commutative closed binary operations on a set $B$ where $|B|=5$.

Answer: 5^{25} - 5^{15}


Question: Determine the maximum order of any element in the alternating group $A_{10}$.

Answer: 21


Question: Determine the center of the Lie algebra $\mathfrak{gl}_n(\mathbb{F})$ when $\mathbb{F}$ is a field with positive characteristic $p$ and $n$ is an integral multiple of $p$. Provide your answer in terms of $n$ and $p$.

Answer: \mathbb{F} I_n


Question: Determine whether the following statement is true or false: If \( \frac{a}{d}\equiv\frac{b}{d}\pmod{\frac{n}{d}} \), then \( \frac{a}{e}\equiv\frac{b}{e}\pmod{\frac{n}{e}} \) for any positive integer \( e \) that divides \( d \).

Answer: True


Question: Determine whether two dense sets in a metric space can have an empty intersection, even if they are not both open.

Answer: Yes


Question: Determine the minimal polynomial over \(\mathbb{Q}\) for \(\sin 18^{\circ}\).

Answer: 4x^2 + 2x - 1


Question: Consider the sequence $Z_n = \frac{X_n^2}{\log(1+X_n^2)}$, where $X_n$ is a Gaussian random variable with mean $0$ and variance $\frac{1}{n}$. Determine the limit $\lim_{n \to \infty} \mathbb{E}\left[ \frac{X_n^2}{\log(1+X_n^2)} \right]$. \(\boxed{?}\)

Answer: 1


Question: Evaluate the integral \( \int _0^1\:\frac{2-x^2}{(1+x)\sqrt{1-x^2}}\,dx \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit of the Fejer kernel on the real line, defined as \( \mathcal{F}_R(t) = R\left(\frac{\sin(\pi t R)}{\pi t R}\right)^2 \), as \( R \to \infty \).

Answer: \delta(t)


Question: Let \( M \subset \mathbb{R}^n \) be a manifold with geodesic injectivity radius \( r_x \) at each point \( x \). Assume \( r = \inf_{x \in M} r_x > 0 \). Determine if there exists a \( \delta > 0 \) such that for any points \( x, y \in M \) with \( \|x - y\|_2 < \delta \), it follows that \( d(x, y) < r \).

Answer: Yes


Question: Let \( \alpha \) be a root of the polynomial \( X^3 + X^2 - 2X + 8 = 0 \). Determine if the expressions \( \alpha^2 \), \( \alpha \cdot \frac{\alpha + \alpha^2}{2} \), and \( \left(\frac{\alpha + \alpha^2}{2}\right)^2 \) can all be expressed in the form \( a + b\alpha + c\frac{\alpha + \alpha^2}{2} \), where \( a, b, c \in \mathbb{Z} \).

Answer: Yes


Question: Let $\{f_n\}_{n \in \mathbb{N}}$ be a sequence of injective and continuous functions from the closed interval $[0,1]$ to the set of real numbers $\mathbb{R}$. Consider two sequences $\{x_n\}_{n \in \mathbb{N}}$ and $\{y_n\}_{n \in \mathbb{N}}$ in $[0,1]$ such that $\lim_{{n \to \infty}} f_n(x_n) = \lim_{{n \to \infty}} f_n(y_n)$. Does it necessarily follow that $\lim_{{n \to \infty}} x_n = \lim_{{n \to \infty}} y_n$?

Answer: No


Question: Find the Fourier cosine series of the even extension of the function \( g:(0,\pi) \rightarrow \mathbb{R} \) defined by \( x \longmapsto \cos x \).

Answer: \cos x


Question: The circle $x^2+y^2=16$ intersects the hyperbola $xy=16$ at four points $(x_i,y_i)$ for $i=1,2,3,4$. Calculate the value of $\sum_{i=1}^4 (x_i+y_i)$. \( \boxed{?} \)

Answer: 0


Question: Let $a, b \in \mathbb{R}$ with $a < b$, and let $f \in C^{1}([a, b])$. Prove that \( \lim \limits_{n \rightarrow \infty} \int \limits_{a}^{b} f(x) \sin (n x) \, dx = 0. \)

Answer: 0


Question: Given that $||\vec{a}|| = ||\vec{c}|| = 5$, $||\vec{b}|| = 1$, $\alpha = \angle(\vec{a}, \vec{b}) = \frac{\pi}{8}$, and $||\vec{a} + \vec{b} + \vec{c}|| = ||\vec{a} - \vec{b} + \vec{c}||$, find the angle $\theta = \angle(\vec{b}, \vec{c})$. Express your answer in radians.

Answer: \dfrac{7\pi}{8}


Question: Evaluate the integral \( I = \int_0^{2\pi} (1 + 2\cos t)^n \cos nt\ \mathrm{d}t \) over the interval \([0, 2\pi]\).

Answer: 2\pi


Question: Determine whether all lines in \(\mathbb{R}^2\), defined as subsets isometric to \(\mathbb{R}\), must be of the form \(y=ax+b\) or be vertical. Justify your answer.

Answer: Yes


Question: Evaluate the functions \(j(y)\) and \(k(y)\) for \(b=1.5\) and \(y=2\), and determine if \(j(y) > k(y)\).

Answer: j(y) > k(y)


Question: In a path-connected metric space with more than one point, determine if every open ball contains a path-connected subset with more than one point.

Answer: Yes


Question: Find a positive real number $R$ such that for all complex numbers $z$ with $|z| > R$, the inequality $|z^{10} + 100z + 9| > 1$ holds.

Answer: 2


Question: For discrete probability distributions \( P \) and \( Q \) defined on the same probability space \( \mathcal{X} = \{0, 1, \ldots, n\} \), where \( P = [p, 0, \ldots, 0, 1-p] \) and \( Q \) follows a binomial distribution \( Q = \binom{n}{k} q^k (1-q)^{n-k} \) for all \( k \in \{0, 1, 2, \ldots, n\} \), determine the parameter \( q \) that minimizes the Kullback-Leibler divergence \( D_{\text{KL}}(P \| Q) \).

Answer: 1 - p


Question:
Determine whether the following expression is true for a continuous random variable $X$ and a discrete random variable $Y$ taking values $0$ and $1$:
$$E[(E[X|Y])^{2}]= [(E[X|Y=1])^{2}]	imes P(Y=1)+ [(E[X|Y=0])^{2}]	imes P(Y=0)$$

Answer: True


Question: Let $p_1, \ldots, p_k$ be $k$ distinct prime numbers in $\mathbb{N}$ and let $n > 1$. Determine whether the equality $[\mathbb{Q}(\sqrt[n]{p_1}, \ldots, \sqrt[n]{p_k}):\mathbb{Q}] = n^k$ holds, given that all roots are in $\mathbb{R}^+$. Provide a justification for your answer.

Answer: n^k


Question: Given the differential equations $f'(x) = g(x)$ and $g'(x) = -f(x)$ for all real $x$, with initial conditions $f(5) = 2$ and $f'(5) = 2$, determine the value of $f^2(10) + g^2(10)$. Provide your answer as a single number.

Answer: 8


Question: For any countably infinite set $M$, consider the sets $X_{M}$, $Y_{M}$, and $Z_{M}$ defined as follows: $X_{M}$ is the set of all finite subsets of $M$, $Y_{M}$ is the set of all countably infinite subsets of $M$ whose complement is finite, and $Z_{M}$ is the set of all countably infinite subsets of $M$ whose complement is also countably infinite. Given that $f: X_{M} \to Y_{M}$ defined by $f(T) = M \setminus T$ is a bijection, and $|X_{M}| = |Y_{M}| = \aleph_{0}$, can we conclude that the power set $P(M)$ is uncountable due to the existence of $Z_{M}$?

Answer: Yes


Question: Find the dimension of the field extension \( \mathbb{Q}(\alpha,\beta) \) over \( \mathbb{Q} \), where \( \alpha^3=2 \) and \( \beta^2=2 \).

Answer: 6


Question:
Evaluate the limit of the integral as \( n \to \infty \):

\[ \lim_{n\to\infty}\int_0^\infty\frac{n\cos^2(x/n)}{n+x^4}dx \]

Determine if the integral converges to a finite value, diverges, or approaches infinity.

Answer: \infty


Question: Determine whether there exists a function on the interval $[0,1]$ that has unbounded variation and is everywhere differentiable.

Answer: Yes


Question: Does there exist a magma \((S,\cdot)\) such that for every \(a,b\in S\), there exists \(x\in S\) such that \(a\cdot x=b\), but there exist \(x_1, x_2, x_3\in S\) such that \(x_1\cdot(x_2\cdot x_3)\neq(x_2\cdot x_1)\cdot x_3\)?

Answer: Yes


Question: Given $n$ distinct objects, determine the number of ways to color these objects using at most $n$ distinct colors, where colorings that differ only by a permutation of colors are considered identical. For example, the coloring $1,1,1,1$ is the same as $2,2,2,2$, and $1,1,2,3$ is the same as $4,4,1,2$, but $1,1,2,3$ is different from $2,3,1,1$. Find the total number of such distinct colorings.

Answer: B_n


Question: Solve the system of equations: \[\begin{cases} \left|x_1+x_2\right|=\left|x_2+x_3\right|=...=\left|x_{2018}+x_1\right|,\\ x_1+x_2+...+x_{2018}=2018. \end{cases}\]

Answer: 1


Question: Determine whether there exists a polynomial ring over an infinite field such that its quotient by a maximal ideal is also an infinite field.

Answer: Yes


Question: Calculate the volume of the solid bounded by the cylinder $x^2+(y-1)^2 \leq 1$ and the planes $z = 0$ and $z = 2 - x - y$. Provide your answer in cubic units.

Answer: \pi


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

Answer: 6


Question: Evaluate the limit: \[ \lim_{\beta\to+\infty}\frac{\int_{0}^2 I_0(\sqrt{t(2-t)})e^{-\beta t}(t-1)dt}{\int_{0}^2 I_0(\sqrt{t(2-t)})e^{-\beta t}dt} \] where \( I_0(x) \) is the modified Bessel function of the first kind.

Answer: -1


Question: Evaluate the limit: $$\lim_{h \to -\frac{1}{2}} \int_0^{1+2h}\frac{\sin(x^2)}{(1+2h)^3}dx$$ Determine if the limit exists and find its value if it does.

Answer: \dfrac{1}{3}


Question: Find the area of the portion of the conical surface defined by the equation \(x^2 + y^2 = z^2\) that is bounded between the planes \(z = 0\) and \(x + 2z = 3\).

Answer: 2\sqrt{6}\pi


Question: Evaluate the double integral \( \iint_R y^3x^3 \, dx \, dy \) over the region \( R = \{ (x,y) : y^{1/2} \le x \le y^{1/6} \} \) by changing the order of integration.

Answer: \dfrac{1}{84}


Question: Determine whether the following statement is true or false: Any feasible solution of a linear programming problem can be expressed as the convex combination of its Basic Feasible Solutions (BFS).

Answer: False


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

Answer: a = e^{-b}


Question: Evaluate the limit using the Cesáro-Stolz theorem: $$\lim_{n \rightarrow \infty} \frac {\sum_{k=1}^{n} \log \left(2k+n\right)}{n}$$

Answer: \infty


Question: In a game where two players take turns placing coins of different sizes (penny, quarter, loonie, twoonie) on a rectangular table without overlapping, if the first player places a coin in the center and mirrors the opponent's moves symmetrically, will the first player always win?

Answer: Yes


Question: Determine whether there exists a gap-1 square-free infinite word using the alphabet \{A, B, C, D, E\}. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: A topological group is necessarily Hausdorff if it is totally disconnected.

Answer: A


Question: Determine the radius of convergence for the Taylor series of the function \( \frac{z}{e^z-1} \) centered at \( z=0 \).

Answer: 2\pi


Question: Determine whether the sequence \((a_n)_{n \in \mathbb{N}} = \frac{n^n}{n!}\) converges, and if it does, find its limit.

Answer: \infty


Question: Determine whether the statement is true or false: If $X^A = Y^B$ but $X \neq Y$ or $A \neq B$, then $f^\flat$ is not well-defined.

Answer: A


Question: Find the constant $c \neq 0$ such that the integral $\int_{-1}^\infty{\frac{e^{\frac{x}{c}}}{\sqrt{|x|}(x+2)}dx}$ is convergent.

Answer: c < 0


Question:
Calculate the limit of the expression as \( n \to \infty \):

\[ \lim_{n\to\infty}\frac{(2(n+1))!n!n^n}{(2n)!(n+1)!(n+1)^{n+1}}\left(1+\frac{1}{n}\right )^{-n} \]

Answer: \dfrac{4}{e^2}


Question: Given \[\cos \left[\sqrt{\left(\sin x + \cos x\right)\left(1 - \sin x \cos x \right)}\right] = \sqrt{\cos \left(\sin x + \cos x \right) \cos \left(1 - \sin x \cos x\right)}.\] Find \(\sin^5 x + \cos^5 x.\)

Answer: 1


Question: Find all functions \( f(x) \) that satisfy the functional equation \( f(x + t) + f(x - t) = 2f(x) + 4t^2 \) for all real numbers \( x \) and \( t \).

Answer: 2x^2 + bx + c


Question: Determine whether the fractional $p$-Laplace operator $(-\Delta_p)^s$, defined by the principal value integral for $0<s<1<p<\infty$, is rotationally invariant under orthogonal maps.

Answer: Yes


Question: Determine the number of rational numbers \( x = \frac{p}{q} \) such that \( |\sqrt{3} - x| < \frac{1}{q^4} \) where \( q \) is a positive integer.

Answer: 2


Question: Determine the number of ways to choose subsets $T_1, T_2, \dots, T_K$ of a set with $P$ elements such that the union of all the subsets is the original set.

Answer: (2^K - 1)^P


Question: Let $X$ be a Hilbert space, and let $X_1$ and $X_2$ be subspaces of $X$. Suppose an operator $B: X \to X$ is bounded on both $X_1$ and $X_2$. Is $B$ also bounded on the intersection $X_1 \cap X_2$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{\ln\left[\dfrac{(1-3x)(1+x)^3}{(1+3x)(1-x)^3}\right]}{x^3}$$ without using L'Hôpital's rule, Taylor series, or differentiation (or integration).

Answer: -16


Question: Given i.i.d Bernoulli random variables $X_1, X_2, \ldots$ with parameter $p$, and a Poisson random variable $N$ with parameter $\lambda$, calculate the probability that the sum $\sum_{i=1}^{N} X_i$ equals zero.

Answer: e^{-\lambda p}


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

Answer: \dfrac{4}{e}


Question: In triangle $ABC$, the centroid is denoted as $G$. Points $M$ and $N$ are located on sides $AB$ and $AC$, respectively, such that $M$, $G$, and $N$ are collinear. Given that $\frac{AM}{MB} = \frac{5}{2}$, find the value of $\frac{AN}{NC}$.

Answer: \dfrac{5}{3}


Question: Calculate the residue of the function \( f(z) = \frac{1}{z^2 \sin(z)} \) at \( z = 0 \), given that it is a pole of order 3.

Answer: \dfrac{1}{6}


Question: Evaluate the integral \( \int_{0}^{1}\int_{0}^{1} \frac{x-y}{(x+y)^3} \, dx \, dy \).

Answer: -\dfrac{1}{2}


Question: Consider the sequence of functions \( f_n(x) = x^n \) defined on the set \( \{1, \frac{1}{2}, \frac{1}{3}, \ldots\} \). Determine whether there exists a subsequence of \( f_n \) that converges uniformly on this set.

Answer: Yes


Question: Let $A_p$ and $B_p$ be sequences of symmetric positive-definite $p \times p$ matrices such that their spectral norms satisfy $\|A_p - B_p\|_2 \leq \frac{1}{p}$. Assuming the spectra of $A_p$ and $B_p$ are contained in a compact interval $(0, \infty)$ for all $p$, and $\operatorname{tr}(B_p) = p$ for all $p$, determine if there exists a constant $K$ such that the inequality $$\left\|\frac{pA_p}{\operatorname{tr}(A_p)} - B_p\right\|_2 \leq \frac{K}{p}$$ holds for all $p$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of homomorphisms from the group $\mathbb{Z}/n\mathbb{Z}$ to the additive group of rationals $\mathbb{Q}$.

Answer: 1


Question:
Consider the function \(f(x)=e^{-x}\sin(x)g(x)\), where \(g(x): \mathbb{R^+} \rightarrow (-M, M)\) for some positive real \(M\). Determine the nature of the solutions for the equation \(\sum_{j=1}^{k}|f(n_j)|=1\), where all \(n_j\) are natural numbers. Choose the correct option:

A) Infinite for some values of \(k\) only
B) Finite for all values of \(k\)
C) Can be infinite for all values of \(k>1\)
D) None of these

Answer: B


Question: Given a natural number $n$ and a matrix $A \in \mathbb{R}^{n \times n}$, determine if it is true that $\det(A + A) = 2^n \det(A)$ for all such matrices $A$. Answer with "True" or "False".

Answer: True


Question: Determine whether there are exactly 4 elements of order 5 in the group of units modulo 1331.

Answer: Yes


Question: Determine whether the Möbius strip is a manifold.

Answer: Yes


Question: Find the dimension of the space of global sections of the tangent sheaf of $\mathbf{P}^3$ using the Euler sequence.

Answer: 15


Question: Determine the minimum value of \( k \) such that the inequality \( \int_{0}^{1}f^2(x)\,dx \leq k\left(\int_{0}^{1}f(x)\,dx\right)^2 \) holds for all integrable functions \( f(x) \) satisfying \( 1 \leq f(x) \leq 2 \) for \( x \in (0,1) \).

Answer: \dfrac{9}{8}


Question: In the symmetric group $S_n$ with $n \geq 5$, determine the size of the smallest subgroup that contains all 3-cycles.

Answer: \frac{n!}{2}


Question: Determine the number of positive integer solutions for the system of equations: \(x^3 = yz\) and \(x + y + z = 72\).

Answer: 0


Question: Calculate the area of the quadrilateral formed by the intersection of two circles with radii \(a\) and \(b\), intersecting at an angle \(\theta\).

Answer: ab \sin \theta


Question:
Evaluate the double integral \( \iint_{D} f(x,y) \, dA \) over the region \( D \) bounded by the lines \( x=0 \), \( y=0 \), and \( x+y=1 \), where \( f(x,y) \) is defined as:

\[f(x,y) = \begin{cases} x^2 & x \geq y \\ y^2 & y > x \end{cases}\]

Answer: \dfrac{7}{48}


Question:
Consider the partial differential equation (PDE) given by \( \frac{\partial^2u}{\partial x^2} = \frac{\partial u}{\partial t} \) where the solution \(u(x,t)\) approaches zero as \(t \to \infty\). Given the initial condition \(u(x,0) = \cos(x)\), determine which of the following Fourier series representations is correct:

1. \( u = \sum_{n=1}^{\infty}a_n\sin(nx+b_n)e^{-nt} \) with arbitrary constants \(a_n\) and \(b_n\).
2. \( u = \sum_{n=1}^{\infty}a_n\sin(nx+b_n)e^{-n^2t} \) with arbitrary constants \(a_n\) and \(b_n\).
3. \( u = \sum_{n=1}^{\infty}a_n\cos(nx+b_n)e^{-nt} \) with \(a_n\) not all zero and \(b_n = 0\).
4. \( u = \sum_{n=1}^{\infty}a_n\cos(nx+b_n)e^{-n^2t} \) with \(a_1 \neq 0\), \(a_n = 0\) for \(n > 1\), and \(b_n = 0\) for \(n \geq 1\).

Answer: 4


Question: Given the function \(f(x) = x^3 + 3x^2 + 4x + a \sin x + b\cos x\) for all \(x \in \mathbb{R}\), determine the greatest value of \(a^2 + b^2\) such that \(f(x)\) is injective.

Answer: 1


Question: Calculate \(15^{123456789012345} \mod 2500\).

Answer: 1875


Question:
Evaluate the limit:

$$\lim_{n\to\infty} \sin^2\left(\frac{1}{n^{48}}\right)\left[\left(n+\frac{1}{n^3}\right)^{100} - \left(n^{32}+n^{10}+1\right)^3 - n^{100}\right]$$

Answer: 99


Question: Let $(\Omega, \mathcal{A}, \operatorname{P})$ be a probability space, $(\mathcal{F}_t)_{t \geq 0}$ a filtration on $(\Omega, \mathcal{A})$, and $E$ a Banach space over $\mathbb{R}$. Consider an $E$-valued process $(X_t)_{t \geq 0}$ on $(\Omega, \mathcal{A}, \operatorname{P})$ that is $\mathcal{F}$-Lévy, meaning it is $\mathcal{F}$-adapted, $X_0 = 0$, increments $X_{s+t} - X_s$ are independent of $\mathcal{F}_s$ for all $s,t \geq 0$, and $X_{s+t} - X_s$ is identically distributed with $X_t$ for all $s,t \geq 0$. Let $(Y_t)_{t \geq 0}$ be another $E$-valued $\mathcal{F}$-Lévy process, either on the same or a different probability space, with $X_t \sim Y_t$ for all $t \geq 0$. Determine if $X$ and $Y$ have the same finite-dimensional distributions, i.e., if \[ \operatorname{P}\left[X_{t_1} \in B_1, \ldots, X_{t_k} \in B_k\right] = \operatorname{P}\left[Y_{t_1} \in B_1, \ldots, Y_{t_k} \in B_k\right] \] for all $B_1, \ldots, B_k \in \mathcal{B}(E)$, $k \in \mathbb{N}$, and $0 \leq t_1 < \ldots < t_k$. Provide a justification for your answer.

Answer: Yes


Question: Using Rouché's theorem, determine the number of zeros of the function \( F(z) = (z^2 + 2)(z^2 + 1) - iz(z^2 + 2) \) in the upper half-plane \( D = \{ z \mid \Im(z) > 0 \} \).

Answer: 2


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

Answer: e^4


Question: In a scalene triangle \(\bigtriangleup ABC\), the area is 45 square units. Points \(P_1\) and \(P_2\) lie on side \(AB\) such that \(AP_1 = P_1P_2 = BP_2\). Similarly, points \(Q_1\) and \(Q_2\) are on side \(AC\) with \(AQ_1 = Q_1Q_2 = CQ_2\). The intersection area of triangles \(BQ_1Q_2\) and \(CP_1P_2\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers. Determine the value of \(m+n\).

Answer: 95


Question: Suppose $p_i$ is an odd prime for all $i=1,\cdots,r$ and there is an integer $m$ such that $2$ divides $m$. Can $m$ be written as $m=\prod_{i=1}^{r}{p_i}^{a_i}$ where $a_i$ is a power of primes and $r$ denotes the number of distinct primes? Answer with "Yes" or "No."

Answer: No


Question: Given a twice differentiable function $k: \Bbb R \to \Bbb R$ such that $k(a) = k(e) = 0$, $k(b) = k(d) = 2$, and $k(c) = 1$, with $a < b < c < d < e$, determine the minimum number of solutions for the equation $(k'(x))^2 + k(x)k''(x) = 0$ within the interval $[a, e]$. Find the minimum number of solutions.

Answer: 4


Question: Given a real sequence \( \{x_n\} \), if the sequence \( \{2x_{n+1}+\sin x_n\} \) is convergent, does it follow that \( \{x_n\} \) is also convergent? Answer true or false.

Answer: True


Question:
Consider a $p \times p$ random matrix $S = \frac{1}{n}\sum_{t=1}^n V_tV_t'$, where $V_t$ are i.i.d $p$-dimensional random vectors drawn from $N(0, W)$. Given that $E(S) = W$ and defining $M = \sqrt{n}(S - W)$, determine which of the following formulas is correct for computing the covariance matrix of $M$:

(a) $E(M_{ij}M_{uv}) = nE\big((S_{ij} - W_{ij})(S_{uv} - W_{uv})\big)$

(b) $Var(M) = E(MM')$, where the $(i^{th}, j^{th})$ entry is $E\big(\sum_{k=1}^p M_{ik}M_{jk}\big) = n\sum_{k=1}^p E\big((S_{ik} - W_{ik})(S_{jk} - W_{jk})\big)$

Choose the correct formula.

Answer: a


Question: Find the remainder when the binomial coefficient \( {}^{72}C_{36} \) is divided by 73 using Fermat's Little Theorem.

Answer: 1


Question: Consider a sufficiently regular (say Lipschitz) bounded domain $\Omega\subset\Bbb R^n$. Determine if a function $u\in H^1_0(\Omega)$ can be extended by zero to the whole of $\Omega$, such that the extension lies in $H^1(\Omega)$. Provide a justification for your answer.

Answer: Yes


Question: Find the last two digits of the number \(7^{7^{7^{10217}}}\).

Answer: 43


Question: Determine the values of $p$ such that the inequality \( \frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q \) holds for all \( q>0 \). Express your answer in interval notation in decimal form.

Answer: [0, 3)


Question: How many distinct permutations exist for the digits $2, 3, 4, 5, 2, 3, 4, 5$ such that no identical digits are adjacent?

Answer: 864


Question: Let $R$ be a finite ring and $x \in R$ be a zero divisor. Determine if it is possible for $x^n = x^m \neq x$ to hold for $1 < m < n$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the summation \( \sum \limits_{i=1}^{1999} f\left(\frac{i}{1999}\right) \) for the function \( f(x) = \frac{4^x}{4^x + 2} \).

Answer: \dfrac{2999}{3}


Question: Evaluate the integral: $$\int_{0}^{1} \left((1-x^7)^{\frac{1}{3}}-(1-x^3)^{\frac{1}{7}}\right)\,dx$$

Answer: 0


Question:
Determine the limit of the difference between consecutive roots of the equation \(x = \tan(x)\):

$$\lim_{n\to\infty}(x_{n+1} - x_n)$$

where \(x_1, x_2, \dots, x_n\) are the roots such that \(x_1 < x_2 < \dots < x_n\).

Answer: \pi


Question: Is every finite unramified extension of $\Bbb Q_p$ complete with respect to the $p$-adic norm?

Answer: Yes


Question: Determine whether there exists a positive continuous function $f(x)$ defined on $[a, \infty)$ such that $\int_a^\infty f(x)\,dx$ converges, and for every $0 < c \le p \le 1$, $\int_a^\infty f^p(x)\,dx$ also converges.

Answer: Yes


Question: Determine the radius of convergence of the power series \( \sum^{\infty}_{n=1} \left(\frac{1}{n^3}+(-3)^n\right)z^n \).

Answer: \dfrac{1}{3}


Question: Let $a$, $b$, and $c$ be real numbers such that $a^2 + b^2 + c^2 = 3.$ Determine the minimum value of the expression $5a - 2b + 3c.$

Answer: -\sqrt{114}


Question: Find the limit of the sequence \(a_n=\frac{\alpha(n)}{n}\), where \(\alpha(n)\) is the total number of distinct prime factors of \(n\). For example, \(\alpha(8)=1\) because 2 is the only prime factor of 8. Determine \(\lim_{n\to \infty}a_n\).

Answer: 0


Question: Let \( G \) be a non-abelian group of order \( p^3 \), where \( p \) is a prime. Determine the number of distinct conjugacy classes in \( G \).

Answer: p^2 + p - 1


Question: Verify Gauss' theorem for the vector field \( \mathbf{F} = y\mathbf{j} + \mathbf{k} \) and for the volume \( V \) enclosed by the planes \( x = 0 \), \( y = 0 \), \( z = 0 \), and \( 3x + y + 2z = 6 \). Determine the limits of integration and perform the necessary calculations to confirm the theorem.

Answer: 6


Question: Let $A$ be a finite-dimensional commutative algebra over $\mathbb{R}$ without zero divisors. Determine the degree of the minimal polynomial of an element $a \in A \setminus \mathbb{R}$.

Answer: 2


Question: Determine whether every invertible linear operator on a finite-dimensional vector space over a finite field of characteristic $p$ satisfies $T^{p(p-1)} = I$, where $I$ is the identity operator.

Answer: False


Question: If $A$ and $B$ are $n \times n$ matrices such that $AEB = CED$, does it imply that $AB = CD$ when $E$ is invertible? Provide a justification for your answer.

Answer: No


Question: In a Noetherian commutative ring, if $I \subseteq P$ where $P$ is a prime ideal and $\sqrt{I} = P$, does there exist an integer $n$ such that $P^n \subseteq I$? Provide a proof or counterexample.

Answer: Yes


Question: Given a tuple of vectors $S$, where each vector has at least two overlaps with every other vector, determine if it is always possible to transform $S$ into an orthogonal set of vectors by changing any nonzero element in any vector to a complex value with magnitude $1$. Answer with 'Yes' or 'No'.

Answer: No


Question: In the floor topology on \( \mathbb{R} \), determine whether every point is a limit point of \( \mathbb{R} \).

Answer: Yes


Question: Calculate the expected value of the kinetic energy \( W = \frac{m v^2}{2} \) for a molecule in a uniform gas at equilibrium, given that the density function of the speed \( v \) is \( f_V(v) = a v^2 e^{-b v^2} \), where \( v > 0 \).

Answer: \dfrac{3 m}{4 b}


Question: Find the limit of $X_t$ as $t$ approaches 1 in the solution of the Brownian bridge stochastic differential equation (SDE).

Answer: b


Question: Determine whether an odd perfect number can be expressed as the product of distinct odd prime factors.

Answer: No


Question: Given the series \(y = \frac{2}{5} + \frac{1\cdot3}{2!} \left(\frac{2}{5}\right)^2 + \frac{1\cdot3\cdot5}{3!} \left(\frac{2}{5}\right)^3 + \cdots\), find the value of \(y^2 + 2y\). Recognize that this series resembles a binomial expansion.

Answer: 4


Question: Calculate the surface integral \( \int_N x_2 \, d\sigma \) over the closed lower-hemisphere \( N=\{ (x_1,x_2,x_3) \in \mathbb{S}^2 \mid x_3 \le 0 \} \) of the unit sphere \( \mathbb{S}^2 \) in \( \mathbb{R}^3 \), where \( d\sigma \) is the standard spherical measure.

Answer: 0


Question: In a non-complete metric space $X$, does every Cauchy sequence converge within the space?

Answer: No


Question: Find the radius of convergence of the series \( \sum_{n=1}^{\infty}b_nx^n \) given that \( \frac{1}{\sqrt{n}} \leq b_n \leq \frac{1}{\sqrt{n+1}} \).

Answer: 1


Question: Determine whether the following statement is true or false: In a Tychonoff space, the closure of the intersection of a closed set and a dense subset is equal to the closed set itself.

Answer: B


Question: Let $G$ be a unary recursive (total computable, on natural numbers) function. Define $F$ by the formula $F(x) = \min_{y \ge x} G(y)$. Does $F$ have to be recursive? Provide a justification for your answer.

Answer: No


Question: Suppose $X$ is a bounded subset of $\mathbb{R}^n$ with Lebesgue measure $0$. Determine whether $X$ can be decomposed into a countable union of pairwise disjoint sets $X_n$ such that the closure of each $X_n$ also has Lebesgue measure $0$. Provide a justification for your answer.

Answer: Yes


Question: Find a non-constant function \( f(x) \) such that \( f(x) = f(\pi + x) = f(\pi - x) \).

Answer: \cos(2x)


Question: Can there exist two distinct irrational numbers $x$ and $y$ such that $y$ is a final segment of $x$?

Answer: Yes


Question: Calculate the mass of the solid $E$ that lies below the plane $z=8$ and above the cone $z=\sqrt{x^2+y^2}$, with a density function $\rho(x,y,z)=z$. Use cylindrical coordinates to set up and evaluate the triple integral for the mass.

Answer: 1024\pi


Question: Determine whether the improper integral \( \lim \limits_{n\to\infty} \int_0^1 \sin\left(\frac{1}{x}\right) \sin(nx)dx \) converges. If it converges, find its limit.

Answer: 0


Question: Let $G$ be a group with subgroups $H$ and $K$. If $H \cap K = \{1\}$ and $HK = G$, determine the order of the external direct product $H \times K$. \( \text{Express your answer as a number.} \)

Answer: |G|


Question: Evaluate the limit: $$\lim_{x\to 0}\left(\frac{1}{(\sin^{-1}(x))^2} - \frac1{x^2}\right)$$ without using L'Hôpital's rule.

Answer: -\dfrac{1}{3}


Question: Evaluate the limit of \( b_n = \frac{n!}{n^{1000}} \) as \( n \to \infty \).

Answer: \infty


Question: Let $X$ be a smooth, rationally connected variety over $\mathbb{C}$. Given any two points $p, q \in X$, is it possible to ensure that there exists a smooth rational curve $C$ containing both $p$ and $q$?

Answer: Yes


Question: Calculate the value of the product \( \tan 1^\circ \times \tan 2^\circ \times \cdots \times \tan 89^\circ \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{b \to \infty}\int_{0}^{\infty}\frac{y}{1 + y^{2}}e^{-by}dy. \]

Answer: 0


Question: Given $a, b, c, d \in \mathbb{R}$ such that $a^2 + b^2 = 4$, $c^2 + d^2 = 4$, and $ac + bd = 0$, find the least possible value of $ab - cd$.

Answer: -4


Question: Approximate the value of \( \lim_{n\to \infty}\sum_{k=0}^{n}\frac{1}{\sqrt{n^2+k}} \).

Answer: 1


Question: Find the value of $b$ that minimizes the function \( z = h(u) = u^3\left(\frac{1}{A} - \frac{B}{A^2}\right) + u^2\left(\frac{\epsilon B}{A^2} - 1 - \frac{\epsilon}{A} + \frac{B}{A}\right) - \frac{u}{A} + \frac{b}{B} \) subject to the conditions \( A > 0 \), \( B > 0 \), \( \epsilon > 0 \), \( \epsilon \ll 1 \), and \( b > 0 \).

Answer: 0


Question: In triangle ABC, let AD be the bisector of angle A, where D lies on BC. Given that AB = AC, AD + BD = BC, and BC = 2AD, determine the measure of angle A.

Answer: 90^\circ


Question: Determine whether the maximum determinant of a symmetric $n \times n$ matrix $A$, where the diagonal entries are $+1$ and the off-diagonal entries are between $-1$ and $+1$ inclusive, is equal to the determinant of the identity matrix $I_n$. Is $\max_{A \in S} \det A = \det(I_n)$?

Answer: Yes


Question: Compute the limit: $$ \lim_{z \to 0} \frac{(1-\cos z)^2}{(e^z-1-z)\sin^2z} $$

Answer: \dfrac{1}{2}


Question: Find the integer value of \(x\) such that \(7^{30} \equiv x \pmod{100}\).

Answer: 49


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty} n \left[ \left( n^5 + 5n^4 \right)^{\frac{1}{5}} - \left( n^2 + 2n \right)^{\frac{1}{2}} \right]$$

Answer: -\dfrac{3}{2}


Question: Find the limiting probability as \( N \to \infty \) that if two natural numbers \( x \) and \( y \) (from 0 to \( N-1 \)) are chosen at random, their sum of squares, \( x^2 + y^2 \), is divisible by 10.

Answer: \dfrac{9}{50}


Question: Determine the cardinality of the set \(A=\left\{f\in C^1[0,1]:f(0)=0,\ f(1)=1,\ \left|f'(t)\right|\le 1\ \forall\ t\ \in [0,1].\right\}\).

Answer: 1


Question: Determine the ideal $J_{z^2}$ for $z = \frac{\overline{Y}}{\overline{X}} \in K(V)$, where $V = V(Y^2 - X^2(X + 1)) \subset \mathbb{A}^2$ and $\overline{X}, \overline{Y}$ are the residues of $X, Y$ in $A(V)$, the coordinate ring of $V$. Provide your answer as a specific ideal in the coordinate ring.

Answer: (1)


Question: Determine the number of real solutions for the equation \(2\log_2\log_2x+\log_{\frac{1}{2}}\log_2(2\sqrt{2}x)=1\).

Answer: 1


Question: Find the limit of the function \( f(x,y) = \frac{x^3y^2}{x^4+y^6} \) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: Find the eigenvalues of the linear map $g: V_3 \rightarrow V_3$ defined by $p(X) \mapsto p'(X)$, where $V_3$ is the vector space of polynomials with degree at most 3.

Answer: 0


Question: Solve the Diophantine equation \(m^3 - 5m + 10 = 2^n\) for all integer pairs \((m, n)\).

Answer: (2, 3)


Question: Determine whether an infinitely differentiable function \( f \) is identically zero if \( f^{(k)}(0) = 0 \) for all non-negative integers \( k \). Can this conclusion be drawn based on Taylor's theorem?

Answer: No


Question: Determine the maximum possible value of the leading coefficient of a real polynomial \( P(x) \) of degree 6, given that \( 0 \leq P(x) \leq 1 \) for all \( -1 \leq x \leq 1 \).

Answer: 16


Question: Consider a discrete random walk where a particle starts at $x=0$ and moves with equal probability to the left or right. The walk continues indefinitely. Determine the probability that the particle will never leave the interval $[-n, n]$, where $n$ is a positive integer.

Answer: 0


Question:
Evaluate the limit of the expression as \((x,y)\) approaches \((0,0)\):
\[\lim_{x,y\to 0}\frac{(x^3y+xy^3)\ln(1+x^2+y^4)}{x^4+6x^2y^2+y^4}\]

Answer: 0


Question: Find the volume of the region bounded by the surfaces: \(x-y+z=6\), \(x+y=2\), \(x=y\), \(y=0\), and \(z=0\).

Answer: \dfrac{16}{3}


Question: Find all real numbers \(x\) such that \(10^x + 11^x + 12^x = 13^x + 14^x\).

Answer: 2


Question: Does the existence of continuous partial derivatives of order 1 with respect to $x$ and $y$ at a point $(x_0, y_0)$ guarantee the existence of the directional derivative in any direction at that point?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty}\exp\left(\frac{\ln\left(2^n+4^n\right)}{n}\right) \]

Answer: 4


Question: Can the BBP algorithm be used to calculate the digits of \(\pi\) in bases other than 16?

Answer: No


Question: Evaluate the limit \( \lim \limits_{n\to \infty} \int_3^4 (-x^2+6x-8)^\frac{n}{2} \, dx \).

Answer: 0


Question: Determine the maximum value of \( x(G) \), the chromatic number of a graph \( G \) with \( n \) vertices, where adjacent vertices share a common color and non-adjacent vertices have no common colors.

Answer: \left\lfloor \dfrac{n^2}{4} \right\rfloor


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{x\sin(x)}{x^2+1} \, dx \) using complex analysis.

Answer: \dfrac{\pi}{e}


Question: In the projective plane $\mathbb{P}^2$, consider the curves defined by $f = x^2 - yz$ and $g = (x+z)^2 - yz$. Compute the intersection multiplicity of these curves at the point $p = [-2:1:4]$. Provide your answer as a single integer.

Answer: 1


Question: Let $F$ be a field and $\bar{F}$ be its algebraic closure. Consider $K = \bigcup_{f \in F[x]} K_f$, where $K_f$ is the extension field containing the zeros of $f(x) \in F[x]$. Prove or disprove that $K = \bar{F}$.

Answer: K = \bar{F}


Question: Let $R$ and $R'$ be rings such that $R$ has unity $1$ and $R'$ has no zero divisors. Let $\phi: R \to R'$ be a non-zero ring homomorphism. Does $R'$ necessarily have a unity? Provide a justification for your answer.

Answer: Yes


Question: Find a point on the plane \(\pi:2x-3y-z+1=0\) that is equidistant from the points \(A(2,3,0)\), \(B(5,2,0)\), and \(C(1,0,0)\).

Answer: (3, 1, 4)


Question: Consider a positive matrix $X \in M_n(\mathbb{C})$, meaning it has positive eigenvalues and is self-adjoint. Determine if there exists a vector $\xi \in \mathbb{C}^n$ such that $X = \xi \xi^*$, where $\xi^*$ is the conjugate transpose of $\xi$. Answer with 'Yes' or 'No'.

Answer: No


Question: Consider the set $E = \{(x,y) \mid y = 0 \text{ if } x \text{ is irrational and } 0 \leq y \leq 1 \text{ if } x \text{ is rational}\}$. Determine whether the Lebesgue outer measure of $E$ is $0$ or $1$. Justify your answer.

Answer: 0


Question: Find the last two digits of \(14^{14^{14}}\) using modular arithmetic.

Answer: 36


Question: Determine whether the following limit holds: \[ \lim_{t\rightarrow\infty}\frac{\nu_t(t!)}{t}=0 \] where for any positive integer $t>1$, $\nu_t(n)$ is defined as the maximal integer $k$ such that $t^k\mid n$. Use Legendre's theorem to assist in your evaluation.

Answer: 0


Question: Evaluate the limit \( \lim_{n\to\infty} \oint_{\partial S_n} \frac{f(w)}{w-z}\, dw \) for a fixed \( z\in\mathbb{C} \) that is not a pole of \( f \), where \( S_n \) is the square centered at \( n + \frac{1}{2} \) with side length 1 in the complex plane, and \( f(z) = \frac{1}{z\tan(\pi z)} \).

Answer: 0


Question: Find the limit superior of the sequence \(\left(\frac{\sin n}{n}+\cos\frac{\pi n}{3}\right)\) as \(n\) approaches infinity. Consider the sub-sequences of the form \(n_k = 6k + b\), where \(0 \leq b \leq 5\), due to the 6-periodicity of the cosine function and the boundedness of the sine function.

Answer: 1


Question: Evaluate the integral \( \int |z| \, dz \) along the path defined by the semicircle \(|z|=1\) with \(-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{2}\), starting from \(z=-i\).

Answer: 2i


Question: Let $\alpha = \sqrt[16]{2}$ be a positive real number, and consider the algebraic extension $K = \mathbb{Q}[\alpha]$ over $\mathbb{Q}$. Determine the number of intermediate fields $F$ such that $\mathbb{Q} \subseteq F \subseteq K$. \( \boxed{} \)

Answer: 5


Question: Calculate the variance of the random variable $X$ given that $Y=25$ for the joint probability density function $f(x,y) = \frac{1}{1250}$, where $0 < x < y < 50$. Provide your answer as a single numerical value.

Answer: \dfrac{625}{12}


Question: Evaluate \( \lim_{n \rightarrow \infty} \int_0^1 \frac{nx}{1 + n^2x^4} \, dx \).

Answer: \dfrac{\pi}{4}


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

Answer: \sqrt{e}


Question: Determine whether the following statement is true or false: If $\Omega\subset\mathbb{R}^n$ is bounded and $u,v\in W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$, then $uv\in W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$ and $$\nabla(uv)=u\nabla v+v\nabla u.$$

Answer: True


Question: Let $T$ be a compact subset of $\mathbb{R}^n$, and let $B_1^n$ be the closed ball of radius 1 centered at the origin. Determine whether the boundary of the Minkowski sum $T \oplus B_1^n$ has Lebesgue measure 0.

Answer: Yes


Question:
Given $Y = \lambda + (\sigma X + \mu)^2$, where $X$ follows a standard normal distribution ($X\sim\mathcal{N}(0,1)$), determine which of the following transformations results in a valid non-central chi-squared distribution:

1. $(Y - \lambda) \sim \mathcal{X'}^2_1(\mu^2/\sigma^2)$, based on $(\sigma X + \mu)^2 \sim \mathcal{X'}^2_1(\mu^2/\sigma^2)$.
2. $(Y - \lambda)/\sigma^2 \sim \mathcal{X'}^2_1(\mu^2/\sigma^2)$, derived from $Y = \lambda + \sigma^2(X + \mu/\sigma)^2$ where $(X + \mu/\sigma)^2 \sim \mathcal{X'}^2_1(\mu^2/\sigma^2)$.

Which transformation is correct according to the properties of non-central chi-squared distributions?

Answer: 2


Question: Let $S_1,S_2\subseteq X$. Suppose $S_1 \times S_2$ is dense in $X^2$. Are $S_1$ and $S_2$ each dense in $X$? Answer true or false.

Answer: A


Question: Given that the set of rational numbers, \(\mathbb{Q}\), is countable, consider a sequence containing all rational numbers, denoted by \(\{q_n\}_{n \in \mathbb{N}}\). Does there exist a subsequence \(\{q_{n_k}\}_{k \in \mathbb{N}}\) that is dense in the real numbers, \(\mathbb{R}\)?

Answer: Yes


Question: Determine whether the sequence \((z_n)\), where \(z_n := \frac{1}{2n + 3} + \frac{1}{2n + 4} + \ldots + \frac{1}{4n}\), converges or diverges.

Answer: \ln 2


Question: Determine if the function \( h(x) = \begin{cases} x^2, & x \in \mathbb{Q} \\ x^3, & x \in \mathbb{R} \setminus \mathbb{Q} \end{cases} \) is differentiable at \( x = 0 \).

Answer: 0


Question: Calculate Kendall's Tau for the random variable vector $(X_1, X_2)$ given the copula: \[ C(u_1, u_2) = \frac{1}{2}W(u_1,u_2)+\frac{1}{2}\Pi(u_1,u_2). \]

Answer: -\dfrac{5}{12}


Question: Calculate the sum of the series \( \sum_{n=1}^{\infty} \frac{n^2}{n!} \).

Answer: 2e


Question: In a group, is it possible for the order of the product of two elements to be infinite if each element individually has a finite order?

Answer: Yes


Question: Determine the radius of convergence of the power series \(\sum_{n=0}^{\infty}\frac{\sin (n)+11^n+6^n}{(2n)!}x^{2n}\).

Answer: \infty


Question: Determine the dimension of the solution space for the Cauchy-Euler equation \(x^3y''' - 6x^2y'' + 11xy' - 6y = 0\).

Answer: 3


Question: Determine the mixing time of a discrete random walk on an N-cycle, where the walk moves to the right or left with equal probability of \( \frac{1}{2} \).

Answer: \Theta(N^2)


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{2^x\sin\left(\frac \pi {2^x}\right)}{\frac \pi {2^x}}.$$

Answer: 0


Question:
Evaluate the limit, if it exists:
\[\lim_{x\rightarrow\infty}\Delta^x_{k=1}\sin\left(\frac{k\pi}{2x}\right)\]

Answer: 0


Question: Let $X$ and $Y$ be independent random variables, each following a standard normal distribution, $X, Y \sim \mathcal{N}(0,1)$. Calculate the conditional expectation $E(X \mid X^2 + Y^2)$. Provide your answer as a function of $X^2 + Y^2$.

Answer: 0


Question: Calculate \( \lim\limits_{n\to\infty}\lim\limits_{x\to\ 0}I^{(n)}_{-\frac{1}{2}+n}(\sqrt{x}) \) using the power series representation of the Bessel function.

Answer: 0


Question: Given a continuous function $f$ defined on $[0, \pi]$, determine if it is true that $f(x) \equiv 0$ if $\int_{0}^{\pi} f(x) \sin(nx)\,dx = 0$ for all integers $n \geq 1$. Provide a justification for your answer.

Answer: True


Question: Find the equation of the orthogonal projection of the space curve defined by \( z = x^y \) onto the x-y plane, given that the curve is symmetrical about the plane \( y = x \) and lies to the right of the line \( y = e \) and above the line \( y = x \), where \( x^y > y^x \).

Answer: x^y = y^x


Question: Let $V$ be the vector space of all polynomials over a field $F$. Consider a subspace $U$ of $V$ with $2 \leq \dim U \leq n$. Given distinct points $x_1, x_2 \in F$, determine the dimension of the subspace $W \cap U$, where $W$ is the subspace of polynomials that vanish at $x_1$ and $x_2$. Provide your answer as a single integer.

Answer: \dim U - 2


Question: In an inscribed triangle $\triangle ABC$ with a circle of radius $1$ and center $O$, the angle $\angle AOM$ measures $150^\circ$. Point $M$ is the midpoint of side $BC$. Determine the length of $BC$ that results in the minimum length of the orthocenter $H$ to the center $O$. Choose the correct answer from the following options: A) $\sqrt{15}$, B) $\frac{\sqrt{13}}{2}$, C) $\frac{\sqrt{3}}{2}$, D) $\frac{\sqrt{13}}{4}$. Express your answer in the form of the given options.

Answer: B


Question: If $X$ and $Y$ are continuous random variables such that $\mathbb{P}(X=x)=\mathbb{P}(Y=x)=0$ for all $x\in\mathbb{R}$, does it follow that $\mathbb{P}(X-Y=c)=0$ for any constant $c$? Answer yes or no.

Answer: No


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^{1} \frac{e^{-(n^2 + x)}}{n + x^2} \, dx \]

Answer: 0


Question: Find the maximum value of \( \frac{2}{\sin{2x}} - \frac{x}{2} \) such that the equation \( \left(\frac{2}{\sin{2x}} - \frac{x}{2}\right)^2 = 3 - k^2 \) has real roots.

Answer: \sqrt{3}


Question: Find the maximum value of \(|z^n + a|\) for complex numbers \(z\) with \(|z| \leq 1\), where \(n\) is a positive integer and \(a\) is a complex number.

Answer: 1 + |a|


Question: Evaluate the integral \( \int_{0}^{1} [mx] \, dx \), where \( [x] \) denotes the floor function. Determine the value of \( m \) for which the integral equals \( \frac{3}{2} \).

Answer: 4


Question: Evaluate the contour integral \( \int_{\gamma} \frac{g'(z)}{g(z)} \, dz \), where \( \gamma \) is a simple closed curve traversed anti-clockwise, \( g(z) \) is analytic inside and on \( \gamma \), and \( g(z) \) has \( m \) zeros inside \( \gamma \).

Answer: 2\pi i m


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

Answer: \dfrac{3}{4}


Question: Determine whether a monotone function $f \in L_{1}([a, \infty))$ implies that $\lim\limits_{x\to \infty}xf(x)=0$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the complex integral \( \int_0^{2\pi} e^{e^{ix}} \, dx. \)

Answer: 2\pi


Question: Determine the number of positive integer solutions to the equation \(x^4 + 4y^4 + 16z^4 + 64 = 32xyz\).

Answer: 0


Question:
Given a group $G$ and an infinite sequence of independent and identically distributed (i.i.d.) random elements $\{X_n\}_{n=1}^{\infty}$ in $G$, where for any subgroup $H \leq G$,
\[P(X_1 \in H) = \begin{cases} \frac{1}{[G:H]} & \text{if $[G:H]$ is finite} \\ 0 & \text{if $[G:H]$ is infinite} \end{cases}\]
Determine if the following holds:
\[\lim_{n \to \infty} P(\forall i,j\leq n, \ [X_i, X_j] = e)^{\frac{1}{n}} = P(X_1 \in Z(G)) \ ? \]

Answer: No


Question: Given a linearly independent set of vectors $\{v_1, v_2, \ldots, v_k\}$ with $\|v_i\| \leq 1$ for all $1 \leq i \leq k$, determine if it is true that for any real numbers $\alpha_1, \alpha_2, \ldots, \alpha_k$ satisfying $\|\sum_{i=1}^k \alpha_i v_i\| = 1$, there exists a constant $M > 0$ such that $|\alpha_i| \leq M$ for all $1 \leq i \leq k$. Provide a proof or counterexample.

Answer: True


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \delta(x-a)\delta(y-bx) \, dy \, dx \) over the entire plane \( \mathbb{R}^2 \).

Answer: 1


Question:
Solve the transport equation given by:

$$ \begin{cases} \frac{d}{dt}u +cu = 0\qquad \mbox{ in } \mathbb{R}^n\times (0,\infty) \\
 u(x,0) = 0 \qquad \mbox{ on }\mathbb{R}^n\times \{t=0\} \end{cases} $$

where $c=(c_1,c_2,\ldots,c_n)$ is a constant vector.

Answer: 0


Question: Determine whether there exists a ring in which a finite degree polynomial has infinitely many roots.

Answer: Yes


Question: Are symmetric matrices with constant diagonals, where all diagonal entries are 1 and non-diagonal entries are between 0 and 1, guaranteed to have all eigenvalues greater than or equal to 0, making them positive semidefinite?

Answer: No


Question: Compute the depth of the polynomial ring $k[x,y]$ with respect to the ideal $(x,y^2)$, where $k$ is a field.

Answer: 2


Question: Differentiate the infinite series function: $$f(x) = \sum_{n=1}^\infty \frac{(-1)^{n-1} x^{2n-1}}{(n-1)!(2n-1)}.$$

Answer: e^{-x^2}


Question: Determine whether the improper integral \( \int^{+\infty}_0 \frac{\arctan x}{x^a} \sin x \, dx \) converges, where \( a \in \mathbb{R} \).

Answer: (0, 3)


Question: Determine whether the series \( \sum_{n=0}^{\infty} \frac{z^n}{2+z^n} \) converges using the ratio test.

Answer: |z| < 1


Question: Compute the integral \( \int_\gamma \frac{1}{(z+a)^n} \, \mathrm{d}z \), where \( \gamma(t) = re^{it} \) for \( t \in [0, 2\pi] \) and \( |a| > r \).

Answer: 0


Question: Evaluate the integral \( \int_C \frac{1}{(z^2-1)(z^2-4)}dz \) using the Residue Theorem, where \( C \) is the contour \( |z| = 3 \) traversed in the counterclockwise direction.

Answer: 0


Question: Evaluate the integral: $$\int_0^{2\pi} e^{\cos \theta} \sin ( \sin \theta) \cos (\cos \theta) d\theta$$

Answer: 0


Question: Determine the maximum temperature on the surface of a sphere with radius 1, where the temperature at any point \((x, y, z)\) on the surface is given by the function \(T(x, y, z) = 4x^2 + yz + 15\).

Answer: 19


Question: Evaluate the limit: $$ \lim_{n\to\infty} \frac {{n!}^{1/n}}{n}. $$

Answer: \dfrac{1}{e}


Question: Is there a group where two elements $a$ and $b$ have finite orders, but the order of their product $ab$ is infinite?

Answer: Yes


Question: Evaluate the limit using the Laplace Method: \[ \lim_{n \to \infty} \int_{0}^{1} nx^n e^{-x^2} \, dx \]

Answer: \dfrac{1}{e}


Question: Evaluate the limit of the infinite series: $$ \lim_{n \to \infty} \sum_{k=0}^n \left( \sum_{j=0}^k \binom{k}{j} \left(-\frac{1}{3}\right)^j \right). $$

Answer: 3


Question: Evaluate the limit \( \lim_{n \to \infty} f_n(x) \) where \( f_n(x) = n^2 x (1-x)^n \) for \( 0 < x < 1 \).

Answer: 0


Question: Find the limit of the integral \( \int_{-1}^1 (x^2+1)\cos(\lambda x)\,dx \) as \( \lambda \to \infty \) using the Riemann-Lebesgue Lemma.

Answer: 0


Question: Consider the closed ball \( B(0,1) \) in \( \mathbb{R}^n \) with respect to the Euclidean norm. Let \( f, g: B(0,1) \rightarrow \mathbb{R}^n \) be continuous functions such that \( f(x) \neq g(x) \) for all \( x \in B(0,1) \). Determine whether there exists an \( \epsilon > 0 \) such that \( B(f(x), \epsilon) \cap B(g(x), \epsilon) = \varnothing \) for all \( x \in B(0,1) \).

Answer: Yes


Question: Evaluate the limit: $$\lim_{{n \to \infty}} \frac{{(2 n)! \cdot e^n \cdot n^n}}{{n! \cdot (2 n)^{2 n}}}.$$

Answer: \sqrt{2}


Question: Calculate the sum of the series \( \sum_{j=0}^{\infty} (-1)^j \binom{2\alpha}{j} \), where \( \alpha > 0 \).

Answer: 0


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

Answer: -\dfrac{1}{45}


Question: True or False: If a square matrix $A = [A_{*1}, A_{*2}, ..., A_{*n}]$ satisfies $AA_{*j} = 0$ for every $j$ ($1 \leq j \leq n$), where $A_{*j}$ is the $j$th column of $A$, then $A$ must be the zero matrix.

Answer: B


Question: If the sum of all eigenvalues raised to the power of \( k \) for a square matrix is zero for all positive integers \( k \), does it imply that the matrix is nilpotent? Provide a justification for your answer.

Answer: Yes


Question: Given an odd function $f(x)$ that is twice differentiable, with $f(-1)=f(2)=f(3)=-1$, express $f(x)$ in the form $f(x)=a_{2n+1}x^{2n+1}+a_{2n-1}x^{2n-1}+\\cdots+a_{1}x$, where $a_{2n+1}, a_{2n-1}, \\ldots, a_{1}$ are constants. How many real solutions does the differential equation $(f'(x))^2+f''(x)=0$ have?

Answer: 2


Question: Provide an example of a sequence $(u_n)$ such that \(\lim \limits_{n\to \infty}(u_{n+1}-u_n)=l\) and \(\lim \limits_{n\to \infty}\frac{u_n}{n}=l\), where \(l\) is a real number.

Answer: u_n = l n


Question:
Consider a linear state system with the state matrix $\boldsymbol{A}$ and output matrix $\boldsymbol{C}$ given by:
$$ \boldsymbol{A} = \begin{bmatrix} μ & 1 & 0 & \dots & 0 \\ 0 & μ & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 &\ldots & 1 \\ 0 & 0 & 0 &\ldots & μ \end{bmatrix}, \quad \boldsymbol{C} = \begin{bmatrix} c_1 & c_2 & \dots & c_n \end{bmatrix} $$
The observability matrix $\boldsymbol{\mathcal{O}}$ is defined as:
$$ \boldsymbol{\mathcal{O}} = \begin{bmatrix} C\\ CA\\ CA^2\\ \vdots\\ CA^{n-1} \end{bmatrix} $$
Prove that the system is observable (i.e., the rank of $\boldsymbol{\mathcal{O}}$ is $n$) if and only if the first element of $\boldsymbol{C}$, $c_1$, is nonzero.

Answer: c_1 \neq 0


Question:
Consider the set $X=\left\{ \left\{x_n \right\}_{n=1}^{\infty} \mid x_n \to 0 \right\}$, where $\{x_n\}$ is a real sequence, equipped with the metric $d(x_n, y_n)=\sup_{n \in \mathbb{N}} |x_n - y_n|$. Define $E=\left\{ \left\{x_n \right\}_{n=1}^{\infty} \mid x_n \in X, \forall n \in \mathbb{N}, \ x_n \geq 0 \right\}$. Determine which of the following statements is true about $E$:

1. $E$ is not connected.
2. $E^\circ = \emptyset$ (where $E^\circ$ denotes the interior points of $E$).
3. $E' \neq E$ (where $E'$ denotes the limit points of $E$).
4. $E$ is not closed.

Answer: B


Question: Let $V$ be a vector space over a field $\mathbb{F}$ with dimension $k$. If $|\mathbb{F}| < k$, determine whether the dimension of $\text{End}(V)$ over $\mathbb{F}$ is greater than $k$. Justify your answer.

Answer: Yes


Question: Determine the existence and value of the limit \( \lim_{x \to 0^+} \frac{1+3x}{3+f(\frac{1}{x})} \), given that \( \lim_{x \to -\infty} f(x) = 4 \) and \( \lim_{x \to \infty} f(x) = 6 \).

Answer: \dfrac{1}{9}


Question: Is the weak$^*$ topology on the dual of a Banach space metrizable if the Banach space has a countable base?

Answer: No


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \( |f(x)-f(y)| \le |x-y|^3 \) for all \( x,y \in \mathbb{R} \) and \( f(2)=5 \). Find \( f(10) \).

Answer: 5


Question: Evaluate the limit \( M = \lim_{a \to \infty} E_x\left[\frac{1}{a + x\sqrt{a}}\right] \) where \( x \sim N(0, 1) \). Prove that \( M = 0 \).

Answer: 0


Question:
Let $k \in \mathbb{N}$ and consider positive real numbers $x, y,$ and $z$ such that $x + y + z = 1$. Determine the minimum value of the function $f(x, y, z)$ defined as:

$$f(x, y, z) = \frac{x^{k+2}}{x^{k+1} + y^k + z^k} + \frac{y^{k+2}}{y^{k+1} + z^k + x^k} + \frac{z^{k+2}}{z^{k+1} + x^k + y^k}$$

Answer: \dfrac{1}{7}


Question: Determine all real numbers \(x\) for which the series \(\sum_{n=1}^\infty\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)\frac{\sin(nx)}{n}\) converges.

Answer: \mathbb{R}


Question: Find the smallest positive real number $g$ such that for every positive integer $n$, the minimum value of $|c - d\sqrt{3}|$ for nonnegative integers $c$ and $d$ with $c + d = n$ is less than or equal to $g$. Provide a proof for your answer.

Answer: \dfrac{1 + \sqrt{3}}{2}


Question: Consider the region $D$ in $\mathbf{R}^3$ that lies above the paraboloid $z = x^2 + y^2 - 1$ and below the semi-sphere $z = \sqrt{1 - x^2 - y^2}$. Determine the volume of $D$ by evaluating the integral $\iiint z \, dV$. The intersection of the surfaces can be found by equating them: $$\sqrt{1 - x^2 - y^2} = x^2 + y^2 - 1$$ Squaring both sides to eliminate the square root: $$1 - x^2 - y^2 = (x^2 + y^2 - 1)^2$$ Expanding and simplifying: $$-1 = x^4 + y^4 - 2x^2 - 2y^2 + 1$$ Rearranging the terms: $$x^4 + y^4 - 2x^2 - 2y^2 = 0$$ This equation represents the intersection of the surfaces. Calculate the volume of the region $D$.

Answer: \dfrac{\pi}{12}


Question: Suppose \(\alpha\) is a real number within the range \(0\leq\alpha\leq\pi\), and the function \(f(x) = \cos x + \cos(x+\alpha) + \cos(x+2\alpha)\) attains a constant value \(c\) for all \(x \in \mathbb{R}\). Determine the value of \(\lfloor c+\alpha \rfloor\).

Answer: 2


Question: Calculate the remainder when $7^{12341}$ is divided by 1000 using Euler's theorem and modular arithmetic.

Answer: 7


Question: Determine whether the product of infinitely many rational numbers can be irrational.

Answer: Yes


Question: Let $R$ and $S$ be rings, and let $f: R \to S$ be a monomorphism. Determine whether $f$ is necessarily injective.

Answer: Yes


Question: Evaluate the limit \(\lim_{x \to 0} \frac{x\cosh(x)-x\sinh(x)}{\sqrt{1+x^4}-\sqrt{1-x^4}}\) using Taylor series expansions.

Answer: \infty


Question: Find the limit of the sequence: $$\lim_{n\to\infty}\sum_{k=1}^n\frac{k^r}{n^{r+1}}.$$

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


Question: Evaluate the infinite series: $$ \sum_{n=0}^\infty \frac{n(n+1)}{3^n} $$

Answer: \dfrac{9}{4}


Question: Find the bounded harmonic function \( h \) in the domain \( H = \{(x,y) \in \mathbb{R}^2 : y > 0\} \setminus B((1,0),r) \) for \( 0 < r < 1 \), such that \( h(x,0) = 1 \) for all \( x \in \mathbb{R} \), using the Cayley transform.

Answer: 1


Question: Determine the Galois group of the polynomial \( g(x) = x^4 - 2 \) over the field of rational numbers \( \mathbb{Q} \).

Answer: D_4


Question: Evaluate the expression \( \sec{(\arcsec{(-\frac{2}{\sqrt{3}})}+\arctan{(-\frac{1}{\sqrt{3}})})} \).

Answer: -2


Question: Given the permutation \(a=(1\ 6)(2\ 5\ 7)(3\ 8\ 4\ 9)\) in the symmetric group \(S_9\), determine the number of permutations \(b\) in \(S_9\) that commute with \(a\), i.e., satisfy \(a \star b = b \star a\).

Answer: 24


Question: Evaluate the integral \( \int x^2y^3\, \mathrm{d}V \) over the region bounded by a right circular cone with height \( h \) and base radius \( r \), where the cone's vertex is at the origin and its axis is along the \( z \)-axis.

Answer: 0


Question: Determine whether the following statement is true or false: If the sequence $\left\{ \sqrt[n]{c_n} \right\}$ converges in $\mathbb{R} \cup \{ \pm \infty \}$, then the sequence $\left\{ \frac{c_{n+1}}{c_n} \right\}$ also converges in $\mathbb{R} \cup \{ \pm \infty \}$ for any sequence $\left\{ c_n \right\}$ of positive real numbers.

Answer: B


Question: Given that $abc=8$ and $a,b,c >0$, find the minimum possible value of $(2+a)(2+b)(2+c)$. Express your answer as a single number.

Answer: 64


Question: Let \((a_n)_{n \ge 1}\) be a sequence of real numbers such that the sequence \(1 + \dfrac{1}{2} + \dfrac{1}{3} + \ldots + \dfrac{1}{n} - a_n \ln n\) is bounded for \(n \ge 1\). Determine the limit of the sequence \((a_n)_{n \ge 1}\).

Answer: 1


Question: If a subset $A$ of the real numbers $\mathbb{R}$ has an outer measure $m^*(A) > 0$, does it necessarily follow that $A$ contains at least one interior point? Answer yes or no and justify your answer.

Answer: No


Question: Calculate the variance of a Laplace distribution with median 0 and scale parameter \( \alpha \) using its characteristic function.

Answer: 2\alpha^2


Question: Determine the number of real solutions to the equation \( \sqrt{x+7-6\sqrt{x-2}}+\sqrt{x+12-8\sqrt{x-2}}=1 \).

Answer: 1


Question: Consider a convergent sequence \((a_n)\) with limit \(A\). Let \(f: \mathbb{N} \to \mathbb{N}\) be a bijective function from the set of natural numbers to itself. Define a new sequence \((b_n)\) as \(b_n = a_{f(n)}\). Determine if the limit of \((b_n)\) as \(n\) approaches infinity is equal to \(A\). Justify your answer with a proof or a counterexample.

Answer: A


Question: Find the limit as \( n \to \infty \) of the function \( f(n) = \frac{\ln(n+1)}{\ln(n)} \frac{\ln(n!)}{\ln((n+1)!)} \).

Answer: 1


Question: Given a simplicial complex with vertices $v_1, v_2, v_3, v_4, v_5$, and 1-simplices $e_1, e_2, \ldots, e_{10}$, construct the chain complex $C_2 \xrightarrow{\partial_2} C_1 \xrightarrow{\partial_1} C_0$ and compute the first homology group $H_1$. What is $H_1$?

Answer: \mathbb{Z}^6


Question: Determine the values of $v$ for which the rank of the matrix \( \begin{bmatrix} -\sin u + \sin (u+v) & \sin(u+v) \\ \cos u - \cos(u+v) & -\cos(u+v) \\ 1 & 0 \end{bmatrix} \) is equal to 2.

Answer: v \in \mathbb{R}


Question: Evaluate the limit of the sequence as \( n \to \infty \): \[ \lim_{n \to \infty} \frac{\sqrt{n}}{\left|c+\frac{d}{\sqrt{n}}\right|+|c|}\left(\left| c+\frac{d}{\sqrt{n}}\right|-|c|\right) \] where \( c, d \in \mathbb{R} \) and \( c \neq 0 \).

Answer: \dfrac{d}{2c}


Question:
Solve the system of congruences:

\[\begin{cases}
 x \equiv 2 \pmod{5} \\
 x \equiv 8 \pmod{11} \\
 x \equiv 5 \pmod{17}
\end{cases}\]

Answer: 107


Question: Determine the Galois group of the splitting field of the polynomial $x^4 - 2x^2 - 2$ over the finite field $F_7$, where $F_7$ is the field with 7 elements.

Answer: \mathbb{Z}/4\mathbb{Z}


Question: Evaluate the line integral \(\int _{\gamma }\:(x+y)dx-(x-y)dy\) where \(\gamma\) is the curve defined by \(y=1-|1-x|\) for \(x\in [0,2]\). Parameterize the path and compute the integral.

Answer: 4


Question: Solve for \( z \) in the equation \( z^3 = 52 + 47i \), given that \( z \) lies in the first quadrant of the complex plane.

Answer: 4 + i


Question: If $x < 0$, find the value of \( \frac{(1-x^2)^{\frac{3}{2}}}{x^2} \) given that \( \cot^{-1} \left(\frac{1}{x}\right)+\cos^{-1}(-x)+\tan^{-1}(x)=\pi \).

Answer: 2


Question: Let $f:(G,*) \rightarrow (H,\circ)$ be a homomorphism between groups $(G,*)$ and $(H,\circ)$. If $a \in G$ and $f(a)$ has an inverse in $H$, does it follow that $a$ has an inverse in $G$? Provide a justification for your answer.

Answer: Yes


Question:
Determine whether the following statement is correct for random variables $X$ and $Y$:

$$Var[aX+bY|Y]=Var[aX|Y] +Var[bY|Y]+2Cov[aX,bY|Y]$$

where $Var[bY|Y]=0$. Provide a justification for your answer.

Answer: Correct


Question: Evaluate the limit: $$\lim_{n \to \infty } \int_0^{\infty} x^{\frac{n}{n+1}} e^{-x^2} \, dx.$$

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \iint_D \sqrt{1-y^2} \, dx \, dy \) where \( D \) is the disk centered at \((1, 0)\) with radius 1.

Answer: \dfrac{8}{3}


Question: Show that the equivalence relation $A \sim A'$ defined by $A' = \pm A$ is compatible with the group structure of $SL_2(\mathbb{Z})$. In other words, prove that if $A \sim A'$ and $B \sim B'$ for $A, B, A', B' \in SL_2(\mathbb{Z})$, then $AB \sim A'B'$.

Answer: AB \sim A'B'


Question: Given a linear invertible map \( L \) on \( \mathbb{Q}^n \) such that \( L^{-1} = L^2 + L^4 \), determine if \( 5 \mid n \).

Answer: 5 \mid n


Question: Given the parabola \( y^2 = 2px \) with \( p > 0 \), a chord \( AB \) passes through the focus \( F\left(\frac{p}{2}, 0\right) \). The slope of \( AB \) is 2. If the distance between the endpoints of \( AB \) is 15, find the value of \( p \).

Answer: 6


Question: Consider an infinite series of coin tosses, where each heads results in a gain of $1 and each tails results in a loss of $1. Let $X_i$ denote the net gain after $i$ tosses, with $X_{i+1} = X_i + d_i$, where $d_i$ is a random variable taking values 1 or -1 with equal probability and is independent of all other $d_j$ for $i \neq j$. Define a Markov time $\tau_M$ as $\tau_M = \min \{ i \mid X_i = M \}$ for some $M \in \mathbb{Z}_+$. Prove that $P(\tau_M < \infty) = 1$ for all $M$. (In other words, show that with probability 1, the net gain will eventually reach any predetermined positive integer $M$.)

Answer: 1


Question:
Given the equations:

1. \( a^3 + b^3 + c^3 = 25 \)
2. \( \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ac} = 1 \)
3. \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{25}{9} \)

Find the value of \( a + b + c \).

Answer: \dfrac{25}{3}


Question: Given that matrices $A$, $B$, and $AB-I$ are invertible, find the inverse of the matrix $A-B^{-1}$. Provide a systematic approach to determine the inverse.

Answer: B(AB - I)^{-1}


Question: Let $S$ be a set of $n$ distinct real numbers, and $A_s$ be the set of arithmetic means of two distinct numbers from $S$. For a given $n \geq 2$, determine the minimum number of elements in $A_s$.

Answer: 2n - 3


Question: For what values of $m$ is the function \[ f\left(\begin{array}{c} x\\ y \end{array}\right)=\begin{cases} \frac{y^{m}\left(1-\cos x\right)}{x^{4}+2y^{2}} & \left(\begin{array}{c} x\\ y \end{array}\right)\neq\left(\begin{array}{c} 0\\ 0 \end{array}\right)\\ 0 & \left(\begin{array}{c} x\\ y \end{array}\right)=\left(\begin{array}{c} 0\\ 0 \end{array}\right) \end{cases} \] continuous at the point \(\left(\begin{array}{c} 0\\ 0 \end{array}\right)\)?

Answer: m > 1


Question: In a first-price sealed-bid auction with two bidders, each having either a low valuation \(V_L\) or a high valuation \(V_H\), determine the minimum bid \(\underline{b_i}\) for a bidder with a low valuation \(V_L\). Assume valuations are unknown to the other bidder, ties are resolved by a coin flip, and bidders use mixed strategies over the domain \([\underline{b_i},\overline{b_i}]\).

Answer: 0


Question: Determine whether the following statement is true or false: A measure space $(X, \mathbb{A}, \mu)$ is $\sigma$-finite if and only if there exists a measurable function $f : (X, \mathbb{A}) \to (\mathbb{R}, B(\mathbb{R}))$ which is strictly positive, piecewise constant, and has a finite integral with respect to $\mu$. Provide a justification for your answer.

Answer: True


Question: Determine whether the function \( f(x) = \begin{cases} x & x< \frac{1}{4} \\ \frac{1}{x} & x\geq \frac{1}{4} \end{cases} \) is Riemann integrable on the interval \([-1,1]\).

Answer: Yes


Question: Consider an operator $T$ on a finite-dimensional vector space that is upper-triangular with respect to some basis. Is $T$ also upper-triangular with respect to any orthonormal basis obtained from that basis using Gram-Schmidt orthogonalization?

Answer: Yes


Question:
Compute the probability $P(Y<X<Z)$ for the random variables $X$, $Y$, and $Z$ with the joint density function:

$$f(x,y,z) = \begin{cases}e^{-x-y-z}&\text{if }x,y,z>0\\ 0&\text{otherwise}\end{cases}$$

Answer: \dfrac{1}{6}


Question: Consider a circle with $2^n$ items on its periphery, each holding either 0 or 1. For each item, examine the sequence of n items to the right. Is it possible for this set of $2^n$ sequences to include all possible n-length combinations of 0s and 1s?

Answer: Yes


Question: Consider a rectangle with sides of length $X$ and $Y$, where $X$ and $Y$ are independent and exponentially distributed with parameter $\lambda = 1$. Calculate the probability that a circle with radius $r$ fits entirely within the rectangle.

Answer: e^{-4r}


Question: Can the set of vectors $\mathbb{Q}^2$ be generated by the vectors $(a,b)$ and $(c,d)$, where $a,b,c,d \in \mathbb{Q}$, if $ad-bc= \pm\frac{1}{2}$?

Answer: Yes


Question: Determine the cardinality of the image of the group homomorphism \( \varphi : SL_2 (\Bbb Z) \longrightarrow SL_2 (\Bbb Z/ 3 \Bbb Z) \) defined by \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \begin{pmatrix} \overline {a} & \overline {b} \\ \overline {c} & \overline {d} \end{pmatrix} \).

Answer: 24


Question: Calculate the expected value of the product $E[B_t \cdot B_v]$ for a Brownian motion process, given that $0 < t < v$. Provide your answer as a single value.

Answer: t


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

Answer: 1


Question: Find the derivative of the expression $b^tAc$ with respect to the matrix $A$ in matrix notation. What is the resulting derivative?

Answer: b c^T


Question: In a one-dimensional real inner-product space, can the commutator operation of a Lie algebra be defined as the inner product?

Answer: No


Question: Evaluate the series: \( \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!} \)

Answer: -1


Question: Determine the quotient group $\mathbb{C}^*/P$, where $\mathbb{C}^*$ is the group of non-zero complex numbers under multiplication, and $P$ is the subgroup of positive real numbers.

Answer: S^1


Question: Given that $a, b, c \geq 1$ and $a + b + c = 15$, determine the maximum value of the expression $abc - (a + b + c) + 3$. Provide your answer as a single number.

Answer: 113


Question: Let $(G, \cdot)$ be a group of order $n$. For a positive integer $p$, define the set \( M = \{(g_1, g_2, \dots, g_p) \in G^p : g_1 \cdot g_2 \cdots g_p = e_G \} \). Determine the number of elements in the set $M$.

Answer: n^{p-1}


Question: Determine whether there exists a countable set \( U \) of real numbers such that for all distinct \( a, b \in U \), the condition \(|a-b|<k\) holds for some constant \( k>0 \).

Answer: Yes


Question: Determine whether the Global Approximation Theorem holds for domains with piecewise $C^1$ boundaries that satisfy the segment condition.

Answer: Yes


Question: Let \( g(x) = \int_{0}^{x^{2}} e^{t} \, dt \). Find \( g''(2) \).

Answer: 18e^4


Question: Suppose a curve $C$ is defined by $F(X,Y) = 0$, where $F$ is a polynomial, and a line $l$ is tangent to $C$ at a point $P$. Let $M$ be an invertible linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$. Define a new curve $C'$ as the image of $C$ under $M$ and a new line $l'$ as the image of $l$ under $M$. Is $l'$ necessarily tangent to $C'$?

Answer: Yes


Question: Let $f$ be a real continuous function such that $f(0) = 0$ and satisfies the equation \[ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = nf(x) + \frac{1}{2} \] for all natural numbers $n$. Find the value of $f(1729)$.

Answer: 1729


Question: Evaluate the limit \( \lim_{x \to 0^+} \frac{\sqrt{x} - \int_0^{\sqrt{x}} e^{-t^2} \, dt}{\sqrt{x^3}} \) using L'Hôpital's rule.

Answer: \dfrac{1}{3}


Question: What is the meaning of $4\mathbb{R}$ in the context of the quotient ring $\mathbb{R} / 4\mathbb{R}$?

Answer: \mathbb{R}


Question: Determine the values of $a$ for which the set \( M_a = \{(x^1, \ldots, x^n, x^{n+1}) \in \mathbb{R}^{n+1} : (x^1)^2 + \cdots + (x^n)^2 - (x^{n+1})^2 = a\} \) is a submanifold of \( \mathbb{R}^{n+1} \) using the regular value theorem.

Answer: a \neq 0


Question: Determine the cardinality of a Hilbert space $H$ that has an orthonormal basis with cardinality $\mathfrak{c}$.

Answer: \mathfrak{c}


Question: In a group $G$, does the equality $ab = cd$ imply $axb = cxd$ for all elements $a, b, c, d, x$ in $G$, given that $a \neq c$ and $b \neq d$?

Answer: No


Question: Evaluate the integral \( \int _{0}^{4} \frac{1}{(2x+8)\, \sqrt{x(x+8)}}\, dx. \)

Answer: \dfrac{\pi}{24}


Question: Evaluate the limit: $$\lim_{x\to\infty}(x-\ln\cosh x)$$ where $$\cosh x=\frac{e^x+e^{-x}}{2}."

Answer: \ln 2


Question: Determine whether there exists a divergent sequence \(a_n\) such that the sequence \(b_n = \frac{\ln(-2a_n - 1)}{a_n + 1}\) is convergent.

Answer: Yes


Question: Given the implicit equation \( \ln(1+y) + \sin(xy) = \ln(5) \), find the second derivative of \( y \) with respect to \( x \), \( y''(0) \), at \( x = 0 \).

Answer: 280


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

Answer: 0


Question: Let $X$ be a separable Banach space and $X^{*}$ its dual space. Consider a weak* closed subspace $M$ of $X^{*}$. Given a linear functional $\varphi \in X^{*} \setminus M$, determine if there exists a $\psi \in M$ such that the infimum of the operator norms of the difference between $\varphi$ and elements of $M$ is attained, i.e., $$\inf\{\|\varphi - m\| : m \in M\} = \|\varphi - \psi\|.$$ Here, $\|\cdot\|$ represents the operator norm of linear functionals on $X^{*}$. Answer with 'True' if such a $\psi$ exists, otherwise 'False'.

Answer: True


Question: Find the integral part of \( \sqrt{2018+\sqrt{2018+\sqrt{\ldots+2018}}} \), where the number of occurrences of 2018 is 2018.

Answer: 45


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{\left\lfloor \dfrac{n+1}{2} \right\rfloor!}{n!}.$$

Answer: 0


Question: Calculate the expected value of the random variable \( S = \sum_{i=1}^{N} \frac{X_i}{2^i} \), where \( X_1, X_2, \ldots \) are i.i.d. random variables uniformly distributed on \([0,1]\), and \( N \) is the smallest positive integer \( k \) such that \( X_k < X_{k+1} \). If no such \( k \) exists, then \( N = \infty \). Given that \( \Pr(N = n) = \frac{n}{(n + 1)!} \), find \( \mathbb{E}[S] \).

Answer: 2\sqrt{e} - 3


Question: Evaluate the double integral \( \iint_{\mathbb{R}^2_+} e^{-(x^2+y^2)} \, dx \, dy \), where \( \mathbb{R}^2_+ \) denotes the first quadrant of \( \mathbb{R}^2 \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the double limit of the integral: $$\lim_{n \rightarrow \infty} \lim_{R \rightarrow \infty} \int_0^R \sin\left(\frac{x}{n}\right) \sin(e^x) \, dx$$

Answer: 0


Question: Determine the limit \( \lim_{x\to \infty}\frac{\pi_1(x)}{\pi_3(x)} \), where \( \pi_1(x) \) is the number of primes less than \( x \) that are congruent to 1 modulo 4, and \( \pi_3(x) \) is the number of primes less than \( x \) that are congruent to 3 modulo 4.

Answer: 1


Question: Find the upper Riemann integral of the function \( f(x) = \begin{cases} x, & x \in [0,1] \cap I \\ 0, & x \in [0,1] \cap Q \end{cases} \) over the interval \([0,1]\). Justify your answer.

Answer: \dfrac{1}{2}


Question: Evaluate \( \lim_{n \to \infty} \int_0^n \frac{dx}{1 + n^2 \cos^2 x} \).

Answer: 1


Question: Determine whether all Banach algebra norms on a commutative Banach algebra are equivalent. Answer with True or False.

Answer: False


Question: Determine the weak convergence of the sequence of probability measures $(\mu_n)_{n \in \mathbb{N}}$ on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, where $\mu_n = f_n \lambda$ and $f_n(x) = \sqrt{\frac{n}{2\pi}}e^{-\frac{nx^2}{2}}$. Here, $\lambda$ is the Lebesgue measure.

Answer: \delta_0


Question: Determine the norm of the operator \( T: f \rightarrow \int_{0}^{x} f(y) \, dy + \int_{x}^{1} f(y) \, dy \) defined on \( L^2([0,1]) \).

Answer: 1


Question: Identify a non-abelian group \( G \) such that its abelianization \( G/[G, G] \) has a proper, non-trivial homomorphic image.

Answer: D_4


Question: Solve the inequality for $x > 0$: $$ \Gamma(x) \leq \frac{1}{x} $$

Answer: (0, 1]


Question: Find the number of spanning trees in the complete graph $K_{12}$ that have exactly two vertices with degree 6.

Answer: 16632


Question: Determine the number of solutions to the equation \(x^{3} + \sin(x) = 20\) using the concept of monotonicity.

Answer: 1


Question: Determine whether it is possible to construct a field with 16 elements by modding a ring out by an irreducible polynomial over \( \Bbb Z_2 \).

Answer: Yes


Question: Find the number of solutions to the equation $x^n = c$ in a finite field $\mathbb{F}_q$, given that $n$ and $q-1$ are relatively prime.

Answer: 1


Question: For a graph $G$ with $n$ vertices, how many subgraphs are both induced and spanning?

Answer: 1


Question: Determine whether the following statement is true or false: Given a convergent series $\sum a_n$ with $a_n > 0$, there exists a sequence $b_n = \frac{1}{n^{1+\alpha}}$, where $\alpha > 0$, such that $\lim_{n\to\infty} \frac{a_n}{b_n} = L > 0$. Provide a justification for your answer.

Answer: False


Question: Determine the complex number obtained by multiplying together all the solutions of the equation $z^6 = -1$. Use the factorization $z^6 + 1 = (z - z_0)(z - z_1)(z - z_2)(z - z_3)(z - z_4)(z - z_5)$, where $z_0, z_1, z_2, z_3, z_4, z_5$ are the solutions.

Answer: 1


Question: Consider a circle with radius 1 and $n$ points randomly distributed on its circumference. Let $k$ be an even number. If we randomly select $k/2$ pairs of points to form a matching, what is the expected sum of the distances between the paired points?

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


Question: Evaluate the limit: $$\lim _{x\to \infty }\left[\left(\frac{x+2}{x+3}\right)^b-\cos\left(\frac{2}{x}\right)\right]^x$$

Answer: 0


Question: Evaluate the integral \( \int_{1}^{\infty} \frac{(x-1)^{51}}{x^{53}} \, dx \) using the substitution method and the binomial theorem.

Answer: \dfrac{1}{52}


Question: Evaluate the limit: $$\lim_{x\to \infty}\left(\frac{x}{x^2+1}+\frac{x}{x^2+2}+\cdots +\frac{x}{x^2+x}\right)$$ without using the squeeze theorem.

Answer: 1


Question: Solve the equation: \( \sqrt[3]{10-x} + \sqrt[3]{30-x} = \sqrt[3]{15-x} + \sqrt[3]{25-x} \). Find the value(s) of \( x \) that satisfy this equation.

Answer: 20


Question: Find the minimum value of the function \( f(x, y) = \frac{1}{x} + \frac{1}{y} + 2x + 2y \) for \( x, y > 0 \) using the theorem that states: If \( f:A\rightarrow\mathbb{R} \) is convex and \( \xi \) has a directional derivative in every direction with \( \nabla f(\xi)=0 \), then \( f(\xi)=\min\{f(x):x\in A\} \).

Answer: 4\sqrt{2}


Question: As the order of trees tends to infinity, does the ratio of the number of trees of order \(2n\) with a perfect matching to the total number of trees of order \(2n\) converge to zero?

Answer: 0


Question: Find the slant asymptote of the function \( y = x \cot^{-1}(x) \). Assume the slant asymptote is of the form \( y = ax + b \) and satisfies \( \lim_{x\to \infty} (f(x) - (ax + b)) = 0 \).

Answer: y = \pi x + 1


Question: Determine whether there exists a function $f$ that is holomorphic in the punctured disk $\{0<|z|<1\}$ such that $$\lim_{z\to 0} \frac{f(z)^2}{z^2}=1.$$

Answer: Yes


Question: Evaluate the limit \( \lim\limits_{n \to \infty }\cos \cos \dots\cos n \) for \( n \) cosine functions.

Answer: 0.739085


Question: Suppose $F$ is a field where every element $x \in F$ satisfies $x^k = 1$ for some positive integer $k$. Does this imply that the characteristic of $F$ is strictly positive?

Answer: Yes


Question: For a vector space $V$ of dimension $n$, does a change of basis matrix $T$ induce an isomorphism from $V$ to itself? Answer yes or no and justify your answer.

Answer: Yes


Question: Determine whether the subspace of continuous functions $C[0,1]$ with $f(0)=0$ is closed under the metric $$ d(f,g) = \max|f(x)-g(x)|. $$

Answer: Yes


Question: Calculate the length of the curve given by the parametric equations: \( x(t) = 1 + 2\cos(t) + \cos(2t) \), \( y(t) = 2\sin(t) + \sin(2t) \), for \( 0 < t < 2\pi \).

Answer: 16


Question: Determine if the function \[ f(x)= \begin{cases} x^2\cos\left(\frac{\pi}{x^2}\right) &\quad\text{if} \hspace{5mm} x\in (0,1]\\ 0 &\quad\text{if} \hspace{5mm} \text{x = 0} \end{cases} \] is differentiable at the point \(x = 0\) using the limit definition of the derivative.

Answer: 0


Question: Determine the values of \( x \) for which the series \( \sum_{n\ge 1}\frac{2^{n-3}}{x^{n}(x+n-3)^{0.5}} \) converges.

Answer: (2, \infty)


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \lfloor n \cdot \frac{\sin x}{x}\rfloor + \lfloor n \cdot \frac{\tan x}{x} \rfloor \right) \] where \( n \in \mathbb{N} \).

Answer: 2n - 1


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{a(1-e^{-x})+b(e^x-1)}{\sqrt{a(e^{-x}+x-1)+b(e^x-x-1)}}$$ where \(a\) and \(b\) are integer constants.

Answer: \sqrt{2(a + b)}


Question: Evaluate the limit \( \lim_{x \to 1^{-}} (1 - x^2)^{\frac{1}{\log(1 - x)}} \) without using L'Hôpital's rule.

Answer: e


Question: Determine whether the following statement is true or false: If the Cesàro mean of a sequence converges, then the original sequence must also converge.

Answer: B


Question: Determine the radius of convergence of the series \( \sum_{n=0}^{\infty}{4^n\cdot x^{2^n}} \).

Answer: 1


Question: Is a commutative ring $A$ regular if it is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$, the dimension of the localization $A_{\mathfrak{m}}$ equals the dimension of $A$?

Answer: No


Question: Suppose $f: [a, b] \rightarrow \mathbb{R}^n$ is Lipschitz continuous and differentiable on $(a,b)$. Let $g: [a, b] \rightarrow \mathbb{R}^n$ be a continuous function such that $g(x)$ is the derivative of $f$ at $x$ for all $x \in (a,b)$. Is $f$ necessarily differentiable at $b$ with the derivative $g(b)$?

Answer: Yes


Question: Determine the maximum number of positive integers that can be selected such that the sum of any 1002 of them is not divisible by 1002.

Answer: 2002


Question: Given the function \( f(x) = (60073-x^{10})^{1/10} \), find the positive integer \( a \) such that \( f'(a) = \frac{1}{f'(2)} \) and \( f(2) = 3 \).

Answer: 3


Question: Evaluate the integral \( \oint \frac{\mathrm{d}z}{q(z)} \) where \( q(z) \) is a polynomial of degree \( m \geq 2 \) with no repeated roots, and the closed path is large enough to enclose all the roots of \( q(z) \).

Answer: 0


Question: Compute the volume of the convex polyhedron with vertices \((0,0,0)\), \((1,0,0)\), \((0,2,0)\), \((0,0,3)\), and \((10,10,10)\) using a triple integral.

Answer: \dfrac{110}{3}


Question: Provide an example of a function $f(x)$ such that the integral \( \int_{1}^{\infty} f(x) \, dx \) diverges, while the integral \( \int_{1}^{\infty} f(x^2) \, dx \) converges.

Answer: \dfrac{1}{x}


Question: Determine the eigenvalues of the matrix: $$ A = \begin{pmatrix} 1&1&1 \\ 1&w&w^2 \\ 1&w^2&w^4 \end{pmatrix} \begin{pmatrix} 1&0&0 \\ 0&-1&0 \\ 0&0&0 \end{pmatrix} \begin{pmatrix} 1&1&1 \\ 1&1/w&1/w^2 \\ 1&1/w^2&1/w^4 \end{pmatrix} $$ where $w$ is a non-real cube root of unity. Choose the correct option: (a) $1,-1,0$ (b) $\frac{1}{3},-\frac{1}{3},0$ (c) $1,w,w^2$ (d) $3,-3,0$.

Answer: D


Question: Four points A, B, C, and D lie on a circle in that order. The extensions of AB and DC meet at point E, and the extensions of AD and BC meet at point F. Tangents EP and FQ are drawn from points E and F to the circle, touching the circle at points P and Q, respectively. If EP = 60 and FQ = 63, find the length of segment EF.

Answer: 87


Question: Determine whether the following statement is true or false: A Riemann integrable function $f:[a,b] \rightarrow \mathbb{R}$ with $\int_a^b f(x) \, dx = 0$ must be zero almost everywhere on $[a,b]$. Provide a justification for your answer.

Answer: B


Question: Determine whether there exists a countable subset $C$ of polynomials with rational coefficients such that every continuous function on a closed interval $[a, b]$ can be uniformly approximated by polynomials from $C$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{1}{n}\int_{0}^{n\pmod {2\pi}} \arcsin(\sin(t))dt \), where \( n\pmod {2\pi} \) represents the remainder when \( n \) is divided by \( 2\pi \).

Answer: 0


Question: Determine the number of subgroups of order 5 in the symmetric group $S_5$. Provide your answer as a single integer.

Answer: 6


Question: Solve the equation for x: $$x^{\log_{10}5}+5^{\log_{10}x}=50$$

Answer: 100


Question: Evaluate the double integral \(\int^{1}_{0}\int^{\sqrt{1-x^2}}_{0}xy(x^2+y^2) \, dy \, dx\) using a suitable substitution.

Answer: \dfrac{1}{12}


Question: Determine the values of $s \in \mathbb{R}$ for which the integral \[ \iint_{(a,b)^2}\frac{1}{|x-y|^s}\mathrm{d}x\mathrm{d}y \] converges, where $(a,b)$ is a finite open interval of $\mathbb{R}$ and $s > 0$. Provide the range of $s$ for which the integral is convergent.

Answer: (0, 1)


Question: Determine the number of distinct two-dimensional subspaces in a four-dimensional vector space over a finite field with 5 elements.

Answer: 806


Question: Find the centroid of the part of a spherical shell centered at the origin with radius 30 that lies in the first octant. The shell is defined by the equation \( z = \sqrt{30^2 - x^2 - y^2} \). Assume symmetry in the first octant.

Answer: (15, 15, 15)


Question: Find all natural numbers \(p\) and \(q\) such that the function \(f(x,y)\) defined by \[ f(x,y) = \begin{cases} \frac{x^p y^q}{x^2 + y^2} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y) = (0,0) \end{cases} \] has continuous partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\), and \(f\) is differentiable at \((0,0)\).

Answer: p + q \geq 4


Question: Given the differential equation \( \dddot{y} + 3\ddot{y} + 5\dot{y} = \dddot{x} + 4\ddot{x} + 6\dot{x} + 8x \), where all initial conditions are zero, determine the transfer function \( G(s) = \frac{Y(s)}{X(s)} \). Then, calculate \( y(t) \) as \( t \to \infty \) when \( x(t) = \delta(t) \).

Answer: \dfrac{8}{5}


Question: Evaluate the limit \( \lim_{n\rightarrow \infty}\frac{1}{\ln (n)}\sum^{n}_{k=1}\frac{a_{k}}{k} \) using the Stolz-Cesàro theorem, given that \( \lim_{n\rightarrow \infty} a_{n} = a \) for a sequence \( a_{1}, a_{2}, a_{3}, \ldots, a_{n} \).

Answer: a


Question: Find all matrices $B$ such that $B = -B^T$ and $B + B^T = 2I$. Provide your answer as a set of matrices.

Answer: \emptyset


Question: Compute the limit \( \lim\limits_{n\to\infty}\int_{[0,1]}\frac{nx^{n-1}}{1+x} \, d\lambda(x) \) without using the Lebesgue dominated convergence theorem.

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ L = \lim_{x \to \infty} \mathrm{e}^{-x} \int_{0}^{x} f(y) \mathrm{e}^{y} \, \mathrm{d}y \] where \( f(x) \) is a continuous function defined on \((0, \infty)\) and \( \lim_{x \to \infty} f(x) = 1 \).

Answer: 1


Question: A ship $S_1$ departs from port A at 10:00, moving in a straight line with its position at time $t$ hours after 10:00 given by $r=t(10, 20)$. A speedboat $S_2$, capable of traveling at three times the speed of $S_1$, must meet $S_1$ by traveling the shortest possible distance. Determine the latest time that $S_2$ can leave port B, located at the point $(70, 30)$. Provide your answer in hours after 10:00.

Answer: \dfrac{28}{15}


Question: Determine whether there is at least one solution to Pell's equation for each prime number.

Answer: Yes


Question:
Determine whether there exists an operator $T$ on a Banach space $X$ such that:

1. The point spectrum of $T$ and the spectrum of $T$ are both equal to $\{0\}$.
2. The kernel of $T$ is of finite dimension.
3. The range of $T$ is a closed and finite-codimensional subspace of $X$.

Provide a justification for your answer.

Answer: Yes


Question:
Given angles $\alpha$ and $\beta$ in a triangle, where $\alpha, \beta \in (0, \pi)$, determine if $\alpha = \beta$ given that:

$$\left(\frac{\sin \alpha}{\sin \beta}\right)^{100}= \frac{\sin^2 \frac {\alpha}{2}}{\sin^2 \frac{\beta}{2}}$$

Answer: \alpha = \beta


Question: Determine whether there exists a surjective map from the space of continuous functions $C([a,b],X)$, where $X$ is a Hilbert space, to the Hilbert space $X$ itself.

Answer: Yes


Question: Let $\gamma:[0,1]\to \mathbb S^2$ be a smooth curve. Consider the set $$ \bigcup_{t\in [0,1]}\mathrm{span}(\gamma(t))\subset \mathbb R^3. $$ Determine if this set has Hausdorff dimension at most 2, and consequently, if it has zero 3-dimensional Lebesgue measure.

Answer: Yes


Question: Determine whether every isomorphism between the spaces of continuous functions on a compact Hausdorff space $X$ and itself is continuous when considering $C(X)$ as a $C^*$-algebra with the usual operations.

Answer: Yes


Question: Find the area of the triangle formed by the line \(x + y = 3\) and the angle bisectors of the pair of straight lines \(x^2 - y^2 + 2y = 1\).

Answer: 2


Question: Suppose that $X$ and $Y$ are independent standard normal random variables. Calculate the probability $P(X^2 + Y^2 < 1)$. Express your answer as a decimal rounded to four decimal places.

Answer: 0.3935


Question: Solve the equation over the real numbers: \((x^2+x+1)^{1/3}+(2x+1)^{1/2}=2\)

Answer: 0


Question: Determine whether there exists a subadditive bijection \( f \) of the positive reals \((0,\infty)\) such that either \( \liminf_{x\to 0^+}f(x)=0 \) or \( \limsup_{x\to 0^+}f(x)=1 \).

Answer: Yes


Question:
Evaluate the limit as \( x \to 0 \) of the function:
\[ \frac{|x|}{\sqrt{x^{4}+4x^{2}+7}} \sin \left(\frac{1}{3 \sqrt{x}}\right) \]

Answer: 0


Question: Evaluate the limit \(\lim_{x\to\infty}\frac{\int_0^{2x}\sqrt{1+t^2}dt}{x^2}\).

Answer: 2


Question: Given the differential equation \( y'(x) = y(x) + \int_0^1 y(x) \, dx \) with the initial condition \( y(0) = 1 \), find the value of \( y\left(\ln\left(\frac{11-3e}{2}\right)\right) \).

Answer: 4


Question: Given the functional equation \(4f(x^2 + y^2) = (f(x) + f(y))^2\), determine the value of \(f(3)\) if \(f: \mathbb{N} \rightarrow \mathbb{N}\) satisfies \(f(0) = 0\) and \(f(1) = 4\).

Answer: 36


Question: Determine the cardinality of the set of all sequences of natural numbers, where each sequence can have a different finite length.

Answer: \aleph_0


Question: Determine whether a meromorphic function defined on the entire complex plane with at most one pole can have an infinite number of fixed points.

Answer: Yes


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

Answer: 3


Question: Is the definition of the improper integral \( \int_{a}^{\infty}f(x)dx \) as \( \lim\limits_{R \to \infty} \int_{a}^{R}f(x)dx \) correct?

Answer: Yes


Question: Determine the limit of the imaginary part of the non-trivial zeros of the Riemann zeta function as the index $n$ approaches infinity.

Answer: \infty


Question: Evaluate the integral \( \int_{-\pi }^{\pi } \frac{1}{25\sin^2x+144} \, dx \).

Answer: \dfrac{\pi}{78}


Question: Evaluate the limit: \[ A = \lim_{{(x,y) \to (0,0)}} \frac{{x^4 + y^4}}{{\sqrt{{x^2 + y^4}}}} \]

Answer: 0


Question:
In the equation for the incomplete Gamma function given by:

$$ \Gamma(a,x)=\frac{\rho^{-x}x^{a}}{\Gamma(1-a)} \int_0^\infty \frac{e^{-t} t^{-a}}{x+t} dt $$

for \(a < 1\) and \(x > 0\), identify the correct value of \(\rho\) if it is a typographical error.

Answer: e


Question:
Find all polynomial functions \( f(x, y) \) of two variables that satisfy the following conditions:
1. \( f(1, 2) = 2 \)
2. The functional equation \( yf(x, f(x, y)) = xf(f(x, y), y) = (f(x, y))^2 \) holds for all \( x \) and \( y \).

Answer: xy


Question: Let $k$ be a fixed natural number and consider the matrix $$A=\begin{bmatrix}a_1&*&*&...&*\\0&a_2&*&...&*\\0&0&a_3&...&*\\ \vdots & \vdots & \vdots & \ddots & \vdots\\0&0&0&...&a_n\end{bmatrix}$$ where $a_i^k=1$ for every natural $i$. The $a_i$ are pairwise distinct and $*$ denotes a random complex number. Determine if $A^k=E$, where $E$ is the identity matrix.

Answer: A^k = E


Question: Determine whether there is a closed-form expression for the limit $C$ of the series $\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$, where $C \approx 1.8695973$. Provide a justification for your answer.

Answer: C


Question: Evaluate the limit: $$\lim_{n\to \infty}{\sqrt{(1-\cos(1/n))\sqrt{(1-\cos(1/n))\dots}}}$$

Answer: 0


Question: Find a function that is not bounded below and whose second derivative is bounded above.

Answer: -x^2


Question: Evaluate the limit: $$\lim_{x\to\infty}\frac{f^{-1}(x)}{\ln(x)}$$ where $f(x)=e^x+x^3-x^2+x$.

Answer: 1


Question: Determine the interval on which the polynomial \( y = -\frac{3\pi}{2} + x - \frac{(2x-3\pi)^3}{48} \) best approximates the function \( y = \cos(x) \).

Answer: [\pi, 2\pi]


Question: Determine the maximum number of Jordan Canonical Form blocks for a $5 \times 5$ matrix with a minimal polynomial $(x-2)^2(x-3)$, given that the geometric multiplicity of the eigenvalue $2$ is $2$. Assume there are two linearly independent eigenvectors associated with the eigenvalue $2$. What is the maximum number of Jordan blocks?

Answer: 4


Question: Evaluate the double integral \( \int_{0}^{\infty}\int_{0}^{\infty}|\sin(x^2+y^2)|\,dx\,dy \).

Answer: \infty


Question: Calculate the volume of the solid bounded by the surfaces $y^2 + z^2 - 2z = 0$, $x = y^2 + z^2$, and $x = 0$ using triple integrals. Determine the appropriate intervals of integration.

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


Question: Find the value of $\arctan(a) + \arctan(b) + \arctan(c)$, where $a$, $b$, and $c$ are the roots of the polynomial $P(x) = x^3 - 4x^2 + x + 1$. Express your answer in terms of known constants or functions.

Answer: \dfrac{\pi}{2}


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

Answer: 1


Question: Find the minimum value of the expression $x^2 + 8y^2 + 27z^2$ given that $x, y, z$ are positive real numbers satisfying $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$. Provide your answer as a single number.

Answer: 216


Question: Determine the values of \(\theta\) for which the set \(S = \{x \in \mathbb{R}^n : |x-a| \leq \theta|x-b|\}\) forms a ball.

Answer: [0, 1)


Question: Determine whether the following statement is true for all natural numbers $m$ and $n$: \[ \sum_{k=1}^{m+n} \binom {m+n}{k} k^m (-1)^k = 0. \]

Answer: True


Question: Evaluate the double integral \( \int \int _D \sqrt{x^2+y^2} \, dA \), where \( D \) is the interior of the circle defined by \( x^2+y^2=2x \).

Answer: \dfrac{32}{9}


Question: Determine whether there exist rational numbers \( q \in (0,1) \cap \mathbb{Q} \) such that \( \sin\left(\frac{\pi}{2}q\right) \in \mathbb{Q} \). If so, list all such numbers.

Answer: \dfrac{1}{3}


Question: Evaluate the contour integral \( I = \frac{1}{2\pi i} \oint_{\gamma} z^7 \cos\left(\frac{1}{z^2}\right) dz \) using the residue theorem, where \( \gamma = \{z \in \mathbb{C} : |z| = 2\} \).

Answer: \dfrac{1}{24}


Question: Evaluate the limit: \( \lim_{n\to \infty} \{(3+2\sqrt{2})^n\} \), where \( \{x\} = x - \lfloor x \rfloor \) denotes the fractional part of \( x \).

Answer: 1


Question: Determine whether it is true that for any prime \( p \geq 3 \) and integer \( m \) dividing \( p - 1 \), the prime \( p \) does not divide the sum \[ \sum_{t=0}^{\frac{p-1}{m}-1}\sum_{s=0}^{\frac{p-1}{m}-1}\binom{tm}{sm}. \]

Answer: False


Question: Calculate the value of \( S = \lim_{n \rightarrow \infty} \frac{2\pi}{n} \sum_{k=0}^{n} \left| \cos(k\pi(3-\sqrt{5})) \right| \).

Answer: 4


Question:
Determine if the function \( g: \mathbb{R}^2 \rightarrow \mathbb{R} \), defined by

\[ g(x,y) = \frac{|x| y}{|x| + |y|} + 3x - 2y, \]

has a linear approximation at the point \((1,1)\).

Answer: Yes


Question: Determine the smallest number \( k \) of elements \( s_1, \ldots, s_k \in M \), where each \( s_i = A_i\text{SO}(3) \) for some \( A_i \in \text{SL}(3,\mathbb{R}) \), such that the simultaneous stabilizer of all \( s_i \) in \( \text{SL}(3,\mathbb{R}) \) is the identity matrix.

Answer: 3


Question: After 1977 pourings, how many liters of water remain in the first urn if it initially contains 1 liter, and during each alternate pouring, a fraction of the water is transferred between the first and second urns according to the following rule: 1/2 of the water in the first urn is poured into the empty second urn, then 1/3 of the second urn's water is returned to the first, followed by 1/4 of the first urn's contents to the second, and so on, with the denominator increasing by 1 with each pouring?

Answer: \dfrac{1}{2}


Question: Determine the coefficient of \(\frac{1}{x^2}\) in the Laurent series expansion of \(\frac{1}{(x-1)\cdot\cos x}\) about \(x=0\).

Answer: 0


Question: In a compact, Hausdorff uniform space $(X, \mathcal{U})$, let $S$ be a closed set contained within an open set $W$. Define $U[x]=\{y: (x, y)\in U\}$ and $U[S]=\cup_{x\in S}U[x]$. Determine whether there exists a $U\in \mathcal{U}$ such that $S\subseteq U[S]\subseteq W$. Provide a justification for your answer.

Answer: Yes


Question: Let $(u_{n})_{n},(v_{n})_{n}$ be two strictly positive real sequences satisfying the following condition: There exist a positive integer $N$ and a positive real number $a$ such that for all $n > N$, $$\frac{u_{n}}{v_{n}} \leq a.$$ Determine whether it is true that $$\frac{u_{n}}{v_{n}} \leq a \iff \frac{1}{a} \leq \frac{v_{n}}{u_{n}}.$$

Answer: True


Question: Determine the value of \( \lim_{N \rightarrow \infty} \frac{f(N)}{N^2} \), where \( f(N) \) is the number of points \((x, y, z)\) in the positive integer plane satisfying \( x + y + z = N \) and \( x \leq y \leq z \). For the unconstrained case, \( f(N) = \binom{N-1}{2} \). How does the constraint \( x \leq y \leq z \) affect the solution? Is the solution simply \( \frac{\binom{N-1}{2}}{3!} \)?

Answer: \dfrac{1}{12}


Question: Evaluate the limit: \[ \lim _{x \to 3} \int^{f(x)}_{3} \frac{2 t^3}{x-3} dt \] where \( f \) is a differentiable function such that \( f(3)=3 \) and \( f'(3)= \frac {1}{2} \).

Answer: 27


Question:
Given a two-dimensional random variable (X, Y) with joint probability density function:

$$ f(x,y) = \left\{ \begin{array}{ll} 2x^2 & \textrm{when $-1<x<1$ and 0<y<|x|$}\\ 0 & \textrm{otherwise}\\ \end{array} \right. $$

Compute the conditional expectation: $$E(X-Y|X+Y=-0.5)$$

Answer: -\dfrac{31}{28}


Question: Let $T$ be an invertible linear operator on a finite-dimensional vector space over a finite field. Determine whether there exists a positive integer $n$ such that $T^n = I$, where $I$ is the identity operator. Provide a justification for your answer.

Answer: Yes


Question: Given the function \( f(x) = \frac{x}{1 + \log_2(x)} \), find an asymptotic expression for its inverse function, \( f^{-1}(y) \), as \( y \to \infty \). Determine if \( f^{-1}(y) \) is in the order of \( O(y \log y) \) or \( O(y \log^k y) \) for some integer \( k \).

Answer: O(y \log y)


Question: Find the points in the complex plane where the function \( f(z) = |z|^2 + i\bar{z} + 1 \) is differentiable.

Answer: -i


Question: Determine whether Tarski Monster Groups exist for the primes $3$ or $5$. Provide a yes or no answer.

Answer: No


Question:
Find the coefficient of \(x^{28}\) in the expansion of \((1+x+x^{2}+x^{3}+
\ldots+x^{27})(1+x+x^{2}+x^{3}+
\ldots+x^{14})^{2}\).

Answer: 224


Question: Evaluate the limit of the expression \((-1)^{n+1}n!\left(1-e\sum_{k=0}^n\frac{(-1)^k}{k!}\right)\) as \(n\) approaches infinity.

Answer: 0


Question: Given the function $L(x) = x - \frac{x^2}{4}$, let $a_n$ be the result of applying $L$ to itself $n$ times, starting with $\frac{17}{n}$ as the initial value. Determine the limiting value of $n \cdot a_n$ as $n$ approaches infinity.

Answer: \dfrac{68}{21}


Question: Given the differential equation \( F'' + 2(\xi F)' = 0 \) and the condition \( F'(0) = 0 \), deduce that the first integral of the equation is \( F' + 2\xi F = 0 \).

Answer: F' + 2\xi F = 0


Question: Determine whether the set \( E' = \{x \in [0,1] : |x - \frac{p}{q}| < q^{-2} \text{ for finitely many } p,q \in \mathbb{N} \} \) has Lebesgue measure 0 or 1.

Answer: 0


Question: Let $B \subseteq \mathbb{R}$ be a non-empty set such that $1$ is a limit point of $B$. Is the set $WB := \{lb : l \in \mathbb{Q}, b \in B\}$ necessarily dense in $\mathbb{R}$? Provide a justification for your answer.

Answer: Yes


Question: Given a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x) = 0$ for $|x| \geq 10$, consider the function $g(x) = \sum_{k \in \mathbb{Z}} f(x+k)$. Which of the following statements is true about $g$?\n\n1. $g$ is differentiable and $g'$ has infinitely many zeros.\n2. $g$ is continuous and $g'$ has no zeros.\n3. $g$ is differentiable and $g'$ has no zeros.\n4. $g$ is differentiable and $g'$ has only finitely many zeros.\n\nSelect the correct statement and justify your choice.

Answer: 1


Question: How many unique 7-tuples of integers (1, 2, or 3) exist such that the sum of the integers is 12, and two tuples are considered equivalent if one can be obtained by circularly shifting the other?

Answer: 38


Question: Find the monic polynomial of least degree in \( \mathbb{R}[x] \) that is a square root of \(-4\) modulo the ideal generated by \((x^2 + 1)^2\).

Answer: x^3 + 3x


Question:
Find the maximum likelihood estimator (MLE) for \( \theta \) given that \( X_1, \ldots, X_n \sim_{iid} X \) with \( x \in [-1,1] \), \( \theta = -1, 1 \), and the probability density function:
\[
f_X(x;\theta)= \begin{cases} \frac{1}{2} & \text{if $\theta=1$} \\
\frac{1}{2(1-x^2)} & \text{if $\theta=-1$}\end{cases}\]

Answer: -1


Question: Suppose $a$ and $b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183$ and $a\sqrt{b} + b\sqrt{a} = 182$. Find the value of $\frac{9}{5}(a+b)$.

Answer: 73


Question: Find the residue of the function \( f(z) = (z-2) \cos\left(\frac{1}{z-1}\right) \) at \( z_0 = 1 \).

Answer: -\dfrac{1}{2}


Question: Find a polynomial with integer coefficients that has the primitive 8th roots of unity as its roots.

Answer: x^4 + 1


Question: Determine if the probability of achieving the 30th failure before the 3rd success is equal to the sum of probabilities of having 0, 1, or 2 successes before the 30th failure, given the probability mass function \( \mathbb{P}(Y=y) = \begin{pmatrix} y+k-1\\ y \end{pmatrix} (1-p)^k p^y \).

Answer: Yes


Question:
Let $N$ be the number of sequences $a_1, a_2, a_3, \ldots, a_{24}$ that satisfy the following conditions:
1. For all $1 \le i \le 24$, $1 \le a_i \le 24$.
2. For all $i \equiv 0 \pmod{2}$, $a_1 + a_2 + \cdots + a_i \equiv 1 \pmod{2}$.
3. For all $i \equiv 0 \pmod{3}$, $a_1 + a_2 + \cdots + a_i \equiv 2 \pmod{3}$.

If $N = p_1^{a_1} p_2^{a_2}$ where $p_1$ and $p_2$ are prime numbers, find $p_1 + a_1 + p_2 + a_2$. \( \boxed{} \)

Answer: 81