Question: Calculate the sum \(\sum_{k=0}^n \binom{n}{k} ^2\) using Parseval's identity for the function \(f(x) = (1+e^{i x})^n\). Given that \(f(x) = \sum_{k=0}^n \binom{n}{k} e^{i k x}\) and the Fourier coefficients of \(f\) are \(f_k = \binom{n}{k}\) for \(0 \leq k \leq n\) and \(0\) otherwise, find \(\int_0^{2 \pi} |f(x)|^2 dx\) to apply Parseval's identity.

Answer: \dbinom{2n}{n}


Question: Given $n=pq$ and $p-q = \sqrt{(p+q)^2 -4n}$, determine the smallest possible value of $p+q$ when $n$ is very large.

Answer: 2\sqrt{n}


Question: In a group $G$, given that $a^5 = e$ and $aba^{-1} = b^2$ for elements $a$ and $b$, find the order of the element $b$, denoted by $o(b)$. Provide your answer as a single integer.

Answer: 31


Question: Determine whether it is always possible to find a permutation of the numbers from 1 to n such that the sum of any two adjacent elements in the permutation is a composite number, for n ≥ 5.

Answer: Yes


Question: Evaluate the limit \( \lim\limits_{n\to\infty}\int_0^a\frac{x^n} {x^n+1}\,\mathrm dx \), where \( a>0 \).

Answer: \max(a - 1, 0)


Question: Evaluate the integral \( S = \int_{0}^{2\pi} \int_{-1}^{1} \sqrt{AC - B^{2}} \, du \, dv \) where the functions are given by \( A = 1 + \frac{1}{16}v^{2} \), \( B = 1 \), and \( C = 1 \). Determine if changing the order of integration affects the result and calculate the value of \( S \).

Answer: \pi^2


Question: Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(x+f(y))=f(x)+f(f(y))+3xy^6+3x^2f(y) \) for all \( x,y \in \mathbb{R} \).

Answer: f(x) = x^3


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( f(x) = x^3 - f(2x) \).

Answer: \dfrac{x^3}{9}


Question: Evaluate the expression: \(\cos\left(\frac{\pi}{11}\right)\cos\left(\frac{2\pi}{11}\right)\cos\left(\frac{3\pi}{11}\right)\cdots\cos\left(\frac{10\pi}{11}\right) = -2^n \). Find the value of \(n\).

Answer: -10


Question: Determine whether the projective closure of an affine plane of order $n$ also has order $n$. Provide a justification for your answer.

Answer: Yes


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

Answer: 0


Question: Given the parabola \(y^2 = 4(x - 1)\), its reflection in the line \(x + y = 2\) can be represented by the curve \(Ax + By = x^2\). Find the value of \(A + B\).

Answer: 0


Question: Determine whether there exists a CW-structure on a connected topological space $Y$ that is covered by a connected CW-complex $X$ via a covering map $p: X \to Y$. Provide a justification for your answer.

Answer: Yes


Question: Let $a_{1}=1$ and $a_{n}=n(a_{n-1}+1)$ for $n=2, 3, \ldots$. Define \[ P_{n}=\left(1+\frac{1}{a_{1}}\right)\left(1+\frac{1}{a_{2}}\right) \cdots\left(1+\frac{1}{a_{n}}\right) \] for $n=1, 2, \ldots$. Find $\lim _{n \rightarrow \infty} P_{n}$.

Answer: e


Question: Given three independent, identically distributed continuous random variables $X_1$, $X_2$, and $X_3$, each following an Exponential Distribution, calculate the probability that the median of these variables is closer to the smallest value $X_1$ than it is to the largest value $X_3$. Provide your answer as a probability value.

Answer: \dfrac{2}{3}


Question: Given that $a, b, c, d$ are real numbers satisfying $a^2 + b^2 + c^2 + d^2 = 4$, determine the maximum value of $a^3 + b^3 + c^3 + d^3$. Provide your answer as a single number.

Answer: 8


Question: If $f\colon A\to\mathbb R$ is uniformly continuous and $A$ is a bounded set, is $f$ necessarily bounded? Provide a justification for your answer.

Answer: Yes


Question: What is the probability that the giant component of a random graph $G_{n,m}$, where $m=\frac{cn}{2}$ and $c>1$, is isomorphic to a specific connected graph $H$?

Answer: 0


Question: Calculate the limit: \( \lim _{x\to 0^+}\frac{x^2}{1-\sqrt[3]{\frac{1-x}{1+x}}} \)

Answer: 0


Question: Find the value(s) of \(\beta\) such that the differential equation \((x^3+2)y''+\beta y = 0\) has a solution in the form of a linear polynomial.

Answer: 0


Question:
Determine the number of values of $n$ at which the sequence $a_n = \frac{1000^n}{n!}$ attains a maximum. Choose the correct option:

(a) Only one value
(b) Exactly two values
(c) More than two values, but finite
(d) Infinitely many values

Answer: B


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

Answer: 1


Question: Determine if there is a solution to the Blue-Eyed Islander puzzle.

Answer: Yes


Question: Determine whether the following statement is true or false: "If a real-valued function $f$ is Riemann integrable on $[a,b]$, and the limit $\lim_{c\to\infty}\int_{-c}^{c}f(x) dx$ exists and is finite, then $f$ has a convergent improper integral $\int_{-\infty}^{\infty}f(x) dx$."

Answer: B


Question: Determine whether the product of a complete vector field and a bounded function on a manifold is always a complete vector field. Answer with true or false.

Answer: True


Question: Determine whether there exists a graph with a chromatic number greater than 2013, where all cycles have a length greater than 2013.

Answer: Yes


Question: Calculate the surface area of the cylinder defined by the equation \(x^2+y^2=2x\), centered at \((1,0)\), and bounded by the cone \(z=\sqrt{x^2+y^2}\) and the plane \(z=0\).

Answer: 8


Question: Consider a function $f(x,y)$ that is anti-symmetric in $y$, meaning $f(x,-y) = -f(x,y)$. Determine if the double integral of $f$ over a region symmetric in $y$ is zero. Assume the symmetry requirement applies only to the $y$-coordinate, with no constraint on symmetry in $x$. Provide a justification for your answer.

Answer: 0


Question: Let $F$ be a function of bounded variation on $[a,b]$. Determine if $\int_{a}^{b}{|F'(x)|dx} = T_{F}(a,b)$ implies that $F$ is absolutely continuous.

Answer: Yes


Question: Solve the equation $5^x - 4^x = 1$ for $x$.

Answer: 1


Question: Find all increasing functions \( g: \mathbb{Z}^+ \to \mathbb{Z} \) such that \( g(1) = 1 \) and \( g(ab) = g(a) + g(b) + g(a)g(b) \) for all positive integers \( a \) and \( b \).

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


Question: Evaluate the limit \( \lim_{x\to 0} \frac{\sin(x)-\arctan(x)}{x^2\ln(1+2x+x^2)} \) using Taylor series or other appropriate techniques.

Answer: \dfrac{1}{12}


Question: Determine whether the solution \((p, x, y) = (7, 2, 5)\) is the only set of prime and natural numbers that satisfies the following system of equations: \[ p + 1 = 2x^2 \] \[ p^2 + 1 = 2y^2 \]

Answer: (7, 2, 5)


Question: Determine whether the following statement is true or false: If $f(x)$ is a positive continuous function on $[a, \infty)$ such that $\int_a^\infty f(x) \, dx$ converges, then there exists $0 < c < 1$ such that for every $c \le p \le 1$, $\int_a^\infty f^p(x) \, dx$ converges.

Answer: B


Question:
If \( \int f(x) \, dx = g(x) \), what is \( \int f^{-1}(x) \, dx \) equal to? Choose the correct option:

1. \( g^{-1}(x) \)
2. \( xf^{-1}(x) - g(f^{-1}(x)) \)
3. \( xf^{-1}(x) - g^{-1}(x) \)
4. \( f^{-1}(x) \)

Given that \( f(x) = y \Rightarrow f^{-1}(y) = x \) and \( \int f^{-1}(f(x)) \, dx = g(f(x)) \), determine the correct choice.

Answer: 2


Question: As \( x \) approaches infinity, determine which expression is larger: \( x^{\log(x)} \) or \( (\log(x))^x \).

Answer: (\log(x))^x


Question: Determine whether there exists a noncyclic group that has a totally ordered subgroup lattice.

Answer: Yes


Question: Determine the number of distinct conjugacy classes in a finite group whose order is a multiple of 8, considering only the case where the order is even.

Answer: 5


Question: Let $A$ be an abelian subgroup of a finite group $G$. Assume that $G$ is generated by the sets $A$ and $g^{-1}Ag$ for some element $g$ in $G$. Does it follow that $G = Ag^{-1}Ag$?

Answer: No


Question: Determine the limit of the sequence \((a_n)\), where \(a_n = \sqrt[n]{3^n + 5^n + 7^n}\).

Answer: 7


Question: Determine the number of roots of the equation \( z^4 - 5z + 1 = 0 \) that lie within the annulus defined by \( 1 \leq |z| \leq 2 \).

Answer: 3


Question: Determine whether the following statement is true or false: If $\nu$ is a signed measure defined over $(X,\mathcal{M})$ such that $\nu(E)\in\mathbb{R}$ for all $E\in\mathcal{M}$, then $\nu$ is bounded.

Answer: A


Question: Let $f(x)$ be a twice-differentiable function over $(0,+\infty)$ such that $f(0)=0$ and for all $x>0$, $2f(x)\leq f(2x)$. If $f''(1) = 0$, does it imply that $f(x)$ is convex?

Answer: Yes


Question: Given a surface $S_{0,4}$ (a disk with three holes) and two curves, $A$ (red) and $B$ (blue), which fill $S_{0,4}$, consider an element $\phi \in MCG(S_{0,4})$ (the mapping class group of $S_{0,4}$). Let $A' = \phi(A)$ and $B' = \phi(B)$. Do $A'$ and $B'$ also fill the surface $S_{0,4}$? Provide a justification for your answer.

Answer: Yes


Question:
Evaluate the expression:

$$\sum_{k_1=0}^n\sum_{k_2=0}^n\cdots\sum_{k_m=0}^n\binom{n}{k_1}\binom{n-k_1}{k_2}\cdots\binom{n-k_1-\cdots-k_{m-1}}{k_m}$$

where $m$ is a positive integer.

Answer: (m+1)^n


Question: Determine whether \(c^c - a^a\) is divisible by \(2^a b^2\) when \(a = 2k\), \(b = 2k+1\), and \(c = 2k+2\) for some integer \(k\).

Answer: Yes


Question: Find the value of the 100th term, \(x_{100}\), in the sequence of numbers that can be expressed as a sum of one or more distinct powers of 3, listed in increasing order. The sequence begins with \(x_1=1\), \(x_2=3\), \(x_3=4\), and so on.

Answer: 981


Question: Let $P:[0,1]\to \mathbb{R}^2$ be a simple continuous curve on the plane, i.e., $P$ is continuous and injective. Can $P$ always be extended to be a sub-curve of a simple closed curve?

Answer: Yes


Question: Evaluate the double integral \(\iint _B (x^2+y^2)\,dx\,dy\) where \(B\) is the region bounded by the parabolas \(y=x^2\) and \(y=2x-x^2\).

Answer: \dfrac{43}{210}


Question: Evaluate the limit: $$\lim_{x\rightarrow 0} \, \frac{\sqrt[3]{x} \ln(|\ln x|)}{\sqrt[3]{(2x+3)\ln x}}$$

Answer: 0


Question: Determine whether $C_0(\mathbb{R})$, the subspace of continuous real-valued functions vanishing at infinity in the Banach space $C(\mathbb{R})$ of continuous real-valued functions defined on $\mathbb{R}$ with the supremum norm, is itself a Banach space.

Answer: Yes


Question: In a school with 100 students forming 450 clubs, any two clubs have at least 3 students in common, and any five clubs have no more than 1 student in common. Determine whether it is necessary that some four clubs have exactly 1 student in common.

Answer: Yes


Question: Calculate the surface integral \( \iint_S \vec{F} \cdot \vec{n} \, dA \) for the vector field \( \vec{F} = [x^2, z, -y]^T \) over the unit sphere \( S = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 = 1\} \).

Answer: 0


Question: For the vector \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), find the expression for \( \nabla \times (\mathbf{k} \times \mathbf{r}) \).

Answer: 2\mathbf{k}


Question: Find the residue of the sum \( S = 1^{965}+2^{965}+\cdots+2018^{965} \) when divided by \(2021\). Express your answer as \(S \equiv x \ (\text{mod}\ 2021)\). What is the value of \(x\)?

Answer: 1012


Question: Consider a sequence of random variables $X_n$ where $X_n \sim \mathrm{Normal}\left(\frac{1}{n}, \frac{1}{n}\right)$. Does $X_n$ converge in distribution to a random variable $X$? If so, identify the distribution of $X$. Use the Weak Law of Large Numbers in your solution.

Answer: 0


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

Answer: 277


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

Answer: 49\sqrt{7}


Question: Evaluate the line integral \( C = \oint_K \mathbf{dr} \cdot \mathbf{A} \), where \( \mathbf{A} = \frac{1}{2} \mathbf{n} \times \mathbf{r} \) and \( \mathbf{n} \cdot \mathbf{n} = 1 \). The curve \( K \) is a circle with radius \( R \) in a plane with normal \( \mathbf{n} \). Determine the value of \( C \).

Answer: \pi R^2


Question: Are the expected values of a function $f(X)$ conditioned on $X \leq Y$ equal to the unconditional expected value of $f(X)$, given that $X$ and $Y$ are independent random variables?

Answer: No


Question:
Given the function \( f(x) = \begin{cases} x-1 & x \in \mathbb{Q} \\ 0 & x \in \mathbb{Q}^{c} \end{cases} \), determine if the following conditions are sufficient to conclude that for any sequence \((x_n)_n\) of real numbers with \(\lim_{n\to\infty}x_n = 1\), \(\lim_{n\to\infty}f(x_n) = f(1)\):
1. If \((x_r)_n\) is a sequence of rational numbers with \(\lim_{n\to\infty}x_r = 1\), then \(\lim_{n\to\infty}f(x_r) = f(1)\).
2. If \((x_i)_n\) is a sequence of irrational numbers with \(\lim_{n\to\infty}x_i = 1\), then \(\lim_{n\to\infty}f(x_i) = f(1)\).

Answer: Yes


Question: Let $\alpha, \beta, \gamma$ be the roots of the equation $x^3 + 4x - 1 = 0$. Given that $\frac{1}{\alpha+1}, \frac{1}{\beta+1}, \frac{1}{\gamma+1}$ are the roots of the equation $6x^3 - 7x^2 + 3x - 1 = 0$, find the value of the expression $\frac{(\beta+1)(\gamma+1)}{\alpha^2} + \frac{(\gamma+1)(\alpha+1)}{\beta^2} + \frac{(\alpha+1)(\beta+1)}{\gamma^2}$.

Answer: 79


Question: Determine the values of $b > 0$ for which the expected value of the random variable $Y_1$, with distribution $\mathbb{P} (Y_{1} \leq x) = 1 - \frac{1}{x^b}$ for $x \in \mathbb{N}$, is finite.

Answer: b > 1


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

Answer: 0


Question: Find the sum of the infinite series: \[ \sum_{n=1}^\infty \frac{n(n+2)}{4^n}. \]

Answer: \dfrac{44}{27}


Question: Determine the metric $d(A,B)$ for the sets $A = \{ x \mid x^2 \leq x + 1 \}$ and $B = [-1, 1]$ in the measure space $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$. The metric should be a real-valued non-negative function, symmetric, and satisfy the triangle inequality. What is the value of $d(A,B)$?

Answer: 1


Question: Determine whether the partial derivative \( \frac{\partial f}{\partial y} \) of the function \( f(x,y) = \begin{cases} \dfrac{xy^2}{\sqrt{x^2+y^2}} & \text{if $(x,y)\neq(0,0)$ } \\[2ex] 0 & \text{if $(x,y)=(0,0)$ } \\ \end{cases} \) is continuous at the point \((0,0)\).

Answer: Yes


Question: Find the value of $x$ at the inflection point of the function $f(x) = 10 \frac{\ln(\ln(x))}{\ln(x)}$. \(x > 1\)

Answer: 26.7


Question: Determine whether the problem of finding the partition function of a 3D Ising model is in the complexity class #P or NP.

Answer: \#P


Question: Given that $Y$ is an $m \times m$ idempotent matrix with $\text{rank}(Y) = m-n$, calculate the rank of the matrix $I_m - Y$. \( \text{Express your answer as a single number.} \)

Answer: n


Question: Consider the space \(\mathbb{R}^3\). Let \(L\) be the set of all lines and \(P\) be the set of all planes in \(\mathbb{R}^3\). Define topologies \(\tau_1\) and \(\tau_2\) on \(\mathbb{R}^3\) generated by the subbases \(L\) and \(P\), respectively. Determine the relationship between \(\tau_1\) and \(\tau_2\). Are these topologies equivalent, or is one finer than the other?

Answer: \tau_1 = \tau_2


Question: Let $g(x)$ be a non-negative function integrable on $(0,1)$ such that \( \int_0^1 g(x) \, dx = 1 \). If \( g(x) = \frac{1}{x} \int_0^x g(t) \, dt \) almost everywhere on \((0,1)\), determine whether $g(x)$ must be identically equal to 1.

Answer: g(x) \equiv 1


Question: Evaluate the limit \( \lim_{k \to \infty} \int_{0}^{\infty}k^{3/2}\left(\int_{0}^{x}\frac{\sin(t)}{t\sqrt{t}}dt\right)e^{-kx} dx \) using the Laplace transform.

Answer: \sqrt{\pi}


Question: Consider two ordinary differential equations with different initial conditions: \( y_1'(t) = f(t, y_1(t)) \) with \( y_1(0) = 0 \), and \( y_2'(t) = -f(t, y_2(t)) \) with \( y_2(0) = 1 \). If the solutions \( y_1(t) \) and \( y_2(t) \) intersect at some point, does this imply that the time it takes for \( y_1(t) \) to reach 1 is equal to the time it takes for \( y_2(t) \) to reach 0?

Answer: No


Question: Given an odd integer $n$, determine the number of critical points of the polynomial function $p_n(x) = 1 + 2x + 3x^2 + \\cdots + (n+1)x^n$. Provide your answer as a single integer.

Answer: 0


Question: Find the function $b(x) > 0$ that satisfies the condition $\int_0^\infty b(x) \, dx = 1$ and minimizes the functional \[ G(b) = \int_0^\infty e^{-kx} \ln(b(x)) \, dx. \]

Answer: k e^{-kx}


Question: Evaluate the limit using L'Hopital's rule: $$\lim _{x\rightarrow \infty}\{x-|x|^{\frac{2}{3}}(6+x)^\frac{1}{3}\}$$

Answer: -2


Question: Evaluate the double integral \( \int_{0}^{\infty}\int_{0}^{u}ue^{-u^2/y}\,dy\,du \).

Answer: \dfrac{1}{2}


Question: Determine whether there exists a sequence \((a_n)_{n \geq 1}\) of positive rational numbers without repetition such that \(\sqrt[n]{a_n}\) converges.

Answer: Yes


Question: Prove that \( \lim_{n \rightarrow \infty} \left ( 1+ \frac{1}{n} \right )^n = e \), where \( e \) is defined as the unique positive number such that \( \int_1^e \frac{dx}{x} = 1 \).

Answer: e


Question: Let $A$ be a complex matrix that is Hermitian, positive definite, and unitary. Prove that $A$ is the identity matrix.

Answer: I


Question:
Determine whether the function \( y = \ln(xy) \) is a solution to the differential equation:
\[(xy - x)y'' + xy'^2 + yy' - 2y' = 0\]

Answer: Yes


Question: Evaluate the path integral \( \oint_{|z|=1} \frac{1}{(z-a)(z-b)(z-c)} \, dz \), where \(|a|, |b|, |c| < 1\) and \(a \neq b \neq c\).

Answer: 0


Question: Determine the radius of convergence for the complex series \(\sum_{n=0}^\infty c_nz^n\), where the coefficients \(c_n\) are given by \(c_n=(1/n!)\int_1^\infty t^{-1}e^{-t}(\log t)^ndt\).

Answer: \infty


Question: Determine the isometry group of the Euclidean plane that preserves a square. Which permutations of the square's vertices does this group maintain?

Answer: D_4


Question: Let $a, b, c$ be distinct real numbers. Determine the number of real solutions to the equation $(x − a)^5 + (x − b)^3 + (x − c) = 0$. Provide your answer as an integer.

Answer: 1


Question: Evaluate the limit \( \lim_{n \to \infty} n \cos (2 \pi \cdot e \cdot n!) \).

Answer: \infty


Question: Simplify the expression \(\arctan 8 + \arctan 2 + \arctan\frac{2}{3}\) in terms of \(\pi\).

Answer: \pi


Question: Consider a sequence of continuous functions $f_n:[0,1]\to[0,1]$ such that $\sum_{n\ge 1}f_n(x)=1$ for all $x\in[0,1]$. Given a sequence of numbers $a_n\in[0,1]$, does the function $S:[0,1]\to [0,1]$, defined by $S(x)=\sum_{n\ge 1}a_nf_n(x)$, attain its maximum value on the interval $[0,1]$?

Answer: Yes


Question: In a metric space $(M, d)$, let $A$ and $B$ be two closed subsets such that $A \cap B = \emptyset$. Determine if it is possible for the distance $d(A, B)$ to be zero.

Answer: Yes


Question: Evaluate the integral \( \int_{r' \le R} \frac{y'}{|\vec{r}-\vec{r}'|}d^3\vec{r}' \), where \( d^3\vec{r}'=dx'dy'dz' \), \( y' = r'\sin \theta \sin \phi \), and \( |\vec{r}-\vec{r}'|^{-1}=(r'^2+r^2-2rr'\cos \theta)^{-1/2} \). Determine if special functions are necessary for the solution.

Answer: 0


Question: Determine if the limit \( \lim_{x \rightarrow a} \frac{f(x) - f(2a -x)}{2(x-a)} \) always exists, given that a function \( f(x) \) is continuous in a neighborhood of \( a \) and \( f'(a) \) exists.

Answer: f'(a)


Question:
Find the function $g(x)$ that satisfies the integral equation:

$$g(x) = \int_0^x (x-t) e^{x-t} g(t) \, dt$$

Answer: 0


Question: Determine whether there exists an analytic function $g: \mathbb{C} \to \mathbb{C}$ such that $g(x) = \cos(\sqrt{x})$ for all $x \in \mathbb{R}^{+}$ and $g'(0)$ exists.

Answer: Yes


Question: A cylindrical tank initially contains 20 liters of water. Water is drained from the tank at a rate proportional to the square root of the volume of water in the tank. If the water is discharged initially at a rate of 1 liter per minute, determine the time required for the tank to empty completely.

Answer: 40


Question: Find all possible values of $b$ such that the area under the curve $y = 1 + \frac{8}{x^2}$ from $x = 2$ to $x = 4$ is divided into two equal parts by the line $x = b$. Express your answer in terms of $b$.

Answer: 2\sqrt{2}


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

Answer: 1


Question: In group theory, consider a finitely generated nilpotent group $G$. Are all terms of its lower central series, defined by $\gamma_i G = [\gamma_{i-1} G, G]$ with $\gamma_0 = G$, finitely generated?

Answer: Yes


Question: Let $f(x)$ be a real-valued function defined for all real numbers such that $|f(x) - f(y)| \leq \frac{1}{2} |x - y|$ for all $x$ and $y$. Determine the number of points where the graph of $y = f(x)$ intersects the line $y = x$. \( \text{Provide your answer as a single number.} \)

Answer: 1


Question: Let \( E = \bigcup_{n = 1} ^ \infty \left(\frac{1}{n+1}, \frac{1}{n}\right) \). Prove that the closure of \( E \), denoted \( \bar{E} \), is the interval \([0,1]\).

Answer: [0,1]


Question: Calculate the average value of the function \( f(x, y, z) = x^2 + y^2 + z^2 \) over the region \( D \) defined by \( |x| + |y| + |z| \leq 1 \). Express the average value as \( \frac{1}{\text{volume of } D}\iiint_D F\,dV \), where \( F \) represents the integrand.

Answer: \dfrac{3}{10}


Question: Determine whether it is possible to ascertain if a given univariate polynomial with integer coefficients has algebraic roots.

Answer: Yes


Question: Determine the number of ways to assign the numbers 1 to 12 to the edges of a cube such that each vertex has the same weight, with the condition that no two edges have the same number.

Answer: 0


Question: Find the locus of the point of intersection of the tangents to the parabolas \(y^2 = 4(x + 1)\) and \(y^2 = 8(x + 2)\) that are perpendicular to each other. Express the equations of the tangents in the form \(y = m_1(x + 1) + \frac{1}{m_1}\) and \(y = m_2(x + 2) + \frac{2}{m_2}\), where \(m_1\) and \(m_2\) are the slopes of the tangents. Use the condition \(m_1 \cdot m_2 = -1\) to find the locus.

Answer: x = -3


Question: Let $A$ and $B$ be $n \times n$ real matrices such that $AB = BA = 0$ and $A + B$ is invertible. If the nullity of $A$ is $k$, determine the nullity of $B$. Express your answer in terms of $n$ and $k$. \( \boxed{} \)

Answer: n - k


Question: Find all polynomials \( p(x) \) that satisfy the equation \((x-1)(p(x+1))=(x+2)(p(x))\) and the condition \( p(2) = 12 \).

Answer: 2x^3 - 2x


Question: Let $\mu$ be a signed measure on a measurable space $(X, \mathcal{M})$. If $A, B \in \mathcal{M}$ are almost disjoint sets (i.e., $\mu(A \cap B) = 0$), does it hold that $\mu(A \cup B) = \mu(A) + \mu(B)$? Answer yes or no and justify your answer.

Answer: Yes


Question: Evaluate the limit: $$ \lim_{n\to \infty} \frac{\displaystyle\log_a\Bigg(\sum_{\substack{k\in \mathbb{N}\\k\leq n~(1-\frac{1}{a})}}\binom{n}{k} (a-1)^k\Bigg)}{n} $$ where \( a > 1 \).

Answer: 1


Question: Calculate the residue of the function \( \frac{4-3z}{z(z-1)} \) at \( z = 0 \).

Answer: -4


Question: Let $f(x)$ be an invertible increasing function such that $f''(x) + 2f'(x) \geq 2$ and $f'(0) = 2$. Determine the maximum value of $(f^{-1})'f(x)$. Provide your answer as a single value.

Answer: 1


Question: Determine whether the statement is true or false: The Alexander-Conway polynomial of any unlink is equal to 0.

Answer: False


Question: Find the orthogonal projection of the function \( f(x,y) = x^2y \) onto the subspace \( V = \{f \in L^2((0,1)^2): \exists g \in L^2(0,1): f(x,y)=g(y)\} \) of \( L^2((0,1)^2) \).

Answer: \dfrac{y}{3}


Question: Find the mass of an object that lies above the disk $x^2 + y^2 \le 1$ in the $x$-$y$ plane and below the sphere $x^2 + y^2 + z^2 = 4$, given that its density is $\rho(x, y, z) = 2z$.

Answer: \dfrac{7\pi}{2}


Question: Let $A \in M(2,\mathbb{Z})$. Define $\sin A := \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}$, where this series is known to converge absolutely in the Banach space $\mathcal{L}(\mathbb{R}^2)$ under the operator norm. If $\sin A \in M(2,\mathbb{Z})$, is it true that $A^2 = O$?

Answer: Yes


Question: Calculate the second-order mixed partial derivative \(\frac{\partial^2g}{\partial u \, \partial v}(1,1)\) for \(g(u,v)=uf(u^2, 2uv)\) given that \(f(1,2)=4\), \(\nabla f(1,2)=(3,-1)\), \(\frac{\partial^2f}{\partial x^2}(1,2)= \frac{\partial^2f}{\partial y^2}(1,2)=1\), and \(\frac{\partial^2f}{\partial x \,\partial y}(1,2)=-1\).

Answer: -4


Question:
Consider a sequence of $n$ independent and identically distributed (i.i.d.) random variables $X_1, X_2, \ldots, X_n$, each following a uniform distribution on the interval $(0, 1)$. Determine the limit as $n$ approaches infinity of the probability that the sum of these random variables is less than or equal to $\frac{n}{2} + n^{3/4}$:

$$\lim_{n\rightarrow \infty} P\left(\sum_{i=1}^n X_i \leq \frac{n}{2} + n^{3/4}\right)$$

Answer: 1


Question: Given a cofinite subset $S \subseteq \mathbb{R}$ and a rational function $r : S \rightarrow \mathbb{R}$, let $a \in S^c$ be a pole of $r$. Determine whether there necessarily exists a polynomial function $p : \mathbb{R} \rightarrow \mathbb{R}$ such that the limit \(\lim_{x \rightarrow a}\left|r(x)-p\left(\frac{1}{x-a}\right)\right|=0\) exists and equals zero.

Answer: Yes


Question: Consider a smooth curve \(\gamma: [0,1] \rightarrow \mathbb{S}^2\). Determine if the set \(\bigcup_{t\in [0,1]} \text{span}(\gamma(t)) \subset \mathbb{R}^3\) has a Hausdorff dimension of at most 2, implying it has zero 3-dimensional Lebesgue measure.

Answer: Yes


Question: Evaluate the integral \(\int_{1}^{10} [2x] \{x/2\} \, dx\), where \([2x]\) is the greatest integer function and \(\{x/2\}\) is the fractional part function.

Answer: \dfrac{807}{16}


Question: Each of $n \ge 2$ people puts his or her name on a slip of paper (no two have the same name). The slips of paper are shuffled in a hat, and then each person draws one (uniformly at random at each stage, without replacement). Calculate the standard deviation of the number of people who draw their own names.

Answer: 1


Question: If there exists a solution for the equation \(x_1v_1 + ... + x_kv_k = w\) for any \(w \in \mathbb{R}^n\), does it imply that \(k = n\)?

Answer: No


Question: Evaluate the integral \( \int_3^{11} \frac{dx}{(x+5)\sqrt{1+x}} \).

Answer: \dfrac{\pi}{12}


Question: Let \(f(z)\) be a cubic polynomial with distinct roots \(r\), \(s\), and \(t\). If \(g(z)=f(cz)\) for some complex number \(c\), and \(g(z)\) also has distinct roots, find the number of possible values of \(|c|\).

Answer: 1


Question: Determine the limit of the sequence defined by \( b_1 = 2 \) and \( b_{n+1} = b_n + b_n^3 \) for all \( n \geq 1 \) as \( n \) approaches infinity.

Answer: \infty


Question: Determine whether the sum \( \sum_{x\in\mathbb{Z}^n} e^{-x^Tx} \) is less than \( 2^n \).

Answer: Yes


Question: Let $\{ Y_n \}_{n \in \mathbb{N_0}}$ be a sequence of random variables with $\mathbb{P}(Y_0=0)=1$. For $n \in \mathbb{N}$, $Y_n$ is uniformly distributed on the interval $(y_{n-1}, y_{n-1}+2)$ given $\{ Y_1=y_1, \ldots, Y_{n-1}=y_{n-1} \}$. Find $\mathbb{E}(Y_n)$. \( \text{Express your answer as a single number.} \)

Answer: n


Question: In the graph $G=(V,E_1\cup E_2)$, where $T_1=(V,E_1)$ and $T_2=(V,E_2)$ are trees, does there exist a vertex with degree at most 3, even if $E_1\cap E_2\neq\varnothing$? Provide a justification for your answer.

Answer: Yes


Question: Determine the number of triples \((a, b, c)\) of real numbers such that \(a, b, c\) are the roots of the polynomial equation \(x^3 + ax^2 + bx + c = 0\).

Answer: 4


Question: Evaluate the limit \( \lim_{n\rightarrow\infty}\int_0^1 x^n g(x)\,dx \), where \( g(x) \) is a polynomial function.

Answer: 0


Question:
Given a $3\times3$ matrix $B$ with different rows, the determinant of $B$ is $-2$. If the rows of $B$ are denoted as $x$, $y$, and $z$, find the determinant of the matrix:

$$\begin{bmatrix} -y+z \\ -z+x \\ -x+y \end{bmatrix}$$

Answer: 0


Question: Determine whether a homeomorphism on \(\bar{\mathbb{R}}:=\mathbb{R}\cup\{\pm\infty\}\) or \(\bar{\mathbb{Z}}:=\mathbb{Z}\cup\{\pm\infty\}\) has zero topological entropy.

Answer: 0


Question: Given that $\{W(t) | t \geq 0\}$ is a Brownian motion, find the conditional expectation $\mathbb{E}[W^{2}(4) | W(1), W(2)]$. Provide your answer as a function of $W(1)$ and $W(2)$.

Answer: W(2)^2 + 2


Question: Determine whether there exists a composite number $m$ such that for every integer $n$, the congruence $n^m \equiv n \pmod{m}$ holds true.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}\int_{0}^{x}\frac{\cos(t^3)}{t+x}dt.$$

Answer: \ln 2


Question: Find the probability density function (PDF) of the random variable \( U = \frac{X}{X+Y} \), where \( X \) and \( Y \) are independent random variables, each following an exponential distribution with parameter 1.

Answer: 1


Question: Find \( g'(2) \) for the function \( g(x) = \int_0^{x^2} y \tan(\pi y) \, dy \).

Answer: 0


Question: Evaluate the limit: $$ \lim_{n\to\infty} \frac{n!\times(2n)!}{(3n)!} $$ using factorial properties and manipulation, without applying L'Hôpital's rule.

Answer: 0


Question: Calculate the limit: $$ \lim_{(x,y)\to(0,0)}\frac{1-\sqrt{1-x^2-y^2}}{(2x^2+y^2)} $$ given the restriction $|y|<x^2$.

Answer: \dfrac{1}{4}


Question: Determine the range of values of $x$ for which the Maclaurin series of $f(x) = \ln(1+x)$, given by \[ f(x) = \sum_{n=1}^\infty (-1)^{n-1}\frac{x^n}{n}, \] converges to the function $f(x)$. Use Taylor's inequality to justify your answer.

Answer: (-1, 1]


Question: Find the sum of the infinite series: $$\sum\limits_{i=1}^{\infty}{\frac{2i-1}{{2^{i-1}}\cdot{(i-1)}!}}$$

Answer: 2\sqrt{e}


Question: Determine whether the total variation of a differentiable function $f$ on the interval $(0,1]$ is given by $TV(f) = \int_0^1 |f'|$. Provide a justification for your answer.

Answer: Yes


Question: Given the function $f:\mathbb{R}\rightarrow \mathbb{R}^+$ with the conditions $f(x+2)\cdot f(x+3)=16$ and $f(x)+f(-x)=8$, evaluate the integral $\int_{-8}^8 f(x-2019)\,dx$. \(\text{Provide your answer as a single number.}\)

Answer: 64


Question: Determine the asymptotic behavior of the series \( \sum_{k=1}^n \frac{1}{\sqrt{k}} \) and compare it to the series \( \sum_{k=1}^n \frac{1}{k} \), which is approximately \( \log(n) \).

Answer: 2\sqrt{n}


Question: Let $\psi:V\times \cdots \times V\to \mathbb R$ be an antisymmetric $k$-tensor on an $n$-dimensional vector space $V$. Prove that if $k>n$, then $\psi$ is the trivial tensor that sends every element to $0$. \(\text{True or False?}\)

Answer: A


Question: Calculate the surface integral \( \int_{F} a \cdot dn \) for the vector field \( a = [-z^2 - 2z, -2xz + 2y^2, -2xz - 2z^2]^T \) over the area \( F \) on the cylinder \( x^2 + z^2 = 4 \), which is above the ground plane \( z = 0 \), in front of the plane \( x = 0 \), and between the planes \( y = 0 \) and \( y = 2 \).

Answer: -\dfrac{136}{3}


Question:
Find the lowest odd integer value for $x$ that satisfies the following system of equations, given an odd integer $n$ where $1 < x \leq n$:
\[\begin{cases} y = \frac{-\cos\left(\frac{\pi}{2}x\right)}{\log(x)} \\
 y = -\cos\left(\frac{\pi}{2}\cdot\frac{n}{x}\right) \\
 y = 0 \end{cases}\]
For example, solve the system when $n = 15$. Provide the value of $x$.

Answer: 3


Question: Find the maximum value of $a+b+c+d+e+f$, where $a, b, c, d, e, f$ are nonnegative real numbers satisfying $a^2 + b^2 + c^2 + d^2 + e^2 + f^2 = 6$ and $ab + cd + ef = 3$. \( \text{Express your answer in the form } \boxed{\text{value}}. \)

Answer: 6


Question: Calculate the average value of the function \( f(x,y,z) = 3x - 4y + 5z \) over the region defined by the triangle (simplex) \( \{ (x,y,z) \mid x+y+z=1 \text{ and } 0 \leq x,y,z \leq 1 \} \).

Answer: \dfrac{4}{3}


Question: Does there exist a sequence of real-valued functions \(\{f_n\}\) with domain \(\mathbb{R}\) that is uniformly convergent on some subset of \(\mathbb{R}\) to a continuous function, but each \(f_n\) is continuous only at points that form a proper dense subset of \(\mathbb{R}\)?

Answer: Yes


Question: Consider two matrices $A, B \in \mathbb{R}^{3 \times 3}$ with characteristic polynomials $p_A(t) = t^3 + 5t^2 - 2t$ and $p_B(t) = t^3 - 2t^2 + 4t - 8$. Determine the possible dimensions of $\operatorname{null}(BA)$. \(\text{Provide your answer as a list of possible dimensions.}\)

Answer: [1]


Question: Is the existence of a plane $P$ through a point $p = (x, y, f(x, y))$ on the surface $S$, such that for every smooth curve $\gamma: I \rightarrow S$ through $p$, the tangent line to $\gamma(I)$ at $p$ lies in $P$, sufficient for the function $f$ to be differentiable at $(x, y)$?

Answer: Yes


Question: Determine whether there exists a function that is continuous at all irrational numbers, discontinuous at all rational numbers, and has bounded variation on the interval $[0,1]$. Provide a justification for your answer.

Answer: Yes


Question:
Determine whether the following inequality holds for all continuous functions $f$ and $g$ on $[0,1]$ in the sup-norm metric:

$$\sup_{x\in[0,1]}\int_{0}^{x}|f(t)-g(t)|\ dt \leq \sup_{x\in[0,1]}|f(x)-g(x)|$$

Answer: Yes


Question:
Consider the set $A = \left\{ \sum_{i=0}^\infty \frac{a_i}{5^i} : a_i = 0, 1, 2, 3, 4 \right\}$. Determine which of the following statements is true about $A$:

1) $A$ is a finite set.
2) $A$ is countably infinite.
3) $A$ is uncountable but does not contain an open interval.
4) $A$ contains an open interval.

Answer: 4


Question: Find a number \( a \in [0, 50] \) such that \( a \equiv (105^{70} + 1)^{15} \pmod{51} \).

Answer: 16


Question: Find the Fourier series expansion of the function \( f(x) = 0 \) for \( 0 < x < 1 \), using the eigenfunctions \( w_{n} = \cos \left[\left( n - \frac{1}{2}\right)\pi x \right] \), where \( n = 0, 1, 2, \ldots \).

Answer: 0


Question: Find a vector $X$ such that $U \cap W = \text{span}\{X\}$, where $U = \langle (1,1,3),(-5,-3,-15) \rangle$ and $W = \langle (4,4,9),(-44,-34,-120) \rangle$. Provide the vector $X$.

Answer: (14, 9, 42)


Question: Determine the value of \( c \in \mathbb{R} \) such that the inequality \( \frac{1}{2}(e^x+e^{-x}) \leq e^{cx^2} \) holds for all \( x \in \mathbb{R} \).

Answer: \dfrac{1}{2}


Question: Let $F$ be a field and $x$ be an element algebraic over $F$ with a minimal polynomial of odd degree. Prove that $F(x) = F(x^2)$.

Answer: F(x) = F(x^2)


Question: Determine the number of integer values of $n$ that satisfy the equation \( \frac{1}{\sin \frac{3\pi}{n}} = \frac{1}{\sin \frac{5\pi}{n}} \), where \( n \in \mathbb{Z} \).

Answer: 2


Question: Determine the group $H = \{z \in \mathbb{C} \mid z^3 \in C_n \}$, where $C_n$ is the group of all n-th roots of unity.

Answer: C_{3n}


Question: Let $A$ and $B$ be two $3 \times 3$ matrices. All entries of $A$ are distinct, and all entries of $B$ are distinct. Furthermore, all entries in both matrices have absolute values greater than 3. Determine whether it is possible for the product $AB$ to be the zero matrix. Answer "True" if $AB \ne 0$ is always true, or "False" if $AB = 0$ is possible.

Answer: False


Question: Determine the Noether Normalization of the ring \( A = \mathbb{C}[x,y,z]/I \), where \( I = (x^2y - z^2, x^3 - y^2, z^3 - xy) \).

Answer: \mathbb{C}


Question: Find the preimage of the set \( \{w \mid |w|=1\} \) under the bilinear transformation \( w=\frac{z+2i}{2iz-1} \). In other words, determine the set of points \( z \) that map to the unit circle \( |w|=1 \).

Answer: |z| = 1


Question: Find the coefficient of \(x^6\) in the expansion of \((1-2x+3x^2-4x^3+5x^4)^4\).

Answer: 1544


Question: Consider the recursive sequence \((q_n)_n\) defined by \(q_0(x) = 1\) and \(q_{n+1}(x) = q_n(x) - \frac{1}{3}(x + q_n(x)^3)\). Does \(q_n(x)\) converge uniformly to \(-\sqrt[3]{x}\) for \(-1 \leq x \leq 0\) as \(n \rightarrow \infty\)?

Answer: Yes


Question:
Solve the system of congruences using the Chinese Remainder Theorem:
\[ 2x \equiv 1 \pmod{7} \]
\[ 9x \equiv 1 \pmod{10} \]
\[ 11x \equiv 1 \pmod{13} \]

Answer: 669


Question:
Calculate the partial derivative of the function \( f(x,y) \) with respect to \( x \) at the point \( (0,0) \), where \( f(x,y) \) is defined as follows:

\[
f(x,y) = \begin{cases}
x \arctan\left[\frac{x}{y}\right] & \text{if } y \neq 0 \\
0 & \text{if } y = 0
\end{cases}
\]

Determine the value of the partial derivative using the limit:

\[
\lim_{h\to 0} \frac{f(0+h,0)-f(0,0)}{h}
\]

Answer: 0


Question: Evaluate the integral: $$\int_0^\pi \frac{\sin\frac{21x}{2}}{\sin \frac x2} dx$$

Answer: \pi


Question: Given the function \( h(t) = f(t^2 - 4t, 5 - 4t^3) \) where \( f \) is differentiable and \( f_x(-3,1) = 2 \) and \( f_y(-3,1) = 6 \), find the derivative \( h'(1) \) using the chain rule.

Answer: -76


Question: Let $p$ and $q$ be distinct primes. Determine the maximum number of distinct solutions to the congruence $x^2 \equiv a \pmod{pq}$, where $a$ is an integer and solutions are considered distinct modulo $pq$. Provide your answer as a single number.

Answer: 4


Question: Let \( B \) be a nilpotent \( n \times n \) matrix with complex entries. Given \( A = B - I \), where \( I \) is the identity matrix, find the determinant of \( A \).

Answer: (-1)^n


Question: Solve for $c$ in the equation $4 - \cos(-1) = c + 5\ln(c-1)$. You may use the transformation $e^{4 - \cos(1) - c} = (c-1)^5$ to assist in finding the solution.

Answer: 2.269


Question: Let $U \subset \mathbb{R}^n$ be a bounded open convex set such that every hyperplane passing through the origin divides $U$ into two sets of equal volume (with respect to the $n$-dimensional Lebesgue measure). Does it follow that $U$ is symmetric with respect to the origin, i.e., $U = -U$?

Answer: U = -U


Question: Evaluate the limit: $$\lim_{x\rightarrow0}\frac{\ln\cos2x}{\left(2^{x}-1\right)\left(\left(x+1\right)^{5}-\left(x-1\right)^{5}\right)}$$

Answer: 0


Question: Define a function \(B(x)\) for \(x>0\) as the maximum integer \(n\) satisfying \(n+ \left \lfloor \tfrac{n}{5} \right \rfloor \le x\). Determine the value of \( \lim_{x \to \infty} \dfrac{B(x)}{x} \).

Answer: \dfrac{5}{6}


Question:
If a quadratic equation with real coefficients, \(p(x) = 0\), has purely imaginary roots, what can be said about the roots of the equation \(p(p(x)) = 0\)? Choose the correct option:

(a) Only purely imaginary roots.
(b) All real roots.
(c) Two real and two purely imaginary roots.
(d) Neither real nor purely imaginary roots.

Answer: D


Question: Find a prime number \( p \) such that the number \((2 \times 3 \times 5 \times 7 \times \ldots \times p) + 1\) is not a prime number.

Answer: 13


Question: Find a set $B \subseteq \mathbb{R}$ such that for every closed set $F$ containing $B$, the intersection $|B \cap F|$ is infinite.

Answer: \mathbb{N}


Question: Evaluate the sum \(\sum_{n=0}^{\infty} \frac{n^{2}-5n+2}{n!}\).

Answer: -e


Question: Given that for a function \( g(t) \), \( \int_{-\infty}^{\infty} g(t)e^{-j\omega t}dt=\omega e^{-2\omega ^2} \) for any real value \( \omega \), and \( y(t)=\int_{-\infty}^{t}g(\tau)d\tau \), calculate the value of \( \int_{-\infty}^{\infty} y(t)dt \).

Answer: -j


Question: Let $E$ be an arbitrary closed subset of $\mathbb{R}$. Determine whether there exists a function $f:\mathbb{R}\to \mathbb{R}$ such that the set of all points where $f$ is not bounded is precisely $E$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exists a random variable $X$ such that the moment generating function of $X + 2X$ can be expressed as the product of the moment generating functions of $X$ and $2X$. Provide a justification for your answer.

Answer: Yes


Question: Determine the values of the parameter \( \alpha \in \mathbb{R} \) for which the function \( f(x, y) = \begin{cases} |xy|^\alpha & \text{for } xy \neq 0 \\ 0 & \text{for } xy = 0 \end{cases} \) is differentiable at every point \((x, y) \in \mathbb{R}^2\).

Answer: (1, \infty)


Question: Determine the number of terms required in the Maclaurin series expansion of \( f(x) = \log \frac{1+x}{1-x} \) to approximate \( \log 2 \) with 6 exact decimal places when \( x = \frac{1}{3} \).

Answer: 6


Question: Evaluate the limit: $$\lim_{x\to\infty} \frac{ 1}{x \sqrt{x} } \int_{0}^{x} \ln \frac{t^{3} + 2}{t^{2} +1 } dt $$

Answer: 0


Question: Compute the limit \( \lim_{x\to0}{\frac{e^{-1/x^2}}{x^2}} \) without using L'Hopital's rule.

Answer: 0


Question: Find the value of \(\lambda\) such that the line given by the equations \(2\lambda x + 3 = 4y - 1 = \lambda z + 2\) does not intersect the plane \(\lambda x + 2y - 3z + 4 = 0\).

Answer: 3


Question: Does there exist a sequence \( \{x_n\} \) that converges to 0 as \( n \to \infty \) such that the function \( g_n(x) = n^{\sin x - \cos x} \) has a local maximum at \( x_n \) for every positive integer \( n \)?

Answer: No


Question: Evaluate the limit: $$\lim_{(x,y)\to(0,0)}\frac{\cos(x^2y^2)}{(x^2+y^2)^{3/2}}$$

Answer: \infty


Question: Evaluate the integral \( \int_0^{2\pi} |e^{i3\theta} - 1|^2 \, d\theta \).

Answer: 4\pi


Question: Calculate the sum of the series \( S = \sum_{k=1}^{2015} (-1)^{\frac{k(k+1)}{2}} \cdot k \).

Answer: 0


Question: Consider an urn with $k$ balls, numbered from $1$ to $k$. A step involves drawing one ball uniformly at random, removing all balls with labels greater than the drawn ball's number, and returning the drawn ball to the urn. The process continues until the ball labeled $1$ is drawn. What is the expected number of steps needed to draw the ball with label $1$?

Answer: 1 + \sum_{i=1}^{k-1} \frac{1}{i}


Question: Find the degree of the splitting field of the polynomial \( x^3 - 5 \) over \( \mathbb{Q} \).

Answer: 6


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty }\left ( \frac{1}{n} + \frac{e^{\frac{1}{n}}}{n} + \frac{e^{\frac{2}{n}}}{n} + \frac{e^{\frac{3}{n}}}{n}+.....+ \frac{e^{\frac{n-1}{n}}}{n}\right ).$$

Answer: e - 1


Question: Determine the Morley rank of the type $x = x$ in the theory of Algebraically Closed Fields of characteristic zero (ACF$_{0}$).

Answer: 1


Question: Find the maximum value of the function \(f(x,y) = x^2 - y^2 + 2xy - 2x - 2y + 1\) subject to the constraint \(x^2 + y^2 \leq 2x\).

Answer: \sqrt{2}


Question: Determine the maximum number of distinct elements that can be generated by the powers of an element $x$ in a Galois Field $\mathbb{F}$ of order 8, using the multiplication operator $\cdot$ in the field.

Answer: 7


Question: Determine the integer values of \(a\) for which the polynomial \(5x^{10} + 3x^8 - 2x^6 + ax^2 + 5\) has integer solutions.

Answer: -11


Question: Determine whether the set $X = C_{[a,b]}^1$, consisting of continuously differentiable functions on the interval $[a, b]$, equipped with the metric $d(f,g) = \max_{x \in [a,b]} |f'(x) - g'(x)| + |f(a) - g(a)|$, is a metric space.

Answer: Yes


Question: Calculate the limit: $$\lim_{x \to 0} \frac{\mathrm e^{x^2}+\mathrm e^{-x^2}-2}{\sin^2\dfrac x2-\dfrac{x^2}4}$$

Answer: -48


Question: Find a closed-form expression for the series: \( \sum_{k=0}^{\infty} \frac{z^k}{(k!)^2} \).

Answer: I_0(2\sqrt{z})


Question: Given a function $f(x)$ defined on the interval $[0,1]$ such that \(\int^{1}_{0}(f(x))^2\,dx=4\) and \(\int^{1}_{0}f(x)\,dx=\int^{1}_{0}x\cdot f(x)\,dx=1\), find the value of \(\int^{1}_{0}(f(x))^3\,dx\).

Answer: 10


Question: Two processes $X_1$ and $X_2$ run simultaneously on a computer, where $X_1$ and $X_2$ represent the time each process works, respectively. Both $X_1$ and $X_2$ follow an exponential distribution with a mean of 60 seconds. Let $T$ be the time of the process that worked longer. Calculate the expected value of $T$, denoted as $E(T)$. Provide your answer in seconds.

Answer: 90


Question: If $(M_k)$ is a sequence of random variables that count the number of events occurring in time intervals $[k, k+1)$ and follows a Poisson distribution for each integer $k$, do the waiting times between consecutive events, denoted by $\\sigma_i$, follow an exponential distribution?

Answer: No


Question: Suppose we have a sequence $g_n$ in $L^2([0,1])$ such that $\sum||g_{n+1}-g_n||_2<\infty$. Does this imply that $g_n$ converges pointwise almost everywhere?

Answer: Yes


Question: Find the equation representing the projection in the xy-plane of the intersection between a sphere with radius \(\sqrt{3}\) and the paraboloid \(z = \frac{1}{2}(x^2 + y^2)\).

Answer: x^2 + y^2 = 2


Question: Use the divergence theorem to evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) for the vector field \( \mathbf{F} = (xy^2, yz^2, x^2z) \), where \( S \) is the sphere of radius 1 centered at the origin.

Answer: \dfrac{4\pi}{5}


Question: In a bank with 1000 customers, the number of withdrawal requests per month follows a Poisson distribution with an average of 60 requests. Each withdrawal amount is exponentially distributed with an average of 700 euros. Calculate the probability that the total amount withdrawn in a given month exceeds 50,000 euros.

Answer: 0.149


Question: Is it true that the null space of $n(p_i(T))^y$ is $T$-invariant for all $1 \leq y \leq e_i$, where $y$ is a natural number, given that the minimal polynomial of a transformation $T$, denoted by $m(x)$, can be factored as $m(x)=p_1(x)^{e_1}p_2(x)^{e_2}\cdots p_s(x)^{e_s}$?

Answer: True


Question: Evaluate the definite integral \(\int_0^{2\pi} \frac{1}{13-5\sin\theta}\,\mathrm{d}\theta\) and choose the correct answer from the options: (a) \(-\frac{\pi}{6}\) (b) \(-\frac{\pi}{12}\) (c) \(\frac{\pi}{12}\) (d) \(\frac{\pi}{6}\)

Answer: D


Question: Determine the parameters \((p, d, q)\) for the \(ARIMA(p, d, q)\) process of the time series \(Y_t\), given the stationary process \(X_t\) defined by the equation \(X_t = \phi X_{t-1} + Z_t + Z_{t-2}\), where \(Z_t\) is white noise with mean 0 and variance \(\sigma^2\), and the process \(Y_t\) defined as \(Y_t = Y_{t-1} + X_t - \phi^2 X_{t-2}\).

Answer: (0, 1, 3)


Question: Calculate the surface integral \(\iint_S (\nabla\times F)\cdot dS\) for the vector field \(F(x,y,z)=(-y,yz^2,x^2z)\) over the surface \(S\) defined by \(S=\{x^2+y^2+z^2=4, z\geq 0\}\), oriented according to the exterior normal vector.

Answer: 4\pi


Question: Is a continuous and surjective map $f: X \to Y$ between topological spaces, where each fiber of $f$ contains finitely many points, necessarily a closed map?

Answer: No


Question: Consider a real-valued function $g$ that is integrable on the interval $[a, b]$. If $g$ is non-zero on a subset of $[a, b]$ with positive measure, does it follow that \( \int_a^b |g(x)| \, dx > 0 \)? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Suppose $A, B, C \sim U(-1, 1)$ are mutually independent random variables. Calculate the probability $P(A + B + C < 0)$. Express your answer as a decimal or a fraction.

Answer: \dfrac{1}{2}


Question: Is the set \(C^0\), consisting of all bounded continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\) with the sup norm, path-connected? In other words, can any bounded continuous function be continuously deformed into another within the space \(C^0\)?

Answer: Yes


Question: Let \( g \) be a real-valued function defined on the interval \((-1,1)\) such that \[ e^{-x}(g(x)-2e^x)=\int_0^x \sqrt{y^4+1}\,dy \] for all \(x\in (-1,1)\). Let \( f \) be another function such that \( f(g(x))=g(f(x))=x \). Find the value of \( f'(2) \).

Answer: \dfrac{1}{3}


Question: Find a particular solution to the differential equation: \[ t^2 u''(t) + 2tu'(t) - 2u(t) = \frac{1}{t}, \quad t \in (0,\infty). \]

Answer: -\dfrac{1}{2t}


Question: Calculate the volume enclosed by the paraboloid \( z = x^2 + y^2 \) and the plane \( z = 10 \) using a double integral in the Cartesian coordinate system.

Answer: 50\pi


Question: Calculate the distance between an element $x \in \ell^2(\mathbb{R})$ and a dense subspace $S$ of $\ell^2(\mathbb{R})$. What is this distance?

Answer: 0


Question: Determine the spectrum of the operator $S: l^q \to l^q$, where $1 \leq q \leq \infty$, defined by $$ S(x_1, x_2, x_3, x_4, \ldots )=(x_2, -x_1, x_4, -x_3, \ldots). $$

Answer: \{i, -i\}


Question: Determine whether the Lebesgue measure of the boundary of a bounded Lipschitz domain in \(\mathbb{R}^n\) is zero.

Answer: Yes


Question: Determine the degree of the field extension \(\mathbb{Q}(\beta)\) over \(\mathbb{Q}\), where \(\beta = \zeta_7^2 + \zeta_7^5\) and \(\zeta_7\) is the seventh root of unity.

Answer: 3


Question: Find the maximum value of $x_1x_2x_3$ subject to the constraint $\frac{1}{1+x_1}+\frac{1}{1+x_2}+\frac{1}{1+x_3}=2$, where $x_1,x_2,x_3>0$.

Answer: \dfrac{1}{8}


Question: Evaluate the Lebesgue integral of the function \( f(x) = \begin{cases} 1 & x \in C \cap \mathbb{Q} \\\ 2 & x \in C \cap (\mathbb{R} \setminus \mathbb{Q}) \\\ 3 & x \in [0, 1] \setminus C \end{cases} \), where \( C \) is the Cantor ternary set.

Answer: 3


Question: In a grand hall, there are 1000 people having dinner. One person is initially sick, while the other 999 are healthy. Each minute, every person talks to one other person in the room at random, ensuring they do not talk to someone they have previously talked to. If a sick person talks to a healthy person, the healthy person becomes sick. Once a person becomes sick, they remain sick. Determine the maximum number of minutes until all 1000 people in the hall are sick.

Answer: 999


Question:
Minimize the function \( f(x, y) = 2x + xy + 3y \) subject to the constraints:
\[ \begin{cases} x^2 + y \geq 3 \\ x \geq 0 \\ y \geq 0 \end{cases} \]
Determine the minimum value of \( f(x, y) \) and the corresponding values of \( x \) and \( y \).

Answer: 2\sqrt{3}


Question: A shape with area \( T \) is located on one of two intersecting planes that form an angle of \( b \). Determine the area of the shape's projection onto the other plane in terms of \( T \) and \( b \).

Answer: T \cos b


Question: Let \( \pi:X\to Y \) be a continuous map and \( \mathcal{G} \) be a sheaf on \( Y \). There is a natural isomorphism \( f_p:(\pi^{-1}\mathcal{G})_p \to \mathcal{G}_{\pi(p)} \). The \( f_p \)'s induce a map \( f: \bigsqcup_{p\in X}(\pi^{-1}\mathcal{G})_p \to \bigsqcup_{q\in Y}\mathcal{G}_{q} \). Is this map \( f \) continuous with respect to the "space of section" (espace etale) topology, whose base consists of sets \( \{(x,s_x):x\in U\} \), where \( U\subset X \) is open and \( s\in \mathcal G(U) \)?

Answer: Yes


Question: Given an infinite ultimately periodic word $\eta = pq^{\omega}$, where $p, q \in X^*$, determine if it is possible to choose $p$ and $q$ such that their concatenation $pq$ is a primitive word.

Answer: Yes


Question: Suppose X and Y are continuous random variables with joint density function f(x, y). What is the probability that X equals Y?

Answer: 0


Question: Find the limit of the sequence \(\{x_n\}\) as \(n\) approaches infinity, where \(x_n\) satisfies the equation \(x_n^3 - (n^2 + 3n)x_n - 3n^2 = 0\) for all \(n \in \mathbb{N}\).

Answer: -3


Question: Is the maximal extension of a given subatlas \( \{ (U_\alpha,\phi_\alpha)\}_\alpha \) of a manifold \( \mathcal M \) unique?

Answer: Yes


Question: Find the seven solutions to the congruence $x^7 \equiv 1 \pmod{29}$, given that $2$ is a primitive root modulo $29$. Provide your answer as a list of integers.

Answer: [1, 7, 16, 20, 23, 24, 25]


Question: Determine the value of $x$ for which the series \(\sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n^2+4}}-\frac{x}{n+2}\right)\) converges. Use the condition \(\left|\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_n}\right|<1\), where \(a_n=\frac{1}{\sqrt{n^2+4}}-\frac{x}{n+2}\) and \(a_{n+1}=\frac{1}{\sqrt{(n+1)^2+4}}-\frac{x}{(n+1)+2}\).

Answer: 1


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{3\tan^3(x) - \sin(2x)}{\ln(1+x^4) - \cos^2(x)}$$ using Taylor series expansions.

Answer: 0


Question: Solve for $x$ in the equation: $$ 0 = bx - a \cdot \log(1 - dx) $$ without using special functions.

Answer: 0


Question: Evaluate the limit \( \lim\limits_{x \to 1^-}{(x^2-3x+2)^{\frac{\ln(1-x)}{\ln(x)}}} \) using the substitution method.

Answer: 0


Question:
Determine whether the following limit approaches zero as \( n \) tends to infinity for all positive integers \( k \):

$$\lim_{n \to \infty} n\left[(n-1)^{-\frac{1}{k}}-n^{-\frac{1}{k}}\right]$$

If it does, provide a proof.

Answer: 0


Question: Calculate \(\lim_{{x\rightarrow 0}} \left(\frac{1}{x^2} - \frac{x}{\sin^3(x)}\right)\) without using L'Hôpital's rule.

Answer: -\dfrac{1}{2}


Question: Given a matrix $A$ with a null trace, determine the minimum number of linear measurements, defined as $A\vec{x}_i \cdot \vec{h}_j$ for specified $\vec{x}_i$ and $\vec{h}_j$, necessary to uniquely determine the matrix $A$. Provide your answer as a single integer.

Answer: 3


Question: Evaluate the limit: $$\lim_{x\rightarrow 0}\left[ \frac{\ln(\cos x)}{x\sqrt{1+x}-x} \right]$$

Answer: -1


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)} \) using residue theory from complex analysis.

Answer: \pi \sqrt{2}


Question: Determine whether there exists a finite topological space whose fundamental group is isomorphic to $\mathbb{Z}_2$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of real solutions to the equation \(4\sin^2x + 10x^2 = \cos x\).

Answer: 2


Question: Find all functions \( f: \mathbb{Q} \to \mathbb{Q} \) that satisfy the functional equation \( f(x+f(x)+2y)=2x+2f(f(y)) \) for all \( x,y \in \mathbb{Q} \).

Answer: f(x) = x


Question: Evaluate the limit \( \lim_{{x\to 0}}\left(\frac{\tan x}{x}\right)^{\frac{1}{1-\cos x}} \) without using differentiation.

Answer: e^{\frac{2}{3}}


Question: Consider a function $g:[0,1]\rightarrow \mathbb{R}$ that satisfies the following conditions: (1) $g$ is bounded and (2) $g$ is integrable on $[a,1]$ for any $0<a<1$. Determine whether the improper integral $\int_0^1 g(x)dx$ necessarily exists.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty}\frac{n^{1/2}+n^{1/3}+\cdots+n^{1/n}}{n}.$$

Answer: 1


Question: Provide an example of a group of order 6 that is not cyclic.

Answer: S_3


Question: Are finite rank operators on a Hilbert space $H$ dense in the set of all bounded operators $B(H)$ with respect to the weak operator topology?

Answer: Yes


Question: Find the integer $x$ such that $x^{701} \equiv 3 \pmod{139}$ using Fermat's Little Theorem.

Answer: 88


Question: Find the coefficient of \(x^{17}\) in the expansion of \((1+x^5+x^7)^{20}\).

Answer: 3420


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(3^n-2^n\cdot n\cdot\cos^2{\frac{n}{3}}\right)^{\left(\frac{1}{3n-n\sin{\frac{1}{n}}}\right)}$$

Answer: \sqrt[3]{3}


Question:
Solve for \( r_2 \) in the inequality:

\[ x \leq y \]

where

\[ x = \left[-(R^4-r_2^4) + 4\left\{\frac{r_2^2(1-\alpha)+R^2\alpha-r_1^2}{(1-\alpha)\ln(r_2)+\alpha \ln(R)-\ln(r_1)}\right\}\left\{\frac{R^2 \ln(R)-r_2^2 \ln(r_2)}{2} - \frac{(R^2 - r_2^2)}{4}\right\} + 2\left\{\frac{(1-\alpha)(R^2 \ln(r_2)-r_2^2 \ln(R))-R^2 \ln(r_1)+r_1^2 \ln(R)}{(1-\alpha)\ln(r_2)+\alpha \ln(R)-\ln(r_1)}\right\}(R^2 - r_2^2)\right] \]

and

\[ y = \left[-(R^4-r_1^4) + 4\left\{\frac{R^2-r_1^2}{\ln(R)-\ln(r_1)}\right\}\left\{\frac{R^2 \ln(R)-r_1^2 \ln(r_1)}{2} - \frac{(R^2 - r_1^2)}{4}\right\} + 2\left\{\frac{r_1^2 \ln(R)-R^2 \ln(r_1)}{\ln(R)-\ln(r_1)}\right\}(R^2 - r_1^2)\right] \]

Answer: r_2 = r_1


Question: Determine if the integral \( \int_{h}^{R}\int_{0}^{2\pi}\int_{0}^{\sqrt{R^2-z^2}}A(2-\sin^2\theta)r \, dr \, d\theta \, dz \) can be expressed as the product of the integrals \( \int_{h}^{R}\int_{0}^{\sqrt{R^2-z^2}}r \, dr \, dz \) and \( \int_{0}^{2\pi}A(2-\sin^2\theta) \, d\theta \).

Answer: Yes


Question: Evaluate the surface integral \( \int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S} \) where \( \vec{F}(x,y,z) = y^2\vec{i} + 2xy\vec{j} + xz^2\vec{k} \) and \( S \) is the surface of the paraboloid \( z = x^2 + y^2 \) bounded by the planes \( x = 0, y = 0 \), and \( z = 1 \) in the first quadrant, with the surface normal pointing upward.

Answer: \dfrac{2}{7}


Question: Evaluate the limit \( \lim_{n \to \infty} \sum_{k=0}^{n} \frac{\binom{n}{k}}{n^k (k+3)} \).

Answer: e - 2


Question: If $A-I$ is positive definite, does it imply that $A$ is invertible? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the group $\Bbb Q/ \Bbb Z$ has a proper subgroup that is not finite. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Find the upper bound for the index of the core of a subgroup $H$ of finite index in a group $G$.

Answer: n!


Question: Determine whether every subset $F$ of the interval $[a,b]$ with zero length is necessarily open. Answer 'yes' or 'no'.

Answer: no


Question: Derive the formula for the $k^{\mathrm{th}}$ factorial moment of a Poisson random variable with parameter $\lambda$ using the identity $E[(X-\lambda)g(X)] = \lambda E[g(X+1)-g(X)]$. Provide your answer as an expression.

Answer: \lambda^k


Question: Let $g:\mathbb{R} \to \mathbb{R}$ be a differentiable function such that $g'(x) < x^3$ for every $x > 0$. Determine the value of $$\lim_{x \to \infty} g(x).$$

Answer: -\infty


Question: Let $X$ be a topological space with a non-trivial topology. If $f: X \to X$ satisfies $f^{-1}(U) = U$ for all open sets $U$, is $f$ necessarily a homeomorphism?

Answer: Yes


Question:
Solve for $x \in \mathbb{R}$ given the equation:
$$(x+m)^m - \binom{m}{1}(x+m-1)^m + \binom{m}{2}(x+m-2)^m + \cdots + (-1)^m\binom{m}{m}x^m = m!$$

Answer: \mathbb{R}


Question:
Find the function \( F(x) = \lim_{y \to 0} f(x,y) \) where \( f: \mathbb{R}^2 \to \mathbb{R} \) is defined as follows:
\[
f(x,y) = \begin{cases} \dfrac{e^{-x^2}-e^{-y^2}}{x^2+y^2} &\text{if } (x,y) \neq (0,0), \\
 k &\text{if } (x,y) = (0,0). \end{cases}
\]
Determine the value of \( k \) such that \( F(x) \) is continuous at \( x = 0 \).

Answer: -1


Question: Determine the values of $b$ for which the expression \((x,y) = x_1y_1 + x_2y_2 + x_3y_3 + bx_1y_2 + bx_2y_1\) defines an inner product on $\mathbb{R}^3$. Provide your answer in terms of $b$.

Answer: -1 < b < 1


Question: In a classroom, every group of three students always includes two who are friends, and every group of four students always includes two who are not friends. What is the largest possible number of students in the classroom?

Answer: 8


Question: Given two functions \( f \) and \( g \), if \( f \circ (f \circ g) = f \circ (g \circ f) \), does this imply that \( f = g \)? Answer 'yes' or 'no'.

Answer: no


Question: In a compact and connected metric space $X$ containing a point $p$, is it possible for any two distinct points in $X \setminus \{p\}$ to be joined by an arc, while no arc in $X$ contains the point $p$?

Answer: Yes


Question: Given 2014 real numbers $x_{i} \in [-11,5]$ such that $x_{1} + x_{2} + \cdots + x_{2014} = 0$, find the maximum value of $x^2_{1} + x^2_{2} + \cdots + x^2_{2014}$, where $(x_{i} - 5)(x_{i} + 11) \leq 0$ for all $i = 1, 2, \cdots, 2014$. Provide your answer as a single number.

Answer: 110710


Question: Find a sufficient statistic for \( \theta \) given that \( y_1, y_2, \ldots, y_n \) are independent and identically distributed as \( U([0,\theta]) \), where \( \theta > 0 \) is unknown.

Answer: \max\{y_1, y_2, \ldots, y_n\}


Question: If $A_n$ is irrational for some $n$, does it imply that $A_m$ is also irrational for all $m > n$? Answer yes or no and justify your answer.

Answer: No


Question: Determine whether a $2 \times 2$ symmetric matrix can have eigenvectors $u$ and $v$ such that the angle between $u$ and $v$ is not $90^\circ$. Provide a justification for your answer.

Answer: Yes


Question: Given the recursive sequence defined by $a_0 = y$ and $a_n = \left \lfloor{ \frac{5 a_{n-1}}{2}} \right \rfloor$, determine if the sequence will eventually contain an even number for any odd natural number $y$. Provide a justification for your answer.

Answer: Yes


Question: Is the set \( N_{b}=\{ f \in {\rm L}^{2}[0,1]:f(x)=0, \text{ for almost every } x \in [b,1]\} \) a closed subspace of \( {\rm L}^{2}[0,1] \) for \( 0 \leq b \leq 1 \)?

Answer: Yes


Question: Determine whether the statement "A prime element in any ring is irreducible" is true or false.

Answer: False


Question: Suppose $X$ and $Y$ are independent random variables, both following the same geometric distribution with probability mass function $\mathbb{P}(X=k)=\mathbb{P}(Y=k)=pq^{k-1}$ for $k\geq 1$, where $q=1-p$. Determine the conditional probability $\mathbb{P}(X=k \mid X+Y=n+1)$ for $n\in\mathbb{N}$ and identify the distribution of this conditional probability.

Answer: \dfrac{1}{n}


Question: Evaluate the limit: $$\lim_{x\rightarrow0^{+}} \frac {\ln(e^{x}  - 1)} {x^{-0.3}}$$

Answer: 0


Question: Evaluate the infinite series: \[ \sum \limits_{k=1}^{\infty} \frac{2k-1}{k\cdot(k+1)\cdot(k+2)} \] Transform the series into a telescoping series and find its sum.

Answer: \dfrac{3}{4}


Question: Determine whether there exists a sequence \((a_n)_{n \geq 1}\) that contains all positive rational numbers without repetition, such that \(\sqrt[n]{a_n}\) converges.

Answer: Yes


Question: Determine whether the ideal generated by $y$ in the integer ring $\mathcal{O_k}$ of the number field extension $\mathbb{Q}(y)$, where $y^{2}=3$, is prime, given that the norm of $y$ is 3.

Answer: Yes


Question:
Given two square matrices $A$ and $B$, determine if the following conditions imply that $AB = BA$:
1. $A + B^{-1} = I$, where $I$ is the identity matrix.
2. $A^2 = 0$.

Answer: AB = BA


Question: Find the minimum positive root of the equation \(\sin{3x}=b\) given that the difference between the two smallest positive roots is \(\frac{\pi}{4}\).

Answer: \dfrac{\pi}{24}


Question: Determine whether the following statement is true or false: If a matrix \( A \) has a minimal polynomial \( x^2 \), then \( A^2 \) has a minimal polynomial \( x \).

Answer: True


Question: Evaluate the limit: $$\lim_{x\to \infty} \left(\cfrac{2x^2-1}{2x^2+3}\right)^{4x^2+2} $$

Answer: e^{-8}


Question: Find the smallest possible value of the expression \( a^2 + 4a\sin b - 4\cos^2 b \).

Answer: -4


Question: Calculate the surface integral \( \iint_S z \sin \gamma \, dS \) over the surface \( S \) of the unit sphere centered at the origin, where \( \gamma \) is the angle between the normal to the surface at any point and the z-axis.

Answer: 0


Question: Determine whether the 1-norm of a vector decreases when it is multiplied by a sub-stochastic matrix, where the columns of the matrix sum to less than 1.

Answer: Yes


Question: Calculate the limit: $$\lim_{x\to 0}\dfrac{1}{1-\cos(x^2)}{\sum_{n=4}^\infty {n^5 x^n} }$$

Answer: 2048


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

Answer: \dfrac{1}{2}


Question: Given a set $E \subset \mathbb{R}$ with positive Lebesgue measure, determine whether there necessarily exists an $x \in E$ such that $x = \frac{1}{2}(y + z)$ for distinct $y, z \in E$. Provide a justification for your answer.

Answer: Yes


Question: Let $S$ be the set of $3 \times 3$ matrices $\rm A$ with integer entries such that $\rm AA^{\top} = I_3$. Determine the cardinality of the set $S$. What is $|S|$?

Answer: 48


Question: Determine whether there exists a set $S$ with a binary operation $\cdot$ that is commutative, possesses an identity element $e$, and has the property that every element has an inverse, but the operation is not associative.

Answer: Yes


Question: Find a particular solution to the differential equation \( y(x) = xy' + \ln y' \) given the initial condition \( y(-1) = -1 \).

Answer: y = -1 - \ln(-x)


Question: Evaluate the limit: $$ \lim_{t\to 1^-} (1-t) \sum_{n=0}^\infty \frac{t^n}{1+t^n} $$

Answer: \ln 2


Question: Determine the conditions on the parameters $b$ and $c$ such that the matrix $A = \begin{bmatrix}1&1&b\\ 0&3&c\\ 0&0&1\end{bmatrix}$ is diagonalizable. Consider the matrix $A - I = \begin{bmatrix}0&1&b\\ 0&2&c\\ 0&0&0\end{bmatrix}$ and find the necessary conditions for the geometric multiplicity of the eigenvalue $\lambda = 1$ to match its algebraic multiplicity.

Answer: c = 2b


Question: Calculate the value of the integral \( \int_0^1 \frac{dx}{\sqrt{x(1-x)}} \).

Answer: \pi


Question: Compute the conditional probability \( P(X=k|X^2=x^2) \) for a continuous random variable \( X \) following a uniform distribution \( U[-1,1] \), given that \( x^2=k^2 \).

Answer: \dfrac{1}{2}


Question: Evaluate the convergence of the limit: $$\lim_{n \to \infty} \left\{ \frac{n!}{\prod_{k=1}^{n} H_k} \right\}^{\frac{1}{n}}$$ where $H_k = \sum_{i=1}^{k} \frac{1}{i}$, the $k$th harmonic number.

Answer: \infty


Question: Evaluate the limit: $$\lim_{n\to \infty} \frac{\ln^{\ell-1}(n)}{e^{-\left(\frac{\ln(|\lambda_{i}|)}{\ell}\right)n}},$$ where $|\lambda_{i}|<1$ and $\ell$ is a positive integer.

Answer: 0


Question: Evaluate the limit \(\lim_{x \to 0} \frac{1}{x e^{ \frac{1}{2x^2}}}\). Provide a rigorous proof of your result.

Answer: 0


Question: Calculate the Taylor series of \( \left(\frac{1}{t^3} + \frac{3}{t^2}\right)^{1/3} - \sqrt{\left(\frac{1}{t^2} - \frac{2}{t}\right)} \) at \( t = 0 \) and determine the limit as \( t \) approaches 0.

Answer: 2


Question: Given that \(\tan\alpha\) and \(\tan\beta\) are the roots of the quadratic equation \(x^2 + px + q = 0\), evaluate the expression \(\sin^2(\alpha + \beta) + p\sin(\alpha + \beta)\cos(\alpha + \beta) + q\cos^2(\alpha + \beta)\).

Answer: q


Question: Suppose \( f \in L^1(\mathbb{R}) \) and its Fourier transform \( \hat{f} \) is continuous, bounded, and satisfies the Riemann-Lebesgue lemma. If \( f \) is also continuous and \( \lim_{x \to \pm\infty} f(x) = 0 \), does it follow that \( \hat{f} \in L^1(\mathbb{R}) \)?

Answer: No


Question: Determine whether the quotient ring $A=\mathbb{R}[x]/\langle x^2 - x + 1 \rangle$ is isomorphic to the complex numbers $\mathbb{C}$. Provide a justification for your answer.

Answer: \mathbb{C}


Question: Evaluate the limit: $$\lim_{n \to \infty}\dfrac{\sqrt{1}+\sqrt{3}+\sqrt{5}+\cdots+\sqrt{2n-1}}{\sqrt{2n^3}}.$$

Answer: \dfrac{2}{3}


Question: Determine whether the limit \( \lim_{x \rightarrow 0} \frac{\sin\left(x \sin \left( \frac{1}{x} \right) \right)}{x \sin \left( \frac{1}{x} \right)} \) exists. If it exists, find its value.

Answer: 1


Question: Evaluate the limit \( \lim_{n \to \infty} \int_n^{n+2} e^{-x^3} \, dx \).

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to \frac{\pi}{2}} \tan(x) \log(\sin(x)).$$

Answer: 0


Question: Consider a bounded domain $\Omega$ with a smooth boundary in $\mathbb{R}^n$. Let $u_0 \in H^1(\Omega) \setminus \{ 0\}$. Define the functional $$ J(u):= \int_{\Omega} \langle A(x) \nabla u(x), \nabla u(x)\rangle \, dx,$$ for $u \in K:=\{ v \in H^1(\Omega); v-u_0 \in H^{1}_{0}(\Omega)\}$, where $A(x) = a_{ij}(x)$ is a matrix with smooth coefficients. Assume there exist constants $0<\lambda_1 <\lambda_2 < +\infty$ such that $$\lambda_1 |\xi|^2 \leq \langle A(x) \xi,\xi\rangle \leq \lambda_2 |\xi|^2 $$ for all $x \in \Omega$ and $\xi \in \mathbb{R}^n$. Does the functional $J$ admit a minimum?

Answer: Yes


Question: Determine the number of elements of order \( p^2 \) in the group \( \mathbb{Z}_{p^2} \times \mathbb{Z}_{p^2} \).

Answer: p^4 - p^2


Question: Let $f$ be a function defined on $(a,b)$, such that it is integrable on any $[c,d]$ with $a<c<d<b$, but not necessarily integrable in the standard Riemann sense on the entire $[a,b]$. Define the function $F$ as an improper integral: $$F(u)=\int_{a}^{u}f(x)dx=\lim_{c\rightarrow a}\int_{c}^{u}f(x)dx.$$ Assuming that $F(u)$ is well defined for any $u\in (a,b)$, determine if $F(u)$ converges to $0$ as $u$ converges to $a$ from the right, i.e., $$\lim_{u\rightarrow a^+}F(u)=0.$$ Consider whether the assumption that $f$ is non-decreasing would be useful.

Answer: 0


Question: Determine whether there exist sequences $\{x_n\}$, with $0 \leq x_n \leq 1$ for all $n$, such that $\lim_{n\to\infty} x_n = 1$ and \[ \lim_{n\to\infty}(x_n^{a_n} - x_n^{b_n})=1. \]

Answer: Yes


Question: Let $G$ be a group and $a, b \in G$. If $\langle a, b \rangle$ is finite and $\langle a \rangle$ is a normal subgroup of $\langle a, b \rangle$, does it follow that $a$ and $b$ have finite order? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the contour integral \( \oint_C \frac{dz}{z+1} \) where \( C \) is the circle \( |z+2|=3 \).

Answer: 2\pi i


Question: Determine whether there exists a matrix $B$ over the complex numbers such that $B^{2018} = A$, where $A$ is a given diagonalizable matrix. Use the Cayley-Hamilton Theorem in your reasoning.

Answer: Yes


Question: Let $A$ and $P$ be $n \times n$ matrices with $P$ a self-adjoint projection (i.e., $P = P^* = P^2$) and $\Vert A \Vert = 1$, where the norm is the operator norm. Suppose that $A$ and $P$ almost commute, meaning the norm of their commutator $[A, P]$ is small. Is there a matrix $B$ close to $A$ in norm such that $[B, P] = 0$?

Answer: Yes


Question: In triangle $XYZ$, points $X'$ and $Y'$ are located on sides $XY$ and $XZ$, respectively, such that $XX' = YY'$. Let $P$ and $Q$ be the midpoints of $XY$ and $XZ$, respectively. If line $PQ$ intersects line $X'Y'$ at point $R$, and $XY = 2XZ$, determine the ratio $XR:RY'$. Express your answer as a simplified ratio.

Answer: 1:1


Question: Determine if the number \(17^{21} + 19^{21}\) is divisible by any of the following numbers: 36, 19, 17, or 21. Use mathematical methods to establish divisibility.

Answer: 36


Question: Let \( \psi: \mathbb{C}[X] \to \mathbb{R} \) be a homomorphism such that \( \psi(X) = 2 \). Determine the ideal generated by \( \ker \psi \).

Answer: (X - 2)


Question: Given a sequence \( x \in \ell^2(\mathbb{N}) \), where \( \ell^2(\mathbb{N}) \) is the space of square-summable sequences, and \( s_n=\sum_{i=1}^n x_i \), determine whether the sequence \( \frac{1}{\sqrt{n}}s_n \) converges.

Answer: 0


Question: Given a bilinear map $B: V \times W \to X$, prove that there exists a unique linear map $T: V \otimes W \to X$ such that $B = T \circ \phi$, where $\phi$ is the canonical map from $V \times W$ to $V \otimes W$. Show the existence and uniqueness of $T$.

Answer: T


Question: Determine whether a measure-preserving transformation of a probability space with a strictly positive density can be non-invertible. Provide a yes or no answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln (\cos(x^3-x))} $$ without using L'Hôpital's rule.

Answer: -2


Question: Suppose two runners start a race at the same time and finish the race in a tie. Let $f(t)$ and $g(t)$ represent the positions of the runners at time $t$. Using Cauchy's Mean Value Theorem, determine if there is a time $t_0$ when the two runners have the same velocity, even if their velocity functions are different.

Answer: Yes


Question: Determine the number of ring homomorphisms from the ring \( \mathbb{Z}[x,y]/(x^3+y^2-1) \) to \( \mathbb{Z}_7 \).

Answer: 11


Question: Evaluate the limit: $$\lim_{x \rightarrow 0} e^{-\frac{1}{x^2}}\cdot \frac{2}{x^3}$$

Answer: 0


Question: Let $M$ be a connected topological 2-manifold, i.e., a Hausdorff space locally homeomorphic to $\mathbb{R}^2$. Suppose $\pi_1(M)$ is a finitely generated free group. Is $M$ necessarily homeomorphic to a closed surface with finitely many punctures?

Answer: Yes


Question: Determine the values of $t$ for which the integral \( \int_{0}^{\pi} |\cos x|^{-t} \, dx \) converges. Specifically, analyze the behavior of the integral for \( t \in (-\infty,1) \) and \( t \in [1,\infty) \).

Answer: (-\infty, 1)


Question: Is it possible to construct a geometric figure with an area that is an irrational number using only a straightedge and compass?

Answer: Yes


Question: Given the equations \(x+y+z=7\) and \(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}=\frac{7}{10}\), find the value of \(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}\).

Answer: \dfrac{19}{10}


Question: Calculate the surface integral of the function \( f(x, y, z) = x^2 + y^2 + z^2 \) over the upper hemisphere of the sphere defined by \( x^2 + y^2 + (z - 1)^2 = 1 \).

Answer: 6\pi


Question: Given the cubic equation \(E_1 : x^3 + px - q = 0\) where \(p, q \in \mathbb{R}\), find the value of the expression \((\alpha + \beta)(\beta + \gamma)(\alpha+\gamma)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)\) where \(\alpha, \beta, \gamma\) are the roots of the cubic equation \(E_1\).

Answer: -\dfrac{p^2}{q}


Question: Evaluate the limit: $$\lim\limits_{(x,y)\to (0,0)}(\cos x)^y$$.

Answer: 1


Question: Determine the limit of the sequence \(x_n=\frac{1}{n}\left(n+\frac{n-1}{2}+\frac{n-2}{3}+\dots+\frac{2}{n-1}+\frac{1}{n}-\log(n!)\right)\).

Answer: \gamma


Question: Calculate \(67^{26^{42^{23}}} \mod 990\).

Answer: 661


Question: Find all values of $k$ for which $M(k) = 100^{k-1} + \dotsc + 100 + 1$ is a prime number.

Answer: 2


Question: Let $\{N(t), t \geq 0\}$ be a Poisson process with rate $\lambda$. Given that $E[N(t)|N(s) = 2] = 2 + \lambda(t - s)$, find $E[N(s)|N(t) = 3]$ for $s < t$. Provide your answer in terms of $\lambda$, $s$, and $t$. \(\boxed{\text{Answer}}\)

Answer: \dfrac{3s}{t}


Question: Evaluate the limit: $$\lim _{n\to \infty }\left(n\left(\frac{\left(\frac{\left(\ln\left(n\right)\right)^{10}}{n^{1.1}}\right)}{\frac{\left(\ln\left(n+1\right)\right)^{10}}{\left(n+1\right)^{1.1}}}-1\right)\right)$$

Answer: \dfrac{11}{10}


Question: Find the sum of the power series: $$\sum_{n=2}^{\infty} \frac{x^{n-1}}{n-1}.$$

Answer: -\ln(1 - x)


Question: Determine whether the set $B' = \{ \mathbb{R} \} \cup \{ (a,b) \cap \mathbb{I} \mid a < b, \ a,b \in \mathbb{I} \}$ forms a base for a topology on $\mathbb{R}$, where $\mathbb{I}$ is the set of irrational numbers.

Answer: Yes


Question: Consider a group homomorphism $f$ between two free abelian groups, one generated by $X$ and the other by $Y$. Does $f$ always map elements of $X$ into $Y$? Provide a justification for your answer.

Answer: No


Question: Determine the number of distinct prime ideals in the ring \( \mathbb{Q}[x]/I \), where \( I \) is the ideal generated by \( x^4-1 \).

Answer: 3


Question: Let $f: M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be a $\mathbb{C}$-linear map. Determine whether there exist matrices $A_1, \dots, A_d \in M_n(\mathbb{C})$ and $B_1, \dots, B_d \in M_n(\mathbb{C})$ such that \[f(X) = \sum_{j = 1}^d A_jXB_j,\text{ }\forall\,X \in M_n(\mathbb{C}).\]

Answer: Yes


Question: Can the center of a nonabelian group of order $2p$, where $p$ is an odd prime, have an order of $2$ or $p$? Provide a justification for your answer.

Answer: No


Question: Determine whether a complex, Hermitian, unsymmetric tridiagonal matrix with nonzero elements in its upper-right and lower-left corners can be transformed into a symmetric matrix using a similarity transform.

Answer: Yes


Question: If a random variable $X$ is in $L^q$ space, does it imply that $X$ is also in $L^p$ space for any natural number $p < q$?

Answer: Yes


Question: Calculate \( \lim _{n\to \infty }\:\left(\int _0^{\infty \:}\left(\sin\left(x\right)\right)^n e^{-x}\:dx\right) \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{h\to0}\left\{\frac{h e^{h}(e^{h}+1)+2(1-e^h)e^h}{(1-e^h)^3}\right\} \]

Answer: -\dfrac{1}{6}


Question: Consider a function $f$ whose graph is a semicircle $S$ with endpoints $(a,0)$ and $(b,0)$, where $a < b$. Evaluate the improper integral $\int_{a}^{b} f(x) f'(x) \, dx$ and determine which of the following statements is true: (A) necessarily zero, (B) possibly zero but not necessarily zero, (C) necessarily nonexistent, (D) possibly nonexistent but not necessarily, (E) none of the above.

Answer: A


Question: For a Hermitian matrix $A$, determine if the 2-norm of the absolute value of $A$ is equal to the maximum of the absolute value of the inner product of $Ax$ and $x$ over all unit vectors $x$. Provide a justification for your answer.

Answer: Yes


Question: Let \(\alpha\) be the closed curve along the square with vertices at \(1, i, -1, -i\). Provide an explicit parametrization for \(\alpha\) and compute the value of \[ \frac{1}{2\pi i}\int_\alpha\frac{dz}{z}. \]

Answer: 1


Question: Determine the smallest positive value of the expression $a^3 + b^3 + c^3 - 3abc$ for positive integers $a, b, c$. List all sets of integers $(a, b, c)$ that achieve this smallest value.

Answer: 4


Question: Determine the maximum absolute interpolation error when the function \( f(x) = x^4 \) is approximated by a quadratic polynomial passing through the points \((-1, 1)\), \((0, 0)\), and \((1, 1)\) within the interval \([-1, 1]\).

Answer: \dfrac{1}{4}


Question: Determine whether the following statement is true or false: For a topological space $(X, T)$ and a subset $A \subset X$, the boundary of the interior of $A$ is always a subset of the boundary of $A$. Provide a proof or counterexample.

Answer: True


Question: Evaluate the expression \( 4\sum^{30}_{n=1} n \cdot T(n) \), where \( T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) +\cos^2(30^\circ +n) \).

Answer: 1395


Question: Let $X_1, X_2, \ldots, X_n$ be a random sample from a continuous distribution with unknown mean $\mu$ and known standard deviation $\sigma$. If $[X_{\min}, X_{\max}]$ is used as a confidence interval for $\mu$, determine the confidence level in terms of the standard normal distribution.

Answer: 1 - 2[\Phi(0)]^n


Question: If a square matrix $B$ has $k$ distinct nonzero eigenvalues, what is the minimum possible nullity of $B$? Express your answer as a single number.

Answer: 0


Question: Determine the maximum number of elements in a subset \(X\) of \(\mathbb{R}^n\) such that any two distinct points in \(X\) are exactly one unit apart.

Answer: n + 1


Question: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function and $a_n\rightarrow a, f(a_n)\rightarrow b$. Does the existence of $\lim\limits_{n\rightarrow\infty}^{}f(a_n)$ imply $f(a)=b$? Answer yes or no.

Answer: No


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{\cos^2\left(1-\cos^2\left(1-\cos^2\left(1-\cdots\cos^2(x)\cdots\right)\right)\right)}{\sin\left(\pi\left(\frac{\sqrt{x+4}-2}{x}\right)\right)}$$ where the cosine function is iterated infinitely.

Answer: \sqrt{2}


Question: Let $X$ and $Y$ be two sets such that there is a one-to-one correspondence between their power sets, $P(X)$ and $P(Y)$. Does this imply that there is a one-to-one correspondence between $X$ and $Y$?

Answer: No


Question:
Determine the number of real roots of the equation:

\[
\frac{1}{x-a} + \frac{1}{x-b} + \frac{1}{x-c} = \frac{1}{(x-a)(x-b)} + \frac{1}{(x-b)(x-c)} + \frac{1}{(x-c)(x-a)}
\]

where $a, b, c$ are distinct real numbers.

Answer: 2


Question: Evaluate \( \lim_{x \to 0} \frac{\ln(\cos(-5x))}{\ln(\cos(-3x))} \) without using L'Hôpital's rule.

Answer: \dfrac{25}{9}


Question: An insurance company pays out claims following an exponential distribution. When a deductible \( d \) is applied, the expected claim payment decreases by 10\%. Calculate the percentage reduction in the variance of the claim payment.

Answer: 1\%


Question: What is the maximum distance between any two positions of a Rubik's Cube, given that the maximum number of moves required to reach the solved state from any position is 20 moves?

Answer: 20


Question: Determine whether a tangent to a cubic curve can intersect the curve orthogonally at another point. Provide a justification for your answer.

Answer: Yes


Question: Express the size $m$ of a plane graph in terms of its order $n$ and an exterior boundary $k$-cycle, given that all interior regions have a triangular boundary and $k \geq 3$. Provide the expression for $m$.

Answer: 3n - k - 3


Question: In an Artinian local ring $A$ with a unique maximal ideal $M$, is the set of zero-divisors in $A$ equal to $M$?

Answer: Yes


Question: Evaluate the integral \( \int_0^{2\pi} \frac{1-a^2}{1+a^2-2a\cos{t}} \, dt \), where \( a \) is a constant with \( |a|<1 \).

Answer: 2\pi


Question: For a strictly increasing unbounded sequence $a_n$, if the series \( \sum_{n=1}^{\infty} \frac{a_{n+1} - a_n}{a_{n+2}} \) converges, does the sequence \( a_n \) necessarily converge? Provide a justification for your answer.

Answer: No


Question: Given the differential equation \( \frac{dy}{dx} = 1-y^2 \) and the condition that the function \( y \) is symmetric about the point \( (\ln(9),0) \), find the value of \( y(\ln(4)) \).

Answer: -\dfrac{65}{97}


Question: Calculate the probability of encountering a run of 3 consecutive heads before encountering 3 consecutive tails in a sequence of coin tosses.

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: If an infinite subset \( A \) of a topological space \( X \) is contained within a basic set \( B \), then \( A \) has at least one limit point.

Answer: B


Question: Evaluate the limit \( \lim_{x\rightarrow -1} \frac{1}{x+1}(e^{2\pi i(x+1)}-1) \) without using l'Hopital's rule.

Answer: 2\pi i


Question: Find the limit of the sequence: $$\lim_{n\to\infty} \frac{n^{n}}{ e^{n}}$$

Answer: \infty


Question: Determine the number of solutions to the system of equations: \[\begin{align*} x^2-y^2&=z, \\ y^2-z^2&=x, \\ z^2-x^2&=y. \end{align*}\]

Answer: 4


Question: Evaluate the limit: $$\lim_{n\to\infty} \int^\infty_0 \frac{1}{1+(\frac{x}{a})^n} \, dx$$

Answer: a


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

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral \( \oint_C \frac{1}{z^2+4} \, dz \), where \( C \) is the contour that starts at \( 2i \), goes in a straight line to \( 0 \), then goes in a straight line to \( -2i \), and finally returns to \( 2i \) in a straight line.

Answer: 0


Question: Solve the system of congruences: \(\begin{align} x & \equiv 11\pmod{36}\\ x & \equiv 7\pmod{40}\\ x & \equiv 32\pmod{75} \end{align}\)

Answer: 407


Question: Find the distance from the origin to the curve defined by the equations \( \frac{x^2}{4} + y^2 + \frac{z^2}{4} = 1 \) and \( x + y + z = 1 \) in \( \mathbb{R}^3 \).

Answer: 1


Question: Determine the number of distinct subrings (with unity) of cardinality $p$ in the field $\mathbb{F}_{p^2}$.

Answer: 1


Question: Show that the sequence of functions \( f_n(x) = n^2 x^n (1-x) \) converges pointwise to 0 on the interval \([0, 1]\).

Answer: 0


Question: Find the value of the constant \( \Phi \) such that the integral \( \int_0^\infty \left( \frac{1}{\sqrt{x^2+4}} - \frac{\Phi}{x+2} \right) dx \) converges.

Answer: 1


Question: Compute the limit \(\lim_{n\to\infty}\frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}\) using the Stolz-Cesaro theorem.

Answer: 1


Question: Find the largest integer \( n < 999 \) such that \( (n-1)^2 \) divides \( n^{2016} - 1 \).

Answer: 673


Question: Given an n-dimensional cube, where the distance between any two vertices is defined as the minimum number of edges connecting them, consider a subset S of the vertices. Denote the full set of vertices as T. Can the minimum sum of distances from a vertex $x_A \in T$ to all vertices in a partition A of S, and from a vertex $x_B \in T$ to all vertices in a partition B of S, be computed using a simple calculation given the distance between every pair of vertices? Answer yes or no.

Answer: No


Question: Evaluate the integral \( \displaystyle \int_{0}^{\frac{\pi}{2}} \frac{1}{1+\tan(x)^{\sqrt{2}}}dx \) using the substitution \( x \to \frac{\pi}{2}-x \).

Answer: \dfrac{\pi}{4}


Question: Compute the second moment of the chi-squared distribution with $n$ degrees of freedom.

Answer: n(n + 2)


Question: For a square matrix $A$, if its characteristic polynomial is $p(t) = t^2 + 1$, determine whether the determinant of $A$ is always equal to 1.

Answer: 1


Question: Evaluate \( \lim_{x\to \frac{\pi}{2}} \frac{2^{-\cos x}-1}{x-\frac{\pi}{2}} \).

Answer: \ln 2


Question: Let $W$ be the set of all real sequences $\{b_j\}_{j=1}^\infty$ that satisfy the recurrence relation $b_j = Db_{j-1} + Eb_{j-2} + Fb_{j-3}$ for all $j \geq 4$, where $D,E,F$ are real constants. Determine if $W$ is a vector subspace of the vector space of all real sequences and find its dimension.

Answer: 3


Question: Determine the number of isomorphism classes of function fields of degree 3 over the rational function field $F = \mathbb{C}(t)$, given that they are ramified only at the points 1 and -1.

Answer: 1


Question: Given differentiable functions $f, g : \mathbb{R} \to \mathbb{R}$, both non-constant, satisfying $f'(x) - g'(x) = g'(x)(f(x))^2 - f'(x)(g(x))^2$ for all $x \in \mathbb{R}$, with initial conditions $f(1) = 1$ and $g(1) = \frac{1}{3}$, and no $x$ values where $f(x) = -2$ or $g(x) = 2$, find the minimum positive value that $(f(x) - g(x) + 3)$ attains.

Answer: 2\sqrt{5} - 1


Question: Determine whether the function \( d^2(.,C):\mathbb{R}^n \rightarrow \mathbb{R} \), defined as \( d^2(x,C)=\inf_{c \in C}||x-c||^2 \) where \( C \) is a nonempty closed convex subset of \( \mathbb{R}^n \), is convex. Note that \( d(x,C) \) is known to be convex and \( d^2(x,C)=||x-P(x,C)||^2 \), where \( P(x,C) \) denotes the projection mapping onto \( C \).

Answer: Yes


Question:
Let \( c \) be a complex number, and consider the equation that holds for every complex number \( z \):
\[(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct)\]
where \( r \), \( s \), and \( t \) are distinct complex numbers. Determine the number of distinct possible values of \( c \).

Answer: 4


Question: Determine the smallest rational number $r$ such that the sum of the first $n$ terms, $S_n = 1 + \frac{1}{1+2} + \frac{1}{1+2+3} + \ldots + \frac{1}{1+2+3+\ldots+n}$, is always less than $r$ for every natural number $n$. Provide your answer as a rational number.

Answer: 2


Question: Can the Green-Tao theorem be used to generate prime numbers? Provide a justification for your answer.

Answer: No


Question: Determine whether the vector $A^Tb$ always belongs to the column space of $A^TA$, regardless of the rank of $A^TA$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: If $H$ and $K$ are subgroups of a group $G$ and their product $HK$ is also a subgroup of $G$, then the orders of $H$ and $K$ ($|H|$ and $|K|$) are not coprime. Provide a proof or a counterexample.

Answer: False


Question: Calculate the limit using the relationship between the Beta and Gamma functions: $$\lim_{x \to \infty} \frac{B\left(\frac{x+1}{2},\frac{x}{2}\right)}{B\left(\frac{x}{2},\frac{x+1}{2}\right)}$$

Answer: 1


Question:
Evaluate the following expression for $b_n$ and determine its value when $n=1$:

$$b_n = 30 \left[\frac{\sin(\pi - n\pi )}{\pi - n\pi } - \frac{\sin(\pi + n\pi )}{\pi + n\pi}\right] - 200\left(\frac{\sin (n\pi )}{(n \pi)^2}\right) + 200\left(\frac{1}{n \pi}\right)$$

Specifically, explain why the expression is 0 if $n \neq 1$ and 1 if $n=1$. Provide a detailed justification for the case when $n=1$.

Answer: 1


Question: Let $E$ be the region in $\\mathbb{R}^3$ between the paraboloid $z = x^2 + y^2$ and the plane $z = 2$. Given the vector field $\\vec{F} = \\langle 2x, 2y, 0 \\rangle$, compute the flux integral $\\iint_S \\vec{F} \\cdot \\mathrm{d}\\vec{r}$ over the surface $S$ using the divergence theorem.

Answer: 8\pi


Question: Evaluate the limit: $$\lim_{x\rightarrow 0} \frac{\tan(x+\sqrt{x^2+1})-\sin x}{(3^{x^2}-1)(\cos (e^{x^2}-1))}$$ using Taylor series expansions.

Answer: \infty


Question: Given the functional equation \(2f(x) - 3f\left(\frac{1}{x}\right) = x^2\) and the conditions \(f(1) = f(-1) = -1\), find the value of \(f(2)\).

Answer: -\dfrac{7}{4}


Question: In a Hilbert space $X$, let $\varphi \in X'$ be a non-zero linear functional. Define the set $D = \{x \in X : \varphi(x) = 2\}$. Determine the orthogonal complement $D^{\perp}$ in $X$.

Answer: \{0\}


Question: Let $P(x)$ be a polynomial with integer coefficients such that $P\left(\sqrt{2}+\sqrt[3]{3}\right)=0$. If $R(x)$ is the remainder when $P(x)$ is divided by $x^2-2x-1$, find the sum of the coefficients of $R(x)$. \(\boxed{\text{Answer}}\)

Answer: -44


Question: Evaluate the limit: $$ \lim_{x\to\infty} \biggl(\frac {e^x-x^2 }{e^x-5x}\biggr)^{\log x} $$

Answer: 1


Question: Determine whether the following statement is true or false: The limit of the derivative of an increasing and bounded function is always 0.

Answer: B


Question: Evaluate the integral \( \int_{|z-i|=3} \frac {dz}{(z^2+2)(z+1)} \) using the residue theorem. The path is a circle centered at \(i\) with a radius of 3. Identify the singularities and determine which residues to consider for this contour integral.

Answer: 0


Question: Find the minimum integer value of $n \geq 3$ such that $p^n$ divides $(p^n-1)!$, where $p$ is a prime number.

Answer: 3


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

Answer: \dfrac{27}{40}


Question: Determine the equation of a plane that passes through the points (1, 1, 2) and (-1, 1, 1) and is tangent to the surface defined by the function \( f(x, y) = xy \).

Answer: x + 6y - 2z = 3


Question:
Given the equations:

\[ ax + by = 3 \]
\[ ax^2 + by^2 = 7 \]
\[ ax^3 + by^3 = 16 \]
\[ ax^4 + by^4 = 42 \]

where \(a, b, x, y\) are real numbers, find the value of \(ax^6 + by^6\).

Answer: 1316


Question: Evaluate the limit: \[ \lim_{n\to \infty} \frac{f(n)}{g(n)} \] where the functions $f(n)$ and $g(n)$ are defined as follows: \[ f(n) = 2^{\frac{1}{\sqrt{5}}\left[\left(\frac{1 + \sqrt{5}}{2}\right)^{n} - \left(\frac{1 - \sqrt{5}}{2}\right)^{n}\right]} \] \[ g(n) = 2^{\left(\frac{1 + \sqrt{5}}{2}\right)^{n+1}} \]

Answer: 0


Question: Consider the interval [0,1]. Calculate the probability that a randomly chosen number from this interval is transcendental.

Answer: 1


Question: Determine whether there exists a real-valued function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that is continuous everywhere but differentiable at only one point.

Answer: Yes


Question: Determine the value of \( m \in \mathbb{R} \) for which the system of inequalities \( x^2 - y + m \leq 0 \) and \( y^2 - x + m \leq 0 \) has a unique solution \((x, y)\).

Answer: \dfrac{1}{4}


Question: Assuming the Continuum Hypothesis and the Axiom of Choice, consider the smallest uncountable ordinal number, denoted by $S$. Let $C(S)$ be the collection of all functions from $S$ into the continuum ($2^{\aleph_0}$), such that the subset of $S$ not mapped to zero is countable at most. Determine if the cardinality of $C(S)$ is equal to the cardinality of $S$.

Answer: Yes


Question: Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all real numbers $x$ and $y$, the equation $$f(x^2 + f(y)) = y + (f(x))^2$$ holds.

Answer: f(x) = x


Question: Evaluate the integral \( \int_a^\infty \frac{1}{x} e^{-x} \,dx \).

Answer: E_1(a)


Question: Calculate the limit: $$\Large \lim_{x\to0^+}\frac{1}{\sin^2x}\int_{\frac{x}{2}}^x\sin^{-1}t\,dt $$

Answer: \dfrac{3}{8}


Question: Find the sum of all possible values of the constant $k$ such that the graph of the parametric equations \( x = 2+4\cos s, \) \( y = k-4\sin s, \) intersects the graph of the parametric equations \( x=1+\cos t, \) \( y=-3+\sin t \) at only one point.

Answer: -12


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

Answer: 0


Question: Determine whether the following statement is true: \[ \sum_{j=1}^{n-1}\binom{3n}{3j}=4p, \text{ where } p\in\mathbb{Z}. \]

Answer: True


Question: Evaluate the improper integral \( \int_0^\infty \frac{\ln(1/x)}{1-x^2}\,dx \).

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


Question: Calculate the limit: $$\lim_{x\to0}\left(\ln(1+x)^x+\ln(1+2x)^{2x}+...+\ln(1+px)^{px}\right)$$

Answer: p


Question: Determine the coefficient of \(x^{33}\) in the expansion of \((x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}\). Use the identity \((1-x)^{-k} = \binom{k+n-1}{k-1}x^n\) to assist in your calculation.

Answer: 83019300


Question: Calculate the Lebesgue measure of the set of natural numbers, \( \mathbb{N} \).

Answer: 0


Question: Determine whether the statement is true or false: Every infinite Polish group has infinitely many closed subgroups.

Answer: A


Question: For the infinite series \(\sum_{n=0}^{+\infty}x^n\) which converges to \( \frac{1}{1-x}\), determine if this convergence property holds true when \(x\) is replaced by an \(N \times N\) matrix. Provide the conditions under which the series converges.

Answer: \rho(A) < 1


Question: On a $1000 \times 1000$ board, each cell is initially set to $0$. Determine the minimum number of operations required to make all cells have the same non-zero number, where each operation consists of setting all cells in a $130 \times 130$ subboard to the same number.

Answer: 64


Question: Determine if the following statement is true: There exists a unique holomorphic function $f$ on the open disc $D(0,3/2)$ in the complex plane with center $0$ and radius $3/2$, satisfying the conditions: $|f(z)|=1/2$ when $|z|=1$, $z=0$, $z=1/2$, and $z=-1/2$ are the only simple zeros of $f$ in $D(0,1)$, and $f'(0)=1/8$. The candidate function for $f$ is given by $f(z)=-\dfrac{z}{2}\left(\dfrac{z-1/2}{1-z/2}\right)\left(\dfrac{z+1/2}{1+z/2}\right).$

Answer: True


Question: Determine if the surface defined by the equation \( x+y+z-\sin(xyz)=0 \) can be locally expressed as \( z=f(x,y) \) near the point \((0,0)\) with \( f(0,0)=0 \).

Answer: Yes


Question: Suppose there exists some $x$ such that $xn \sim n^2$ as $n \to \infty$. Determine if it can be concluded that $x \sim n$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of perfect squares between $1$ and $N$ that can be expressed as $\sqrt{N^2 + k^2}$, where $k$ ranges from $1$ to $\left\lfloor \frac{N}{2} \right\rfloor$. Provide your answer as a single integer.

Answer: 0


Question: Given a full-dimensional simplex $S \subset \mathbb{R}^n$ formed by vectors $v_1, v_2, \ldots, v_{n+1}$, let $v_c = \frac{\sum_{i=1}^{n+1} v_i}{n+1}$ be its centroid. For any vector $v \in S$, is it true that $\|v - v_c\|_2 \leq \max \{\|v_i - v_c\|_2\}$ for $i = 1, 2, \ldots, n+1$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty} \left(\frac{\sqrt{3!!}\cdot \sqrt[3]{5!!} \ldots \sqrt[n]{(2n-1)!!}}{n^n}\right)^{\frac{1}{n}}$$ without using the Stolz-Cesaro theorem.

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


Question: Evaluate the limit: $$\lim_{x \to +\infty} \frac{e^{\sin \frac{1}{x}}-1-\frac{1}{x}}{\ln \left(1+\frac{x^2}{(1+x)^3} \right )-\frac{x^2}{(1+x)^3}}$$ and determine if the solution using the substitution $y=\frac{1}{x}$ and Maclaurin polynomials, leading to $$\displaystyle \lim_{ y\to 0^{+}} \left ( -\frac{1}{6} \frac{y + o(y^3)}{\frac{1}{(1+y)^6}+o\left (\frac{y}{(1+y)^3}\right )} \right ) = 0,$$ is correct.

Answer: -1


Question: Let $A$ and $B$ be two $n \times n$ real matrices with strictly positive eigenvalues. Can the matrix $A + B$ have a strictly negative eigenvalue?

Answer: Yes


Question: Calculate the work done by the force field \( \mathbf{F} = x^3\,\mathbf{i} + y^3\,\mathbf{j} \) when an object is moved from point \( P(1, 0) \) to point \( Q(2, 2) \).

Answer: \dfrac{31}{4}


Question: Let $f:\mathbb{R} \to \mathbb{R}$ be a function such that $\displaystyle \lim_{x\to \infty} (f(x+1)-f(x))=1$. Determine if it is true that $\displaystyle \lim_{x\to \infty} \frac{f(x)}{x}=1$. Justify your answer.

Answer: 1


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_{e^n}^{n^2} \frac{x}{e^{x^{2017}}} \, dx$$

Answer: 0


Question: Determine the rank of the first homology group of the double branched cover of a knot in $S^3$.

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to \infty} \left[\sin\left(\ln(x+1)\right) - \sin\left(\ln(x)\right)\right].$$

Answer: 0


Question: Determine the smallest integer $n$ for which the number of pairs $(x, y)$ satisfying $x \ge y$, $\operatorname{lcm}(x, y) = n!$, and $\gcd(x, y) = 2023$ is greater than 2023. Note that $2023 = 7 \times 17^2$. What is this value of $n$?

Answer: 41


Question: Consider a subset of the plane with the property that the intersection of any two closed connected subsets is also connected. Does this property imply that the complement of the subset in the plane is path-connected? Answer yes or no.

Answer: Yes


Question: Evaluate the infinite sum: \[ \sum_{n=1}^\infty \frac{1}{(n+1)(n+2)(n+3)(n+4)(n+5)} \]

Answer: \dfrac{1}{480}


Question: Let $B$ be a ball in $\mathbb{R}^n$ and $f$ a diffeomorphism mapping $B$ onto a bounded region in $\mathbb{R}^n$. Can a diffeomorphism $g$ of $\mathbb{R}^n$ be found such that $g$ coincides with $f$ on $B$?

Answer: Yes


Question: Consider an infinite series $\sum a_n$ where each term $a_n$ is non-negative. If the series diverges, is it true that $\sum a_n = +\infty$? Provide a justification for your answer.

Answer: Yes


Question: Let $X$ be a metrizable space with metric $d$. If $x \in X$ is disjoint from a closed subset $A$ of $X$, is there always an $\epsilon > 0$ such that the $\epsilon$-ball around $x$ is disjoint from $A$?

Answer: Yes


Question: Let $L = \mathbb{Q}(\sqrt[4]{2}, \omega)$ where $\omega = e^{\frac{2\pi i}{8}}$. Determine the degree of the field extension $[L: \mathbb{Q}(\sqrt[4]{2})]$. \( \boxed{\text{Your answer here}} \)

Answer: 2


Question: Does there exist a sequence \((x_n)\) of real numbers such that for every \(d\) in \((0,1]\) and every \(S \subseteq \mathbb{N}\) of natural density \(d\), the subsequence \((x_i\, : \, i \in S)\) is divergent?

Answer: Yes


Question: Calculate the limit \[ \lim_{b\to \infty}\frac{\vert \text{erf}(\sqrt{a+\mathrm{i} b})\vert^{2}}{\sqrt{a^2+b^2}} \] where \[ \text{erf}(a+\mathrm{i} b)=\frac{2}{\sqrt{\pi}}\int_{0}^{a+\mathrm{i} b}e^{-t^2}~dt \] is the error function.

Answer: 0


Question: Simplify the limit: \[ \lim_{y\to\infty}\frac{f^{-1}(8y)}{y^{1/3}} \]

Answer: 2


Question: Determine whether a surjective function can exist between a simply connected domain and a non-simply connected domain. Provide a justification for your answer.

Answer: Yes


Question:
Let $g(x)$ be a differentiable function that satisfies $0 < g(x) < 1$ and $|g'(x)| < 1$ for all $x$ on the interval $0 \le x \le 1$. Determine the number of solutions to the equation

$$ \underbrace{g(g(g( \ldots g}_{2023~\text{times}}(x) \ldots) =x $$

on the interval $0\leq x\leq 1$. \( \boxed{} \)

Answer: 1


Question: Determine whether every element of $SU(n)$ is conjugate to a block-diagonal matrix with blocks from $SU(2)$ and $SU(1)$. Provide a justification for your answer.

Answer: No