Question: Given that \(x = \frac{\pi}{5}\), the expression \(y = \frac{1}{\sqrt{2}}(\sqrt{(1+\cos2x)}+\sqrt{(1+\cos4x)}+\sqrt{(1+\cos6x)}+\ldots+\sqrt{(1+\cos12x)})\) can be written as \(\frac{5}{4}(\sqrt{p}+q^2)\). Determine the value of \(\frac{5}{6} [p + q^2]\).

Answer: 5


Question: Let \(p\) be a prime number, and let \(\mathbb{F}_{p}\) be a finite field of order \(p\). Consider the unitriangular group \(U_{n}\) of \(n \times n\) upper triangular matrices with ones on the diagonal over \(\mathbb{F}_{p}\). Let \(H\) be an abelian subgroup of \(U_{n}\) with order \(p^{m}\). Must \(H\) be elementary abelian of rank \(m\), i.e., is \(H \simeq (\mathbb{F}_{p})^{m}\)?

Answer: No


Question: Assuming the hour, minute, and second hands of a clock are of equal length, how many times do they form an equilateral triangle in a 12-hour period?

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to0} \frac{\ln(x+1)-x}{1-\cos(x)} \]

Answer: -1


Question: Determine the interval of convergence for the power series \( \sum_{n=0}^{\infty}\left(\frac{1}{\sqrt{3n}}-\frac{1}{n!}\right)(x+2)^n \).

Answer: [-3, -1)


Question: Consider the C* algebra $C_0(\mathbb{N})$, where $\mathbb{N}$ is equipped with the discrete topology. Is this space the same as the space $c_0$, which consists of all sequences that converge to zero?

Answer: Yes


Question: Calculate the limit: \[ \lim_{x\to \infty } \left ( \frac{x^2+2}{x^2-4} \right)^{9x^2} \]

Answer: e^{54}


Question: Given the field $K = \mathbb{Q}(2^{1/5} e^{2i\pi/5})$, consider its automorphism. Can this automorphism be extended to an automorphism of $\mathbb{C}$? If so, determine whether the extended automorphism is the identity or a non-identity automorphism of $\mathbb{C}$. Provide your answer as either "identity" or "non-identity".

Answer: non-identity


Question: Let \( F(x), G(x) \in \mathbb{Z}[x] \) with \( G(x) \) being irreducible in \( \mathbb{Z}[x] \). If \( G(x) \mid F(x) \) in \( \mathbb{Q}[x] \), can we conclude that \( G(x) \mid F(x) \) in \( \mathbb{Z}[x] \)?

Answer: Yes


Question: Find the expected value of the sum of the dots when \( k \) 6-sided dice are rolled using the theory of generating series.

Answer: \dfrac{7k}{2}


Question: Does the manifold $S^1$ have a boundary?

Answer: \emptyset


Question: In a combinatorics class, there are 5 people who need to complete 4 tasks. Each task must be completed by a team of 2 people. Determine the number of ways to assign teams to tasks such that no person is left without participating in any task. Use the inclusion-exclusion principle to solve the problem.

Answer: 4320


Question: Given a quadratic equation \( ax^2 + bx + c = 0 \), where \( a, b, c \) are positive integers in an arithmetic progression, and the roots \( \alpha \) and \( \beta \) are integers, find the value of \( \alpha + \beta + \alpha\beta \).

Answer: 7


Question: Calculate the probability that exactly 2 out of 4 couples sit next to each other when seated randomly in a row of 8 seats. Assume each couple consists of two people, and the seating arrangement is completely random.

Answer: \dfrac{1}{5}


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

Answer: \sqrt{e}


Question: Determine the domain of the operator $|\Delta|^{1/2}$, where $U$ is a bounded open subset of $\mathbb{R}^N$ and $\Delta: H^2(U) \subset L^2(U) \to L^2(U)$ is the Laplace operator.

Answer: H^1(U)


Question: Find the exact value of the integral: $$\int_{x}^{+\infty}\frac{e^{-t}}{t}dt$$

Answer: E_1(x)


Question: Is the boundary of an open ball in \(\mathbb{R}^{n}\) homeomorphic to \(\mathbb{R}^{n-1}\)?

Answer: No


Question: Determine whether the limit \( \lim_{n\to\infty}\int_a^b f(x)\sin^3{(nx)} \,dx \) exists, given that \( f:[a,b]\to\mathbb{R} \) is continuous.

Answer: 0


Question: Find the value of \( 17^{{{17}^{17}}^{17}} \pmod{25} \).

Answer: 2


Question: Evaluate the limit \( \lim_{n\to\infty} \left( n \int_0^{\pi/4} (\tan x)^n \, dx \right) \).

Answer: \dfrac{1}{2}


Question: Determine whether there exist three integers \(a\), \(b\), and \(c\) such that \(a^{p_1} + b^{p_2} = c^{p_3}\), where \(p_1\), \(p_2\), and \(p_3\) are distinct odd prime numbers.

Answer: Yes


Question: Given the functions \(f(x) = e^{6x}\) and \(g(x) = ax^2\) where \(a > 0\), find the condition on \(a\) such that there are exactly two lines, each tangent to both functions.

Answer: a > 9e^2


Question: A parabola has its vertex at \(\left(\frac{1}{4},\frac{-9}{8}\right)\) and is described by the equation \(ax^2 + bx + c = y\), where \(a > 0\) and \(a + b + c\) is an integer. Determine the minimum possible value of \(a\).

Answer: \dfrac{2}{9}


Question: Let \( A, B, C, D, E \) be five points marked in clockwise direction on the unit circle in the plane with center at the origin. Given the function \( f(P) = \alpha x + \beta y \) for a point \( P = (x, y) \), if \( f(A) = 10, f(B) = 5, f(C) = 4, \) and \( f(D) = 10 \), find the value of \( f(E) \).

Answer: 5


Question: A broken clock moves the minute hand each minute randomly one minute ahead or one minute behind with equal probabilities, independently of previous movements. The hour hand is synchronized with the minute hand. The clock starts running at noon. Let T denote the first time that the clock shows either 1 o'clock or 11 o'clock, whichever occurs first. Compute the expected value of T, E(T).

Answer: 3600


Question: Evaluate the limit: $$\lim_{x\to 0}\left\{\frac{e^{-x^2}\sin|x|-|x|}{|x^3|}\right\}$$

Answer: -\dfrac{7}{6}


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \sum_{j=n}^{4n} \binom{4n}{j} \left(\frac{1}{4}\right)^j \left(\frac{3}{4}\right)^{4n-j} \]

Answer: \dfrac{1}{2}


Question: Calculate \( \lim_{x\to \infty}\left(x + \ln\left(\frac{\pi}{2} - \arctan(x)\right)\right) \) using L'Hopital's rule.

Answer: \infty


Question: Evaluate the limit: \[ \lim\limits_{(x,y)\to(0,0)}\frac{e^{x^2+y^2}-x^2-y^2-1}{(x^2+y^2)^2} \]

Answer: \dfrac{1}{2}


Question: Determine whether every compact complex manifold of complex dimension greater than or equal to two possesses a nontrivial rank 2 holomorphic vector bundle.

Answer: Yes


Question: Let $X$ and $Y$ be jointly continuous random variables with the joint density function given by: $$f_{X,Y}(x,y) = \begin{cases} \frac{1}{3}(x+y), & \text{for } x \in (0,1) \text{ and } y \in (0,2) \\ 0, & \text{otherwise}. \end{cases}$$ Compute the expected value $\mathbb{E}[\sin(\pi XY)]$. Express your answer as an integral if necessary.

Answer: \dfrac{1}{\pi}


Question: Find the supremum of the set \( \{x \geq 0 : \sum_{n=1}^\infty x^{\sqrt{n}} < \infty \} \).

Answer: 1


Question: Does the $\sigma$-algebra generated by clopen sets in the Stone–Čech compactification $\beta \Gamma$ of a countable discrete group $\Gamma$ equal the Borel $\sigma$-algebra on $\beta \Gamma$?

Answer: No


Question:
Calculate the limit of the sum of binomial coefficients given by:
\[ S_{n} = \sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right). \]

Answer: 0


Question: Evaluate the limit: \[ \lim _{n \to \infty }\left[n^2\left(\left(1+\frac{1}{n\left(n+2\right)}\right)^n-\frac{n+1}{n}\right)\right] \]

Answer: -\dfrac{3}{2}


Question: Evaluate the integral \( \int_{\mathbb{T}} z^2 \, d\mu \), where \( \mathbb{T} = \{z \in \mathbb{C} : |z| = 1\} \), \( \phi:[0,1] \to \mathbb{T} \) is defined by \( \phi(t) = e^{2 \pi i t} \), \( m \) is the Lebesgue measure, and \( \mu(E) = m(\phi^{-1}(E)) \).

Answer: 0


Question: Are all compact subsets of \( \mathbb{R}^{\mathbb{R}} \) separable? Consider \( \mathbb{R} \) with its usual topology and \( \mathbb{R}^{\mathbb{R}} \) with the topology of pointwise convergence. Provide a justification for your answer.

Answer: No


Question: Evaluate the line integral \( \int_C zdx + xdy + xydz \) where \( C \) is the path of the helix \( \mathbf{r}(t) = (4\cos t)\mathbf{i} + (4\sin t)\mathbf{j} + t\mathbf{k} \) for \( 0 \le t \le 2\pi \).

Answer: 24\pi


Question: Given the implicit function $f(x,y) = 2x^3 - 4xy + 2y^3$, where $y = f(x)$, use the implicit function theorem to find the second derivative $y''(1)$ at $x = 1$. Assume $y(1) = 1$ and $y'(1) = -1$. Evaluate $y''(1)$.

Answer: -16


Question: Consider a binomial random variable $X \sim \operatorname{Bin}(12, p)$. Compare the mean squared errors of the estimators $\hat{p}_1 = \frac{X}{10}$ and $\hat{p}_2 = \frac{X}{12}$. Determine the range of $p$ for which the mean squared error of $\hat{p}_1$ is greater than that of $\hat{p}_2$. Assume $\operatorname{MSE}(\hat{p}_2) = \frac{1}{12}(p - p^2)$ and $\operatorname{MSE}(\hat{p}_1) = \frac{12p - 8p^2}{100}$. Find the range of $p$ where $\operatorname{MSE}(\hat{p}_1) > \operatorname{MSE}(\hat{p}_2)$. \(0 \leq p \leq 1\)

Answer: (0, 1]


Question: Find the last two digits of the number $9^{9^9}$.

Answer: 89


Question: Let $S$ be a commutative monoid and $f(x_1, \dots, x_n)$ be a function from $S^n$ to $S$. Given a partition $\alpha$ of $[n]$, we say that $f$ factors with respect to $\alpha$ if for each $A \in \alpha$ there exists a function $f_A$ (which only depends on the variables $x_i$ for $i \in A$) such that $f=\prod_{A \in \alpha} f_A$. Given two partitions $\alpha$ and $\beta$ of $[n]$, $\alpha \wedge \beta$ is the partition of $[n]$ whose sets are the non-empty sets of the form $A \cap B$ for $A \in \alpha$ and $B \in \beta$. Is it true that if $f$ factors with respect to both $\alpha$ and $\beta$, then $f$ also factors with respect to $\alpha \wedge \beta$?

Answer: Yes


Question:
A wheel of radius \(r\) inches starts moving over an incline from point \(O\), which is a tangential point. The wheel moves up the incline due to an external force and stops at point \(T\), sweeping an angle \(\theta\). Given the expression \(\left(\sqrt{\theta^2+1}\right)\sin \beta\), determine its value from the following options:

\[\begin{array}{ll}
1.&\cos\alpha-\theta\sin \alpha\\
2.&\sin \alpha+\theta\cos \alpha\\
3.&\cos\alpha+\theta\sin \alpha\\
4.&\sin\alpha - \theta \cos\alpha\\
\end{array}\]

Answer: 3


Question: Find the derivative \( \frac{Df(A)}{DA} \) of the function \( f(A) = \frac{1}{2} \text{trace}(A^\top A) \), where \( A \) is a matrix.

Answer: A


Question: Find the exact coefficient of \(x^{12}\) in the expansion of \((2+x^3+x^6)^{10}\).

Answer: 71040


Question: Given point $A(-3, 1)$, the equation of the median through $B$ as $L_1: 2x + y - 3 = 0$, and the equation of the angle bisector of angle $C$ as $L_2: 7x - 4y - 1 = 0$, find the equation of side $AB$ in the triangle.

Answer: 4x + 7y + 5 = 0


Question: Determine the cardinality of the set \( \mathcal{R}[0,1] \) of all Riemann integrable real functions on the interval [0,1].

Answer: 2^{\mathfrak{c}}


Question: Determine whether there exists a subset of \( \mathbb{R} \) that is Lebesgue-null, right-perfect, and left-perfect.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\int_{0}^{1}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}\,dx \]

Answer: 0


Question: Determine whether the following statement is true or false: For each polynomial of the form \( p(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1} + z^n \), the inequality \( \sup\{ |p(z)| \mid |z| \le 1 \} \ge 1 \) holds.

Answer: True


Question: Evaluate the integral \( \int_{C} \frac{1}{\sin(z)} \, dz \) where \( C \) is the circle centered at the origin with radius 1, using complex analysis methods.

Answer: 2\pi i


Question: Suppose two metrics induce the same bornology. Do they necessarily induce the same topological space?

Answer: No


Question: Solve the congruence equation: \(636^{369} \equiv x \pmod{126}\).

Answer: 90


Question: Evaluate the limit \( \lim_{x \to \infty} x \log{\left|\frac{x+2}{3-x}\right|} \).

Answer: 5


Question: Determine whether the dimension of the vector space \( V = \prod_{i \in I} \mathbb{C} \), where \( I \) is a set, is equal to the cardinality of \( I \). If not, describe the dimension of \( V \) as an infinite cardinal.

Answer: 2^{|I|}


Question:
Let $p$ and $q$ be prime numbers such that $3 \leq p < q$. Define the set $\langle p, q \rangle = \{ mp + nq \mid m, n \in \mathbb{N} \}$. Compute the radius of convergence of the power series:
\[ S(z) = \sum_{s \in \langle p,q \rangle} z^s. \]

Answer: 1


Question: Consider the sequence of sets defined by $A_1 = \{ 5 \}$ and $A_{n+1} = \{ A_n \}$ for each $n \in \mathbb{N}$. Let $A = \bigcup_{n=1}^\infty \mathcal{P}(A_n)$. Determine whether $A \in A$. Provide a justification for your answer.

Answer: A \notin A


Question:
Evaluate the limit of the following sum involving binomial coefficients, where $n$ and $q$ are non-negative integers, and $n > q$:
\[ \lim_{m \to \infty} \frac{1}{2^m} \sum_{k} \binom{m}{nk+q} \]
Does this limit exist? If so, find its value.

Answer: \dfrac{1}{n}


Question: Calculate the size of the centralizer \(\left|C_{S_{12}}(\sigma)\right|\) where \(\sigma = (1\ 2\ 3\ 4\ 5\ 6)(7\ 8\ 9\ 10\ 11\ 12)\).

Answer: 72


Question: If $u(x,t)$ satisfies the Schrödinger equation \( i \frac{\partial }{\partial t}u (x,t )+ \Delta u(x,t) =0 \) with initial condition \( u(0,x)=\phi(x) \), does \( \chi_2(D)u \) also satisfy the Schrödinger equation, where \( \chi_2(D)u = \mathcal{F}^{-1} \left(\chi_{2}(\xi) \hat{u} \right) \) and \( \chi_{2}(\xi) = \chi_{[2, 4]}(\xi) \)?

Answer: Yes


Question: Let $ABCDEFG$ be a regular heptagon inscribed in a unit circle centered at $O$. A line $\ell$ is tangent to the circle at point $A$, and $P$ is a point on $\ell$ such that triangle $AOP$ is isosceles. Determine the value of $p^2$, where $p = AP \cdot BP \cdot CP \cdot DP \cdot EP \cdot FP \cdot GP$. \(p^2\) is the square of this product.

Answer: 113


Question: Consider an infinite graph $G$ with vertices labeled as $v_{i,j}$, where $i$ and $j$ are positive integers. Each vertex $v_{i,j}$ is connected to all vertices $v_{k,i+j}$ for every positive integer $k$. Determine if the chromatic number of the graph $G$ is necessarily infinite.

Answer: \infty


Question: Solve the differential equation \( \frac{dy}{dx} = \frac{6}{x+y} \) with the initial condition \( y(0) = 0 \). Find the value of \( y \) when \( x+y = 6 \).

Answer: 6 \ln 2


Question: Determine whether the following statement is true or false: For every positive integer $d$, $\sqrt{d}$ is either a positive integer or an irrational number.

Answer: A


Question: Let \( f: [a,b] \to \mathbb{R} \) be a continuous function on \( [a,b] \). Prove that there exists a point \( c \in [a,b] \) such that: \[ \int_a^c f(x) \, \mathrm{d}x = \int_c^b f(x) \, \mathrm{d}x. \]

Answer: c


Question: Is it possible for a function $f$ to have a derivative $f'(c)$ that exists at some point $c$, but for $f'$ to not be continuous at $c$? Provide a justification for your answer.

Answer: Yes


Question: Given the equations $x + y = 5xy$, $y + z = 6yz$, and $z + x = 7zx$, find the value of $x + y + z$. Express your answer as a single number.

Answer: \dfrac{13}{12}


Question: Evaluate the limit: $$\lim_{x \to 0}\frac{1}{x^3}\int^x_0\frac{t\ln(1+t)}{t^4 + 4} \, dt$$

Answer: \dfrac{1}{12}


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} \dfrac{P_{n-1}\cdot P_{n+1}}{P_{n}^{2}} \] where \( P_{n} \) is the product of the numbers in the \( n \)-th row of Pascal's Triangle.

Answer: e


Question: Consider the function $f:[1,3]\to\mathbf{R}$ given by $f(x)=-x^4+8x^3+ax^2+bx+d$, where $a$, $b$, and $d$ are real constants. Determine the values of $d$ such that $f$ has exactly 3 stationary points in the interval $(1,3)$ and satisfies $f(1)=f(3)=0$. Provide your answer as a set of values for $d$.

Answer: (-15, -9)


Question: Let \( f(x) \) be a continuous function on the interval \([0, 1]\) satisfying the equation \(4xf(x^2) + 3f(1-x) = \sqrt{1-x^2}\) for all \(x \in [0, 1]\). Calculate the integral \( I = \int_0^1 f(x) \, dx \).

Answer: \dfrac{\pi}{20}


Question: Does there exist a rational number that can be expressed as the product of an infinite number of distinct irrational numbers, such that no finite sub-product is rational, and neither the inverse nor any integer multiple of any irrational number in the product is included? If so, provide an example or prove its existence.

Answer: 2


Question: Given two real polynomials $f(x)$ and $g(x)$ of degrees $m$ $(m \ge 2)$ and $n$ $(n \ge 1)$ respectively, satisfying the equation $f(x^2 + 1) = f(x)g(x)$ for every $x \in \mathbb{R}$, determine the number of roots of the polynomial $f(x)$. What can be concluded about the roots of $f(x)$?

Answer: 0


Question: Let X be an uncountable set and S be a countable set of uncountable subsets of X. Define R(S) as the sigma-algebra generated by S. For elements x and y in X, x is "indistinguishable from y" if every set s in S contains either both x and y or neither. A point x in X is "indistinguishable" if there exists some y in X from which it is indistinguishable. Let T be the subset of X containing all indistinguishable points. Is T an element of R(S)?

Answer: Yes


Question: Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$. Consider a flat family of linear spaces $\Lambda_{\xi, q}$, with $\xi \in C$, parametrized by $C$. Assume that for a general $\xi\in C$, the intersection multiplicity of $\Lambda_{\xi, q}$ and $X$ at $\xi$ is at least $m_1$, and the intersection multiplicity of $\Lambda_{\xi, q}$ and $X$ at $q$ is at least $m_2$. Let $\Lambda_{q, q}$ be the limit linear space of the family $\Lambda_{\xi, q}$ as $\xi\mapsto q$ along $C$. Is the intersection multiplicity of $\Lambda_{q, q}$ and $X$ at $q$ at least $m_1+m_2$?

Answer: Yes


Question: Compute the degree of the morphism \( f: X \to \mathbb{A}_{\mathbb{C}}^1 \), where \( X = \{(t,t^2,t^3) : t \in \mathbb{C}\} \) and \( f(t,t^2,t^3) = t \).

Answer: 1


Question: Determine whether the following statement is true or false: If $|X_n|\xrightarrow{\mathcal{D}}0$, then $X_n\xrightarrow{\mathcal{D}}0$. Justify your answer.

Answer: True


Question: Evaluate the integral \( \int_{S} (x + 2y - z) \, dV \), where \( S \) is the tetrahedron in \( \mathbb{R}^3 \) with vertices \((0,0,0), (1,2,3), (0,1,2), (-1, 1, 1)\). Use a suitable linear transformation as a change of variables to simplify the integration.

Answer: \dfrac{1}{6}


Question: Determine whether the growth of the function \( f(n) = \frac{(n+1)(n+2)\ldots(n+\sqrt{n})}{(n-1)(n-2)\ldots(n-\sqrt{n})} \) is polynomial. Specifically, does there exist constants \( k \) and \( m \) such that \( f(n) < n^k \) for all \( n > m \)?

Answer: Yes


Question: Determine the domain in which the function \( f(z) = z \cdot \text{Im}(z) \) is differentiable and calculate its derivative.

Answer: 0


Question: Determine the number of binary linear \((10,8)\)-codes. A binary linear \((n,k)\)-code is a linear subspace of \(\mathbb{F}_2^n\) with dimension \(k\).

Answer: 174251


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{\displaystyle\int_0^x(x-t)\sin t^2\ dt}{x\sin^3x}. \]

Answer: \dfrac{1}{12}


Question: Find the minimum value of the expression \[ P = \frac{x^2}{\sqrt{x^3+8}} + \frac{y^2}{\sqrt{y^3+8}} + \frac{z^2}{\sqrt{z^3+8}} \] given that \( x, y, z \) are positive real numbers such that \( xy + yz + zx = 3 \).

Answer: 1


Question: Evaluate the limit \( \lim_{n \to \infty} \binom{2n}{n} \left(\frac{1}{4}\right)^n \) using Stirling's approximation. Use the bounds: \( e^{\frac{1}{12n+1}}<\frac{n!}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}<e^{\frac{1}{12n}} \).

Answer: 0


Question: Determine the number of $L$-equivalence classes for the language $L$ consisting of all strings in $(a+b)^*$ that have an even number of letters and do not contain $aaba$ as a substring.

Answer: 9


Question: Find the radius of convergence of the function \( \frac{z+1}{z-i} \) around the point \( z_0 = 2+i \).

Answer: 2


Question: Evaluate the integral \( \int_C \frac{2+3\sin(\pi z)}{z(z-1)^2}\mathrm{d}z \), where \( C \) is the circle of radius 6 centered at 0, positively oriented. Show that the value of the integral is \(-6\pi^2 i\).

Answer: -6\pi^2 i


Question: How many values of \( k \) in the set \( \{0, 1, 2, \ldots, 200\} \) make \( \binom{200}{k} \) divisible by 3?

Answer: 165


Question: Calculate the tensor product \( \mathbb{Z}/2 \otimes_{\mathbb{Z}} \mathbb{Z} \) considered as \( \mathbb{Z} \)-modules. What is the result?

Answer: \mathbb{Z}/2\mathbb{Z}


Question: Let $P(x)$ be a polynomial with non-negative integer coefficients such that $P(0)=33$, $P(1)=40$, and $P(9)=60000$. Find the value of $P(2)$.

Answer: 87


Question: Evaluate the flux integral \( \iint\limits_S \mathbf{F} \cdot \mathbf{n} \, dS \) over the surface \( S \), where \( S \) is the part of the sphere \( x^2 + y^2 + z^2 = 4 \) in the first octant. The vector field is given by \( \mathbf{F}(x, y, z) = \langle 1, 1, 1 \rangle \), and \( \mathbf{n} \) is the unit outward normal vector to the surface \( S \).

Answer: 3\pi


Question: Determine whether there exists an irreducible polynomial over \( \mathbb{Z}[x] \) with coefficients that are setwise coprime, which evaluates to a composite number for all integer arguments.

Answer: Yes


Question: Find the area of the region enclosed by the ellipse given by the equation \(5x^2 + 6xy + 2y^2 + 7x + 6y + 6 = 0\).

Answer: \dfrac{\pi}{2}


Question: Evaluate the sum \( \sum_{i=1}^3 \frac{P(x_{i})-x_{i}^{3}}{P'(x_{i})} \) for the polynomial \( P(x) = x^3 - x^2 - x + a \), where \( a \in \mathbb{R} \) and \( x_1, x_2, x_3 \) are distinct roots of \( P(x) = 0 \).

Answer: -1


Question: Determine the function $f(x)$ that satisfies the functional equation \( f(x^2+y^2)=f(x^2-y^2)+f(2xy) \) for all \( x, y \in \mathbb{R} \), given that \( f(0)=0 \) and \( f''(0)=2 \).

Answer: x^2


Question:
Find the marginal probability density function with respect to $x$, $f_X(x)$, for the joint pdf given by:
$$
 f(x,y) = \frac{2}{x^2 (x-1) y^{(2 x-1)/(x-1)}}
$$
for $x > 1$ and $y > 1$. Determine the correct limits of integration and ensure the integral converges.

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


Question: Given the differential equation \((x^2 - 1)y'' - 2xy' + 2y = 0\) and a known solution \(y_1 = x\), use the method of reduction of order to find a second linearly independent solution.

Answer: x^2 + 1


Question:
Let the incomes in two cities follow the Pareto distributions with probability density functions given by:
\[ f(x) = \frac{2}{x^3}, \quad 1 < x < \infty \]
\[ g(y) = \frac{3}{y^4}, \quad 1 < y < \infty \]
where one unit represents $20,000. A person with income is selected at random from each city. Let X and Y be their respective incomes. Assuming X and Y are statistically independent, compute the probability \( P(X < Y) \).

Answer: \dfrac{2}{5}


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\ln(\sin 3x)}{\ln(\sin x)} \] without using L'Hospital's Rule.

Answer: 1


Question: Determine the value of \( k > 0 \) for which the equation \( \ln x - kx^2 = 0.5 \) has exactly one solution.

Answer: \dfrac{1}{2e^2}


Question: We roll a fair die repeatedly. Let $X$ be the number of rolls needed to obtain a 5, and $Y$ be the number of rolls needed to obtain a 6. Calculate the expected value $E(X|Y=2)$.

Answer: \dfrac{33}{5}


Question: Is there a dimension $n$ such that a sphere packing in $\mathbb{R}^n$ can achieve a higher density than any lattice packing?

Answer: Yes


Question: Find the value of \( \csc^2 \frac{\pi}{7} + \csc^2 \frac{2\pi}{7} + \csc^2 \frac{3\pi}{7} \).

Answer: 8


Question: Given a coin that lands on heads with probability \( p \) such that \( P(p \le x) = x^5 \), and knowing that the first 9 tosses of the coin resulted in heads, what is the probability that the 10th toss will also be a head?

Answer: \dfrac{14}{15}


Question:
Given the joint moment generating function of random variables X and Y:
\[ M_{X,Y}(a,b) = \frac{4}{5}\left[\frac{1}{(1-a)(1-b)}+\frac{1}{(2-a)(2-b)}\right] \]
find the expected value of the product E(XY).

Answer: \dfrac{17}{20}


Question: Given a random sample $X_1, X_2, \ldots, X_{10}$ from a normal population with mean 0 and variance $\sigma^2$, let $\bar{x}$ be the sample mean and $s^2$ be the sample variance. Determine the distribution of the statistic $\frac{s^2}{10\bar{x}^2}$. (The answer is $F_{9,1}$)

Answer: F_{9,1}


Question: How many distinct cellular automata rules are there that use the Moore neighborhood on a two-dimensional Cartesian lattice if each site can be in one of eight states?

Answer: 8^{8^9}


Question:
Consider the functional defined by \( F(\lambda, z) = \int_{\mathbb{R}^n} |f(x, \lambda, z)|^p \, dx \) for \( 1<p<+\infty \), where \( f=f(x, \lambda, z):\mathbb{R}^n\times \mathbb{R}_{+}\times \mathbb{R}^n \). If \( \lambda^{*}, z^{*} \) are minimizers of \( F \), verify whether the following first-order conditions hold:

1. \( \partial_{\lambda} F(\lambda^{*},z^{*}) = 0 \implies p\int |f|^{p-2} f \partial_{\lambda }f \, dx = 0 \)
2. \( \partial_{z}F(\lambda^{*},z^{*}) = 0 \implies p\int |f|^{p-2} f \partial_{z}f \, dx = 0 \)

Are these conditions correct?

Answer: Yes


Question: Determine the value of \( a \) such that the equation \( ax^2 + 4xy + y^2 + ax + 3y + 2 = 0 \) represents a parabola.

Answer: 4


Question: Evaluate the double integral of the function \( f(x,y) = 12\tan^{-1}\left(\frac{y}{x}\right) \) over the region bounded by the lines \( r_1 = 1 \), \( r_2 = 3 \), \( \theta_1 = \frac{\pi}{6} \), and \( \theta_2 = \frac{\pi}{3} \) in polar coordinates.

Answer: 2\pi^2


Question:
Calculate the norm of the matrix \( M \):
\[
M =
\begin{pmatrix}
 1 & 2 & 3\\
 4 & 5 & 6\\
 7 & 8 & 9
\end{pmatrix}
\]

Answer: \sqrt{285}


Question: Let $f: l^{\infty} \rightarrow \mathbb{R}$ be a continuous linear functional, where $l^{\infty} := \{x=(x_n)_{n \in \mathbb{N}} \subset \mathbb{R} : ||x||_{l^{\infty}} := \sup_{n \in \mathbb{N}} |x_n| < \infty\}$. Assume that $f(x) \ge 0$ for all $x=(x_n)_{n \in \mathbb{N}}$ such that $x_n \ge 0$ for all $n \in \mathbb{N}$. Furthermore, let $f(\overline{x}) = 2021$, where $\overline{x} = (1, 1, \ldots)$. Calculate the norm of $f$ in the topological dual of $l^{\infty}$. \[ ||f|| := \sup_{x \in l^{\infty}, x \ne 0} \frac{|f(x)|}{||x||_{l^{\infty}}} \]

Answer: 2021


Question: Compute the conditional probability \( A = P(X_6 > X_2 \mid X_1 = \max[X_1, X_2, X_3, X_4, X_5]) \) for iid random variables \( X_i \).

Answer: \dfrac{7}{12}


Question:
Solve the following system of simultaneous congruences for the integer x:
\[\begin{align*}
3x &\equiv 1 \pmod{7} \\
2x &\equiv 10 \pmod{16} \\
5x &\equiv 1 \pmod{18}
\end{align*}\]

Answer: 173


Question:
Let \( \{a_n\} \) be a strictly increasing sequence of positive real numbers with \( \lim_{n\rightarrow \infty} a_n = (\sqrt{2})^{e} \), and let \( s_n = \sum_{k=1}^{n} a_k \). If \( \{x_n\} \) is a strictly decreasing sequence of real numbers with \( \lim_{n\rightarrow \infty} x_n = (e)^{\sqrt{2}} \), find the value of
\[ \lim_{n\rightarrow \infty} \frac{1}{s_n} \sum_{k=1}^{n} a_k x_k. \]

Answer: e^{\sqrt{2}}


Question: Find an upper bound for the expression \(\left(\frac{2^{64}-1}{2^{64}}\right)^{2^{56}}\).

Answer: e^{-1/256}


Question: Determine the radius of convergence of the series: \[ f(z) = \frac{1}{2z^3} + \frac{1}{12z} - \frac{z}{240}. \]

Answer: \infty


Question: Given the function \( f(x,y) = y \cos(\pi x) - x \cos(\pi y) \), determine a direction from the point (2,1) such that you maintain the same function value. Find a vector \( \vec{u} = (x, y) \) that is orthogonal to the gradient of \( f \) at (2,1).

Answer: (1, -1)


Question: Find the largest integer that can be expressed as the sum of squares of two positive integers in three or more distinct ways.

Answer: 325


Question: Evaluate the limit of the integral: \[ I = \lim_{n \to \infty} \int_0^1 \frac{1 + nx^2}{(1 + x^2)^n} \log(2 + \cos(x/n))\,dx. \] Find the value of \( I \).

Answer: 0


Question: Determine the rank of a subgroup $H$ of finite index $e$ in a free abelian group $G$ of rank $n$. Express the rank of $H$ in terms of $n$ and $e$.

Answer: n


Question: Assume that $v \in C^\infty_c(U)$ and $u \in C^2(U)$. Does the second equality in Green's identity still hold under these conditions?

Answer: Yes


Question:
Is it true that for every prime number \( p \geq 5 \), there exist integers \( i_1, \ldots, i_{m(p)} \) and \( j_1, \ldots, j_{n(p)} \) and \( k_1, \ldots, k_{m(p)} \) and \( l_1, \ldots, l_{n(p)} \) and \( a_1, \ldots, a_{m(p)} \) and \( b_1, \ldots, b_{n(p)} \) such that:
\[
p = \sum_{r=1}^{m(p)}(-1)^{i_r} 2^{k_r}a_{r} + \sum_{s=1}^{n(p)}(-1)^{j_s}3^{l_s}b_{s}
\]
where \( i_1, \ldots, i_{m(p)}, j_1, \ldots, j_{n(p)} \in \{1,2\} \), \( a_1, \ldots, a_{m(p)}, b_1, \ldots, b_{n(p)} \in \{0,1\} \), \( k_1 \cdot \ldots \cdot k_{m(p)} \neq 0 \), \( l_1 \cdot \ldots \cdot l_{n(p)} \neq 0 \), and the terms \( 3-2=1 \) and \( 3^2-2^3=1 \) are absent? Additionally, both sums should be non-zero.

Answer: Yes


Question: Calculate the integral \( \int_{C_2(0)} \frac{z}{(9-z^2)(z+i)} \, dz \), where \( C_2(0) \) is the circle of radius 2 centered at 0.

Answer: \dfrac{\pi}{5}


Question: Let $G$ be a simple linear algebraic group acting on a projective variety $X$ through rational maps. Consider a point $x_0 \in X$ with stabilizer group $H$, and assume that $G/H$ is not compact and carries a $G$-invariant measure $\mu$. Let $z \in X$ be in the boundary of the orbit $G.x_0$, and let $U \subset X$ be an open neighborhood of $z$. Define $G(U) \subset G/H$ as the set of all $gH \in G/H$ such that $g.x_0 \in U$. Determine whether $\mu(G(U)) = \infty$. Provide a justification for your answer.

Answer: \mu(G(U)) = \infty


Question:
Determine the order of the quotient group $G/G'$ for the group $G$ defined by the presentation:
\[ G = \langle a, b \mid a^{2^{n-1}} = 1, b^2 = a^{2^{n-2}}, b^{-1}ab = a^{-1} \rangle. \]

Answer: 4


Question: Determine the radius of convergence of the series \( \sum_{n=0}^{\infty} \frac{n^{n}(2n)!}{(3n)!}x^{n} \).

Answer: \dfrac{27}{4e}


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

Answer: I_n(1)


Question: Let $f_1, \ldots, f_n, g_1, \ldots, g_n$ be polynomials in $\mathbb{Q}[X_1, \ldots, X_n]$. If $g = (g_1, \ldots, g_n)$ satisfies $f_i(g(x_1, \ldots, x_n)) = x_i$ for all $i = 1, \ldots, n$, does it follow that $f$ and $g$ must have degree 1?

Answer: No


Question: Let $X$ be a random variable with the probability density function (pdf) given by $f(x) = ke^{-x^2-7x}$. Find the expected value $E(X^2)$. Assume $k$ is a constant that makes $f(x)$ a valid pdf.

Answer: \dfrac{51}{4}


Question: If \( k_1 \) and \( k_2 \) are the least values of the functions \( f(x)=2\log_{10}(x)-\log_x(0.01) \) and \( g(x)=e^x+e^{-x} \), respectively, find the value of \( \frac{k_1}{k_2} \).

Answer: 2


Question: Assume that a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) coincides with its Taylor series centered at a point \( p \in U \) on an open set \( U \), i.e., \( \exists p \in U, \forall x \in U, f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(p)}{n!}(x-p)^n. \) Is \( f \) analytic on \( U \)? In other words, does \( \forall p \in U, \exists \varepsilon > 0, \forall x \in \mathbb{R}, |p-x| < \varepsilon \Rightarrow f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(p)}{n!}(x-p)^n \) hold?

Answer: Yes


Question: What is the minimum number of points a sample space must contain in order for there to exist $n$ independent events, none of which has probability zero or one?

Answer: 2^n


Question: Evaluate the limit of \( \frac{{{n + d - 1} \choose d}}{n^d} \) as \( n \to \infty \).

Answer: \dfrac{1}{d!}


Question: Let $X_n$ and $Y_m$ be independent Poisson random variables with means $n$ and $m$, respectively. Determine the limiting distribution of \( \frac{X_n - Y_m - (n-m)}{\sqrt{X_n + Y_m}} \) as $n, m \to \infty$. Provide your answer in terms of a standard distribution.

Answer: N(0,1)


Question:
Given two independent random variables \(X_1\) and \(X_2\) with the probability density function (pdf):
\[
f(x_i) = \begin{cases} \frac{1}{2 \theta}, & -\theta < x_i < \theta \\ 0, & \text{otherwise} \end{cases}
\]
If it is known that \(\operatorname{Var}(X_1 X_2) = \frac{64}{9}\), determine the value of \(\theta\).

Answer: 2\sqrt{2}


Question: Let the continuous random variables \(X_1\), \(X_2\), \(X_3\), and \(X_4\) be independent such that \(X_1 \sim N(\mu, 1^2)\), \(X_2 \sim N(\mu, 2^2)\), \(X_3 \sim N(\mu, 3^2)\), and \(X_4 \sim N(\mu, 4^2)\), where \(\mu\) is an unknown parameter. Define the estimator \(\hat{\mu}\) as \(\hat{\mu} = (4X_1 + 3X_2 + 2X_3 + X_4)/10\). Calculate the 95% confidence interval for \(\mu\) using \(\hat{\mu}\) when \(x_1 = 77.47\), \(x_2 = 77.17\), \(x_3 = 79.38\), and \(x_4 = 81.87\).

Answer: (76.20, 80.20)


Question: Given a matrix \( A \) in a 3-dimensional Cartesian coordinate system \( S \), determine the rotation matrix required to diagonalize \( A \) in a new coordinate system \( S' \). Assume the eigenvalues of \( A \) and its diagonalized form are known. What is the rotation matrix that transforms \( S \) to \( S' \)?

Answer: R


Question:
Let \( t \to y(t) \) be a real-valued smooth function on any open interval containing \([0,1]\). Suppose that \( y(t) \) satisfies the differential equation:
\[ y''(t) + w(t)y(t) = \lambda y(t) \]
where \( \lambda \) is a real constant and \( w(t) \) is a given function. Assume that \( y(t) > 0 \) for all \( t \in [0,1] \) and that \( y'(0) = y'(1) = 0 \). Given that \( \int_{0}^{1}\left(\frac{y'}{y}\right)^{2}\,dt = 10 \) and \( \int_{0}^{1}w(t)\,dt = 20 \), find the value of \( \lambda \).

Answer: 30


Question: Given two red points, $r_1$ and $r_2$, and two blue points, $b_1$ and $b_2$, all placed randomly and uniformly in the unit square $[0,1]^2$, each point points to the closest point of the opposite color based on Euclidean distance. If $r_1$ points to $b_1$, what is the probability that $r_2$ also points to $b_1$?

Answer: \dfrac{2}{3}


Question: Given a series of 20 Bernoulli trials with the outcomes: \( T, T, T, H, H, H, H, H, H, H, T, H, H, T, T, T, H, H, H, T \), where there are 12 heads and 8 tails, the probability of the 21st trial being a tail is calculated as \( P(X_{21} = T) = \frac{9}{22} \). Determine the probability that both the 21st and 22nd trials result in tails, \( P((X_{21} = T) \cap (X_{22} = T)) \). Is it \( \left(\frac{9}{22}\right)^2 = \frac{81}{484} \) or \( \frac{9}{22} \cdot \frac{10}{23} = \frac{90}{506} \)?

Answer: \dfrac{90}{506}


Question: If \( x = \sqrt{2} + \sqrt{5} \) is a root of the polynomial equation \( kx^4 + mx^3 + nx^2 + l = 0 \), where \( k, m, n, l \) are integers, find the value of \( |k| + |m| + |n| + |l| \).

Answer: 24


Question: Evaluate the definite integral: \[ \int_{1}^{3} \sqrt{1 + \left(\frac{x^4 - 1}{2x^2}\right)^2} \, dx \] and find its value.

Answer: \dfrac{14}{3}


Question:
Let $X$ be a topological space of infinite cardinality that is homeomorphic to $X \times X$. Determine which of the following statements is true:

(a) $X$ is not connected.
(b) $X$ is not compact.
(c) $X$ is not homeomorphic to a subset of $\mathbb{R}$.
(d) None of the above.

Select the correct option.

Answer: D


Question: Evaluate the integral \( \int_{C}\frac{|z|e^{z}}{z^{2}}dz \), where \( C \) is the circumference of the circle of radius 2 centered at the origin.

Answer: 4\pi i


Question: Calculate the limit: $$\lim_{t \rightarrow 0}\frac{f(t^2+t)-f(t)}{t^2},$$ where the function \(f(x)\) is defined as: $$f(x)=\begin{cases}\exp\left(-\frac{1}{x^2}\right) & x\not=0 \\ 0& x=0\end{cases}.$$ Does this limit equal zero?

Answer: 0


Question: Given a sequence of real numbers \( \alpha_n \), if \( \forall t \in \mathbb{R}, \lim_{n \rightarrow \infty} e^{i \alpha_n t} = 1 \), does it necessarily follow that \( \lim_{n \rightarrow \infty} \alpha_n = 0 \)? Provide a justification for your answer.

Answer: Yes


Question:
Find the exact area of the surface obtained by rotating the curve \( y = \sqrt{9x-18} \) about the x-axis for \( 2 \le x \le 6 \). Use the formula for the surface area of revolution:
\[ \int 2\pi y \sqrt{1+\left(\frac{dy}{dx}\right)^2} \, dx \]

Answer: 49\pi


Question: Given that \( \frac{a-b}{c-d}=2 \) and \( \frac{a-c}{b-d} = 3 \), determine the value of \( \frac{a-d}{b-c} \), where \( a, b, c, d \) are real numbers.

Answer: -5


Question: Evaluate the integral: \[ \int_{0}^{+\infty} \left(\frac{\sqrt\pi}{2} - \int_{0}^{x} \mathrm{e}^{-t^2} \, dt \right) \, dx \]

Answer: \dfrac{1}{2}


Question: Find the minimum value of the product \( ab \) if the roots of the cubic equation \( x^3 - ax^2 + bx - 2 = 0 \) are all positive.

Answer: 18


Question: Find the twice differentiable function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f''(x) = (x^2 - 1)f(x) \) with the initial conditions \( f(0) = 1 \) and \( f'(0) = 0 \).

Answer: e^{-x^2/2}


Question: Let $V$ be a unitary space with finite dimension, and let $T: V \to V$ be a non-negative linear transformation. Assume there exists an integer $n$ such that $T^n = I$, where $I$ is the identity transformation. Is it necessarily true that $T = I$? Justify your answer.

Answer: T = I


Question: Determine the number of integral solutions to the equation \(4y^3 = 3x^2 + 1\).

Answer: 2


Question: Find the integrating factor of the form \(\mu(x,y) = x^m y^n\) that makes the first-order ODE \((2y^2-3xy)dx + (4xy-3x^2)dy = 0\) exact.

Answer: x y^2


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous and differentiable function such that if \( f'(s) = 0 \), then \( s \) is unique and is an inflection point. Is \( f \) necessarily one-to-one? Justify your answer.

Answer: Yes


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

Answer: -2e^{\pi}


Question: Given that the function \[ f(x) = \lim_{n\to\infty} \frac{(1-\cos(1-\tan(\frac{\pi}{4}-x)))(1+x)^n+\lambda\sin((n-\sqrt{n^2-8n})x)}{x^2(1+x)^n+x} \] is continuous at $x=0$, find the value of $f(0) + 2\lambda$.

Answer: 3


Question: Calculate the value of the integral \( \iint_{D_1} e^{-2x^2 -4xy - 4y^2} \,dx\,dy \), where \( D_1 \) is the first quadrant of \( \mathbb{R}^2 \).

Answer: \dfrac{\pi}{16}


Question: Compute the double integral of the piecewise function \( f(x,y) = \begin{cases} 7e^{x^2} & y \leq x \\ 7e^{y^2} & y > x \end{cases} \) over the square region \( D = [0,9] \times [0,9] \).

Answer: 7(e^{81} - 1)


Question: Is it possible for a finite state machine to determine if a bit string of arbitrary length contains an equal number of zeros and ones? Provide a justification for your answer.

Answer: No


Question: Let $t_1$ and $t_2$ be lower semicontinuous semifinite densely defined traces on a $C^*$-algebra $A$. Denote by $\mathcal{R}_1$ and $\mathcal{R}_2$ their ideals of definition, i.e., $\mathcal{R}_i=\{x\in A\ |\ t_i(x^*x)<+\infty \}$. Let $B$ be an involutive dense subalgebra of $A$ such that $B\subset \mathcal{R}_1,\mathcal{R}_2$ and $t_1(b^*b)=t_2(b^*b)$ for any $b\in B$. Is it true that $t_1=t_2$ on the whole algebra $A$?

Answer: Yes


Question: Compute the residue at \( z = a \) of the function \( \frac{f'(z)}{f(z)} \), given that \( f(z) \) is holomorphic on a punctured domain with a pole of order \( n \geq 1 \) at \( z = a \).

Answer: -n


Question: Let $X$ be a real random variable with mean $\mu > 0$ and variance $\mu^2$. Determine whether there exists a non-trivial upper bound on the probability $\Bbb P(X < 0)$, or provide a counterexample showing that this probability can be as high as possible.

Answer: \dfrac{1}{2}


Question: Can a total order be defined on the set of complex numbers \( \mathbb{C} \), regarded as \( \mathbb{R}^2 \), that possesses the least upper bound property?

Answer: Yes


Question: Find a tight upper bound for the matrix norm \( \| I - \frac{x x^T}{\|x\|_2^2} \| \), where \( x \) is a column vector.

Answer: 1


Question: Evaluate the limit: \[ \lim _{x\to 0}\frac{1-\frac{x^2}{2}-\cos\left(\frac{x}{1-x^2}\right)}{x^4} \]

Answer: \dfrac{23}{24}


Question: Determine the values of \(x\) for which the series \(\sum_{n=1}^{\infty} \left[(n^3+1)^{1/3} - n\right] x^n\) converges.

Answer: [-1, 1]


Question: Determine whether the following statement is true or false: If \( f: M \to N \) is a smooth map between smooth manifolds \( M \) and \( N \) with \( \text{dim}(M) > \text{dim}(N) \), then \( f \) cannot be injective. Assume \( M \) and \( N \) are smooth manifolds without boundary.

Answer: True


Question: Evaluate the limit: \[ \lim_{n \to 0} \prod_{\substack{i=nk \\k \in \Bbb Z_{\geq 0}}}^{2-n} \left( 2-i \right) \]

Answer: 0


Question: Determine all differentiable functions \( f \) such that \( f(x) + f(y) = f\left(\frac{x+y}{1-xy}\right) \) for all real numbers \( x \) and \( y \) with \( xy \neq 1 \).

Answer: 0


Question: Find the minimum value of the integral \( \int_{0}^{\infty} |x^3 - a - bx - cx^2|^2 e^{-x} \, dx \) with respect to the real numbers \(a, b, c\).

Answer: 36


Question: Determine whether the function \( G(x)=\sum^\infty_{n=1}\frac{1}{n}\mathbb{1}_{\{\tfrac{1}{n}\}}(x) \) is Riemann integrable on the interval \([0,1]\).

Answer: Yes


Question: Let $X$, $Y$, and $Z$ be independent and identically distributed random variables, each uniformly distributed between $0$ and $2$. Calculate the probability that $X + Y + Z \leq 2$. Express your answer as a fraction.

Answer: \dfrac{1}{6}


Question: Consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with a subalgebra $\mathcal{G} \subseteq \mathcal{F}$. Let $X$ be a real-valued random variable and $X_1, X_2, \ldots$ be $\mathcal{G}$-measurable random variables such that \( \mathbb{E}[Uf(X_n)] \rightarrow \mathbb{E}[Uf(X)] \) holds for all continuous bounded functions $f:\mathbb{R} \rightarrow \mathbb{R}$ and $\mathcal{G}$-measurable bounded functions $U:\Omega \rightarrow \mathbb{R}$. Does this imply that \( \mathbb{E}[Uf(X_n)] \rightarrow \mathbb{E}[Uf(X)] \) for all continuous bounded functions $f:\mathbb{R} \rightarrow \mathbb{R}$ and $\mathcal{F}$-measurable bounded functions $U:\Omega \rightarrow \mathbb{R}$?

Answer: Yes


Question: If $p: E \to X$ is a covering map where $X$ is connected and locally arcwise connected, is the space $E$ necessarily locally connected?

Answer: Yes


Question: Given a set of n natural numbers, determine the number of possible order relations among them. Express your answer in terms of Fubini numbers.

Answer: F_n


Question:
Determine whether the following inequality holds for positive real numbers $a_1, a_2, \ldots, a_n$ and integers $2 \le k \le n$:
\[ \frac{\sum_{\text{cyclic}} \sqrt[k]{a_1 \cdots a_k}}{\binom{n}{k}} \le \frac{\sum_{\text{cyclic}} \sqrt[k-1]{a_1 \cdots a_{k-1}}}{\binom{n}{k-1}}. \]

Answer: True


Question: Find the greatest common divisor of the polynomials \((x+1)^{4n+3} + x^{2n}\) and \(x^3-1\).

Answer: x^2 + x + 1


Question: Determine the radius of convergence of the power series \( \sum\limits_{n=0}^{\infty} \frac{(-1)^n}{\sqrt{n+2} \sqrt[4]{n+4}} \cdot x^n \).

Answer: 1


Question: Solve the congruence \(2^{59}(59!) \equiv x \pmod{61}\).

Answer: 31


Question:
Consider a martingale $Y_n = \sum_{i=1}^n X_i$, where $X_i$ is a martingale difference sequence such that $\mathbb{E}[X_n\mid \mathcal{F}_{n-1}]=0$ for all $n$. We know that for $r=2$, the following equality holds:
\[ \mathbb{E}\big[Y_n^2-Y_{n-1}^2\big]=\mathbb{E}X_n^2. \]

Investigate whether a similar inequality holds for powers $1<r<2$, specifically:
\[ \mathbb{E}\big[|Y_n|^r-|Y_{n-1}|^r\big]\le C\mathbb{E}|X_n|^r, \]
for some constant $C>0$. Determine if such an inequality is valid.

Answer: Yes


Question: Given a Hermitian matrix \( A = \begin{pmatrix} 4 & 3+4i \\ 3-4i & 31 \end{pmatrix} \), determine if \( A \) remains Hermitian when transformed by an orthogonal matrix \( B \). Recall that a matrix \( B \) is orthogonal if \( B^T B = B B^T = I \).

Answer: Yes


Question: Evaluate the limit: \[ \lim_{\beta\to 0^-} \left(-\ln|\beta| + e^{\beta}\right) + \lim_{\beta\to 0^+} \left(\ln|\beta|\right) \]

Answer: 1


Question: Let $S \subset \mathbb{C}$ be a countable set with no accumulation points. Does there exist an analytic function $f(z)$ such that $f(z_i) = 0$ for all $z_i \in S$ and $f(z) \neq 0$ for $z \notin S$?

Answer: Yes


Question: Calculate \( f'(0) \) for the function \( f(x) = \prod_{n=0}^{100} (x-n) \).

Answer: 100!


Question:
Evaluate the limit:
\[ \lim_{x\to\infty} x^2\int_0^x e^{t^3-x^3}\,dt \]
Choose the correct answer:
A. \(\frac{1}{3}\)
B. \(2\)
C. \(\infty\)
D. \(\frac{2}{3}\)

Answer: A


Question:
Consider three different types of computer components with lifetimes following exponential distributions with parameters 1, 2, and 3, respectively. The probability density functions are given by:

\[ f_1(x_1) = e^{-x_1}, \quad f_2(x_2) = \frac{1}{2}e^{-\frac{x_2}{2}}, \quad f_3(x_3) = \frac{1}{3}e^{-\frac{x_3}{3}} \]

where the support sets are \(x_i \geq 0\) for \(i = 1, 2, 3\). One component from each type is randomly sampled, and their lifetimes are recorded. Calculate the probability that \(X_1 > 5X_2 > 10X_3\).

Answer: \dfrac{1}{374}


Question: In the context of Theorem 8.2.5 from Kreyszig's Functional Analysis, consider a linear operator \( A: Y' \to \mathbb{R}^n \) defined by \( Ag = (g(Tx_1), g(Tx_2), \ldots, g(Tx_n)) \) where \( g \in Y' \), the dual space of the normed space \( Y \). Given that \( Y \) is a complex normed space, should the operator \( A \) map from \( Y' \) to \( \mathbb{C}^n \) instead of \( \mathbb{R}^n \)? Provide a justification for your answer.

Answer: \mathbb{C}^n


Question: Let $A$ be an algebra of finite type over $\mathbb{Z}$, meaning $A$ is a finitely generated ring. If $\mathfrak{m}$ is a maximal ideal of $A$, is $A/\mathfrak{m}$ necessarily a finite field?

Answer: Yes


Question: Let $G$ be a group, $H \trianglelefteq G$ a normal subgroup, and $P$ a $p$-Sylow subgroup of $G$. Prove that \( \gcd([H: H \cap P], p) = 1 \).

Answer: 1


Question: Evaluate the line integral \( \int_\gamma (x^2 + iy^2) \, dz \), where \( \gamma \) is the line segment joining the points \( 1 \) and \( i \) in the complex plane. The function is given by \( f(x+yi) = x^2 + iy^2 \).

Answer: -\dfrac{2}{3}


Question: Given a triangle $ABC$, construct a point $P$ such that the distances from $P$ to the vertices $A$, $B$, and $C$ are in the ratio $PA : PB : PC = 1 : 2 : 4$. Describe the steps of the construction.

Answer: P


Question: Find the smallest integer \( n \) such that the symmetric group \( S_n \) contains a cyclic subgroup of order 2019.

Answer: 676


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{x^x - 4}{2^x - x^2} \]

Answer: \infty


Question: Calculate the expectation \( E(e^{-2\sqrt{s} W(s)}) \), where \( W(s) \) is a Wiener process.

Answer: e^{2s^2}


Question: Is there a proper class $X$ such that $X \subsetneq \operatorname{Ord}$?

Answer: Yes


Question: Find the largest integer \(\lambda\) such that the inequality \[ \frac{\lambda (xyz)}{x+y+z} \le (x+y)^2 + (x+y+4z)^2 \] holds for all positive real numbers \(x, y,\) and \(z\).

Answer: 100


Question: Compute the expected value of the stopping time \( E[\tau(a)] \), where \( \tau(a) = \min (t \geq 0 : W(t) \geq a) \) is the first time a standard Brownian motion \( W(t) \) hits the level \( a \).

Answer: \infty


Question: Consider the function \( f(x) = x^2 + h(x) \), where \( h(x) = O(\log x) \). Determine if the inverse function \( f^{-1}(x) \) is asymptotic to \( x^{1/2} \) for large \( x \).

Answer: Yes


Question: Let \(X_1 \sim \text{Exp}(2)\) and \(X_2 \sim \text{Exp}(2)\) be independent random variables. Calculate the probability \(P(X_1 < 3, X_1 + X_2 > 3)\).

Answer: 6e^{-6}


Question: Determine whether the following statement is true: \( \mathcal H(\mathrm U \times \mathrm V) \simeq \mathcal H(\mathrm U) \widehat{\otimes} \mathcal H(\mathrm V) \) for open affine sets \(\mathrm U\) and \(\mathrm V\). Consider the specific case where \(\mathrm U = \mathrm V = \mathbb{C}^\times\).

Answer: True


Question: Determine whether every non-hermitian matrix can be expressed as a sum of hermitian matrices. Specifically, if $X$ is a non-hermitian matrix such that $X^\dagger \neq X$, can $X$ be written as $X = \sum_i \alpha_i Y_i$, where $\alpha_i$ are complex scalars and $Y_i$ are hermitian matrices?

Answer: Yes


Question: Determine whether the following equality holds for a function $f \in L^{1}(d\mu)$ in a complete measure space $(X, \Sigma, \mu)$: \[ \int \limits_{X} f\chi_{\{ f \neq 0 \} } \,d\mu = \int \limits_{X}f \,d(\chi_{\{ f\neq 0 \} }\,d\mu). \]

Answer: Yes


Question: Given a triangle \( \triangle ABC \), which fulfills the equations \( 6\cos{A}+2\cos{B}=7 \) and \( 6\sin{A}-2\sin{B}=\sqrt{3} \), find the value of \( \angle C \).

Answer: 120^\circ


Question: Calculate the conditional expectation \( E(X^2 \mid X+Y=1) \) where \( X \) and \( Y \) are independent and identically distributed standard normal random variables.

Answer: \dfrac{3}{4}


Question: Given a fundamental discriminant \(d \equiv -1 \pmod{8}\) such that the quadratic imaginary number field \(\mathbb{Q}(\sqrt{-d})\) has an odd class number \(h(-d)\), determine if the diophantine equation \(x^2 + dy^2 = 2^{2 + h(-d)}\) can always be solved with odd integers \(x\) and \(y\).

Answer: Yes


Question: Determine the constant $k$ such that the polynomial \( P(x, y, z) = x^5 + y^5 + z^5 + k(x^3+y^3+z^3)(x^2+y^2+z^2) \) is divisible by \( x+y+z \).

Answer: -\dfrac{5}{6}


Question: Let \( f: \mathbb{C} \setminus \{0\} \rightarrow \mathbb{C} \) be a holomorphic function. Given that the integral of \( f \) over the positively oriented circle of radius 1 around 0 is \( 2\pi i \), find the value of the integral of \( f \) over the positively oriented circle of radius 2 around 1.

Answer: 2\pi i


Question: Let \( f \) be a nonnegative, continuous, and strictly increasing function on \([0,1]\). For \( p > 0 \), let \( x_p \) be the number in \((0,1)\) that satisfies \( f^p(x_p) = \int_0^1 f^p(x) \, dx \). Determine \( \lim_{p \to \infty} x_p \).

Answer: 1


Question:
Let $n > 1$ be an odd number. Let $A$ be an $n \times n$ matrix defined as follows:
\[
a_{i, j} = \begin{cases} 1, & \text{for}\ i - j  \equiv \pm 2 \pmod n\\ 2, & \text{for}\ i = j\\ 0, & \text{otherwise}.\end{cases}
\]
Calculate the determinant of matrix $A$. \( \boxed{?} \)

Answer: 4


Question: Suppose that for any \(-1 \leq x \leq 1\), \(|ax^2 + bx + c| \leq 1\). Find the maximum possible value of \(|a| + |b| + |c|\).

Answer: 3


Question: Calculate the expected value of the fourth power of a standard normal random variable, denoted as \( E(Z^4) \), where \( Z \) follows the standard normal distribution with probability density function \( f(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2} \).

Answer: 3


Question:
Find the variance of the random variable \( Y = \int_{0}^{2} X(t) \, dt \), where \( X(t) \) is a stationary process with mean \( E[X(t)] = 3 \) and autocorrelation function \( R_{xx}(\tau) = 9 + 2e^{-\lvert \tau \rvert} \). The variance of \( Y \) is:

\( \text{(A) } 1 \)
\( \text{(B) } 2.31 \)
\( \text{(C) } 4.54 \)
\( \text{(D) } 0 \)

Answer: C


Question: Is it true that a Borel probability measure \( \sigma \) on the complex unit circle \( \mathbb{T} \) is symmetric if and only if \( \hat{\sigma}(n) = \hat{\sigma}(-n) \) for every integer \( n \)?

Answer: Yes


Question: Evaluate the integral \( \int_C \omega \) where \( \omega = \frac{-y \, dx + (x-1) \, dy}{(x-1)^2 + y^2} \) and \( C \) is the cardioid given by \( r = 1 + \cos \varphi \), positively oriented.

Answer: 2\pi


Question: Consider the graph $G_{\mathbb{N}} = (C, E)$, where $C$ is the set of graphs with vertex set $\mathbb{N}$, and $E$ consists of pairs $\{G, H\}$ such that $G, H \in C$, $G \not\to H$, and $H \not\to G$. Does $G_{\mathbb{N}}$ contain an uncountable clique?

Answer: Yes


Question: Evaluate the sum \( \sum_{h = 1}^p e^{\frac{2\pi i h}{p}} \) where \( p \) is a prime number.

Answer: 0


Question: Determine whether there exists a critical covering of \( \omega \) consisting only of infinite sets.

Answer: Yes


Question: Determine the number of distinct eigenvalues of the linear transformation \( T: \mathbb{C}[x] \to \mathbb{C}[x] \) defined by \( T(f(x)) = f(x+1) \) on the complex vector space of polynomials \( \mathbb{C}[x] \).

Answer: 1


Question:
Let $B$ be an infinite-dimensional Banach space and $T: B \to B$ be a continuous operator such that $T(B) = B$ and $T(x) = 0 \Rightarrow x = 0$. Determine which of the following statements is correct:

1. $T$ maps bounded sets into compact sets.
2. $T^{-1}$ maps bounded sets into compact sets.
3. $T^{-1}$ maps bounded sets into bounded sets.
4. $T$ maps compact sets into open sets.

Answer: 3


Question: Evaluate the limit: \( \lim_{x \to \infty} \int_{1}^{x} \ln(4t^3 + 1) \, dt \).

Answer: \infty


Question: Let $b \geq 2$ be an integer and $x \in [0,1)$. Define $T_b(x) = bx \mod 1$. It is known that $x$ is a normal number with respect to $b$ if and only if the sequence $(T_b^n(x))_{n \in \mathbb{N}}$ is uniformly distributed in $[0,1)$. If $x$ is normal with respect to $b$, then the orbit of $x$ under $T_b$, denoted as $\operatorname{orb}(x) = \{T_b^n(x) : n \in \mathbb{N}\}$, is dense in $[0,1)$. Is the converse true? Specifically, do there exist $b$ and $x$ such that $\operatorname{orb}(x)$ is dense, but $x$ is not normal with respect to $b$?

Answer: Yes


Question: Is the union of all $\delta$-Hölder continuous functions for $\delta \in (0,1)$ a dense subset of the space of continuous functions on $[0,1]$ with respect to the uniform metric? In other words, for any function $f \in C^0$, can we find a sequence of Hölder continuous functions $f_n$ such that \(\sup_{x \in [0,1]} |f_n(x) - f(x)| \to 0\) as $n \to \infty$?

Answer: Yes


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

Answer: \infty


Question: Maximize the function \( f(x, y) = x^2y \) subject to the constraint that \( (x, y) \) lies on a circle with radius \( \sqrt{3} \) centered at the origin. Find the maximum value of \( f(x, y) \).

Answer: 2


Question: Consider a differentiable function \( f: U \to \mathbb{R}^m \) where \( U \subseteq \mathbb{R}^n \) is open and convex. For any \( x, x+h \in U \), is it true that there exists some \( A \) in the closure of the convex hull of \( \{ f'(x+th) \mid 0 < t < 1 \} \) such that \( f(x+h) - f(x) = Ah \)? Justify your answer.

Answer: Yes


Question: Let $\Omega$ be an open set in $\mathbb{R}^n$ with a $C^1$ boundary $\partial \Omega$. Suppose $f$ is a $C^1$ function on $\Omega \cup \partial \Omega$, where $C^1$-smoothness of $f(x)$ on $\partial \Omega$ is defined via local $C^1$-diffeomorphism. Define a function $g$ on $\Omega$ as $g(x) = f(x)$ for $x \in \Omega$. Fix $x_0 \in \partial \Omega$. Is it true that $D^{\alpha} g(x) \to D^{\alpha} f(x_0)$ as $x \to x_0$ for each multiindex $|\alpha| \leq 1$?

Answer: Yes


Question: Compute the expected value \( E[W(1)W^2(2)] \) for a Wiener process \( W(t) \).

Answer: 0


Question: How many 8-digit numbers do not contain a sub-string of 3 repeated digits?

Answer: 85096170


Question:
Determine the positive integer $k$ such that the non-constant complex analytic function $f$ on $\mathbb{C}$ satisfies the functional equation:
\[ f(s) = (-1)^{k} f(2-s) \]

Answer: 1


Question:
Solve the following set of simultaneous congruences using the Chinese Remainder Theorem:
\[
\begin{cases}
2x \equiv 1 \pmod{5} \\
3x \equiv 9 \pmod{6} \\
4x \equiv 1 \pmod{7} \\
5x \equiv 9 \pmod{11} \\
\end{cases}
\]

Answer: 653


Question: Determine the Krull dimension of the local ring \( \mathbb{C}[x,y]_p/(y^2-x^7,y^5-x^3) \), where \( p \neq (0,0) \).

Answer: 0


Question: Determine whether the following statement is true or false: A topological space $X$ is Hausdorff if and only if the product space $X \times X$ is Hausdorff. Provide a proof or counterexample to support your answer.

Answer: True


Question: Find the number of solutions to the equation \(x_1 + x_2 + x_3 + \ldots + x_{10} = 27\) where each \(x_i\) is an integer satisfying \(1 \leq x_i \leq 4\).

Answer: 100110


Question: Calculate the limit: \[ \lim_{n\to\infty} \int\limits_1^2 g(nx) \, dx, \] where the function $g$ is continuous on $[0,\infty)$, positive, strongly increasing, and satisfies $\lim_{t\to\infty} g(t) = L$. Provide your answer in terms of $L$.

Answer: L


Question: Let $Y$ be a closed subscheme of a scheme $X$, and let $i: Y \rightarrow X$ be the inclusion morphism. Is the morphism $i$ quasi-compact? Justify your answer.

Answer: Yes


Question: Given the function \( f(x) = x + \int_{0}^1 t(t+x)f(t) \, dt \), find the value of \( f(0) \).

Answer: \dfrac{18}{23}


Question: Determine the interval of convergence for the series \( \sum_{n=1}^{\infty} \left(\frac{n - i}{i - 2n}\right)^{n} (z-2)^{n} \).

Answer: (0, 4)


Question: Evaluate the limit \( \lim_{x\rightarrow1^-}{(1-x)^{\ln(x)}} \) using L'Hospital's rule.

Answer: 1


Question: Find the minimum value of \( M = \frac{x}{x^2+1} + \frac{y}{y^2+1} + \frac{z}{z^2+1} \) given that \( x, y, z \in \mathbb{R} \) and \( x+y+z = xy+yz+xz \).

Answer: -\dfrac{1}{2}


Question: Does there necessarily exist a Lie group $G$ such that for a given Lie algebra $\mathfrak{h}$, the Lie algebra corresponding to $G$, denoted $\mathfrak{g}$, satisfies $\mathfrak{g} \cong \mathfrak{h}$?

Answer: Yes


Question:
Evaluate whether the following limit is equal to Euler's constant \(-\gamma\):
\[ \lim_{n\to \infty}\int_0^{\pi/2}\cos x\cos\left(\frac{x}{n}\right)\ln(\ln\csc x) \, dx = -\gamma \]

Answer: -\gamma


Question: Determine the Galois group of the polynomial \((x^2-1)(x^2-2)\ldots(x^2-p+1)\) over the finite field \(\mathbb{Z}_p\), where \(p\) is an odd prime.

Answer: \mathbb{Z}/2\mathbb{Z}


Question: Compute the derivative of the expression \( \operatorname{trace}(XWW^T X^T) \) with respect to the matrix \( W \), where \( X \) and \( W \) are \( n \times n \) real matrices.

Answer: 2 X^\top X W


Question: Determine the minimal polynomial of \( \overline{x+1} \) over the field \( \mathbb{F}_2 \) in the field \( K = \mathbb{F}_2[x] / (x^3+x+1) \).

Answer: t^3 + t^2 + 1


Question: Let $G$ be an abelian group and $H$ a subgroup of $G$. Suppose for some integer $n$, $H/nH = 0$ and $([G : H], n) = 1$. Is it true that $G/nG = 0$?

Answer: Yes


Question: Calculate the volume of the solid bounded by the cylinder \( z = x^2 \) and the planes \( z=2-x, y=0, y=2 \).

Answer: 9


Question: Determine the spectrum of the operator \( \Lambda: C[0,1] \rightarrow C[0,1] \) defined by \( \Lambda(f)(x) = (1-x) \cdot f(x) \).

Answer: [0,1]


Answer: e^{-2}


Question: Simplify the expression \( \frac{1}{1+2x} - \frac{1-x}{1+x} \) for \( x \) much smaller than 1, using appropriate approximations or series expansions.

Answer: 2x^2


Question: Let \( E \) be the elliptic curve defined by \( y^2 = x^3 + 1 \) over the finite field \( F_{599} \). Using Hasse's theorem, determine the upper bound for the order of \( E(F_{599}) \).

Answer: 648


Question:
Let $k$ be a field and $k[X]$ the polynomial ring in one variable. Consider a torsion $k[X]$-module $M$ and the submodules:
\[ M_1 = \{a \in M \mid X^n a = 0 \text{ for some } n \in \mathbb{N}\}, \]
\[ M_2 = \{a \in M \mid f a = 0 \text{ for some } f \in k[X], \text{ relatively coprime to } X\}. \]
Is it true that $M_1 + M_2 = M$?

Answer: Yes


Question: Is it possible for a number to be divisible by a prime that does not appear in its unique prime factorization? Provide a justification for your answer.

Answer: No


Question: Evaluate the line integral \( \int_{L} \frac{-y \,d x+x \,d y}{x^{2}+y^{2}} \) where \( L \) is the line segment from \((1,0)\) to \((0,1)\) parametrized by \( L(t)=(1-t)(1,0)+t(0,1), \quad 0 \leq t \leq 1 \).

Answer: \dfrac{\pi}{2}


Question: Find the 6-digit integer \( x \) such that \( x + d(x) + d(d(x)) + d(d(d(x))) = 999868 \), where \( d(n) \) denotes the digital sum of a positive integer \( n \).

Answer: 999820


Question: Find the irreducible polynomial of the expression $2\sqrt{2} + \sqrt{7}$.

Answer: x^4 - 30x^2 + 1


Question: Evaluate the contour integral \( \int_{|z|=R} z^n \, dz \) for \( n \geq 0 \), where the contour is the circle centered at the origin with radius \( R \).

Answer: 0


Question:
Determine the type of curve described by the equation \( \int_{0}^{y}\frac{f(v)dv}{\sqrt{y-v}}=4\sqrt{y} \), where the curve \( y=y(x) \) passes through the point \( (\sqrt{3},1) \) and \( f(y)=\sqrt{1+\frac{1}{y^{'2}}} \). Choose from the following options:
1. Straight line
2. Circle
3. Parabola

Provide the correct option number.

Answer: 1


Question: Calculate the number of unique sequences of 6 objects, each of which can be in one of 4 states, where sequences that are mirror images of each other are considered equivalent. For example, the sequences 111112 and 211111 are equivalent, as are 123123 and 321321. What is the total number of such unique sequences?

Answer: 2080


Question: Let T be a solid tetrahedron with four pairwise distinct non-coplanar vertices in three-dimensional Euclidean space. If p is any interior point of T and Q(p) is any plane containing p, does Q(p) always strictly separate two of the vertices of T? Answer yes or no.

Answer: Yes


Question: Calculate the residue of the function \( f(z) = \frac{\mathrm{e}^z}{z^2} \) at the point \( z = 0 \) using the definition of residue.

Answer: 1


Question: Find the supremum of the fraction \( \frac{|c_1^\top z + b_1|}{c_2^\top z+b_2} \) with respect to \( z \in S = \{x \in \mathbb{R}^n: c_2^\top x + b_2 > 0 \} \), where \( c_1, c_2 \in \mathbb{R}^n \). Ignore the trivial cases where \( c_1 = 0 \) and/or \( c_2 = 0 \).

Answer: +\infty


Question:
Find the equation of the tangent line at the point (0,0) to the curve defined by the implicit equation:
\[ \ln(1+x+y)=\left( x^{42} e^y +  \cos(xy)\sin(xy)\right)^{2015} \left( ye^{x\cos y} + 429 \sin(y \cos(x))\right)^{257} + 2x \]

Answer: y = x


Question: Find all extreme values of the function \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the constraint \( x^2 + 2y^2 - z^2 - 1 = 0 \) using the method of Lagrange multipliers.

Answer: \dfrac{1}{2}


Question: Determine the fundamental group of the space \(X=\{(x,y) \in [-b,b]\times [-b,b]: \, (xy-1)(x^2+y^2-a^2)=0, 0<a<b\}\). This space is the union of a circle and a hyperbola.

Answer: \mathbb{Z}


Question: Determine the number of zero divisors in the ring \( \frac{\mathbb{Z}_3[x]}{(x^4 + 2)} \).

Answer: 48


Question: Suppose $f$ is a nonconstant real analytic function on the interval $(a,b)$. Can $f$ be constant on a nonempty subinterval $(c,d)$ of $(a,b)$? Provide a justification for your answer.

Answer: No


Question: Let $G$ be a finite group and $N$ a normal subgroup of $G$ such that the center $Z(N)$ is trivial and the automorphism group $\operatorname{Aut}(N)$ equals the inner automorphism group $\operatorname{Inn}(N)$. Prove that $N$ has a complement $H$ in $G$ and that $H$ is normal in $G$.

Answer: H


Question: Find the minimum value of the expression \( |1+z_1| + |1+z_2| + \cdots + |1+z_n| + |1+z_1z_2\cdots z_n| \) where \( n \) is an even integer and \( z_1, z_2, \ldots, z_n \in \mathbb{C} \).

Answer: 2


Question:
Given a matrix $A \in \mathbb{R}^{m \times n}$ with full rank and $m \le n$, solve the optimization problem:
\[ \mathop{\arg\max}\limits_{\substack{v \in \mathbb{R}^m \\ \| v \| = 1}} v^T A A^T v \]
using the singular value decomposition (SVD) of $A$. Determine the vector $v$ that maximizes the expression.

Answer: u_1


Question: Find an integer $x \in \mathbb{Z}[i]$ such that $(1+2i)x \equiv 1 \pmod{3+3i}$. Provide your answer in the form of a Gaussian integer.

Answer: 2 - i


Question: Given the joint probability distribution \( f(x,y) = \begin{cases} xe^{-x-y}, & x,y>0 \\ 0, & \text{elsewhere} \end{cases} \), compute the probability \( P(X \le Y) \).

Answer: \dfrac{1}{4}


Question: Let $M \in \mathbf{R}^{n \times n}$ be a symmetric matrix that is positive semidefinite. Let $M_i$ be the leading principal $i \times i$ minor of $M$, and assume that $\det(M_i) \neq 0$. If $\det(M_{i+1}) = 0$, is it sufficient to conclude that the rank of $M$ is $i$?

Answer: No


Question: Find a closed-form expression for the sum \( B = \sum_{k \in A} \frac{1}{k} \), where \( A \) is the set of all positive integers with no prime factor larger than 3. Express \( B \) in terms of a double sum \( \sum_{i=0}^\infty \sum_{j=0}^\infty X \). What should \( X \) be to make this equation true?

Answer: \dfrac{1}{2^i 3^j}


Question:
Given the equations:
\[ a = \sqrt{4 + \sqrt{5 + a}} \]
\[ b = \sqrt{4 - \sqrt{5 + b}} \]
\[ c = \sqrt{4 + \sqrt{5 - c}} \]
\[ d = \sqrt{4 - \sqrt{5 - d}} \]
where \(a, b, c, d\) are four distinct positive real numbers, find the value of \(abcd\).

Answer: 11


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

Answer: \ln 2


Question: Determine the expected divisibility $D(n)$ for a positive integer $n$. Define $E(p) = \frac{\sum_{n=1}^p D(n)}{p}$, where $D(n) = \frac{\text{Log}(n)}{\log_2(n)}$ and Log(n) is the total exponent in the prime factorization of $n$. Find the value of \( E = \lim_{p \to \infty} E(p) \).

Answer: 0


Question: Find the smallest natural number \( k \) such that among \( k \) different and pairwise relatively prime natural numbers less than 2021, there is at least one prime number.

Answer: 15


Question: Determine the sub-interval of \([0,1]\) on which the sequence of functions \(f_n(x) = x^{1/n}\) converges uniformly to the constant function 1.

Answer: [a, 1]


Question:
Let $z$ be a point in a fundamental domain of $\Gamma(2)\subset \mathrm{SL}_2(\mathbf{Z})$ in the complex upper half plane. Determine whether there exists an $\epsilon >0$ such that the geodesic distance $d(z,\gamma z)$ satisfies
\[ \cosh( d(z,\gamma z)) > 1+\epsilon \]
for all $\gamma \neq \pm 1$ in $\Gamma(2)$. Note that $\epsilon$ may depend on $z$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the double integral \( \iint \sqrt{x^2+y^2} \, dy \, dx \) over the region \( \{(x,y)\in \mathbb{R^2}\mid x \le x^2+y^2\le2x\} \).

Answer: \dfrac{28}{9}


Question: Let $G$ be a finite group and $p$ a prime number such that $g^p = e$ for all $g \in G$, where $e$ is the identity element. Does this imply that $G$ is abelian?

Answer: No


Question: Let \( f \) be a real-valued continuous function on the interval \([0,2]\) which is differentiable at all points except \( x = 1 \). Given that \( \lim_{x \to 1} f'(x) = 5 \), show that \( f \) is differentiable at \( x = 1 \).

Answer: 5


Question: Calculate the limit: \[ \lim\limits_{(x,y)\to(0,0)} \frac{e^{-1/(x^2+y^2)}}{x^6+y^6} \]

Answer: 0


Question: Calculate the volume of the solid bounded by the elliptic cylinder $9x^2+4y^2 = 36$ and the plane $3x+4y-6z=0$ in the first octant. Express the volume as an integral and find its value.

Answer: 6


Question: Solve for the positive real number $x$ that satisfies the equation: \[ 2^x + 4^x + 2^{\lfloor x \rfloor} + 4^{\lfloor x \rfloor} + 2^{x - \lfloor x \rfloor} - 4^{x - \lfloor x \rfloor} = 50 + \sqrt{50} \]

Answer: \dfrac{5}{2}


Question: You are on a basketball team, and at the end of every practice, you shoot half-court shots until you make one. Each half-court shot, independent of all other shots, has a 0.1 probability of going in. Your team has 100 practices per season. Estimate the probability that you shoot more than 1111 half-court shots after practices this season.

Answer: 0.12


Question: Let $f(x)$ and $g(x)$ be smooth non-constant functions from an interval $(a, b)$ to $\mathbb{R}$ such that $f(x)^2 + g(x)^2 = 1$ for all $x$ in $(a, b)$. Is it true that there exists a smooth function $h(x)$ such that $f(x) = \cos(h(x))$ and $g(x) = \sin(h(x))$?

Answer: Yes


Question: Compute the limit \( \lim_{z\rightarrow 2n\pi i}\dfrac{e^z-1}{z-2n\pi i} \) for a complex number \( z \) and integer \( n \).

Answer: 1


Question: Calculate the Lie bracket \([X_1, [X_1, X_2]]\) for the vector fields \(X_1 = \frac{\partial}{\partial x_1} + x_2\frac{\partial}{\partial x_3} + x_2^2\frac{\partial}{\partial x_4}\) and \(X_2 = \frac{\partial}{\partial x_2}\).

Answer: 0


Question:
Evaluate the integral \( \int_0^{90} f(x) \, dx \), where the function \( f(x) \) is defined as follows:
\[
f(x) = \begin{cases}
(x-3k)^2 & \text{if } 3k-1 \le x < 3k+1 \\
1 & \text{if } 3k+1 \le x < 3k+2
\end{cases}
\]
for \( k \in \mathbb{Z} \).

Answer: 50


Question:
Find the complete solution set of the equation \(\sin^{-1}\left(\frac{x + \sqrt{1 - x^2}}{\sqrt{2}}\right) = \frac{\pi}{4} + \sin^{-1}x\). Choose the correct option:

(A) \([-1,0]\)
(B) \([0,1]\)
(C) \([-1,\frac{1}{\sqrt{2}}]\)
(D) \([\frac{1}{\sqrt{2}},1]\)

Answer: C


Question:
Determine the range of values for $1 \le p \le \infty$ such that a function $u$ belongs to the Sobolev space $W^{1,p}(\Omega)$. Specifically, find the values of $p$ for which the following condition holds:
\[ \int_{\Omega} |u|^p \, dx + \int_{\Omega} |Du|^p \, dx < \infty. \]

Answer: [1, \infty]


Question: Let $A$ be an $n \times n$ complex matrix, expressed as $A = X + iY$, where $X$ and $Y$ are real $n \times n$ matrices. Suppose that for every square submatrix $S$ of $A$, the modulus of the determinant satisfies $|\mathrm{det}(S)| \leq 1$. Is it true that $|\mathrm{det}(X)| \leq 1$?

Answer: Yes


Question: What is the expected number of Bernoulli trials required to achieve \( k \) successes, given that the probability of success in a single trial is \( p \) and the probability of failure is \( q = 1 - p \)?

Answer: \dfrac{k}{p}


Question: Given that \( F(x) = f(xf(xf(x))) \) and the values \( f(1) = 2, f(2) = 3, f '(1) = 3, f '(2) = 4, \) and \( f '(3) = 5, \) find \( F '(1) \).

Answer: 115


Question: Evaluate the limit: \[ \lim_{x \to \infty} x\left(a^{1/x} - 1\right) \]

Answer: \ln a


Question: Let $V$ be a vector space of dimension 6 over a field $F$. Determine whether there exists a linear operator $T$ on $V$ such that $\text{Ker}(T) = \text{Range}(T)$. Answer with "True" if such an operator exists, or "False" if it does not.

Answer: A


Question: Determine the radius of convergence of the power series \( p(x) = \sum b_n x^n \), given that the series \( \sum b_n \) is conditionally convergent but not absolutely convergent.

Answer: 1


Question: Given the p-adic expansion \(5^{-1} + 1 + 2 \cdot 5 + 5^2 + (4 \cdot 5^3 + 2 \cdot 5^4 + 3 \cdot 5^5 + 2 \cdot 5^7 + 5^8) + (4 \cdot 5^9 + 2 \cdot 5^{10} + 3 \cdot 5^{11} + 2 \cdot 5^{13} + 5^{14}) + (4 \cdot 5^{15} + 2 \cdot 5^{16} + 3 \cdot 5^{17} + 2 \cdot 5^{19} + 5^{20}) + \ldots\), where the repeated part is in parentheses, find the rational number associated with this expansion.

Answer: \dfrac{17}{35}


Question: What is the probability of choosing the number 1 from the sample space of natural numbers \( \mathbb{N} \), assuming each number is equally likely?

Answer: 0


Question: Determine whether the following statement is true or false: "If $R$ is a field, then $R[x]$ is an Euclidean Domain (ED)."

Answer: A


Question: Determine the infimum of the set \( S = \{x \in \mathbb{Q} \mid x^2 < 2\} \).

Answer: -\sqrt{2}


Question:
Calculate the trace of the matrix exponential $e^{A}$, where $A$ is the $4 \times 4$ matrix given by:
\[ A = \begin{bmatrix} 0 & 0 & 0 & t \\
 0 & 0 & -t & 0 \\
 0 & t & 0 & 0 \\
 -t & 0 & 0 & 0 \end{bmatrix} \]
with $t = \frac{\pi}{4}$. What is the trace of $e^{A}$?

Answer: 2\sqrt{2}


Question: If $A \in M_3(\mathbb{R})$ is a $3 \times 3$ matrix with $\det(A)=0$ and the square of each element equals its cofactor, does it necessarily follow that $A=0_3$? Here, $a_{ij}^2=A_{ij}$, where $A_{ij}=(-1)^{i+j}M_{ij}$ and $M_{ij}$ is the minor of $a_{ij}$. Provide a justification for your answer.

Answer: A = 0_3


Question: Solve the initial value problem (IVP) for the differential equation: \[ y'' = 2\arctan(y) + \frac{3}{2}y' - \frac{1}{2}\pi \] with the initial conditions \( y(0) = 1 \) and \( y'(0) = 0 \).

Answer: 1


Question: What is the largest possible sum of the interior angles of a triangle on a torus? Consider the geometry of the torus and the placement of the triangle to determine this maximum sum.

Answer: 540^\circ


Question: Find the integer \( n \) such that \( 2^n \) is the closest power of 2 to \( 50! \).

Answer: 214


Question: Verify that \((2 + \frac{10}{3\sqrt{3}})^{1/3} + \frac{2}{3} (2 + \frac{10}{3\sqrt{3}})^{-1/3} = 2\).

Answer: 2


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

Answer: xyz = 9k^3


Question: Evaluate the complex contour integral \( \oint_{C(-i, 1)} \left(\frac{1}{(z+i)^3} - \frac{5}{z+i} + 8\right)\,dz \), where \( C(-i, 1) \) is a contour around the point \(-i\) with radius 1.

Answer: -10\pi i


Question: Simplify the expression \(\langle \operatorname{Ind}^G_1 1, \operatorname{Ind}^G_H\phi\rangle_G\) for a finite group \(G\) and a subgroup \(H\), where \(\phi\) is an irreducible character of \(H\) and \(\mathbb{1}\) is the trivial character of the trivial subgroup. Determine if this simplifies to \([G:H]\phi(1)\).

Answer: [G:H]\phi(1)


Question:
Determine the nature of the roots of the quadratic equation \( ax^2 + (b-c)x - 2b - c - a = 0 \). Given that the expression \( ax^2 + 2bx + c \), where \( a \) is a non-zero real number, has the same sign as \( a \) for every real value of \( x \), choose the correct option:

(A) real and equal
(B) real and unequal
(C) non-real having positive real part
(D) non-real having negative real part

Answer: B


Question: Find the radius of convergence for the Taylor series of the function \( f(z) = \frac{z}{\sin(z^2 + z)} \) at \( z = 0 \), given that \( f(0) = 1 \) to make \( f \) analytic at \( z = 0 \).

Answer: 1


Question: Determine whether the function \( f(x) = \ln(1 + x_1^2 + x_2^2) \) is Lipschitz continuous with respect to the Euclidean 2-norm in \( \mathbb{R}^2 \).

Answer: Yes


Question: A bag contains 100 unique marbles. With replacement, you pick 10 marbles at a time, at random. How many times must you pick the marbles to have a 95% probability of having seen every unique marble at least once?

Answer: 76


Question: Let $f(x)$ be a polynomial function with non-negative coefficients such that $f(1)=f'(1)=f''(1)=f'''(1)=1$. Find the minimum value of $f(0)$. Use the Taylor series expansion to express $f(x)$ and determine $f(0)$. What is the minimum value of the constant term $f(0)$?

Answer: \dfrac{1}{3}


Question: Expand the function \( f(x) = (5x-4)^{-7/3} \) using Taylor's series around \( a = 1 \), and find the radius of convergence of the series.

Answer: \dfrac{1}{5}


Question: If $n$ is a primitive root modulo an odd prime $p$, does there always exist a least residue $t$ such that $n^t \equiv t \pmod{p}$? Answer yes or no.

Answer: No


Question: Find the smallest non-negative integer solution to the equation \( x^{2017^2} + x^{2017} + 1 \equiv 0 \pmod{2017} \). Use the fact that 2017 is a prime number and consider applying Fermat's Little Theorem.

Answer: 1008


Question: Let $\mathbb{P}$ be a poset and $\dot{Q}$ a $\mathbb{P}$-name. Determine if there exists a cardinal $\mu$ in the ground model such that $\Vdash |\dot{Q}| \leq |\mu|$.

Answer: Yes


Question: Is it possible to generate a completely random integer between 1 and 13 using a standard D&D dice set? The set includes dice with the following number of sides: 2, 4, 6, 8, 10, 12, and 20. You may roll any combination or quantity of these dice, and interpret the outcomes as needed, to achieve 13 uniformly distributed outcomes. Assume the dice are perfectly random. Provide a justification for your answer.

Answer: Yes


Question:
Determine the topological structure of the color spectrum in the HSV model after the following identifications:

1. A hue of zero describes the same color as a hue of one, i.e., \((0, s, b) \sim (1, s, b)\) for each \(s, b \in I\).
2. If a color has no saturation, it is uniquely described only by its brightness, i.e., \((h, 0, b) \sim (h', 0, b)\) for each \(h, h', b \in I\).
3. A brightness of zero or one is black or white respectively, i.e., \((h, s, 0) \sim (h', s', 0)\) and \((h, s, 1) \sim (h', s', 1)\) for each \(h, h', s, s' \in I\).

What is the resulting topological structure of \(D^3\) after these identifications?

Answer: S^3


Question: Let $m$ be the minimum value of $|z|$, where $z$ is a complex number such that \( |z-3i | + |z-4 | = 5. \) If $m$ can be written in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find the value of $a+b$. \( \boxed{} \)

Answer: 17


Question: Find the x-intercept of the equation \( e^{-x} - x + 1 = 0 \).

Answer: 1.2785


Question: Evaluate the double integral: \[ \int_0^1 \int_0^{3v} \sqrt{u+v} \ du \ dv \]

Answer: \dfrac{28}{15}


Question: Evaluate the limit \( \lim_{x\to0_+} \frac{e^x - x e^x - 1}{\left(e^x - 1 \right)^2} \) without using L'Hôpital's rule.

Answer: -\dfrac{1}{2}


Question: Find the value of $a$ such that the curves $y = (x+c)^{-1}$ and $y = a(x+k)^{1/3}$ are orthogonal trajectories of each other.

Answer: \sqrt[3]{3}


Question: Find the limit: \[ \lim_{n \to \infty} \frac{\gamma_{n-1}}{\gamma_{n}} \] where \( \gamma_{n} \) is the \( n \)-th Stieltjes Constant.

Answer: 0


Question: Compute the limit \( \lim_{n\to\infty} \frac{n}{\sum_{i=1}^n \frac{1}{a_i}} \) where \( a_n=(2^n+3^n)^\frac{1}{n} \).

Answer: 3


Question: Calculate the limit: \[ \lim_{x\to0^+}\frac{\sqrt{x}-\int_0^\sqrt{x} \mathrm{e}^{-t^2}\,\mathrm{d}t}{\sqrt{x^3}} \] using L'Hospital's rule.

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{\infty} \frac{1}{(2k-1)^{2n}} \] Does this limit converge to 1?

Answer: 1


Question: Determine the number of zeros of the function \( f(z) = \log(z+3) + z \) inside the disk \( D_1(0) \), where \( z \in \mathbb{C} \).

Answer: 1


Question: Compute the limit \( \lim\limits_{(x,y)\to(0,0)} \frac{xy^3}{x^2+y^4} \).

Answer: 0


Question: Determine the values of \( p \) for which the integral \( \int_0^\infty x^p \arctan x \, dx \) converges.

Answer: (-2, -1)


Question: Given a point $K$ inside triangle $\triangle ABC$, determine if it is possible to construct a line through $K$ that bisects the area of the triangle.

Answer: Yes


Question:
Does the Minkowski inequality imply that for functions $f_n$ and $f$ in $L^p$, the following holds?
\[ ||f_n - f||_p \leq ||f_n||_p - ||f||_p \]

Answer: No


Question: Find the limit of the sequence \( \{x_n\}^{+\infty}_{n=0} \) defined by the recurrence relation \( x_0 = 0, x_1 = 1, x_{n+1} = \frac{x_n + nx_{n-1}}{n+1} \).

Answer: \ln 2


Question: Show that \( x_{l,L}X_{L,k} = \delta_{kl} \), where \( x_{l,L} = \frac{\partial x_l}{\partial X_L} \) and \( X_{L,k} = \frac{\partial X_L}{\partial x_k} \).

Answer: \delta_{kl}


Question: Given a differentiable function \( y = f(x) \) such that \[ f(x) = x + \int_0^1 x^2 z f(z) \, dz + \int_0^1 x z^2 f(z) \, dz \] find the value of \( f\left(\frac{-9}{4}\right) \).

Answer: 0


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

Answer: \dfrac{1}{2}


Question: Let $f(x)$ be a polynomial in $\mathbb{Q}[x]$ but not in $\mathbb{Z}[x]$. For an integer $n \geq 2$, can $f(x)^n$ be a polynomial in $\mathbb{Z}[x]$?

Answer: No


Question: Prove using an \(\epsilon - \delta\) argument that \(\lim_{x \to a} (2x^3 - 7x) = 2a^3 - 7a\) for every \(a \in \mathbb{R}\).

Answer: 2a^3 - 7a


Question: Calculate the value of the infinite product \( \prod_{n=0}^\infty a_n \), where the sequence \( \{a_n\} \) is defined recursively by \( a_0 = \frac{1}{2} \) and \( a_n = 1 + (a_{n-1} - 1)^2 \) for \( n \geq 1 \).

Answer: \dfrac{2}{3}


Question: Determine whether the sequence \((\log(n!) \mod 1)_{n \in \mathbb{N}}\) is dense in the interval \([0,1]\).

Answer: Yes


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

Answer: \dfrac{2}{3}


Question: Determine the number of solutions to the congruence equation \(x^2 \equiv 121 \pmod{1800}\).

Answer: 16


Question: Consider the cup product $H^{1}(X,\mathbb{Z}) \otimes H^{1}(X,\mathbb{Z}) \rightarrow H^{2}(X,\mathbb{Z})$. Could $a \cup b$ be a torsion element in $H^{2}(X,\mathbb{Z})$?

Answer: Yes


Question: Consider a commutative ring $A$ without a multiplicative identity. Let $r \in A$ be a nonzero element such that $r$ is contained in every proper prime ideal of $A$. Can we conclude that $r = 0$? Justify your answer.

Answer: No


Question: Evaluate the limit: \[ L = \lim_{x \to 0} \frac{1-\cos(\sin x)+\ln(\cos x)}{x^4}. \]

Answer: -\dfrac{7}{24}


Question: Determine the smallest integer \( n \geq 3 \) for which there exists an \( n \times n \) matrix such that all entries are in the interval \((0, 1)\), each row and column sums to 1, and no other subsets of the entries sum to 1.

Answer: 4


Question: Determine the negative integer \(d\equiv 2\pmod{4} \) for which the ring of integers in \(\mathbb{Q}(\sqrt{d})\) is a unique factorization domain.

Answer: -2


Question: Evaluate the line integral \( \int_C \frac{2xy^2dx-2yx^2dy}{x^2+y^2} \), where \( C \) is the ellipse \( 3x^2 + 5y^2 = 1 \) oriented in the positive direction.

Answer: 0


Question:
Let $X \subseteq \mathbb{R}$ and $f,g : X \rightarrow X$ be continuous functions such that $f(X) \cap g(X) = \emptyset$ and $f(X) \cup g(X) = X$. Determine which of the following cannot be $X$:

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

Answer: A


Question: Find the maximum value of \(|f(z)|\) on the closed complex disk of radius 2, where \(f(z) = \frac{z^4}{z^2+10}\).

Answer: \dfrac{8}{3}


Question: Let $A$, $B$, and $C$ be events in the same probability space. If $A$ is conditionally independent of $C$ given $B$, i.e., $\mathbb{P}(A | B \cap C) = \mathbb{P}(A | B)$, is it true that $C$ is conditionally independent of $A$ given $B$, i.e., $\mathbb{P}(C | B \cap A) = \mathbb{P}(C | B)$? Provide a justification for your answer.

Answer: Yes


Question:
Find the limit of the sequence as \( x \to 1 \) for \( x > 1 \):
\[ \lim_{x\to 1} \frac{\ln(x) - \sin(\pi x)}{\sqrt{x -1}} \]

Answer: 0


Question:
Given matrices $A$, $B$, and $C$ with sizes $m \times m$, $n \times n$, and $n \times m$, respectively, where $\det(A) = 2$ and $\det(B) = 3$, find the determinant of the block matrix:
\[ \det \begin{pmatrix} 0 & A \\ B & C \end{pmatrix} \]

Answer: (-1)^{mn} \cdot 6


Question: Let $A, B \in M_{10}(\mathbb{R})$ such that $A$ and $B$ have ranks 3 and 2, respectively. If $\operatorname{Img} B \subset \operatorname{Coimg} A$, find the rank of $AB$. Here, $\operatorname{Img} B$ denotes the column space of $B$, and $\operatorname{Coimg} A$ denotes the row space of $A$. Determine the rank of the matrix product $AB$. \( \boxed{?} \)

Answer: 2


Question: Determine the second homotopy group of the space \( \mathbb{R}^3 \setminus \{ (0,0,0) \} \).

Answer: \mathbb{Z}


Question: Determine whether there exists a continuous function \( f: \mathbb{T} \rightarrow \mathbb{R} \) such that \( \lim_{n\rightarrow\infty} |S_{n}f(0)| = 1 \), where \( S_n \) denotes the partial sums defined by the Dirichlet Kernel \( D_n \).

Answer: Yes


Question: Given the matrix \( A = \begin{pmatrix} 0 & 2 \\ K & -1 \end{pmatrix} \), if \( A(A^3+3I)=2I \), find the value of \( K \).

Answer: \dfrac{1}{2}


Question: Let $f(x)$ be a polynomial satisfying $\lim_{x\to \infty} \frac {x^4 f(x)}{x^8+1} =3$, $f(2)=5$, $f(3)=10$, $f(-1)=2$, and $f(-6)=37$. Find $f(0)$. Assume $f(x)$ is a 4th degree polynomial.

Answer: 109


Question: Let $G = A_5$ and $H = \langle (1,2,3,4,5) \rangle$. If $g \in N_G(H)$ is an element of order $5$, compute the order of the group $H \langle g \rangle$.

Answer: 5


Question: Given two positive operators $B$ and $B'$ on a Hilbert space $\mathcal{H}$, is the operator $B - B'$ self-adjoint?

Answer: Yes


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty}\frac{a_n}{n}x^n \), where \( \{a_n\}_{n\in\mathbb N} \) is a real sequence with \( a_n \ge 0 \) for all \( n \in \mathbb N \) and \( \sum_{n=1}^{\infty}a_n < \infty \).

Answer: 1


Question: Evaluate the integral \( \int p(\mathbf{x})(\mathbf{x} - \boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x} - \boldsymbol{\mu}) \, d\mathbf{x} \), where \( p(\mathbf{x}) \) is the probability density function of a multivariate Gaussian distribution with mean \( \boldsymbol{\mu} \) and covariance matrix \( \Sigma \).

Answer: d


Question: Let $U$ and $V$ be bivariate random variables with a continuous distribution, and let $f_{U,V}$ be the joint density of $(U,V)$. Suppose that $f_{U,V}(-u,v) = f_{U,V}(u,v)$ for all $u, v \in \mathbb{R}$. Prove that $\text{cov}(U,V) = 0$. Provide a justification for your answer.

Answer: 0


Question: Find the length of the minimum radius vector of the curve \( \frac{a^2}{x^2}+\frac{b^2}{y^2}=1 \).

Answer: a + b


Question: Find a closed form expression for \( h_k \), the number of feasible sequences of stamps with a total value of \( k \) cents, using two types of 2-cent stamps and one type of 3-cent stamp. Assume the recursive relation \( h_k = h_{k-1} + h_{k-2} + (-1)^k \) holds, and explore the connection to the Fibonacci sequence.

Answer: h_k = F_k + (-1)^k


Question: Calculate the limit: \[ \lim\limits_{\rho\rightarrow 0^+}\frac{\log{(1-(a^{-\rho}+b^{-\rho}-(ab)^{-\rho}))}}{\log{\rho}} \] where \(a > 1\) and \(b > 1\).

Answer: 2


Question:
Determine the set of values of \( m \in \mathbb{C} \) for which the equation \((1+iz)^n = m(1-iz)^n\) has all roots real. Choose the correct option:

(a) \( m \in \mathbb{R} \)
(b) \( m \in [-1,1] \)
(c) \( m \in \{-1,1\} \)
(d) \( m \in \{-1,1,-i,i\} \)
(e) \( m \in \{z \in \mathbb{C} \mid |z|=1\} \)
(f) \( m \in \{z \in \mathbb{C} \mid |z| \leq 1\} \)

Answer: E


Question:
Let \( \{a_n\}, \{b_n\}, \{c_n\} \) be sequences such that:

- \( a_n + b_n + c_n = 2n + 1 \),
- \( a_n \cdot b_n + b_n \cdot c_n + c_n \cdot a_n = 2n - 1 \),
- \( a_n \cdot b_n \cdot c_n = -1 \),
- \( a_n < b_n < c_n \).

Find the value of \( na_n \) as \( n \to \infty \).

Answer: -\dfrac{1}{2}


Question: Find the range \([a, b]\) of the expression \(\sum_{n=0}^\infty\frac{\cos nx}{3^n}\). Calculate the value of \(b-a\).

Answer: \dfrac{3}{4}


Question: Let $U = M_{n}(\mathbb{C})$ be the space of $n \times n$ complex matrices, and consider the bilinear form $\xi(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\text{tr}(B)$. Find the dimension of the orthogonal complement $U_{\perp}$, where $U_{\perp} = \{ A \in M_{n}(\mathbb{C}) \mid \forall B \in M_{n}(\mathbb{C}), \ n \cdot \text{tr}(AB) - \text{tr}(A)\text{tr}(B) = 0 \}$.

Answer: 1


Question: Evaluate the triple integral \( \int_{-2}^{-1} \int_{-2}^{-1} \int_{-2}^{-1} \frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz \).

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{n \to \infty}\frac{2^{-n^2}}{\sum_{k = n+1}^{\infty}2^{-k^2}} \]

Answer: \infty


Question: In a metric space $(X,d)$, if every subset of $X$ is bounded, does it imply that the space $X$ itself is bounded?

Answer: Yes


Question: A line passing through the origin intersects the region $R$ in the first quadrant, which is enclosed by the curve $f(x) = x - x^2$ and the $x$-axis. The line divides the region $R$ such that the area of the lower portion is seven times the area of the upper portion. Determine the slope of this line.

Answer: \dfrac{1}{2}


Question: Consider the irreducible polynomial $g = X^4 + X + 1$ over $F_2$. Let $E$ be the extension of $F_2$ with a root $α$ of $g$. How many elements does the field $E$ have?

Answer: 16


Question: Are the roots of all polynomials with rational coefficients dense in the real numbers?

Answer: Yes


Question: Show that if \( \left(1-2p^{2}\sin^2{\omega}\right)^2 + p^{2}\sin^{2} \omega \leq 1 \), then \( 4p^2 \leq 4 \).

Answer: 4p^2 \leq 4


Question: In the space $C[a,b]$, consider the norm defined by \( \left \| f \right \| = \max\left \{ \sup |f|, \sup |f'| \right \} \). Does this norm satisfy the triangle inequality?

Answer: Yes


Question: Find the limit superior of the sequence of sets \( \left\{ \left[0, \frac{1}{m}\right] \right\}_{m=1}^{\infty} \).

Answer: \{0\}


Question: Calculate the definite integral: \[ \int_0^\pi \sin t \cdot \sin^{11}(\cos t) \, dt \]

Answer: 0


Question: Evaluate the integral \( \int \frac{\cos\pi z}{z^2-1}\, dz \) over the rectangle with vertices at \(2+i, 2-i, -2+i, -2-i\).

Answer: 0


Question: Given the joint probability density function \( f_{X,Y}(x,y) = 6e^{-(2x+3y)} \) for \( x \geq 0 \) and \( y \geq 0 \), find the probability that \( X > Y \).

Answer: \dfrac{3}{5}


Question: Determine the radius of convergence of the Taylor series for the function \( \frac{1}{(z^2 - 2z + 3)^2} \) centered at \( z_0 = 1 \).

Answer: \sqrt{2}


Question: Evaluate the integral \( \int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx. \)

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


Question: Determine the number of integers \( k \) such that \( k \cdot 6076 \equiv 5292 \pmod{8722} \).

Answer: 98


Question: Find the spectrum of the operator \( A: C[0,1] \rightarrow C[0,1] \) defined by \( (Af)(x) = f(x) + \int_0^x f(t)dt \).

Answer: \{1\}


Question: Evaluate the integral \( I = \int_{C(0,2)^+} \frac{z^3}{z^3+z^2-z-1} \, dz \) using the residue theorem.

Answer: -2\pi i


Question: Let $R$ be a commutative ring with identity. Suppose $R^n \simeq R^n \oplus M$ as $R$-modules, where $n$ is a fixed natural number. Prove or disprove that $M = \{0\}$.

Answer: M = \{0\}


Question: Compute the integral \( \int_{\gamma} \frac{1}{e^z - 1} \, dz \) where \( \gamma \) is the circle of radius 9 centered at 0, using the residue theorem.

Answer: 6\pi i


Question: Solve the differential equation \(x^2y'' + 2xy' - 2y = 18x^4\) given that \(y(x)\) is finite as \(x \to 0\) and \(y(1) = 0\).

Answer: x^4 - x


Question: Find the coefficient of \(x^{12}\) in the expansion of \((1-x^6)^4 \cdot (1-x)^{-4}\).

Answer: 125


Question: Calculate the line integral \( \int_{C} \vec{F} \cdot \vec{n} \, ds \) where \( \vec{F}(x, y) = x^{10} \vec{i} + (3x - 10x^9y) \vec{j} \) and \( \vec{n} \) is the unit normal vector to the curve \( C \). The curve \( C \) is defined by \( x^2 + y^2 = 1 \) and is restricted to the first quadrant. Provide the value of the integral.

Answer: \dfrac{3}{2}


Question: Given a sequence of i.i.d. random variables \((X_n)_{n \in \mathbb{N}}\) with distribution \(N(0,1)\), determine the limit of the sequence \[ Y_n = \frac{X_1^2+ \ldots + X_n^2}{(X_1 - 1)^2 + \ldots + (X_n - 1)^2} \] as \(n \to \infty\).

Answer: \dfrac{1}{2}


Question: Find an asymptotic bound for the sum \( \sum_{k = 0}^{\lg n} 2^k \lg(\lg(n) - k) \).

Answer: \Theta(n)


Question: What is the Lebesgue measure of the set \(\{(a,b) \in \mathbb{R}^2 \mid a-b \in \mathbb{Q}\}\) in \(\mathbb{R}^2\)?

Answer: 0


Question: Determine the number of solutions to the equation \(2\tan^{-1}|x|\cdot\ln|x|=1\).

Answer: 2


Question: Determine the minimum value of the function \( f(x) = \frac{x^4 - 2ax^3 + 3a^2x^2 - 2a^3x + a^4 + 9}{x^2 - ax + a^2} \) for \(-2 \leq a \leq 2\) and \(a \neq 0\).

Answer: 6


Question: Determine the number of solutions to the congruence equation \(x^2 + y^2 \equiv 1 \pmod{p^2}\), where \(p\) is a prime such that \(p \equiv 3 \pmod{4}\).

Answer: p(p + 1)


Question: Compute the limit \( \lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x} \) using L'Hôpital's rule.

Answer: 0


Question: Let $k$ be a finite field. Consider a smooth, connected, affine algebraic group scheme $G$ of finite type over $k$. If $G(k)$ is trivial, does it necessarily follow that $k = \mathbb{F}_2$ and $G$ is a split torus?

Answer: Yes


Question: In the projective plane over any field, consider a cubic curve defined by a homogeneous polynomial \( f \in k[x,y,z] \) of degree 3, where \( C = \{(x:y:z) \in P(2,k) : f(x,y,z) = 0\} \). Let \( E = \{(x:y:z) \in P(2,k) : ax + by + cz = 0\} \) be a straight line in the projective plane. Is it true that the intersection \( |C \cap E| = 3 \)?

Answer: No


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

Answer: p > 1


Question: Evaluate the line integral \( \int z \bar{z} \, dz \) over the unit circle \( r(t) = \langle \cos t, \sin t \rangle \) for \( 0 \leq t \leq 2\pi \). Additionally, determine if the function \( f(z) = z \bar{z} = x^2 + y^2 \) is analytic.

Answer: 0


Question:
Let \( a_k \) be a sequence defined by:
\[ a_k = \begin{cases} \frac{(-1)^n}{\ln(n)} & \text{if } k = 3n-1, \\ 0 & \text{otherwise.} \end{cases} \]
Find \( \limsup \sqrt[k]{|a_k|} \).

Answer: 1


Question: Evaluate the line integral \( \oint_{\mathcal C}\begin{pmatrix}x_2^2 \cos x_1 \\ 2x_2(1+\sin x_1)\end{pmatrix} \cdot d\vec{x} \) where \( \mathcal C \) is the unit circle in \( \mathbb{R}^2 \) traced counterclockwise starting and ending at the point (1,0). Use the parameterization \( \vec{x}(t) = \begin{pmatrix}\cos t \\ \sin t\end{pmatrix} \) for \( t \in [0, 2\pi] \).

Answer: 0


Question: Determine the common fraction equivalent to the expression \( \frac{\dfrac{1}{1^3}+\dfrac{1}{3^3}+\dfrac{1}{5^3}+\dfrac{1}{7^3}+\cdots}{\dfrac{1}{1^3}+\dfrac{1}{2^3}+\dfrac{1}{3^3}+\dfrac{1}{4^3}+\cdots} \).

Answer: \dfrac{7}{8}


Question: Find the radius of convergence for the power series \( \sum_{n=0}^{\infty} a_n x^n \) given the recursive relation \( \frac{a_{n+2}}{a_n} = \frac{n(n-1)+1}{(n+1)(n+2)} \).

Answer: 1


Question:
Given a discrete random variable $X$ with parameter $a > 0$ and probability mass function:
$$
f_X(x) = \begin{cases}0.2, &x = a\\0.3, &x = 6a\\0.5, &x = 10a\end{cases}
$$
Assume a random sample of size $n$ from this distribution. Determine if $\frac{1}{7}\cdot\overline{X}$ is an unbiased estimator for $a$, where $\overline{X}$ is the sample mean.

Answer: Yes


Question: Solve for \( x \) in the equation: \( 2f(x) + 3f^{-1}(x) = 10 \), where \( f(x) = x^3 + x - 8 \).

Answer: 2


Question:
Determine whether the set \( \mathcal{A} = A_1 \cup A_2 \cup A_3 \cup A_4 \) in \( \mathbb{R}^2 \) has an area in the sense of the Riemann integral, where:

\[ A_1 = \{(q_1, 0) \in \mathbb{R}^2 : q_1 \in [0,1] \cap \mathbb{Q}\} \]
\[ A_2 = \{(1, q_2) \in \mathbb{R}^2 : q_2 \in [0,1] \cap \mathbb{Q}\} \]
\[ A_3 = \{(q_1, 1) \in \mathbb{R}^2 : q_1 \in [0,1] \cap \mathbb{Q}\} \]
\[ A_4 = \{(0, q_2) \in \mathbb{R}^2 : q_2 \in [0,1] \cap \mathbb{Q}\} \]

The set \( \mathcal{A} \) consists of points on the edges of the unit square \([0,1] \times [0,1]\) with rational coordinates. Consider whether the integral \( \int_\mathcal{A} 1_\mathcal{A} \) exists, where \( 1_\mathcal{A} (x,y) = 1 \) if \((x,y) \in \mathcal{A}\) and \( 1_\mathcal{A} (x,y) = 0 \) otherwise.

Answer: 0


Question: Evaluate the integral \( I = \int_{0}^{1} t^2 \cos(2t\pi) \tan(t\pi) \ln[\sin(t\pi)] \, dt \).

Answer: 0


Question: Find the expected time of the first event, \( E[W_1 \mid X(3) = 1] \), where \( W_1 \) is the time of the first event and \( X(3) = 1 \) indicates that one event occurred before time 3 in a Poisson process with rate \( \lambda \).

Answer: \dfrac{3}{2}


Question: Consider a Riemannian manifold $\mathcal{M}$ modeled on a (possibly infinitely dimensional) Hilbert space. Suppose that $\{p_i\} \subset \mathcal{M}$ and $p_i \to p$. A sequence of covectors $v^*_i \in T^*_{p_i}\mathcal{M}$ is said to weakly converge to $v \in T_p^*\mathcal{M}$ if for every smooth function $f$ on $\mathcal{M}$, we have \(\langle v^*_i, df(p_i) \rangle \to \langle v^*, df(p) \rangle.\) Is it true that if $\{v_i^*\}$ is bounded, then it has a weakly convergent subsequence?

Answer: Yes


Question: Find all functions \( f: \Bbb N \to \Bbb N \) such that \( f(f(m) + f(n)) = m + n \) for all \( m, n \in \Bbb N \).

Answer: f(n) = n


Question: Let A be a diagonalizable square matrix with real eigenvalues greater than zero. Is it necessarily true that there exists a matrix S such that \( A = S^2 \)? Justify your answer.

Answer: Yes


Question: Find the value of the tenth derivative of the function \( f(x) = e^x \sin x \) at \( x = 0 \).

Answer: 32


Question: Determine the order of the zero at \( z = 0 \) for the function \( f(z) = \frac{d^{50}}{{dz}^{50}}[z^{100}\sin(z^{10})] \).

Answer: 60


Question:
Evaluate the limit of the sequence:
\[ \lim_{n\rightarrow \infty}\frac{1\cdot 4\cdot \ldots \cdot (3n+1)}{2\cdot 5\cdot \ldots \cdot (3n+2)} \]

Answer: 0


Question: Determine whether the following conjecture is true or false: In a biconnected graph, the minimal cardinality of a vertex $k$-cut is $k$. In other words, can one obtain $k$ connected components by removing fewer than $k$ vertices from a biconnected graph?

Answer: False


Question: Determine whether the sequence \( \frac{n^n}{e^{n^2}} \) converges to 0 as \( n \to \infty \).

Answer: 0


Question:
Determine the nature of the points for the differential equation \( x y'' -\frac{2}{x(x-2)}y' + x^2y = 0 \). Which of the following statements is true?

1. \(x=0\) is a regular singular point of this equation.
2. \(x=0\) is an irregular singular point of this equation.
3. \(x=0\) is an ordinary point of this equation.
4. \(x=2\) is an irregular singular point of this equation.

Answer: 2


Question: Given the matrix \( \begin{pmatrix} -4 & -15 \\ 2 & 7 \end{pmatrix} \), let \( \begin{pmatrix} -4 & -15 \\ 2 & 7 \end{pmatrix}^{100} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \). Find the value of \( 3a + b + 3c + 4d \).

Answer: 7


Question: Determine the radius of convergence of the power series expansion of the function \( f(z) = \frac{\mathrm{e}^z}{z-1} \) about \( z = 0 \).

Answer: 1


Question: How many ordered pairs \((\alpha_1, \alpha_2)\) of permutations in the symmetric group \(S_n\) commute, i.e., satisfy \(\alpha_1 \circ \alpha_2 = \alpha_2 \circ \alpha_1\), where \(\alpha_1, \alpha_2 \in S_n\)?

Answer: n! \cdot p(n)


Question: Determine whether there exist measurable functions $f_n: [0,1] \to [0, \infty)$, $n \in \mathbb{N}$, such that \(\limsup_{n\to\infty} f_n (x) = \infty\) for all $x \in [0,1]$, but \(\lim_{n\to\infty} \int_{[0,1]} f_n (x) \, dx = 0\).

Answer: Yes


Question: Calculate \( \lim_{n\to\infty}\sqrt[n]{|1-z^n|} \) where \( z \in \mathbb{C} \) and \( |z| \neq 1 \).

Answer: \max\{1, |z|\}


Question: Let \( A \in M_{6}(\mathbb{R}) \) be a matrix such that \( A^{3} - 2A^{2} - 15A = 0 \). Given that \( \operatorname{tr}(A) = 4 \), find \( \operatorname{rank}(A) \).

Answer: 4


Question: A fair six-sided die is rolled repeatedly. What is the expected number of rolls needed to get two consecutive sixes?

Answer: 42


Question: Let $D = \mathbb{C} \setminus \{e^{(1+i)t}: t \in [0, \infty)\}$. This is a simply connected domain. Suppose $\ell$ is a branch of the logarithm on $D$, such that $\ell(e^\pi) = \pi$. Determine the value of $\ell(e^{3\pi})$.

Answer: 3\pi


Question: Evaluate the integral: \[ \int_{-\infty}^\infty xe^{-3x^{2}}\cos(x)\,dx \]

Answer: 0