Question: Evaluate the limit \( \lim_{x \to 0}\frac{1}{x^3}\int_0^x \frac{t\ln(1+t)}{t^4+4}\,dt \) using a method other than L'Hôpital's rule.

Answer: \dfrac{1}{12}


Question: Calculate \(25^{(2022^{2022})} \mod 40\) using Euler's theorem and the Chinese Remainder Theorem.

Answer: 25


Question: Determine whether there exists a polynomial with integer coefficients \( Q(x) \) such that \( Q(n)P(n) \) is a perfect cube for all integers \( n \ge 1 \), where \( P(n) \) is a given polynomial.

Answer: Yes


Question: Given that \( \sum_{n=1}^{\infty}a_n \) converges where \( a_n > 0 \), determine the behavior of the series \( \sum_{n=1}^{\infty}a_n^{3} \sin(n) \). Choose one of the following options: (a) converges, (b) diverges, (c) does not exist, (d) None of the above.

Answer: a


Question: Determine a sequence of the digits from 1 to 9, denoted as $a_1, a_2, \ldots, a_9$, such that each two-digit number formed by consecutive digits ($a_ia_{i+1}$) is divisible by either 7 or 13. Express this sequence as a single number $10^{8}a_1 + 10^{7}a_2 + \cdots + 10a_8 + a_9$. This problem is adapted from the Brazilian Mathematics Olympiad (OBM).

Answer: 784913526


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

Answer: 0


Question: Find the residue of the function \( \frac{10z^4-10\sin(z)}{z^3} \) at \( z=0 \).

Answer: 0


Question:
Find the spectral radius of the following $n \times n$ matrix:

$$ \begin{matrix} 1 & 0 & . & . & . &-1 \\ 0 & 1 & . & . & . &0 \\ . & . & . & & &. \\ . & . & & . & &. \\ . & . & & & . &. \\ -1 & 0 & . & . & . &1 \\ \end{matrix} $$

Answer: 2


Question: Determine the asymptotic behavior of the expected number of flips required for the number of heads to exceed the number of tails in a sequence of fair coin flips.

Answer: \infty


Question: Determine the period of the function \( g(x) \) that satisfies the equation: \[ g(x+1) = \frac{g(x) + 2}{g(x) - 1} \]

Answer: 2


Question: In a game of Cheat with 4 players, each player is dealt cards from a deck of 13 cards numbered from 1 to 13. What is the probability that the initial card distribution will result in the game lasting exactly 90 turns?

Answer: 0


Question: Find the center of gravity of a hemispherical shell with radius $a$ and uniform density of 1, positioned with its base on the $x-y$ plane. Use the method of slicing parallel to the $x-y$ plane to determine the center of gravity.

Answer: \dfrac{a}{2}


Question: Determine the number of integer values of $p$ for which the equation $3 p \sin x - (p+\sin x)(p^2-p \sin x + \sin ^{2} x) =1$ has at least one solution for $x$. \(x\) is a real number.

Answer: 4


Question: Evaluate the flux of the curl of the vector field \( \vec{F} = (z-y, 0, y) \) through the portion \( S' \) of the sphere with radius \( \sqrt{2} \) centered at the origin, which lies above the curve \( C \) defined by \( x^2 + y^2 = 1 \) on the plane \( z = 1 \). Assume the surface \( S' \) is oriented outward.

Answer: \pi


Question: Evaluate the double integral \( \int_{y=0}^{3}\int_{x=y}^{\sqrt{18-y^2}} (7x + 3y) \,dx\,dy \).

Answer: 72 + 54\sqrt{2}


Question: Evaluate the limit: \[ L_1 = \lim_{x\to 0}\frac{\text{erf}(x^2)}{\text{erf}(x)}. \]

Answer: 0


Question: Consider two linear transformations $M$ and $N$ between finite-dimensional vector spaces $X$ and $Y$. If $M$ and $N$ have the same null space, which is not equal to the zero vector space, does there exist a scalar $k$ such that $M = kN$? Answer yes or no.

Answer: No


Question: Suppose $R$ is a commutative ring with identity, and let $a$, $b$, $c \in R$. If $a$ divides $b$ and $b$ divides $c$, does $a$ divide $c$? Answer yes or no and justify your answer.

Answer: Yes


Question:
Let $A$ be the matrix defined as follows:

$$A := \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Determine the number of matrices similar to $A$ over the field $\mathbb{Z}/3\mathbb{Z}$.

Answer: 117


Question: Calculate the determinant of the derivative of the function \( h(u,v) = f(u^2 - v^2, uv) \), where \( f \) is a continuously differentiable function from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \), at the point \((1,1)\). It is given that \( f(0,1) = (3,6) \) and \( D_f(0,1) \) is not invertible.

Answer: 0


Question: In a universe $U$ shaped like $\mathbb{R}^3$, points are randomly distributed in a Poisson fashion with density parameter $\lambda$, such that in any measurable set $S$ of volume $V$, the expected number of points in $S$ is $V$. Consider a random line $L$ that intersects the origin. What is the probability that $L$ intersects at least one point in this distribution?

Answer: 0


Question: Find the radius of convergence for the power series \(\sum_{n=0}^{\infty} a_n z^n\) where \(a_n = \frac{n^2 - 5}{4^n + 3n}\).

Answer: 4


Question: Determine whether the following statement is true or false: For any monic polynomial \(f^{(t)}(z)=a_0^{(t)}+\dotsb+a_{n-1}^{(t)}z^{n-1}+z^n\in\mathbb C[z]\) with coefficients \(a_0^{(t)},\dotsc,a_{n-1}^{(t)}\colon\mathbb R\to\mathbb C\) continuous on \(t\), there exists a complex-valued continuous function \(\phi^{(t)}\colon\mathbb R\to\mathbb C\) such that \(f^{(t)}(\phi^{(t)})=0\) for all \(t\in\mathbb R\).

Answer: True


Question: Consider a field extension $L / \mathbb{Q}$ with $[L : \mathbb{Q}] = 8$. If $L$ contains a primitive 8th root of unity, determine if it is possible for $L \cap \mathbb{R}$ to be strictly larger than $\mathbb{Q}$. Provide a yes or no answer.

Answer: Yes


Question: Suppose $b$ is an element of a group $G$ such that $b^5 = e$. Determine whether there exists an element $y \in G$ such that $b = y^4$. Justify your answer.

Answer: Yes


Question: Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be the function defined by \( f(x,y) = e^{2x} + \left( \frac{y}{5} \right)^4 \). Determine if the preimage of the open interval \((-\infty, 1)\) under \( f \) is an open set in the metric space \((\mathbb{R}^2, d)\), where \( d \) is the Euclidean distance.

Answer: Yes


Question: Let $f(x)$ be an irreducible polynomial of degree 4 over a field $F$. If the splitting field of $f(x)$ over $F$ has a Galois group isomorphic to the dihedral group $D_4$, determine the possible number of distinct roots of $f(x)$. \( \boxed{4} \)

Answer: 4


Question: Evaluate the integral \( \oint_{|z|=R} \frac{1}{(z-c)(z-d)} \, dz \) where the circle \(|z|=R\) encloses the points \(c\) and \(d\).

Answer: 0


Question: Consider a complete graph $K_9$. Determine the number of spanning trees such that vertex 1 has a degree of 4.

Answer: 143360


Question: Determine the values of \( p > 0 \) for which the series \( \sum_{n=2}^\infty \frac{1}{(\log 2)^p + (\log 3)^p + \cdots + (\log n)^p} \) converges.

Answer: p > 1


Question: Determine the region of the domain \( D = \{ ( x , y ) \in \mathbb{R}^2 \mid x \ge 1 , y \ge 1 \} \) on which the function \( f(x, y) = \begin{cases} \frac{2x^2 - xy - y^2}{x^2 - y^2} & x \neq y \\ \frac{3}{2} & x = y \end{cases} \) is continuous.

Answer: D


Question: Find the area of one of the regions formed by the part of the folium of Descartes, given by the equation \(x^3+y^3-3xy=0\), that passes through the triangle with vertices \((\sqrt[3]{4}, \sqrt[3]{2}), (0,0), (\sqrt[3]{4}, 0)\). The folium intersects the triangle at the point \((\sqrt[3]{4}, \sqrt[3]{2})\).

Answer: \dfrac{1}{2}


Question: Determine the order of magnitude of the function \( \sinh\left(\frac{1}{\epsilon}\right) \) as \( \epsilon \) approaches zero, expressed in terms of the exponential function.

Answer: e^{1/\epsilon}


Question: Let $(X_n)$ be i.i.d., positive random variables. Compute \[ E\left[\frac{\sum_{i=1}^k X_i}{\sum_{j=1}^n X_j}\right] \] for $k \le n.$

Answer: \dfrac{k}{n}


Question: Let $B_1, B_2 \subset \mathbb{R}^3$ be two topological balls with non-empty interiors in the usual Euclidean topology. Determine whether the connected components of the intersection $B_1 \cap B_2$ are also topological balls.

Answer: Yes


Question: Evaluate \(\lim_{x \to 0^-}(\frac{\tan x}{x})^\frac{1}{x^3}\).

Answer: 0


Question: Let $G$ be a connected cubic graph of order $n > 4$ with a girth of 3. Determine the chromatic number $\chi(G)$. \(\boxed{\chi(G)}\)

Answer: 3


Question: Determine whether the order of the wreath product of $\mathbf{Z}_{p^n}$ and $\mathbf{Z}_{p^m}$ is $p^{m+n}$. Provide a justification for your answer.

Answer: No


Question:
Find the limit of the expression as \(n\) approaches infinity:
\[ L = \lim_{n\to \infty}\frac{\left(2\sqrt[n]{n}-1\right)^n}{n^2} \]

Answer: 1


Question: Consider a sequence of independent and identically distributed (i.i.d.) random variables $U_1, U_2, \ldots$, each uniformly distributed on the interval (0,1). Define a random variable $N((0,s])$ as the supremum of natural numbers $k$ such that the sum of the first $k$ transformed random variables, $\sum_{i=1}^k (-\log(1- U_i))$, is less than or equal to $s$. Determine if $N((0,s])$ is a Poisson distributed variable with parameter $s$. Provide a justification for your answer.

Answer: Yes


Question: Calculate \( \ln(i^i) \).

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


Question: Evaluate the integral \( \int_C \frac{z^3}{(z-2)(z+3)^2} \, dz \) using Cauchy's Residue Theorem, where \( C \) is the circle with radius 4 centered at the origin.

Answer: -8\pi i


Question: Find the difference between the maximum and minimum values of $u^2$, where $u = \sqrt{a^2\cos^2x + b^2\sin^2x} + \sqrt{a^2\sin^2x + b^2\cos^2x}$ for real numbers $a$ and $b$, and $x$ is an angle.

Answer: (|a| - |b|)^2


Question: Determine whether there exists a continuous function $h$ such that \( \int_a^b |f-h|^2 < \epsilon \) for any \( \epsilon > 0 \), given that \( f \in R[a,b] \) and for any \( \epsilon > 0 \), there exists a continuous function \( g \) satisfying \( \int_a^b |f-g| < \epsilon \), and \( f^2 \) is Riemann integrable on \([a,b]\).

Answer: Yes


Question: Determine the number of points at which the functions \( f(x) = x^2 \) and \( g(x) = x\sin(x) + \cos(x) \) intersect.

Answer: 2


Question: Determine whether there exist non-permutation Boolean matrices that have an inverse when addition is defined modulo two. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate the improper integral: $$\int_{2}^{\infty}\frac{dx}{\left(x-1\right)\sqrt{x-2}}$$

Answer: \pi


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{(3k-2n)^{5}}{n^{6}}$$

Answer: -\dfrac{7}{2}


Question: If a matrix A is diagonalizable, does it necessarily imply that its eigenvectors form a basis of the vector space V?

Answer: Yes


Question: Determine the range of the complex variables $z$ and $z^*$ such that the following integral holds: $$\int\int\frac{i}{2}\frac{1}{2\pi}e^{-\frac{|z|^2}{2}}dzdz^*=1$$

Answer: \mathbb{C}


Question: Determine the values of $a > 0$ for which the improper integral \( \int_{0}^{\infty} \frac{\sin x}{x^a} \, dx \) converges.

Answer: (0, 2)


Question: Find all integer solutions to the congruence equation \(x^4 + x + 3 \equiv 0 \pmod{27}\).

Answer: 24


Question: Evaluate the line integral \(\int \limits_{C} F \cdot dr\) where \(F(x,y) = (x, y)\) and \(C\) is the Archimedes's spiral defined by \(r = 1 + \theta\) for \(\theta \in [0, 2\pi]\). Parameterize the curve \(C\) and determine the appropriate limits for the integral.

Answer: 2\pi(1 + \pi)


Question: Consider a rectangle with vertices $2+iT$, $2-iT$, $-1+iT$, and $-1-iT$ where $T>3$ is not an ordinate of zero of the Riemann zeta function $\zeta(s)$. If $N(T)$ is the number of zeros of the Riemann zeta function up to height $T$, show that $$\lim_{T\to \infty}\frac{N(T)}{ \frac{T}{2\pi}\log\left(\frac{T}{2\pi}\right)-\frac{T}{2\pi} }=1.$$

Answer: 1


Question: Given a base $b$ and an irrational number $m$, consider the power series $f(x)$ where the coefficient of $x^n$ is the $n$th non-zero $b$-ary digit of $m$ multiplied by $b^{-i}$, with $i$ being the index of the digit. Determine if there exists a rational number $x \neq 0$ such that $f(x)$ is irrational.

Answer: Yes


Question: Determine whether a bounded linear operator on a Hilbert space can have a spectrum consisting of only one point.

Answer: Yes


Question: Determine the maximum number of real roots of the equation $e^x + P(x) = 0$, where $P(x)$ is a polynomial of degree $n$.

Answer: n + 1


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) for the vector field \( \mathbf{F}(x,y,z) = x\mathbf{i} + y\mathbf{j} + 5\mathbf{k} \). The surface \( S \) is the boundary of the region enclosed by the cylinder \( x^2 + z^2 = 1 \) and the planes \( y = 0 \) and \( x + y = 2 \). Use the positive (outward) orientation for the closed surface.

Answer: 4\pi


Question: Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}$ satisfies the differential equation $\Delta f + \lambda f = 0$, where $\lambda > 0$ is a constant. If $f$ is zero in an open set $U \subset \mathbb{R}^n$, is $f$ identically zero in $\mathbb{R}^n$?

Answer: Yes


Question: Determine the conditions under which the matrix $X = A^\top Y A$ is invertible, given that $A \in F^{m \times n}$ is full rank and $Y$ is positive definite.

Answer: m \geq n


Question:
Compute the quadratic variation $[X,X]_t$ for the Ornstein-Uhlenbeck process $(X_t)_{t\geq 0}$ defined by the integral representation:

\[ X_t = \sigma e^{-\alpha t}\int_0^t e^{\alpha s}\,dB_s, \quad X_0 = 0 \]

The process also satisfies the stochastic differential equation:

\[ dX_t = \alpha X_t\,dt + \sigma\,dB_t, \quad X_0 = 0 \]

Answer: \sigma^2 t


Question: Simplify \( \tanh^{-1}(3i) \) to \( i\tan^{-1}(3) \).

Answer: i \tan^{-1}(3)


Question: Determine the fraction of permutations of the letters A through G that, when inserted into an initially empty binary search tree, will produce the same structure as the binary search tree derived from the sequence A, E, F, G, B, D, C.

Answer: \dfrac{1}{504}


Question: Evaluate the triple integral \( \iiint_E (x+y) \, dV \) over the right circular cone \( E \) with height 10 and diameter 5, centered at the origin with its base on the \( xy \)-plane. Use the symmetry of the cone to simplify the evaluation.

Answer: 0


Question: Determine the sign of the determinant of the matrix \(\begin{bmatrix} A & B \\ C & D \end{bmatrix}\) in terms of the determinant of \(AD - BC\), where \(A, B, C, D\) are commuting \(n \times n\) matrices.

Answer: \det(AD - BC)


Question: Determine whether the closure of a dense subset of a metric space contains every point of the space. Answer with "True" or "False."

Answer: A


Question: Evaluate the limit \( \lim_{n\to\infty} \int_{0}^{1} \frac{n^3 x}{1+(nx)^2} \, \mathrm{d}x \) using the dominated convergence theorem.

Answer: \infty


Question: Find the shortest distance from the point \((1,0)\) to the parabola \(y^2=4x\) using the method of Lagrange multipliers.

Answer: 1


Question: Provide an example of a simple equation with rational coefficients whose graph has no rational coordinates.

Answer: x^2 + y^2 = 3


Question: Evaluate the alternating series: $$\sum^{\infty}_{n=0} (-1)^n\frac{n+1}{n!}$$.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to 0^+} \frac{\sqrt{\log^2(x)-\sin(x)}}{\log(x) + e^x}$$

Answer: -1


Question: Calculate the flux of the vector field \( \vec{r}=x\hat{i}+y\hat{j}+z\hat{k} \) through the surface of the sphere \( x^2+y^2+z^2=R^2 \) using the formula involving the determinant.

Answer: 4\pi R^3


Question:
Given positive numbers \(x, y, z > 1\) satisfying \(x + y + z = xyz\), find the minimum value of the expression:
\[ \large \frac{x - 1}{y^2} + \frac{y - 1}{z^2} + \frac{z - 1}{x^2} \]

Answer: \sqrt{3} - 1


Question: Determine the generator of the ideal \( I = \{a+bi \in \mathbb{Z}[i] \mid a \equiv b \pmod{2}\} \), where \( \mathbb{Z}[i] \) is the ring of Gaussian integers.

Answer: 1 + i


Question: Determine the minimum number of geostationary satellites required to ensure that at least three satellites are visible from any point on Earth at all times.

Answer: 7


Question: Determine the number of distinct ways to rearrange the letters of the word INDIVISIBILITY such that no two 'I's are adjacent.

Answer: 3386880


Question: Determine whether the restriction of a state from the unitization of a $C^*$-algebra is also a state. Answer with true or false.

Answer: False


Question: Determine the number of conjugacy classes of subgroups isomorphic to $D_4$ in the dihedral group $D_{2n}$, where $n$ is divisible by 2 but not divisible by 4.

Answer: 0


Question: Does Zhang's work imply that the next greatest prime number is necessarily less than $2^{57,885,161}-1 + 70,000,000$? Answer yes or no.

Answer: No


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{e^{-bx^2}\cos(mx)}{x}dx \) using the Fourier Transform and the error function.

Answer: 0


Question: Find the largest natural number \( n \ge k \) such that any selection of \( k \) numbers from a set of \( n \) natural numbers results in a composite sum.

Answer: 2k - 1


Question: Evaluate the integral \( f(x) = \int_0^{2\pi} e^{x \sin \theta} \, \text{d}\theta \) and express the result in terms of common functions, if possible.

Answer: 2\pi I_0(x)


Question: Let $z$ be a complex number such that $|z| = 2$. Determine the minimum value of the expression $| (z - 1)^2 (z + 3)^3|$. Provide your answer as a single number.

Answer: 9


Question: Determine whether the limit \(\lim_{(x,y)\to (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2}\) exists.

Answer: 1


Question: Determine whether every bounded, open, and convex subset of Euclidean space satisfies the interior sphere condition at some point on its boundary. Answer with 'True' or 'False'.

Answer: True


Question: Determine the homogeneity degree of the function \(f(x_1,x_2)=\int_{0}^{\sqrt{{x_1}^2+{x_2}^2}}e^{-\frac{w^2}{{x_1}^2+{x_2}^2}}dw\) given that \(f(0,0)=0\), without explicitly evaluating the integral.

Answer: 1


Question: Determine if the expression \( \sqrt{28 \cdot 29 \cdot 30 \cdot 31 + 1} \) can be computed manually without electronic assistance.

Answer: 869


Question: Two individuals, A and B, start at opposite poles of a sphere with radius \( r \). They move randomly along either a latitude or a longitude, but only one individual moves at a time, with a step length of 1 unit. What is the expected time for them to meet?

Answer: \pi^2 r^2


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{\sin^{-1}(\cos^2 x) \cos^{-1}(\sin^2 x)}{x^2-x^4}$$

Answer: \infty


Question: For an idempotent matrix $A$, determine whether the number of eigenvalues equal to 1 is equal to the rank of $A$. Provide a justification for your answer.

Answer: Yes


Question: Given that $x$ and $y$ are real numbers satisfying $x^2 + 2xy - y^2 = 6$, determine the minimum value of $(x^2 + y^2)^2$. Provide your answer as a single number.

Answer: 18


Question: Find the Laurent series expansion of \( f(z)=\frac{ z }{ z^2-z-2 } \) in the annulus centered at \(0\) and containing the point \(2+i\). Determine the radius of convergence of this series.

Answer: 2


Question: Find the limit of \( \frac{\varphi(b_n)}{b_n} \) as \( n \to \infty \), where \( b_n \) is the product of the first \( n \) odd prime numbers.

Answer: 0


Question: Evaluate the limit: $$\lim_{h\to 0}\frac{g(5+h\sin 2h)-g(5+2h\cos h)}{h\cos h-2h\sin 2h}$$ where $g(x)$ is a differentiable function with $g'(x)$ continuous on $(-\infty,\infty)$ and $g'(5) = -3$.

Answer: 6


Question:
Find the pair \((n, m)\) of natural numbers such that:
\[\begin{cases}13n-9m=110\\\text{lcm}(n,m)-\text{gcd}(n,m)=3n+12\end{cases}\]

Answer: (14, 8)


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

Answer: \dfrac{9}{4}


Question:
Evaluate the series expression for \( \sin(i) \) given by:

\[ \sin(i) = i\sum_{j=0}^\infty \frac{1}{(2j+1)!} \]

What does this series converge to?

Answer: i \sinh(1)


Question: Evaluate \( \lim_{x \to 0^+}\left(\frac{x}{\tan x}\right)^{1/x^3} \).

Answer: 0


Question: Determine the sum of the infinite series: \[ S = 1 + \frac{2}{3} \cdot \frac{1}{2} + \frac{2\cdot5}{3\cdot6} \cdot \frac{1}{2^2} + \frac{2\cdot5\cdot8}{3\cdot6\cdot9} \cdot \frac{1}{2^3} + \ldots \] Choose the correct answer from the following options: \( \text{(A) } 2^{1/3}, \text{(B) } 2^{2/3}, \text{(C) } 3^{1/2}, \text{(D) } 3^{3/2} \).

Answer: B


Question: Let $f$ be a function and $V_0$ and $W_0$ be open sets. Suppose $f$ is an isometry on both $V_0$ and $W_0$, and for some $x_0 \in V_0$ and $y_0 \in W_0$, it holds that $d(x_0, y_0) = d(f(x_0), f(y_0))$. Does it follow that $f$ is an isometry on $V_0 \cup W_0$?

Answer: No


Question: Find the limit of the sequence \(a_n = \frac{1}{(n+1)^2} + \frac{1}{(n+2)^2} + \ldots + \frac{1}{(2n)^2}\) using the Stolz-Cesaro theorem.

Answer: 0


Question: Find the area of an equilateral triangle $\triangle ABC$ if there is a point $I$ inside the triangle such that $IA = 8$, $IB = 10$, and $IC = 6$. Express your answer in simplest form.

Answer: 36 + 25\sqrt{3}


Question: Determine whether the extension $\mathbb{Q}(\alpha):\mathbb{Q}$, where $\alpha=e^{2\pi i/3}$, is a normal extension given that it has degree 2.

Answer: Yes


Question: In a normed linear space $F$ with a norm that is not strictly convex, and assuming $F$ contains more than just the zero element, determine if there always exist distinct vectors $f_1$ and $f_2$ in $F$ such that $\lVert f_1 \rVert = \lVert f_2 \rVert = 1$ and $\lVert f_1 + f_2 \rVert = 2$. Provide a justification for your answer.

Answer: Yes


Question: Consider the recurrent sequence $(x_n)_{n \ge 0}$ defined by $x_0 \in \mathbb{R}$ and $x_{n+1} = 2^{\frac{x_n}{2}}$. Determine the interval of initial values $x_0$ for which $\lim\limits_{n \to \infty} x_n = 2$. Choose from the following options: A. $x_0 \in \{2\}$ B. $x_0 \in [-2, 2]$ C. $x_0 \in (-\infty, 2]$ D. $x_0 \in [2, 4)$ E. Other answer.

Answer: E


Question: Find the minimum value of the function \( h(u) = \sqrt[3]{\frac{3u^2+1+u^3+3u}{8}} + \sqrt[3]{\frac{3u^2+1-u^3-3u}{8}} \) for \( u \ge \frac{1}{2} \).

Answer: 1


Question: Find the values of natural numbers $m$ and $n$ such that the limit \( \lim_{(x,y) \to (0,0)} \frac{x^m y^n}{x^4 + y^4} \) exists.

Answer: m + n \geq 5


Question: Determine whether the following statement is true or false: For two disjoint bounded open sets $\Omega_{1}$ and $\Omega_{2}$ in $\mathbb{R}^{N}$, can $H_{0}^{1}(\Omega_{1} \cup \Omega_{2})$ be expressed as $\{ u+v \mid u \in H_{0}^{1}(\Omega_{1}), v \in H_{0}^{1}(\Omega_{2}) \}$?

Answer: True


Question: Verify the Gauss-Divergence theorem for the vector field \( \mathbf{F} = 4x\mathbf{i} - 2y\mathbf{j} + z\mathbf{k} \) over the region bounded by the surfaces \( r = 4 \), \( z = -2 \), and \( z = 2 \). Here, \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are unit vector directions.

Answer: 192\pi


Question: Suppose $V$ is an analytic variety of an open subset $U\subset \mathbb{C}^n$. Let $f:U\setminus V\rightarrow \mathbb{C}$ be a holomorphic function that is $L^2$-bounded in $U$. Determine whether there exists a unique holomorphic extension of $f$ to all of $U$. Provide a justification for your answer.

Answer: Yes


Question: Suppose $f$ is a real-valued function defined on $\mathbb{R}$, and for any $x \in \mathbb{R}$, the limit \(\lim _{n \rightarrow \infty}\left[f\left(x+a_{n}\right)-f\left(x-a_{n}\right)\right]=0\) holds for all sequences \(\{a_{n}\}\) converging to $0$. Is $f$ necessarily continuous at $x$? Provide a justification for your answer.

Answer: No


Question: Determine the cardinality of the set $F$ of all entire functions $f$ such that $f(x) = |x^k|$ for all $x \in (-1, 1)$, where $K$ is an odd integer $\geq 1$. What is the cardinality of $F$?

Answer: 0


Question: Does every smooth curve have a length?

Answer: Yes


Question: Let $p$ be a prime number greater than $3$. Determine the greatest common divisor of $p!$ and $(p-3)! - 1$. Express your answer as a single integer.

Answer: 1


Question: Calculate the limit \(\lim\limits_{x\to\infty} (a^x+b^x-c^x)^{\frac{1}{x}}\) where \(a>b>c>0\).

Answer: a


Question: Determine the number of possible rays that satisfy Snell's law in the lifeguard problem, given only the angle between the line connecting the two people and the normal.

Answer: 1


Question: Evaluate the limit: $$ \lim_{x \to \infty} \frac{\gamma + \log_2(x + \gamma)}{\gamma + \sqrt{\gamma + x} - \sqrt{x}} $$

Answer: \infty


Question: Calculate the value of the integral \( \int_0^2 x f(x) \, dx \), where \( f(x) = \int_x^2 \frac{dy}{\sqrt{1+y^3}} \).

Answer: \dfrac{2}{3}


Question: Evaluate the limit \(\lim_{M\to\infty} \frac{1}{2M+3} \sum_{m=-M}^{M} \left( \frac{1+\sin\left(\frac{\pi}{4}m \right)}{2}\right).\)

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( I = \oint_C \frac{\cos z}{z(z-\pi)} \, dz \), where the contour \( C \) is the circle centered at \((1,0)\) with a radius of 2.

Answer: -2i


Question: Find an upper bound for the series \( \sum_{i=1}^{n}\left(\frac{n}{i}\right)^{i} \).

Answer: n e^{n/e}


Question: Consider the intervals \((p_1, p_2)\) and \((p_1+b, p_2+b)\), where \(p_i, b \in \mathbb{R}\). Determine if the Lebesgue measure of \((p_1, p_2) \cap \mathbb{I}\) is equal to that of \((p_1+b, p_2+b) \cap \mathbb{I}\).

Answer: Yes


Question: What is the value of the partition function \( p(100) \), which represents the number of partitions of the positive integer 100?

Answer: 190569292


Question: Evaluate the double integral \( \int\int_{D} \sin(x) \, dA \), where \( D \) is the unit circle centered at the origin \((0,0)\).

Answer: 0


Question: Consider several fractions \(\frac{a_1}{b_1}, \frac{a_2}{b_2}, \ldots, \frac{a_n}{b_n}\) where \(a_i < b_i\) for \(1 \leq i \leq n\), and all \(a_i > 0\) and \(b_i > 0\). The mediant of these fractions is \(M = \frac{a_1 + a_2 + \cdots + a_n}{b_1 + b_2 + \cdots + b_n}\). The weighted mean of the fractions with respect to the numerator value is \(A = w_1\frac{a_1}{b_1} + w_2\frac{a_2}{b_2} + \cdots + w_n\frac{a_n}{b_n}\), where \(w_i = \frac{a_i}{\sum_{i=1}^{n} a_i}\). Prove or disprove that \(A \geq M\).

Answer: A \geq M


Question: Given that real numbers $a, b, p, q$ satisfy $a^2 + b^2 = p^2 + q^2 = 2$, find the maximum value of $E = (1-a)(1-b) + (1-p)(1-q)$. Provide your answer as a single number.

Answer: 8


Question: For a Poisson random variable $X \sim Pois(\lambda)$, determine the expected value $E(2^X)$ and specify the condition under which this expectation is finite.

Answer: e^{\lambda}


Question: Determine whether the algebraic structure \((N,+)\), where \(N\) is the set of natural numbers defined by the Peano axioms, is an ordered commutative monoid.

Answer: Yes


Question: Compute the distributional Fourier transform of $x \delta(x)$. Provide your answer in terms of standard distributional operations.

Answer: 0


Question: Calculate the residue of the function \( \frac{\sin z}{z^{2}(\pi-z)} \) at \( z=0 \).

Answer: \dfrac{1}{\pi}


Question: Find the smallest positive value of the expression \(a^3 + b^3 + c^3 - 3abc\) for positive integers \(a, b, c\). List all integer tuples \((a, b, c)\) that achieve this minimum value.

Answer: 4


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

Answer: \dfrac{4}{e}


Question: Find the least positive integer \( n \) such that \( 79^{79} \equiv n \pmod{100} \) using the Chinese Remainder Theorem.

Answer: 19


Question: Evaluate the limit: \[ \lim_{n\to \infty} \int_0^n \left(1+\frac{x}{n}\right)^{-n}\log\left(2+\cos\left(\frac{x}{n}\right)\right)\,dx \]

Answer: \log 3


Question: Evaluate the limit \(\lim_{(x,y)\to (0,0)}x\ln(xy)\) along the path defined by \(y = e^{-\frac{1}{x^2}}\).

Answer: -\infty


Question: Find the smallest positive integer \( n \) such that \( 2^n \equiv 1 \pmod{5\cdot 7\cdot 9\cdot 11\cdot 13} \).

Answer: 60


Question: Find the value of \( \lim \limits_{t \to \infty} y(t) \) for the solution of the differential equation \( y'(t) = -y^3 + y^2 + 2y \) with the initial condition \( y(0) = y_0 \), where \( y_0 \) is between -1 and 0.

Answer: -1


Question: Does a set $A$ of natural numbers that has arbitrarily long arithmetic progressions necessarily have positive Banach density?

Answer: No


Question: Given a continuous function $f(x): \mathbb{R} \to \mathbb{R}$, consider the sequence of integrals defined as $a_n = \int_{-\frac{1}{n}}^{\frac{1}{n}} f(x) \,dx$ for all $n > 0$. Determine whether the limit of $a_n$ as $n$ approaches infinity converges to zero.

Answer: 0


Question: Find all pairs of positive integers \((a, b)\) such that \(a - b \ge 2\) and \(p_a - p_b \mid 2(a - b)\), where \(p_n\) denotes the \(n\)-th prime number starting from \(p_1 = 2\).

Answer: (4, 2)


Question: Let $f:C[-1,1]\to C[-1,1]$ be defined by $f(x)=\int_{-1}^0 x(t)\,dt -\int_0^1x(t)\,dt$. Find $\Vert f\Vert$ with respect to the sup norm defined on $C[-1,1]$. \(\Vert f\Vert = \sup_{\Vert x \Vert = 1} \Vert f(x) \Vert \).

Answer: 2


Question: Evaluate the limit: \( \lim_{x\to0^-} (\sin(x))^{\cot^2(x) } \)

Answer: 0


Question: Find a function \( g: \mathbb{N} \to \mathbb{P}(\mathbb{N}) \) such that the set \( A = \{ n \in \mathbb{N} \mid n \notin g(n) \} \) is finite.

Answer: g(n) = \mathbb{N}


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

Answer: 1


Question: Determine the point on the paraboloid \( z = \frac{x^2}{4} + \frac{y^2}{25} \) that is closest to the point \((3, 0, 0)\).

Answer: (2, 0, 1)


Question: Determine the number of distinct triplets \((a, b, c)\) of non-negative integers that satisfy the equation \(3^{a} + 8^{b} = 11^{c}\).

Answer: 1


Question: Evaluate the integral \( \int _c^d g^{'}(x)g^{''}(x) \, \mathrm{d}x \) given that the tangent to the graph of the function \( y=g(x) \) at the points with abscissas \( x=c \) and \( x=d \) forms an angle of \( \frac{\pi}{3} \) with the \( x \)-axis.

Answer: 0


Question: In triangle $ABC$ inscribed in a circle, versines are drawn from the midpoints of the sides, perpendicular to them, with lengths $\alpha=16$, $\beta=20$, and $\gamma=13$. Find the area of triangle $ABC$. \( \text{(Express your answer in square units.)} \)

Answer: 1344


Question: Evaluate the integral: $$\int_{-\infty }^{\infty } \frac{-1+e^{-i k x}}{x^2} \, dx$$

Answer: -\pi |k|


Question: Evaluate the limit: $$\lim_{n\to \infty} \frac{1}{n^4}\left(\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+\cdots+\frac{n}{a_n}\right)$$ where $a_1=1$ and $a_{n+1}=\dfrac{a_n}{1+na_n}$.

Answer: \dfrac{1}{8}


Question: Does the proof of the rank-nullity theorem in Lang's "Linear Algebra" book require the axiom of choice for finite sets? Answer yes or no.

Answer: No


Question: Determine whether the function \( y=\sqrt{x^4-x^2+1} \) has any branch points within a sufficiently large circle \( C \) centered at the origin in the complex plane.

Answer: Yes


Question: Determine the number of roots of the polynomial \(z^4 - 8z + 10\) that lie in the annulus defined by \(1 < |z| < 3\).

Answer: 4


Question: Identify the group defined by the presentation \( \langle x, y: xy=yx, x^5=y^3 \rangle \).

Answer: \mathbb{Z}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sqrt{n}(\sqrt[n]{x} - 1) \] where \( x \) is a positive constant.

Answer: 0


Question: Determine the constant $P$ such that the function $f(x,y) = \frac{3}{\sqrt{\pi}}\int_{0}^\frac{y}{\sqrt{x}} e^{-t^2}dt$ satisfies the equation $$\frac{\partial f}{\partial x} = P\frac{\partial^2f}{\partial y^2}.$$

Answer: \dfrac{1}{4}


Question: Determine the period of the function \( f(x) = \sum_{k=1}^{n}(\cos (kx)+\sin(kx)) \) for \( n \in \mathbb{N} \) and \( n < \infty \).

Answer: 2\pi


Question: Convert the double integral \( I = \int_1^2 \int_0^\sqrt{1-(1-x)^2} \dfrac{x}{x^2+y^2} \ \mathrm dy\ \mathrm dx \) from Cartesian coordinates to polar coordinates and evaluate the integral.

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true or false: If all eigenvalues of the Jacobian matrix \(\frac{\partial f}{\partial x}(x_0)\) have negative real parts, then the equilibrium point \(x_0\) of the autonomous ordinary differential equation \(\dot{x} = f(x)\) is exponentially stable.

Answer: True


Question: Given a square matrix $A$ that is not a scalar multiple of the identity matrix $I$, determine whether there always exists an invertible matrix $B$ such that the element in the first row and first column of the matrix $BAB^{-1}$ is zero.

Answer: Yes


Question: Given that $f(x)$ is defined over $(-\infty, \infty)$ and its Fourier transform $g(w)$ is approximately $1-w^2$ for small $w$ ($w<<1$), can the value of $$\int_{-\infty}^\infty x^2f(x) \,dx$$ be calculated?

Answer: 2


Question: Find the minimum positive root of the equation \(\sin{3x}=a\), where \(a>0\), given that there exist two roots with a difference of \(\frac{\pi}{4}\).

Answer: \dfrac{\pi}{24}


Question: Consider the differential form \( a = p_1 dq_1 + p_2 dq_2 - p_1 p_2 dt \) in the space \( \mathbb{R}^5 \) with coordinates \((p_1, p_2, q_1, q_2, t)\). Compute \( da \) and \( da \wedge da \). Evaluate the integral \( \int_S t \, da \wedge da \) where \( S \) is the 4-dimensional surface (with boundary) in \( \mathbb{R}^5 \) defined by the relations: \( p_1^2 + p_2^2 \le 1, \ q_1^2 + q_2^2 \le 2 \) and \( t = 3 \).

Answer: -12\pi^2


Question: Determine whether it is possible to construct a sequence of continuous, non-negative functions $f_n: H \to \mathbb{R}$, for $n \in \mathbb{N}$, such that $f_n$ converges pointwise to $\mathbf{1}_U$ and $f_n \in C_b(\mathbb{R}^d)$ for each $n$, where $d=d_n$ is allowed to depend on $n$. Provide a justification for your answer.

Answer: Yes


Question: Let $Y \sim N(0,\sigma)$. The cumulative distribution function (CDF) of $Y$ is given by $F_Y(y) = \Pr(Y < y)$. What is the expectation of $F_Y(Y)$?

Answer: \dfrac{1}{2}


Question: Let $X_1, X_2, \ldots, X_n$ be iid random variables from a continuous distribution whose density is symmetric about $0$. Suppose $\mathbb{E}(|X_1|) = 2$. Let $Y = \sum_{i=1}^{n} X_i$ and $W = \sum_{i=1}^{n} |X_i|$. Calculate the covariance between $Y$ and $W$. \( \text{Cov}(Y, W) = ? \)

Answer: 0


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

Answer: No


Question: Find all functions $f(x)$ such that \( \int_{a}^{b} (f(x) - 3x) \, dx = a^2 - b^2 \) for all real numbers $a$ and $b$. \( \)

Answer: x


Question: Given a field extension $K \subseteq F$, where $K(\alpha)/K$ is an extension of degree $4$ and $\alpha^2$ is not a root of the minimal polynomial of $\alpha$ over $K$, calculate the degree of the extension $[K(\alpha^2):K]$. Provide your answer as an integer.

Answer: 2


Question:
Consider the space $C^{\infty}_0$, which consists of smooth functions with compact support. Define the space $C^{\infty,2}_0$ as the direct sum $C^{\infty}_0 \oplus C^{\infty}_0$. An inner product $(\cdot , \cdot)_0$ is given by:
$$ (f,g)_0 = \sum^n_{i=1} (D_i f_1, D_i g_1) + M^2 (f_1,g_1) + (f_2, g_2) $$
where $f,g \in C^{\infty,2}_0$, $f_i$ are the components of $f$, $M^2$ is a positive constant, and $(\cdot, \cdot)$ denotes the $L^2$ scalar product on $\mathbb{R}^n$. Determine if the completion of $C^{\infty,2}_0$ with respect to the induced norm is a subset of $L^2(\mathbb{R}^n)^2$. Provide a justification for your answer.

Answer: Yes


Question: Consider an infinite matrix product \(\prod_{i=1}^\infty (U_i+A_i)\), where each \(\Vert U_i \Vert =1\), the product \(\prod_{i=r}^\infty U_i\) converges for all \(r\), and \(\sum_i \Vert A_i \Vert <\infty\). Does this imply that \(\prod_{i=m}^n (U_i+A_i) \xrightarrow{m,n\to\infty} I\)?

Answer: Yes


Question: Given a function $f(x)$ that is twice differentiable on the interval $[1,5]$ and satisfies the conditions $f(1) = f(5) = 0$ and $f'(x) = f(x) - f''(x)$ for $x \in [1,5]$, find the value of $(f(2) + f(4) - f'(3))$. \( \boxed{} \)

Answer: 0


Question: Determine whether the sequence \(\left(\frac{1}{n}\right)_{n=1}^\infty\) converges to 0 effectively. Use the definition of effective convergence: There exists a computable function \(e:\mathbb{N}\to \mathbb{N}\) such that for any \(N,\) \(k\geq e(N)\) implies \(|\frac{1}{k}|<2^{-N}.\)

Answer: Yes


Question: You have $n^2$ distinct balls, each of which can be colored in $n^2$ different ways. You randomly select $n^b$ balls without replacement. Determine the asymptotic probability that at least $n^a$ balls have the same color, where $0 < a < b \leq 2$. Provide your answer in terms of $n$.

Answer: 0


Question: Can the factorial function be extended to non-integer arguments in more than one way?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to\infty}-x \cdot \tan^{-1} \left( \dfrac1{2x+3} \right) \]

Answer: -\dfrac{1}{2}


Question: In a graph with multiple blocks, if every pair of blocks intersects at a single vertex, does this imply that the vertex is a cut-vertex? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the definite integral \( \int\limits_{1/2}^{2}\frac{1}{x}\sin \left(x-\frac{1}{x}\right)dx \).

Answer: 0


Question: Evaluate the integral \( \int_{-4}^4 \{nx\} \, dx \), where \( n \in \mathbb{R} \) and \( \{\cdot\} \) denotes the fractional part function.

Answer: 4


Question: Determine the values of $x \ge 0$ for which the function \( f(x) = \lim_{t \to \infty} \frac{x^t}{x^t - 1} \) is discontinuous.

Answer: 1


Question: Are $X^2$ and $Y$ independent if $X$ and $Y$ are independent random variables?

Answer: Yes


Question: Provide an example of a polynomial \( f(x) \in \mathbb{Z}[x] \) that has a root in every finite field \( \mathbb{F}_p \), but no root in \( \mathbb{Z} \).

Answer: (x^2 - 2)(x^2 - 3)(x^2 - 6)


Question:
Let $a, b, c \in \mathbb{R}$ satisfy the following equations:
\[ a^2 + b^2 + c^2 = 2018 \]
\[ 2a + 2b - c = 15 \]
\[ 2b + 2c - a = 36 \]
Determine the maximum value of the expression $2c + 2a - b$. Provide your answer as a single number.

Answer: 129


Question: Given that \( a + b + c = 0 \) and \( l^2 = \frac{a^2}{2a^2+bc} + \frac{b^2}{2b^2+ac} + \frac{c^2}{2c^2+ab} \), where \( a, b, \) and \( c \) are real numbers, find the value of \( l \).

Answer: 1


Question: Evaluate the limit \( \lim_{x\rightarrow 0}x\cdot \sum_{k=1}^{15}\left\lfloor \frac{k}{x}\right\rfloor \) by expressing the floor function in terms of its integer and fractional parts.

Answer: 120


Question: Determine the number of ways to express the number 2160 as a product of factors that are relatively prime to each other.

Answer: 5


Question: Consider the differential equation \( y'(x) = y^2(x) + x \). Determine the number of times the solution \( y(x) \) is differentiable.

Answer: \infty


Question: Given the constraints $0 \le x \le 3$ and $0 \le y \le 3$, find the value of $x+y$ such that the equation \( y^2 + 4 + 3\cos(xy + y) = 2y \) has at least one real solution.

Answer: \pi


Question: In a Banach space $\mathfrak{X}$, it is known that weak-* convergent sequences in $\mathfrak{X}^*$ are norm bounded. Determine whether a weak-* convergent net in $\mathfrak{X}^*$ must also be eventually norm bounded. Specifically, is there a point in the net's domain beyond which all elements are norm bounded?

Answer: Yes


Question: Determine the values of $k$ for which the improper integral \( \int_{2}^{\infty} \frac{1}{x^k\ln(x)}\,dx \) converges.

Answer: k > 1


Question: Determine if the set of bijections that map a subset Y of a set X onto itself is a subgroup of the group of all bijections from X to X.

Answer: Yes


Question: Evaluate the integral: $$\int_{\frac{1}{e}}^{\tan x}\frac{t}{1+t^2}dt + \int_{\frac{1}{e}}^{\cot x}\frac{1}{t(1+t^2)}dt$$

Answer: 1


Question: Suppose \( f: \mathbb{C} \setminus \{1\} \rightarrow \mathbb{C} \) is a holomorphic function. If the integral of \( f \) on a positively oriented circle of radius 2 around 0 is \( 4\pi i \), find the value of the integral of \( f \) on a positively oriented circle of radius 1 around 1.

Answer: 4\pi i


Question: Let $(u_n)$ be a sequence that satisfies the following recurrence relation: $u_1 = -2, u_2 = -1,$ and for $n \ge 3$, $u_{n+1} = \sqrt[3]{n(u_n^2 + 1) + 2u_{n-1}}$. Find the value of $u_{2021}$.

Answer: 2018


Question: If $x$ in $\mathbb{R}^d$ lies on the boundary of the convex hull of a set $P$, can $x$ be expressed as a convex combination of $d$ points in $P$? Provide a justification for your answer.

Answer: Yes


Question: Let $p$ be an odd prime and $n \geq 1$. Determine the number of quadratic subextensions of the cyclotomic extension $\mathbb{Q}(\zeta_{p^n})$. Provide your answer as a single integer.

Answer: 1


Question: Determine whether there exists a sequence of integers \((a_n)_{n \in \mathbb{N}}\) such that for some integer \(n_0\), the inequality \( \frac{1}{4} < \frac{a_n}{n} < \frac{1}{2} \) holds for all \(n \ge n_0\).

Answer: Yes


Question: Evaluate the limit: $$\lim_{y \to \infty} \frac{y\log(e^{y/6}+8/y)}{2}. $$

Answer: \infty


Question: Determine whether the ordinary chain rule holds for the product of an Ito process $X(t)$ and a non-stochastic function $p(t)$, such that $d(X(t)p(t)) = dX(t)p(t) + X(t)p'(t)dt$. Provide a justification for your answer.

Answer: Yes


Question: Consider sets $E_1, E_2, \dots, E_n$, each containing 4 elements. Suppose each element in the set $\{1, 2, \dots, k\}$ appears in at most 3 of these sets. If we select one element from each $E_i$, can we ensure that the set of chosen elements contains at most $\frac{3k}{5}$ distinct members?

Answer: \dfrac{3k}{5}


Question: Given the polynomial \( P(x) = x^3 + mx^2 - 3x + 1 \) where \( m \in \mathbb{R} \), find the expression for \( x_1^3 + x_2^3 + x_3^3 \) in terms of \( m \), where \( x_1, x_2, x_3 \) are the roots of the polynomial.

Answer: -m^3 - 9m - 3


Question: Let $X$ be a second countable, locally compact $T_1$ topological space with a compact dense subset $Y$. Is $X$ necessarily compact? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit \( f(x) = \lim_{n \to \infty} \left(n \int_0^{\frac{\pi}{4}} (\tan x)^n \, dx\right) \) using appropriate mathematical techniques.

Answer: \dfrac{1}{2}


Question: Suppose you have a vector space $V$ with a basis $B = \{ \vec{v}_i \}$. Is it always possible to define an inner product $I$ on $V$ such that the basis $B$ becomes orthonormal, i.e., $I(\vec{v}_i, \vec{v}_j) = \delta_{ij}$ for all $\vec{v}_i, \vec{v}_j \in B$?

Answer: Yes


Question: Is a subset $A$ of $\mathbb{R}^n$ necessarily a closed subset if it is homeomorphic to the closed unit disk $D^m = \{x \in \mathbb{R}^{m+1} : ||x|| \leq 1 \}$?

Answer: Yes


Question: Evaluate the contour integral \( \displaystyle \oint_{C} \frac{1}{z^{3} + 9 z} \, d{z} \), where \( C \) is the circle \( \{ z \in \mathbb{C} \,|\, |z| = 4 \} \) oriented counterclockwise.

Answer: 0


Question: Compute the differential form $d\beta$ for the two-form $\beta = z(y^2-x^2)\,dx\,dy - x^2y\,dx\,dz - xy^2\,dy\,dz$ on three-dimensional real space.

Answer: 0


Question: Calculate the volume of the solid generated by rotating the region bounded by \(y=x^2\) and \(y=-x\) around the line \(y=1\) using the method of slicing.

Answer: \dfrac{\pi}{5}


Question: Determine the value of $10N$, where $N$ is the number of integers in the range of the function $f: \mathbb{R} \to \mathbb{R}$ defined by \[ f(x) = \frac{3 + 2\sin x}{\sqrt{1 + \cos x} + \sqrt{1 - \cos x}}. \]

Answer: 20


Question: Find the minimum value of the function \( G(a,b,c) = \frac{b}{a+c}+\frac{c}{b+a}+\frac{a}{2a+5c} \) for \( a, b, c \in \left [ 1, 4 \right ] \).

Answer: \dfrac{46}{65}


Question: Determine the number of real solutions for the equation \( \cos(\cos(\cos(\cos(x)))) = \sin(\sin(\sin(\sin(x)))) \).

Answer: 0


Question: Determine the range of $p \in (0, \infty)$ for which the function $\frac{1}{1-z}$ belongs to the Hardy space $H^p(\mathbb{D})$. Justify your answer.

Answer: (0, 1)


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

Answer: 0


Question:
Evaluate the limit of the following integral as $n$ approaches infinity:
$$\lim_{n\to +\infty}\int_{(0,1)^n}\frac{\sum_{j=1}^n x_j^2}{\sum_{j=1}^n x_j}d\mu$$

Answer: \dfrac{2}{3}


Question: Evaluate the limit: $$\lim_{x \to 1}\frac{\Gamma(1+x)\Gamma(2+x)\Gamma(3+x) - 12}{x-1}$$ without using L'Hôpital's rule.

Answer: 52 - 36\gamma


Question:
Evaluate the limit as \( n \to \infty \) of the expression:
\[
\lim_{n \to \infty} \frac{\sum_{k=0}^n \binom{2n}{2k}3^k} {\sum_{k=0}^{n-1} \binom{2n}{2k+1}3^k}
\]

Answer: \sqrt{3}


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{3} +\frac{1}{4}+\ldots+\frac{1}{n}}{1+\frac{1}{3} +\frac{1}{5}+\frac{1}{7}+\ldots+\frac{1}{2n+1}} \]

Answer: 2


Question: Determine whether the expression \(a = \frac{36^{36} + 41^{41}}{77}\) is an integer, given that the last digit of \(41^{41}\) is 1 and the last digit of \(36^{36}\) is 6.

Answer: Yes


Question: Determine the number of ways to arrange 10 men and 4 women around a circular table such that no three women sit consecutively.

Answer: 5356108800


Question: Evaluate the double integral \( \int_F \arctan\left(\frac{x}{y}\right)\ dx\ dy, \) where \( F \) is the region in the \( xy \)-plane bounded by the curves \( x^2+y^2=1 \), \( x^2+y^2=4 \), \( y=-|x| \), and \( y=|x| \).

Answer: 0


Question: Does the existence of a continuous and differentiable function \(f:[a,b]\longrightarrow \mathbb{R}\) with \(f(b)=f(a)\) on the interval \(]a,b[\) imply the existence of a local extremum?

Answer: Yes


Question: Find the limit of the expression \( \lim_{h \to 0} \frac{\sin\frac{1}{h}}{\frac{1}{h}} \).

Answer: 0


Question: Suppose $X$ is a subset of $\mathbb{R}$. If there exists a continuous function $f: X \to \mathbb{R}$ that is not uniformly continuous, does it imply that $X$ is not closed? Provide a justification for your answer.

Answer: No


Question: Given that the norm of the operator $T$ is 1, determine the norm of the iterated operator $T^2$.

Answer: 1


Question: Determine whether an odd perfect number can be divisible by 165.

Answer: No


Question: Evaluate the limit of the function \( g(z) \sim e^{-z^2} \) as \( z \to \infty \) along the line \( \theta = \pi/8 \).

Answer: 0


Question: Find the number of unitary ring homomorphisms from the ring \( \mathbb{Z}[X]/(X^3-1) \) to \( \mathbb{Z}_{12} \).

Answer: 1


Question: Find the MacLaurin polynomial of degree 7 for the function \( F(x) = \int_0^x \sin(6t^2) \, dt \). Use this polynomial to estimate the value of \( \int_0^{0.75} \sin(6x^2)\,dx \).

Answer: 0.1573


Question: Find the Dirichlet density of the set of primes of the form \( p = 2n^2 + 1 \).

Answer: 0


Question: Find the sum of the real roots of the equation \( \frac{3x^2 - 9x + 17}{x^2 + 3x + 10} = \frac{5x^2 - 7x + 19}{3x^2 + 5x + 12} \).

Answer: 6


Question: Find the order of the quotient group \( \mathbb{Z}_7[X] / (X^3 + 2) \).

Answer: 343


Question: Given a floating-point number $x$ with $e$ bits for the exponent and $m$ bits for the mantissa, and an elementary function $f$ with domain and range in the interval $[1,2)$, consider an algorithm $\psi$ that approximates $f$ in the same floating-point system. Assuming that $f(x)$ is not a floating-point number, determine the minimum accuracy achievable with $\psi$. Express your answer in terms of $e$ and $m$.

Answer: 2^{-(m+1)}


Question: Evaluate the integral \( \oint_C \frac{z^{m-1}}{e^{z/u}} \, du \) using the residue theorem, where \( C \) is a contour containing the point \( u = 0 \).

Answer: -2\pi i z^m


Question: Find a function $g(z)$ defined as a power series $$g(z)=\sum_{n=0}^{\infty}{b_{n}z^{n}}$$ with a positive radius of convergence, such that $g(0)=1$ and $g'(z)=g(z)+1$ for all $z\in\mathbb{C}$.

Answer: 2e^z - 1


Question: Given real numbers $a \geq b \geq c \geq d > 0$ such that $a + b + c + d = 9$ and $a^2 + b^2 + c^2 + d^2 = 21$, determine the minimum value of $a + b$. Express your answer as a single number.

Answer: 5


Question: Find the radius of convergence for the power series \( \sum_{n=0}^{\infty}{(2n)!x^{(2n)!}} \).

Answer: 1


Question: Determine the average number of coin flips required such that the difference between the percentage of heads and tails is less than 0.1% with a probability of at least 99%.

Answer: 6635776


Question:
Let $f$ be a continuous function on $[2,\infty)$ and \(\displaystyle\lim_{x\to \infty}x(\log x)^pf(x)=A\) , where $A$ is a non-zero finite number. Determine the convergence of \(\displaystyle\int_2^{\infty}f(x)\,dx\) based on the value of $p$. Choose the correct option:

(A) convergent for all $p$.
(B) convergent for $p>1$.
(C) convergent for $0<p<1$.
(D) always convergent.

Answer: B


Question: Suppose a group $G$ has an involution. Does $G$ have an odd number of involutions?

Answer: Yes


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

Answer: e - 2


Question: Determine whether the following statement is true or false: If $g \in L_1(X)$, then for every $\epsilon>0$, there exists a set $E$ with $m(E)<\infty$, $\sup_E(|g|)<\infty$, and $\int_{E^c} |g| < \epsilon$. Provide a justification for your answer.

Answer: True


Question: Evaluate the limit: $$\lim\limits_{z \to 0}\left(\dfrac{1-\cos z}{z}\right)^{1/z^2}$$

Answer: 0


Question:
Evaluate the limit of the infinite product as \( n \) approaches infinity:

\[
\lim_{n \to \infty} \left(1 - \frac{2}{3}\right)^{\frac{3}{n}} \left(1 - \frac{2}{4}\right)^{\frac{4}{n}} \left(1 - \frac{2}{5}\right)^{\frac{5}{n}} \ldots \left(1 - \frac{2}{n+2}\right)^{\frac{n+2}{n}}
\]

Answer: e^{-2}


Question: Determine the asymptotic behavior of the series \( \sum_{n=2}^\infty\left(\frac{2}{n}\right)^\sigma \) as \( \sigma \to \infty \).

Answer: 1


Question:
Determine the \(\hat{r}\) component of the following integral in cylindrical coordinates without performing the integration:

$$\int \limits_0^{2\pi} \int \limits_0^a \frac{r^2}{(r^2+h^2)^{3/2}} \hat{r} \, dr \, d\theta$$

Answer: 0


Question: Find the constant $c$ such that the random variable \( Y = \frac{c(X_1 + X_2)}{\sqrt{(X_3)^2 + (X_4)^2}} \) follows a t-distribution, given that $X_1$, $X_2$, $X_3$, and $X_4$ are independent standard normally distributed random variables.

Answer: 1


Question: Determine the cardinality of the set \( Z = \{ \{z_n\} : z_n \in \{0,1\} \text{ and } z_n = 0 \text{ for at most finitely many } n \} \).

Answer: \aleph_0


Question: Determine whether there can exist an uncountable collection of infinite subsets of a countable set, such that none of these subsets is a subset of another.

Answer: Yes


Question: Calculate the determinant of the matrix \(A_{2n}:=\begin{pmatrix} a &0 &\cdots & \cdots &0 &b \\ 0& a & & & b&0 \\ \vdots &\vdots &\ddots &\ddots &\vdots &\vdots \\ \vdots&\vdots & \ddots & \ddots &\vdots &\vdots \\ 0& b & & & a&0 \\ b&0 &\cdots &\cdots &0 &a \end{pmatrix}\in \mathbb{R}^{2n \times 2n}\).

Answer: (a^2 - b^2)^n


Question: In a triangle, the angles satisfy the conditions $\cos A + \cos B + \cos C = 0$ and $\sin A + \sin B + \sin C = 0$. Determine the value of $\sin^3A + \sin^3B + \sin^3C$. Express your answer as a single number.

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \frac{\cos x}{1+x^2} \, dx \) using the residue theorem.

Answer: \dfrac{\pi}{e}


Question: Let $([0, 1], B, m)$ be a probability space where $B$ is the Borel σ-algebra and $m$ is the Lebesgue measure on $[0, 1]$. Construct a sequence of random variables $\{X_i\}_{i \in \mathbb{N}}$ on this probability space such that: (a) for each $i$, $X_i$ takes values $0$ and $1$ with equal probabilities, and (b) for each pair $i \neq j$, $X_i$ and $X_j$ are independent random variables. Using $\{X_i\}$, define a random variable $Y = \sum_{i \geq 1} \frac{X_{2i}}{2^i}$. Determine the distribution of $Y$. \(\text{Express your answer in terms of a known distribution or provide a probability density function.}\)

Answer: f_Y(y) = 1


Question: Does there exist a mathematical structure known as a "silly-ring," which is a near-ring that satisfies two-sided distributivity but has a non-Abelian underlying additive group?

Answer: Yes


Question: Determine the largest possible value of $f(4)$ for a continuously differentiable function $f: \mathbb{R} \to \mathbb{R}$ that satisfies $f(0) = 0$ and $f(x)f'(x) \le 2$ for all $x \in \mathbb{R}$. Justify why this value cannot be exceeded and provide an example of a function that achieves this maximum value.

Answer: 4


Question: Evaluate the integral \( \int_{|z-1|=2} \frac{\sin z}{z^2-z} \, dz \) using partial fractions and Cauchy's Integral Theorem.

Answer: 2\pi i \sin 1


Question: Suppose $S$ and $T$ are linear maps from a vector space $V$ to itself such that $\text{Ker}(S) = \text{Ker}(T)$ and $\text{Range}(S) = \text{Range}(T)$. Additionally, $V$ is the direct sum of $\text{Ker}(S)$ and $\text{Range}(S)$. Does this imply that $S = T$?

Answer: No


Question: Determine the order of the subgroup $K$ of $GL(2,GF(5))$ defined by $K = \left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in GL(2,GF(5)) : ad - bc = 2\right\}$. Provide your answer as a single integer.

Answer: 120


Question: Determine whether there are analytical solutions for finding the intersection between a Bezier curve and a line.

Answer: Yes


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

Answer: 4\sqrt{2}


Question: Find the minimum value of the function \( f(x, y, z) = x^2 + 4xy + 4y^2 + 2z^2 \), given that \( xyz = 32 \) and \( x, y, z > 0 \).

Answer: 96


Question: Evaluate the line integral of the vector field \(\overrightarrow{F}(x,y) = \langle y - \cos{y}, x\sin{y} \rangle\) along the circle \(C\) defined by \((x-3)^2 + (y+4)^2 = 4\), oriented clockwise. Simplify the integration using polar coordinates if possible.

Answer: 4\pi


Question: Evaluate the limit: $$\lim_{(x,y) \to (+\infty,+\infty)}(x^2+y^2)e^{-(x+y)}$$

Answer: 0


Question: Determine if the integral \( \int_{0}^{\infty}\frac {\cos{x}}{(1 + x^2)} dx \) can be evaluated using differentiation under the integral sign or any other method, without involving complex analysis.

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


Question: Evaluate the limit: \[ \lim_{x\to 1} \frac{1-\cos(\sin(x^3-1))}{x^3-1} \] without using L'Hospital's Rule.

Answer: 0


Question: Given $x > 1$, $y > 1$, and $(\ln x)^2 + (\ln y)^2 = \ln(x^2) + \ln(y^2)$, determine the maximum value of $x^{\ln y}$. \(\boxed{\text{Answer}}\)

Answer: e^4


Question: Determine the radius of convergence of the function \( f(z) = \frac{z^2}{1 + \cosh(z)} \) at \( z_0 = 0 \), using the identity \( \cosh(z) = \frac{e^z + e^{-z}}{2} \).

Answer: \pi


Question: Let $G$ and $H$ be locally compact groups that are locally isomorphic through an isomorphism $f: V_G \rightarrow V_H$ defined around their respective identity elements $e_G$ and $e_H$, with $V_G$ and $V_H$ being open neighborhoods. If $G$ is an unimodular group, does it follow that $H$ is unimodular as well? Provide a justification for your answer.

Answer: Yes


Question: Find the spectrum \( \sigma_p(P) \) of an orthoprojection \( P: H \rightarrow H \) where \( P \neq 0 \) and \( P \neq I \).

Answer: \{0, 1\}


Question: Determine whether $AG$ is unique for a matrix $A$ such that $AGA = A$, given that $G$ is not unique. Specifically, if $AG_1A = AG_2A = A$, does it necessarily follow that $AG_1 = AG_2$?

Answer: No


Question: In a metric space, determine whether every bounded sequence with no convergent subsequence satisfies the condition that the distance between any two distinct elements is greater than some positive constant.

Answer: No


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{2^{\log_3 n}}{3^{\log_2 n}}.$$

Answer: 0


Question: Calculate the expected value of \( \gamma = \max(\zeta_1^2, \zeta_2^3) \), where \( \zeta_1 \) and \( \zeta_2 \) are independent random variables uniformly distributed between 0 and 1.

Answer: \dfrac{5}{11}


Question: Calculate the correlation coefficient for the random variables X and Y given the joint density function: \[ f(x, y) = \begin{cases} 4 & 0 < y \le x < 1 \\ 0 & \text{otherwise} \end{cases} \]

Answer: \dfrac{1}{2}


Question: Determine whether the limit \( \lim_{n \to \infty}({p_1\times p_2\times p_3\times\cdots \times p_n})^{\frac{1}{p_n}} \) converges, where \( p_i \) is the \( i \)-th prime. If it converges, find its limit.

Answer: e


Question: Find the intersection of the ideals $I = (y) \cap (x) \cap (x-y)$ in the polynomial ring $k[x,y]$, where $k$ is an algebraically closed field.

Answer: (xy(x - y))


Question: Determine whether the following statement is true or false: Let $f: (a,b)\to\mathbb R$ be a continuous function, where $a$ and $b$ are finite. The function $f$ is uniformly continuous on $(a,b)$ if and only if the limits \( \lim_{x\to a^+}f(x) \) and \( \lim_{x\to b^-}f(x) \) exist finitely.

Answer: True


Question: Given a non-closed subset \( S \) of real numbers, determine whether there always exists a sequence in \( S \) that converges to a point outside of \( S \), but within the closure of \( S \).

Answer: Yes


Question:
Given the equations for a point \((x, y, z)\) in \(\mathbb{R}^3\):

\[ x\cos\alpha - y\sin\alpha + z = 1 + \cos\beta \]
\[ x\sin\alpha + y\cos\alpha + z = 1 - \sin\beta \]
\[ x\cos(\alpha+\beta) - y\sin(\alpha+\beta) + z = 2 \]

where \(\alpha, \beta \in (0, 2\pi)\), find the minimum value of \(x^2 + y^2 + z^2\).

Answer: 2


Question: Determine the structure of the Galois group of the field extension \( \mathbb{Q}(\sqrt[3]{5},\omega):\mathbb{Q} \), where \( \omega = e^{2\pi i/3} \) is a primitive cube root of unity.

Answer: S_3


Question: Evaluate the limit using polar coordinates: \[ \lim_{(x,y) \to (0,0)} \frac{x^2 |y|}{x^2+y^2} \]

Answer: 0


Question: Let $X$ and $Y$ be independent random variables, each uniformly distributed on $[0,1]$. Define $Z = \max(X, Y)$. Calculate the expected value of $X$ given $Z$, expressed as $\mathbb{E}[X|Z]$ in terms of $Z$. Express your answer as a function of $Z$.

Answer: \dfrac{3}{4}Z


Question:
Let \( A = \{x^2 : 0 < x < 1\} \) and \( B = \{x^3 : 1 < x < 2\} \). Determine which of the following statements is true:

1. There exists a one-to-one and onto function from \( A \) to \( B \).
2. There does not exist a one-to-one and onto function from \( A \) to \( B \) that maps rationals to irrationals.
3. There does not exist a one-to-one and onto function from \( A \) to \( B \) which is onto.
4. There does not exist an onto function from \( A \) to \( B \) which is one-to-one.

Answer: 1


Question: Find all functions \( f: \mathbb{Q}_+ \to \mathbb{Q}_+ \) that satisfy the following conditions: \( f(x+1) = f(x) + 1 \) for all \( x \in \mathbb{Q}_+ \) and \( f(x^2) = f(x)^2 \) for all \( x \in \mathbb{Q}_+ \).

Answer: f(x) = x


Question: Calculate the integral \( \int_B \frac{dz}{z^2+4} \), where \( B = \{ z \in \mathbb{C} \mid |z+2| = 3 \} \), in the positive direction.

Answer: 0


Question:
Let $(s_1, s_2, s_3, \ldots )$ be an equidistributed sequence on $[0,1]$. Determine if it is true that for each $0<\delta<\varepsilon<1$, there exists an $N$ such that for all $n\geq N$, the following inequality holds for every interval $[c, c+\varepsilon]$ of length $\varepsilon$ contained in $[0,1]$:

$$ \varepsilon - \delta < \frac{ \left\lvert \{ s_1, \ldots, s_n \} \cap [c, c+\varepsilon] \right\rvert }{ n } < \varepsilon + \delta $$

Answer: True


Question: Find the relationship between the determinants of the matrices \( M_1 = \begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{vmatrix} \) and \( M_2 = \begin{vmatrix} x_1+ay_1 & y_1+bz_1 & z_1+cx_1 \\ x_2+ay_2 & y_2+bz_2 & z_2+cx_2 \\ x_3+ay_3 & y_3+bz_3 & z_3+cx_3 \end{vmatrix} \), where \( a \), \( b \), and \( c \) are constants.

Answer: \det(M_2) = (1 + abc) \det(M_1)


Question: Given a set of linearly independent $d \times d$ complex unitary matrices $\{U_i\}_{i=1}^n \subseteq M_d$ with $n \geq d$, determine if there exists a vector $v \in \mathbb{C}^d$ such that the span of $\{U_1v, U_2v, \ldots , U_nv\}$ is $\mathbb{C}^d$. Answer with 'yes' or 'no'.

Answer: yes


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

Answer: f(x) = x


Question:
Consider the set $U=\{T \in L(\mathbb{R}^5,\mathbb{R}^4):\text{dimnull} T>2\}$. Determine whether $U$ is a subspace of the linear transformation space $L(\mathbb{R}^5,\mathbb{R}^4)$. Let $v_1,v_2,v_3,v_4,v_5$ form a basis for $\mathbb{R}^5$ and $u_1,u_2,u_3,u_4$ be a basis for $\mathbb{R}^4$. Define transformations $T, S \in U$ as follows:

$$
T(a_1v_1+a_2v_2+a_3v_3+a_4v_4+a_5v_5) = a_4u_1+a_5u_2,
$$

$$
S(b_1v_1+b_2v_2+b_3v_3+b_4v_4+b_5v_5) = b_3u_3+b_5u_4.
$$

Find $(S+T)$ and determine if it satisfies the condition for being in $U$. Provide your answer as either "Yes" or "No".

Answer: No


Question: Let \(\mu = e^{2\pi i/9}\). Define \(u_j = (\mu^j, \mu^{2j}, \mu^{3j}, \ldots, \mu^{9j})^T \in \mathbb{C}^9\) for \(j = 1, \ldots, 9\). Let \(V\) be the vector subspace of \(\mathbb{C}^9\) spanned by \(\{u_2, u_3, u_4, u_5, u_6, u_7\}\). It is known that there exists a matrix \(A \in M_{3 \times 6}(\mathbb{C})\) such that \(V = \{(x_1, \ldots, x_9) \in \mathbb{C}^9 : (x_3, x_6, x_9)^T = A(x_1, x_2, x_4, x_5, x_7, x_8)^T\}\). If the statement "On a special kind of 6-dimensional vector subspace of \(\mathbb{C}^9\)" is true, then \(A\) is unique and has the form \(A = \begin{pmatrix} a_1 & a_2 & a_3 & a_4 & a_5 & a_6 \\ a_5 & a_6 & a_1 & a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5 & a_6 & a_1 & a_2 \end{pmatrix}\). Let \(L\) be the smallest subfield of \(\mathbb{C}\) generated by the entries of \(A\). It is known that \(L \subseteq \mathbb{Q}(\mu)\), and hence \(L/\mathbb{Q}\) is a Galois extension. Determine the extension degree \([L : \mathbb{Q}]\). Given that \([\mathbb{Q}(\mu) : \mathbb{Q}] = \phi(9) = 6\), the possible values for \([L : \mathbb{Q}]\) are 1, 2, 3, or 6. Find the exact value.

Answer: 3


Question: Evaluate the triple integral \( \iiint r^2 \cos(\theta) \, dV \) over the solid bounded by the paraboloid \( z = 2 - r^2 \) and the \( xy \)-plane, where the bounds of integration in cylindrical coordinates are \( 0 \leq \theta \leq \pi/2 \), \( 0 \leq r \leq 1 \).

Answer: \dfrac{1}{3}


Question: If $f(x)$ is a polynomial of degree $n$ such that \(1+f(x)=\frac{f(x-1)+f(x+1)}{2}\) for all \(x \in \mathbb{R}\), find the degree \(n\) of the polynomial.

Answer: 2


Question: Evaluate the expression: $$ \log_{10}(\cot(1°)) + \log_{10}(\cot(2°)) + \cdots + \log_{10}(\cot(89°)) $$ where all logarithms are base 10.

Answer: 0


Question: Find the value of \( \lim\limits_{n\to\infty} \sup \exp\left( \frac{k}{n} \ln(n^2) \right) \) for \( k \in \mathbb{N} \).

Answer: 1


Question: Determine whether the sequence \((b_n)\) defined by \[ b_n = e^{-\frac{\log(n!)}{n}} \] converges, and if so, find its limit.

Answer: 0


Question: Determine whether it is possible to arrange the numbers from 1 to 500 such that the absolute value of the difference between any two consecutive numbers is neither a perfect square nor a prime number.

Answer: Yes


Question: Find the value of $x$ that satisfies the modular equation $x^{17} \equiv 20 \pmod{1001}$. Use principles from Cryptology to solve the equation.

Answer: 531


Question: Determine whether the function $f(x) = \cos(1/x)$ can be approximated pointwise on $(0, \infty)$ using a sequence of polynomials.

Answer: Yes


Question: Determine if the function \(f(x,y) = \begin{cases} \frac{x^3y^3}{x^2+y^2} & x^2+y^2 \neq 0 \\ 0 & x = y = 0 \end{cases}\) is differentiable at \((0,0)\).

Answer: Yes


Question: Evaluate the limit \(\lim_{x\to 0}\frac{x-\sin x+x\cos x}{x^3}\) without using Taylor series.

Answer: \infty


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{\sinh x -\sin x}{\cosh x - \cos x}$$

Answer: 0


Question: Find the eigenvalues and their corresponding eigenvectors of the linear operator $T: P(\mathbb{R}) \rightarrow P(\mathbb{R})$ defined by $Tp = p'$, where $P(\mathbb{R})$ is the space of polynomials with real coefficients.

Answer: 0


Question: Find the singular solution of the differential equation \(8ap^3 = 27y\), where \(p = \frac{dy}{dx}\).

Answer: y = 0


Question: Calculate the Liapunov number \(\sigma\) for the quadratic system \(\begin{align*}\ \dot{x}&=\mu x-y+x^2\\ \dot{y}&=x+\mu y+x^2 \end{align*}\) at the origin when \(\mu=0\).

Answer: -\dfrac{1}{4}


Question:
Find a function $f(x,y)$ that satisfies the following integral equations:

$$ \int f(x,y)dx = (1-y) \cdot f(x,y) $$
$$ \int f(x,y)dy = x \cdot f(x,y) $$

Answer: 0


Question: Let $C$ be the curve of intersection of the cylinder $\frac{x^2}{25} + \frac{y^2}{9} = 1$ with the plane $3z = 4y$. Let $L$ be the line tangent to $C$ at the point $(0,-3,-4)$. Determine the x-coordinate of the point where $L$ intersects the plane $2x - 3y - 4z = 27$. \( \boxed{\text{Answer}} \)

Answer: 1


Question: Compute the Cauchy principal value of the integral \( \int_{-\infty}^{\infty} \frac{x}{(x^{2}+4)(x-1)} \, dx \).

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


Question:
Evaluate the limit:

$$\lim_{n\rightarrow\infty} \frac{a_n^3 -2 a_n^2 - a_n - 6}{\sqrt{6 + a_n} - \sqrt{2a_n + 3}}$$

where the sequence $a_n$ converges to $3$ and satisfies $\forall n: a_n > 3$. Find the value of the limit.

Answer: -84


Question: Find the radius of convergence for the series \(\sum_{n=1}^\infty \frac{(2n+1)}{(n+1)^2} (x+2)^{2n}\) using the formula \(R=\frac{1}{\limsup_{n\to\infty}|c_n|^{1/n}}\).

Answer: 1


Question: Evaluate the integral \( I = \int_{-\infty}^{\infty} \frac{\sin^2{x}}{x^2} \, dx \).

Answer: \pi


Question: Given a basis \( B = \{x_1, x_2, \ldots, x_n\} \) for a vector space \( V \), determine the basis for the exterior power \( \wedge^k(V) \) and state how many elements this basis contains.

Answer: \binom{n}{k}


Question: Let \( h(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} \) and let \( k(x) = h^{-1}(x) \). Find \( k^{(4)}(0) \).

Answer: 1


Question: Determine whether the following statement is true or false: For every matrix $A \in M_{2}( \mathbb{C})$, there exists a matrix $X \in M_{2}( \mathbb{C})$ such that $X^2 = A$. Provide a justification for your answer.

Answer: False


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

$$\sum_{n=1}^\infty \frac{(4n)!}{(2n)!^2} x^n$$

Answer: \dfrac{1}{16}


Question:
Given a positive integer $x$, let $x_n$ denote the remainder when $x$ is divided by $n$. Compute the sum of all elements in the solution set of the equation:

$$x^5(x_5)^5 - x^6 - (x_5)^6 + x(x_5) = 0,$$

where $x_5$ is restricted to the interval $[0, 4]$. What is the sum of all such $x$?

Answer: 1300


Question: Evaluate the limit: $$ \lim_{n \to \infty} \frac{1}{n}\sqrt[n]{n^5+(n+1)^5+\ldots+(2n)^5} $$

Answer: 0


Question: Given non-negative numbers $x, y, z$ such that $x + y + z = 4$, find the maximum value of the expression $x^3y + y^3z + z^3x$. Provide your answer as a single number.

Answer: 27


Question: Find all pairs of prime numbers \((p, q)\) such that \(p^{q+1} + q^{p+1}\) is a perfect square.

Answer: (2, 2)


Question: Evaluate the limit \( \lim_{p \to 0} \int_{-p}^{p} \frac{1}{\sqrt{p^{2}-x^{2}}} \, dx \).

Answer: \pi


Question: Given the polynomial equation \(x^4 - 2x^3 - 3x^2 + 4x - 1 = 0\) with four distinct real roots \(x_1, x_2, x_3, x_4\) such that \(x_1 < x_2 < x_3 < x_4\) and the product of two roots equals unity, find the value of \(x_1x_2 + x_1x_3 + x_2x_4 + x_3x_4\). It is known that \(x_2x_4 = 1\) and \(x_1x_3 = -1\).

Answer: 1


Question: Suppose a sequence $y_n$ converges to a positive number $y > 0$, and let $a$ be a real number. Determine whether the limit \( \lim\limits_{n\to\infty} 2^n \arcsin \left(\frac{a}{2^n y_n}\right) \) exists and, if so, find its value.

Answer: \dfrac{a}{y}


Question: Evaluate the limit: $$\lim_{x\to-\infty} \left(\frac{3x+5}{3x-1}\right)^{3x-2}$$

Answer: e^6


Question: Determine the number of elements of order 5 in a group of order 90.

Answer: 4


Question: Does there exist a function defined on the interval $[0,1]$ that is entirely composed of jump discontinuities but is not piecewise continuous? A jump discontinuity at a point $a$ is characterized by the existence of both one-sided limits, $\lim_{x \rightarrow a^{-}}f(x)$ and $\lim_{x \rightarrow a^{+}}f(x)$, which are not equal. A function is piecewise continuous if every finite subinterval contains only a finite number of jump discontinuities.

Answer: Yes


Question: Calculate the volume of the solid bounded by the surfaces \( z = \frac{1}{2}(x^2 + y^2) \) and \( z = 6 - x^2 - y^2 \). The intersection of these surfaces forms a circle in the \(xy\)-plane, and the length of the solid along the \(z\)-axis is 6 cm.

Answer: 12\pi


Question: Let $x, y, z, n$ be positive integers such that $n$ has at least three distinct prime divisors. Determine the minimum value of $x + y + z$ given that $xyz = n$. Provide your answer as a single integer.

Answer: 10


Question: Evaluate the integral \( \int_{-1}^{1} (\lfloor x^3 \rfloor + \{x^3\}) \, dx \).

Answer: 0


Question: Given a variety $X$ and a line bundle $L$ on it, if the dimension of the space of global sections of $L^{\otimes k}$ increases as $k$ goes to infinity, i.e., $\dim \Gamma(X,L^{\otimes k}) \to \infty$, is $L$ necessarily ample? Answer yes or no.

Answer: No


Question: Given a complete separable metric space $(X,d)$, a finite Borel measure $\mu$ on $X$, and a compact subset $K$ of $X$ with $\mu(\partial K)>0$, determine if the set $\{x\in\partial K \cap \operatorname{supp}(\mu) : \limsup_{r\to 0^+}\ \frac{\mu(B_r(x)\cap K^c)}{\mu(B_r(x))}>0\}$ always has $\mu$-measure zero, even if the Lebesgue differentiation theorem doesn't hold in $(X,d)$. Provide a justification for your answer.

Answer: Yes


Question: Find the value of \( g(10) \) if \( g(y) \) is a polynomial of degree 9 that satisfies \( g(y) = \frac{1}{y+2} \) for \( y = 0, 1, 2, \ldots, 9 \).

Answer: \dfrac{5}{66}


Question: Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables with continuous distributions. For $m = 1, 2, \ldots$ and $i \le m$, define $A_{i,m} := (\max\{Y_1, Y_2, \ldots, Y_m\} = Y_i)$ and $B_m := A_{m,m}$. Prove or disprove that $P(B_m) = P(A_{m-1,m}) = \ldots = P(A_{2,m}) = P(A_{1,m})$ and hence $P(B_m) = 1/m.$

Answer: \dfrac{1}{m}


Question: Let $c$ be a positive constant and let $f(t)=\delta(t-c)$. Compute the convolution $f*g$. Express your answer in terms of $g(t)$. \( \boxed{} \)

Answer: g(t - c)


Question: If two normed vector spaces, $V$ and $U$, are isomorphic, does it imply that $U$ is a Banach space if and only if $V$ is a Banach space? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question:
Determine the limit of the sequence \(b_n\) defined by:

\[b_{n}:=\frac{n}{\sqrt[3]{n^{6}-n}}+\frac{n}{\sqrt[3]{n^{6}-n+1}}+\ldots+\frac{n}{\sqrt[3]{n^{6}-2}}\]

Answer: 1


Question: Determine the number of distinct integer values that the expression \( p = 144^{\sin^2 x} + 144^{\cos^2 x} \) can take, where \( x \) is any real number. Use the inequality of arithmetic and geometric means (AM-GM) to assist in your solution.

Answer: 122


Question: Calculate the limit: $$\lim\limits_{n\to \infty }\left(2^{2n+3}-2^{\sqrt{4n^2+1}}\right)$$

Answer: \infty


Question: Evaluate the limit \( \lim_{x \to \infty} \int_0^{\infty} \frac{1-e^{-u^4}}{u^2} \sin(x u) \, du \) without using the Riemann-Lebesgue lemma.

Answer: 0


Question: Calculate the limit \( \lim_{n\to \infty} \frac{2\cdot 4 \cdot \ldots \cdot (2n)}{1\cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)} \) without using convergence tests such as the root test or the ratio test.

Answer: \infty


Question: Given non-negative real numbers $x$, $y$, and $z$ such that $x + y + z = 1$, determine the maximum value of the expression $xy + yz + zx - xyz$. Provide your answer as a single number.

Answer: \dfrac{8}{27}


Question: If $A^2$ is diagonalizable over a field $F$, is $A$ diagonalizable given that $A$ is non-singular? Answer yes or no and justify your answer.

Answer: No


Question: Determine all real values of \( a \) for which the quadratic equation \( x^2 + (a + i)x - 5i = 0 \) has at least one real solution.

Answer: -5


Question: Evaluate the integral \( \int_{D}\int{\sin(x-y)\,dx\,dy} \) where \( D=\{(x,y)\mid x^2+y^2\le1\} \).

Answer: 0


Question: Evaluate the limit: \( \lim_{x \rightarrow 0^+} \sum_{n=1}^{\infty} \frac{1}{n^2 x} \) using the technique of convolution with an approximate identity.

Answer: \infty


Question: Given two upper triangular matrices \( A = (a_{ij})_{n \times n} \) and \( B = (b_{ij})_{n \times n} \), where \( a_{ij} = b_{ij} \) whenever \( i > j \), determine the \((i, i)\) entry of their product \( AB \). Express the result using the properties of upper triangular matrices, and simplify the summation \( \sum_{k=1}^{n} a_{ik}b_{ki} \) accordingly.

Answer: a_{ii}b_{ii}


Question: Determine whether the infinite product \( \prod_{n=1}^{\infty}\left(1+\frac{(-1)^{n}x^{2n}}{2n+1}\right) \) converges for \( x > 0 \).

Answer: (0, 1]


Question: Solve the differential equation \( \frac{dy}{dx}=\frac{y^2-1}{x^2-1} \) with the initial condition \( y(2)=2 \).

Answer: y = x


Question: In an acute-angled triangle ABC, with \(\angle A = 20^\circ\), let D, E, and F be the feet of the altitudes from A, B, and C respectively, and H be the orthocenter. Given that \(AH = 2R\cos A\), \(AD = 2R\cos A + 2R\cos B\cos C\), \(BH = 2R\cos B\), \(BE = 2R\cos B + 2R\cos A\cos C\), \(CH = 2R\cos C\), and \(CF = 2R\cos C + 2R\cos A\cos B\), determine the sum \(\frac{AH}{AD} + \frac{BH}{BE} + \frac{CH}{CF}\).

Answer: 2


Question:
Determine whether the following statement is true or false: For any sequence $\{a_n\}$ converging to $L$, we have:
$$\lim_{n \to \infty} \left(1+\frac{a_n}{n} \right)^n = e^L.$$

Answer: True


Question: Let $F$ be a subfield of the complex numbers. We define $n$ linear functionals on $F^n$ ($n \geq 3$) by $f_k(x_1, \dotsc, x_n) = \sum_{j=1}^n (2k-2j+1) x_j$, $1 \leq k \leq n$. Determine the dimension of the subspace of $F^n$ that is annihilated by the linear functionals $f_1, \dotsc, f_n$. \( \boxed{} \)

Answer: n - 2


Question: Evaluate the limit: $$\lim_{n\to \infty} \left(\frac{1}{\pi}\arctan (u n)+\frac{1}{2}\right)^n$$ where $u$ is a positive constant.

Answer: e^{-\frac{1}{\pi u}}


Question: Calculate the expectation of $Y^2$ where $Y = \exp(X^2)$ and $X$ is a standard normal random variable ($X \sim N(0, 1)$). Assume the moment-generating function of $X$ is known.

Answer: \infty


Question: Calculate the flux of the vector field \( \vec{F}(x, y, z) = \langle x, y, z \rangle \) across the surface of the cone defined by \( z = \sqrt{x^2 + y^2} \) for \( 0 \le z \le 1 \) using the divergence theorem.

Answer: 0


Question:
Let $a, b, c, d \in \mathbb{R}$ with $c^2 + d^2 \neq 0$. Determine the condition under which the Cauchy problem for the linear PDE $au_x + bu_y = e^{x+y}$, with the initial condition $u(x, y) = 0$ on the line $cx + dy = 0$, has a unique solution. Choose the correct option:

(A) $ac + bd \neq 0$

(B) $ad - bc \neq 0$

(C) $ac - bd \neq 0$

(D) $ad + bc \neq 0$

Answer: A


Question: Determine whether the compactness of a topological space under a finer topology implies its compactness under a coarser topology.

Answer: Yes


Question: Determine the condition for the convergence of the improper integral \( \int_{0}^{\frac{\pi}{2}}\tan^\alpha (x)\,dx \) for \( \alpha > 0 \).

Answer: (0, 1)


Question: Evaluate the limit \( \lim_{{x \to 0}} \frac{{x^3 - x}}{{\sqrt[3]{x^2 - 8} - (x - 2)}} \) without using L'Hopital's Theorem.

Answer: 1


Question: Prove that for any prime number $p > 3$, the sum of all distinct positive integer pairs $ij$ such that $1 \le i < j < p$ is divisible by $p$.

Answer: 0


Question: Calculate the upper bound for the integral: $$\int_{0}^{+\infty}e^{-A\sqrt{x^2+B}}\cos(Cx)\frac{x'}{\sqrt{x'^2-B}}dx'$$ where A, B, and C are real numbers, and A and B are positive.

Answer: \dfrac{e^{-A\sqrt{B}}}{A}


Question: Given the equations \(p+q+r=1\), \(p^2+q^2+r^2=2\), and \(p^3+q^3+r^3=3\), find the value of \(p^4+q^4+r^4\).

Answer: \dfrac{25}{6}


Question: Determine whether the following statement is true or false: An infinite subset of a closed set always has a limit point within that set, even if the closed set is not bounded.

Answer: B


Question: In a room lit by two bulbs, the bulbs are replaced only when both burn out. The lifetimes of the bulbs are independent and exponentially distributed with parameter \( \lambda = 1 \). What fraction of the time is the room only half-lit?

Answer: \dfrac{2}{3}


Question: Find the variance of $X^2$ where $X$ is a standard normal random variable $(X \sim \mathcal{N}(0,1))$. Express your answer as a single number.

Answer: 2


Question: Consider an $n \times n$ matrix ${\bf A}$, where $a_{ij} = n$ for $i \neq j$ and $a_{ii} = i$. Determine the determinant of ${\bf A}$. \( \boxed{?} \)

Answer: (-1)^{n-1} n!


Question: Verify if the equation \(x\cdot \frac{\partial u}{\partial x} + y\cdot \frac{\partial u}{\partial y}=4u\) holds true for the function \(u=\frac{x^4-6x^2y^2+y^4}{x^2+y^2}\).

Answer: No


Question: Determine whether the following statement is true or false: In the Zariski topology on $\mathbb{C}^n$, for every non-empty Zariski open set $U$, the intersection $U \cap \mathbb{R}^n$ is open dense in $\mathbb{R}^n$. Provide a justification for your answer.

Answer: True


Question: Calculate the value of the following summation: \[ \sum_{n_{1}+n_{2}+\cdots+n_{k}=n}\binom{n}{n_{1}, n_{2}, \dotsc,n_{k}} \cdot (n_{1}+n_{2}+\cdots+n_{k})^2 \]

Answer: n^2 k^n


Question: Find the limit of the sequence: \[ \lim_{n \to \infty}{\frac{1^3+3^3+5^3+\cdots+(2n-1)^3}{n^4}}. \]

Answer: 2


Question: Calculate the flux of the vector field \( F = (x^2, y^2, z^2) \) through the surface of a cube with side length \( a \) centered at the origin.

Answer: 0


Question: Simplify the fraction: $$\frac{(2^4+2^2+1)(4^4+4^2+1)(6^4+6^2+1)(8^4+8^2+1)(10^4+10^2+1)}{(3^4+3^2+1)(5^4+5^2+1)(7^4+7^2+1)(9^4+9^2+1)(11^4+11^2+1)}$$

Answer: \dfrac{3}{133}


Question: Evaluate the integral \( \int_{\gamma} \left(\frac{1}{z^2-1}\right) dz \), where \( \gamma \) is the circle centered at the origin with radius 2, traversed once counterclockwise.

Answer: 0


Question: Find the maximum value of the integral \( \int_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx \) where \( y \in [0,1] \).

Answer: \dfrac{1}{3}


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

Answer: p > 1


Question: Find the value of $A > 1$ that minimizes the function \( U = \frac {F + (A - F)^2 } {A} \), where \( F = \lfloor A \rfloor \) and \( F \ge 1 \).

Answer: \sqrt{2}


Question: Determine the minimum number of 4-member subsets required from an 18-member set to ensure that each member appears in at least one subset.

Answer: 5


Question: Consider the recursive sequence defined by \(a_{n+1} = \sqrt{a_n - 1} + a_n\) with the initial term \(a_1 = 4\). Determine whether the sequence \(\{a_n\}\) converges or diverges. If it converges, find the limit \(\lim_{{n \to \infty}} a_n\).

Answer: \infty


Question: Determine whether the following statement is true or false: \( \exists_{n} x_n < x + \epsilon \implies \inf x_n \leq x \) for all \( \epsilon > 0 \).

Answer: B


Question: Determine whether the following statement is true or false: If $A$ is a real symmetric matrix of order $n (\geq 2)$, then there exists a symmetric matrix $B$ such that $B^{2k+1} = A$. Justify your answer.

Answer: True


Question: Given a complex function $G(z)$, where $z = x + iy$ and $x, y \in \mathbb{R}$, which is analytic and bounded in the upper half-plane, i.e., \(|G(z)| \leq C\) for all \(z \in \mathbb{C}^{+}\), does it imply that \(G(x)\) is bounded on the real line, i.e., \(|G(x)| \leq C_{o}\) for all \(x \in \mathbb{R}\), given that \(G(x)\) has no poles on \(\mathbb{R}\)? Here, \(\mathbb{C}^{+} = \{z \in \mathbb{C} : \text{Im}(z) > 0\}\).

Answer: Yes


Question: Find the range of the function \( f(x) = \cos^{2018}(x) + \cos^{2018}(5x) \).

Answer: [0, 2]


Question:
Let $X$ and $Y$ be two continuous random variables with values in $\mathbb{R}$. Given a continuous and differentiable function $f: \mathbb{R} \to \mathbb{R}$, determine whether there exists a measurable random variable $C$ such that the mean value theorem holds for $f(X)$ and $f(Y)$, expressed as:

$$f(X) - f(Y) = f'(C) (X - Y)$$

Is the existence of such a $C$ guaranteed?

Answer: Yes


Question: Determine the set $T$ of all $x \in \mathbb{R}$ for which there exists a polynomial $P$ in the set $\mathcal{P}$ such that $P(x) \in (-2, 2)$. The set $\mathcal{P}$ is the smallest set of polynomials containing the constants $-1$ and $1$, and closed under the operation $XP + (1-X)Q$ for any $P, Q \in \mathcal{P}$. What is the set $T$?

Answer: \mathbb{R}


Question: Evaluate the limit of the integral \(I_n = \int_{0}^{1} \left(\frac{t}{t^2+1}\right)^n\ \mathrm dt\) as \(n\) approaches infinity.

Answer: 0


Question: Does a point-wise convergent sequence of bounded functions \(\{f_n\}\) guarantee that the limit function \(f\) is also bounded? Provide a justification for your answer.

Answer: No


Question: Determine the radius of convergence for the complex series \(\displaystyle\sum\limits_{n=0}^\infty \frac{z^{3n}(n+1)!}{(2+n^3)^{n+1}}\).

Answer: \infty


Question: Determine the minimum number of coin flip series required such that the probability of obtaining more than 50 heads in total exceeds 0.8. Each series ends when the first tail appears.

Answer: 60


Question: Calculate the value of \(2^{2^{2^{\cdot^{\cdot^{2}}}}} \mod 2016\), where the base \(2\) is repeated 2016 times. Given that \(2^{11} \equiv 2^5 \mod 2016\) and \(2^{16} \equiv 2^{10} \mod 2016\), determine the next step in the calculation when there are 2012 instances of \(2\) left.

Answer: 1024


Question: Find the limit: \[ \lim_{n \rightarrow \infty} \dfrac{2^{2n}(n!)^2}{(2n+1)!} \]

Answer: 0


Question: Determine the center of the symmetry group $S_n$ for $n \geq 3$.

Answer: \{e\}


Question: Determine whether there exist any simple groups that are non-cohopfian, meaning they have an injective group endomorphism that is not surjective.

Answer: Yes


Question: In quadrilateral $ABCD$, it is given that $AD = BD$, $\angle ADC = 3\angle CAB$, $AB = \sqrt{2}$, $BC = \sqrt{17}$, and $CD = \sqrt{10}$. Determine the length of $AC$. \(AC = ?\)

Answer: 5


Question: Find the limit of the sequence: \( \lim_{n\rightarrow\infty}\left(\frac{1}{n}\sum_{k=1}^{n}\sqrt[k]{k}\right) \).

Answer: 1


Question: Suppose that $f$ and $g$ are positive and continuous functions on $[1,\infty)$. If $\lim_{x\to\infty} \frac{f(x)}{g(x)} = 0$, does it follow that $\lim_{x\to\infty} \frac{\int_x^\infty f}{\int_x^\infty g} = 0$? Provide a justification for your answer.

Answer: 0


Question: Determine the number of solutions to the equation \( f(f(x)) = \int_3^4{f(x)\mathrm{d}x} \) in the interval \([1, 4]\), given that \( f(x) = \frac{4x+3}{x^2+1} \).

Answer: 1


Question: Determine whether for a probability space $(\Omega, \mathcal{F}, \mu)$ and an integrable function $f$, the measure $\mu \{ \omega: |f(\omega)|=\alpha\}$ is zero for all but countably many real numbers $\alpha$.

Answer: Yes


Question: Evaluate the integral \( \int_{\gamma} \cos(z) \, dz \) where \( \gamma \) is a circle of radius 3 centered at the point \(-2i\).

Answer: 0


Question: A group $G$ acts on a set $X$. Given that there are $n$ conjugacy classes of subgroups of $G$, determine the number of transitive $G$-sets of $X$ that exist, up to equivalence.

Answer: n


Question: Determine if the sequence \(1, \frac{1}{3}, \frac{2}{3}, 0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, 0, \frac{1}{5}, \frac{2}{5}, \ldots\) is equidistributed in the interval \([0, 1)\).

Answer: Yes


Question: Evaluate the expression \(\left(\frac{\sum_{n=-\infty}^{\infty}\frac{1}{1+n^2}}{\coth(\pi)}\right)^2\) given that \(\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}\) and \(16\int_0^\pi\int_0^1x^2\sin(y)\,dxdy=\frac{32}{3}\). Find the value of the expression inside the big brackets.

Answer: \pi^2


Question: Find the minimum value of the sum of the lengths of line segments connecting the midpoints of the hypotenuses of $n$ right triangles to a common point, given that the sum of the areas of the triangles is a constant.

Answer: 0


Question: Provide an example of a decreasing function \( f(x) \) such that \( \lim_{x\to +\infty} f(x) = e. \) The function \( f(x) \) should have a simple form.

Answer: e + \dfrac{1}{x}


Question: Find the normalizer of the cyclic permutation group \( G = \langle (123456) \rangle \) in the symmetric group \( S_6 \). The normalizer is the set of all elements \( g \) in \( S_6 \) such that for any element \( a \) in \( G \), the conjugate \( gag^{-1} \) is also in \( G \).

Answer: \langle (123456), (16)(25)(34) \rangle


Question: Find the interval of values for $x$ such that the inequality $(x^2-5x+4)(y^2+y+1)<2y$ holds for all real values of $y$.

Answer: (2, 3)


Question: Calculate the surface integral of the function \( g = x - y \) over the surface \( S \).

Answer: 0


Question: What is the probability that a randomly selected subset of the interval [0, 1] is measurable according to the Lebesgue measure?

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to2}(x-2)^5\ln(x-2)$$

Answer: 0


Question: Determine whether there exists a non-zero functional $g \in X^*$ such that $g(Z) = \{0\}$, where $Z = \{(b_n) \in \ell^2 : b_3 = 3b_2, b_5 = 5b_4\}$.

Answer: Yes


Question: Determine whether there exists a real analytic diffeomorphism $f: \mathbb{R} \to \mathbb{R}$ such that $f(x_i) = a_i$ for a given finite set $F = \{x_1, \dots, x_k\} \subset \mathbb{R}$ with $x_i < x_{i+1}$ and numbers $a_1, \dots, a_k \in \mathbb{R}$ with $a_i < a_{i+1}$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following two definitions of $C(\bar{\Omega})$ are equivalent for an open subset $\Omega$ of $\mathbb{R}^d$: 1) the set of functions $u: \Omega \to \mathbb{R}$ that are uniformly continuous on bounded subsets of $\Omega$, and 2) the set of functions $u: \Omega \to \mathbb{R}$ that have a continuous extension on $\Omega \cup \partial \Omega$. Provide a justification for your answer.

Answer: Yes


Question: Find the limit of the infinite series as \( n \) approaches infinity: \[ \lim_{n\to\infty}\left(\sin^4x + \frac{1}{4}\sin^42x + \frac{1}{4^2}\sin^44x + \cdots + \frac{1}{4^n}\sin^42^nx\right) \] Choose the correct option: (A) \( \sin^4x \) (B) \( \sin^2x \) (C) \( \cos^2x \) (D) Does not exist.

Answer: B


Question: Evaluate the integral \( \int e^{x+e^x} \, dx \).

Answer: e^{e^x} + C


Question: Determine the limiting proportion of odd divisors to total divisors as \( n \) approaches infinity.

Answer: \ln 2


Question: Are all irreducible polynomials of the same degree in the factorization of \(x^{p^n} - x\) distinct?

Answer: Yes


Question: Determine the number of distinct ways to partition the natural numbers into a countable collection of non-empty, disjoint subsets.

Answer: 2^{\aleph_0}


Question: Determine whether it is possible to create two Latin squares that, when superimposed, result in fewer than two repeated combinations.

Answer: Yes


Question: Given the RSA encryption system, the ciphertext C = 6 was obtained using the public key (n = 187, e = 107). Determine the original message M.

Answer: 29


Question: Determine whether there exists a cubic polynomial $f(x) \in \mathbb{Q}[x]$ such that the Galois group of its splitting field over $\mathbb{Q}$ is isomorphic to $\mathbb{Z}_3$. Provide a justification for your answer.

Answer: Yes


Question: Let $X$ be a locally compact, metrizable, and path-connected space that is not compact. Determine if the one-point compactification of $X$ is path-connected.

Answer: Yes


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

Answer: 1


Question: Determine if the last digit of \(2^{2^{n-1}(2^n-1)}-1\) is always 5 for all integers \(n > 3\).

Answer: 5


Question: Given a series of $n$ geomagnetic readings modeled as independent observations from a random variable uniformly distributed on the interval $(0, \theta)$, where $\theta > 1$, and $m$ out of $n$ readings are less than $1$, find the maximum likelihood estimate (MLE) of $\theta$. Assume the remaining $n-m$ readings are greater than $1$.

Answer: \dfrac{n}{m}


Question: Can every left-orderable group be embedded into the group of orientation-preserving homeomorphisms of the real line?

Answer: No


Question: In a finite $T_1$ topological space, determine the intersection of all open sets containing a point $x$. Provide your answer as a set.

Answer: \{x\}


Question: Find the limit \( \lim_{n\to\infty} \frac{a_1 b_n + a_2 b_{n-1} + \cdots + a_n b_1}{n} \) given that \( \lim_{n\to\infty} a_n = a \) and \( \lim_{n\to\infty} b_n = b \).

Answer: ab


Question: Find the polynomial $P(x)$ of degree at least 3 such that $P(2) = 1$, $P'(2) = 3$, $P''(2) = 4$, and $P^{(n)}(2) = 0$ for all $n \geq 3$. Express your answer in the form of a polynomial.

Answer: 2x^2 - 5x + 3


Question: Evaluate the limit using the squeeze theorem: $$\lim_{n\to\infty}\sum_{k=1}^n \cos\left(\frac{2\pi}{\sqrt{n^2+2k}}\right)$$

Answer: \infty


Question: Determine whether there exist irreducible polynomials with coefficients in \(\mathbb{Q}\) such that all their roots have absolute value greater than 1, but do not lie in a radical extension of \(\mathbb{Q}\).

Answer: Yes


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

Answer: \dfrac{2}{3}


Question: Given two line segments $L_1$ and $L_2$ in the plane, each defined by two endpoints, and a point $C$ not on either line segment, determine the minimum angle through which $L_1$ must be rotated about $C$ so that it intersects $L_2$. The direction of rotation is important. What is this minimum angle?

Answer: \theta


Question: Let $G$ be a finite group of order $n$, and let $k$ be a positive integer. Suppose $G = \{x^k : x \in G\}$. Must it be true that the greatest common divisor of $n$ and $k$ is 1?

Answer: Yes


Question: Determine the maximum value of \( k \in \mathbb{R}^{+} \) such that the inequality \[ \frac{a^3}{(b-c)^2} + \frac{b^3}{(c-a)^2} + \frac{c^3}{(a-b)^2} \geq k (a+b+c) \] holds for all distinct positive real numbers \( a, b, c \).

Answer: 1


Question: Consider a pair of random variables \(X, Y\) with a constant joint density on the parallelogram with vertices at \((0, 0)\), \((2, 0)\), \((1, 1)\), and \((3, 1)\). Calculate the probability \(P(X + 2Y > 3)\).

Answer: \dfrac{1}{3}


Question: Find the values of $p$ for which the limit \( \lim_{(x,y)\to (0,0)} \frac{\left|x\right|\cdot\left|y\right|^p}{x^2 + y^2} \) exists.

Answer: p > 1


Question: The 12 edges of a parallelepiped are integers, and their sum is 100. What is the maximum possible volume of the parallelepiped?

Answer: 576


Question:
Consider a vector field $v$ in $\mathbb{R}^3$ such that $\nabla \cdot v = 0$ and $\nabla \times v = \alpha v$, where $\alpha: \mathbb{R}^3 \to \mathbb{R}$. Determine the value of $v \cdot \nabla \alpha$. \[ \text{Given:} \]
1. $\frac{\partial U}{\partial x} + \frac{\partial V}{\partial y} + \frac{\partial W}{\partial z} = 0$ (Divergence of $v$)
2. $(\frac{\partial W}{\partial y} - \frac{\partial V}{\partial z}, \frac{\partial U}{\partial z} - \frac{\partial W}{\partial x}, \frac{\partial V}{\partial x} - \frac{\partial U}{\partial y}) = \alpha (U, V, W)$ (Curl of $v$)

Answer: 0


Question: Determine the values of \( n \) for which the ring \( S_n = C^n([0,1], \mathbb{R}) \) of all continuously \( n \)-times differentiable functions fails to be an integral domain.

Answer: n \geq 0


Question: In a metric space $X$, if every closed and bounded subset is sequentially compact, does this imply that $X$ is a finite-dimensional normed space? Answer yes or no.

Answer: No


Question: Find the conjugate harmonic function \( v(x,y) \) of \( u(x,y) = e^{4x}(x\cos(4y) - y\sin(4y)) \) and express \( w = u + iv \) as an analytic function of \( z = x + iy \).

Answer: z e^{4z}


Question: Determine whether the group $D_{∞}^{(2)}$, consisting of all bijections $f: \mathbb{Z}^2 \rightarrow \mathbb{Z}^2$ that preserve the Euclidean distance, is virtually abelian. In other words, does there exist an abelian subgroup $H$ of $D_{∞}^{(2)}$ with finite index?

Answer: Yes


Question: Let $C \subset \mathbb{R}^2$ be an algebraic curve defined by an irreducible polynomial $P$. Suppose $Q \in \mathbb{R}[x,y]$ is another polynomial such that $Q(x,y) = 0$ on a set $A \subset C$ where $A$ has a limit point. Does it follow that $P$ divides $Q$?

Answer: Yes


Question: In the projective plane $\mathbb{P}^2$, consider the union of the lines $L_1: x=0$, $L_2: y=0$, and $L_3: z=0$. Determine the class group $Cl(U)$ of the complement $U:=\mathbb{P}^2\setminus Y$, given that $U$ is a noetherian integral scheme regular in codimension one.

Answer: 0


Question: Determine the conditions under which the sequence defined by \(x_{n+2}=|x_{n+1}-x_{n-1}|\) has a period of 1.

Answer: x_0 = x_1 = x_2 = 0


Question: Given a set $X$, a $\sigma$-field $\mathcal{F}$ of subsets of $X$, and a probability measure $\mu$ on $X$, suppose there exist random variables $f$ and $g$ mapping $X$ to $\mathbb{R}$ such that the equality \[\int_{\mathbb{R}} h \, d\mu_f = \int_{\mathbb{R}} h \, d\mu_g\] holds for all Schwartz functions $h$. Does it necessarily imply that the push-forward measures $\mu_f$ and $\mu_g$ are equal? Provide a justification for your answer.

Answer: Yes


Question:
Consider the languages $L_1$ and $L_2$ defined as follows:
\[ L_1 = \{ w ∈ \{0,1\}^∗ | \text{w has at least as many (110) substrings as (011) substrings} \} \]
\[ L_2 = \{ w ∈ \{0,1\}^∗ | \text{w has at least as many (000) substrings as (111) substrings} \} \]
Which of the following statements is correct regarding the regularity of $L_1$ and $L_2$?

A) $L_1$ is regular, but $L_2$ is not
B) $L_2$ is regular, but $L_1$ is not
C) Both $L_1$ and $L_2$ are regular
D) Neither $L_1$ nor $L_2$ are regular

Choose the correct statement.

Answer: D


Question: In triangle $ABC$, point $E$ is on side $BC$ such that $BE:EC = 3:2$. Points $D$ and $F$ are on sides $AB$ and $AC$, respectively, with $3AD = 2AF$. The lines $AE$ and $DF$ intersect at point $G$. Given $AB = 7$ and $AC = 9$, find the ratio $DG:GF$. \( \text{Express your answer as a simplified ratio.} \)

Answer: 9:7


Question: Given a real dividend $a$ and a nonzero real divisor $b$, determine whether there exists a unique pair of an integer quotient $q$ and a real remainder $r$ such that $a = bq + r$ with $0 \leq r < \lvert b \rvert$. Answer with 'yes' or 'no'.

Answer: yes


Question: Determine whether for every prime number $p$, there exist infinitely many integers $k$ such that $p$, $p+k$, and $kp+1$ are all prime numbers.

Answer: Yes


Question:
Evaluate the limit of the integral as \( n \to \infty \):
\[
\lim_{n\to\infty} n\int_{-\infty}^{\infty} e^{-2n\vert x\vert}\frac{x+3}{x^4+3x^2+1} \mathrm{d}x
\]

Answer: 3


Question:
Let $n$ and $m$ be positive integers such that $n \mid m$, and consider the ring surjection $\pi: \mathbb{Z}/m \twoheadrightarrow \mathbb{Z}/n$. If $z^* \in \mathbb{Z}/m$ denotes the least non-negative residue equivalent to $z$ modulo $m$, and $x \in \mathbb{Z}$ with $x \geq 0$, determine if it is true that for all $z \in \mathbb{Z}/m$ satisfying $z^2 \equiv 1 \pmod{m}$, the following strict inequality holds:

$$
\left\lfloor\frac{x - \pi(z)^*}{n} \right\rfloor > \left\lfloor\frac{x - z^*}{m}\right\rfloor
$$

Assume that $n$ and $m$ are both square-free integers, and $n < m$. Answer with "True" if the inequality holds for all such $z$, or "False" otherwise.

Answer: False


Question: Calculate the work done by the vector field \(\vec{F}=(6zx^2e^{x^3}+yz)\vec{i}+(xz+y)\vec{j}+(xy+2e^{x^3})\vec{k}\) in moving a particle along the curve \(\mathcal{C}\), which consists of three segments: from \((0,0,0)\) to \((2,4,6)\), from \((2,4,6)\) to \((3,6,2)\), and from \((3,6,2)\) to \((0,0,1)\). Express the work as the line integral \(W=\int_{\mathcal{C}}{\vec{F}\cdot{d\vec{r}}}\). Since \(\vec{F}\) is not conservative, close the curve by adding segment \(\mathcal{C_4}\) from \((0,0,1)\) to \((0,0,0)\). Parameterize the curve segments and apply Stokes' Theorem to evaluate the line integral.

Answer: 2


Question: Find the derivative \( \frac{df}{du} \) of the functional \( f(u)(x) = \int_{\Gamma} g(x,y) u(y) \, dy \) with respect to \( u \), where \( g(x,y) \) is a known kernel.

Answer: g(x,y)


Question:
Given the sequence $a_n$ defined by the recurrence relation for $n \geq 1$:
$$a_{n+k} = \frac{n}{n+k} - \frac{1}{n+k}\sum_{i=k+1}^{n+k}a_i$$
and knowing that $a_j = 0$ for all $j \leq k$, find a closed-form expression for $a_k$.

Answer: 0


Question:
Given a family of positive probability density functions $(\rho_t)_{t\in[0,\infty]}$ on $\mathbb R^d$ satisfying the following conditions:
1. $\rho_t(x) > 0$ for all $x \in \mathbb R^d$ and $t \in [0, \infty)$.
2. $\int_{\mathbb R^d} \rho_t(x) \, dx = 1$ for all $t \in [0, \infty)$.
3. $\int_{\mathbb R^d} \rho_t(x) |\log \rho_s(x)| \, dx < \infty$ for every $t \geq 0$ and $s = 0, t, \infty$.
4. $\|\rho_t - \rho_0\|_{TV} \to 0$ as $t \to 0$, where $\|\cdot\|_{TV}$ denotes the total variation distance.
5. $D_{KL}(\rho_t \| \rho_\infty) \to D_0 \in [0, \infty)$ as $t \to 0$, with $D_{KL}$ being the Kullback-Leibler divergence.

Determine if $D_{KL}(\rho_0 \| \rho_\infty) = D_0$. Provide a justification for your answer.

Answer: D_{KL}(\rho_0 \| \rho_\infty) = D_0


Question: Provide an example of a non-commutative division ring.

Answer: \mathbb{H}


Question: Determine whether all irrational numbers can be represented as the limit of an infinite sum of rational numbers. Provide a justification for your answer.

Answer: Yes


Question: For a holomorphic function $f$, define $h(z)=\sin(f(z^2))$. Calculate the residue of $h(z)$ at $z=0$ in terms of $f(0)$ and $f'(0)$. Express your answer as a single complex number.

Answer: 0


Question: Determine whether the following statement is true or false: Every subspace of a connected space is connected.

Answer: B


Question: Determine the order of the largest cyclic subgroup in the automorphism group of the integers modulo 720, \( \text{Aut}(\mathbb{Z}_{720}) \).

Answer: 12


Question: Provide an example of a continuous function \( f: \mathbb{R} \to \mathbb{R} \) that is closed but not open.

Answer: f(x) = x^2