Question: Given a sequence of independent and identically distributed random variables $X_1, X_2, \dots$ with mean $\mu = 4$ and variance $\sigma^2 = 20$, define $S_n = X_1 + X_2 + \dots + X_n$ and $V_n = X_1^2 + X_2^2 + \dots + X_n^2$. Determine a constant $c$ such that, according to the Strong Law of Large Numbers, $\lim_{n \to \infty} \frac{S_n}{V_n} = c$ with probability $P = 1$. Find the value of $c$.

Answer: \dfrac{1}{9}


Question: Given two graphs $G_1$ and $G_2$ with vertex chromatic numbers 5 and 7, respectively, and an edge between vertices 7 and 8 in both graphs, determine the vertex chromatic number of the graph $G$ formed by taking the union of the edge sets of $G_1$ and $G_2$. Provide your answer as a single number.

Answer: 7


Question: Evaluate \( \mathop{\int\!\!\! \int}_{R_{1}}f(x,y)\,dA \) where \( R_{1} = [0, 5] \times [-4, 4] \), given that \( \mathop{\int\!\!\! \int}_{R_{2}}f(x,y)\,dA = 10 \) for \( R_{2} = [0, 5] \times [0, 4] \), \( \mathop{\int\!\!\! \int}_{R_{3}}f(x,y)\,dA = -24 \) for \( R_{3} = [-5, 0] \times [0, 4] \), and \( f(x,-y) = -f(x,y) \) for all \((x,y)\).

Answer: 0


Question: In a unit square on the Cartesian plane, $N$ points are randomly generated. Calculate the expected average Manhattan distance between all pairs of points, excluding distances of 0.

Answer: \dfrac{2}{3}


Question: Evaluate the integral \( \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2+2xy)}\,dx\,dy \) using the substitution technique.

Answer: \infty


Question: Determine whether there exists a non-trivial, connected, compact, orientable topological manifold with Euler characteristic \( \chi = 1 \).

Answer: Yes


Question: Suppose we have maps of spaces $g_1:X_1\to A$, $g_2:X_2\to A$ and a map $f:Y\to A$. From these, we can form the homotopy pullbacks $Y\times^{h}_{A}X_1$ and $Y\times^{h}_{A}X_2$. A map $f:X_1\to X_2$ such that $g_2\circ f=g_1$ induces a map $$Y\times^{h}_{A}X_1\to Y\times^{h}_{A}X_2.$$ If $f$ is a weak homotopy equivalence, is this induced map also a weak homotopy equivalence?

Answer: Yes


Question: Find the number of distinct terms in the expansion of \((x+2y-z+3w)^{15}\) where exactly two of the variables have exponents greater than 5, and the other two variables have even exponents.

Answer: 48


Question: Evaluate the limit: $$\lim_{x\to \frac{1}{2}^-}\frac{\arcsin{2x}-\frac{\pi}{2}}{\sqrt{x-2x^2}}$$

Answer: -2


Question:
Calculate the radius of convergence \( R \) for the power series \( S(x) \) given by:

\[ S(x)=\sum_{n\geq0} \frac{x^{4n+1}}{4n+1}+ \sum_{n\geq0} \frac{x^{4n+2}}{4n+2}. \]

Determine whether \( R = 1 \).

Answer: 1


Question:
Determine which of the following functions is a solution to the differential equation \( u''+u-u^3=0 \), given that \( y=\tanh(x/\sqrt{2}) \) is a solution:

A) \( u = \tanh(x\sqrt{2}) \)
B) \( u = \coth(x/\sqrt{2}) \)
C) \( u = \coth(x\sqrt{2}) \)
D) None of these.

Answer: B


Question: Determine the order of the group \( \operatorname{SL}_3(\mathbb{F}_7) \), where \( \mathbb{F}_7 \) is the finite field with 7 elements.

Answer: 5630688


Question: Find the maximum value of the function \( f(x,y) = x^2 - y^2 + x^2y^2 \) on the closed disk of radius 2, excluding the critical point (0,0).

Answer: 5


Question: Determine the cardinality of the set of all total functions from \( \mathbb{N} \) to \( \mathbb{N} \).

Answer: 2^{\aleph_0}


Question: Provide an example of a limit ordinal that is not a cardinal number.

Answer: \omega + \omega


Question: Evaluate \( \lim_{n \to \infty} \int_0^1 g_n(x) \, dx \), where the function \( g_n \) is defined on \([0,1]\) as follows: \( g_n(x) = 0 \) for \( x = 0 \), and for \( x \in \left(\frac{k-1}{n}, \frac{k}{n}\right] \), \( g_n(x) = \cos\left(\frac{k\pi}{2n}\right) \) for \( k = 1, 2, \ldots, n \).

Answer: \dfrac{2}{\pi}


Question: Determine the number of integer solutions to the Diophantine equation \(4x^3 - 3 = y^2\) for \(x > 0\).

Answer: 4


Question: Find the derivative of the inverse function \((f^{-1})'(x)\) of \(f(x) = \ln(x)+\arctan(x)\) at \(x=\frac{\pi}{4}\).

Answer: \dfrac{2}{3}


Question: Find the interval of convergence for the integral \( \int_1^{\infty} \frac{\log{x}}{x^p} \,dx \).

Answer: (1, \infty)


Question: Calculate the volume of a torus formed by rotating a circle of radius \( b \) around an axis located at a distance \( a \) (where \( a > b \)) from the circle's center.

Answer: 2\pi^2 a b^2


Question:
For any positive even integer \( n \), determine whether the following limit is true:
\[ \lim_{x \to \infty} \frac{\pi_{2n}(x)}{\pi(x)} = 0 \]
where \( \pi_{2n}(x) \) is the number of primes \( p \) less than or equal to \( x \) such that \( p+2n \) is also prime, and \( \pi(x) \) is the prime counting function.

Answer: 0


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{ \sum_{i=1}^n\lfloor i^4x \rfloor}{ n^5} \) using the Stolz-Cesaro theorem.

Answer: \dfrac{x}{5}


Question: Determine if $f(x) = C \log x$ $(C \in \mathbb{R})$ is the only solution to the equation $\sum_{i=1}^{n} f(x_{i}) = f(\prod_{i=1}^{n} x_{i})$ for $x_{i} \in \mathbb{R}_{+}^{*}$. Provide a justification for your answer.

Answer: f(x) = C \log x


Question: Prove or disprove that for every positive integrable function $f: M \to \mathbb{R}$ over a relatively compact domain $M$, the inequality $A(f) \geq G(f)$ holds, where $A(f) = \frac{1}{\mu(M)}\int_M f$ is the arithmetic mean and $G(f) = \exp\left(\frac{1}{\mu(M)}\int_M \log f\right)$ is the geometric mean.

Answer: A(f) \geq G(f)


Question: Evaluate the complex contour integral \( \frac{1}{2 \pi i} \int_{\mid z \mid = 1} \frac{6z^{98}}{23z^{99}-2z^{81}+z^4-7}dz \) using the residue theorem.

Answer: \dfrac{6}{23}


Question: Does the existence of the limit \(\lim_{h\to0}\dfrac{f(x+h)+f(x-h)-2f(x)}{h}\) for all \(x\in\mathbb{R}\) imply that \(f\) is twice differentiable on \(\mathbb{R}\)? Answer yes or no.

Answer: No


Question: Evaluate the limit: \[ \lim_{n \to \infty} n\int_0^1 \frac{x^n}{4x + 5} \, dx \] where the sequence \((I_n)_{n \geq 1}\) is defined by \(I_n = \int_0^1 \frac{x^n}{4x + 5} \, dx\) and is known to be decreasing and convergent to 0.

Answer: \dfrac{1}{9}


Question:
Consider a simple random walk on a square-shaped graph with vertices labeled A, B, C, and D. The payoff function is given by:

\[f(A) = 2, \quad f(B) = 4, \quad f(C) = 5, \quad f(D) = 3\]

There is no cost for moving between vertices, but there is a discount factor \(a\). Determine the largest value of \(a\) for which the optimal stopping strategy is to stop at all vertices, i.e., the set \(S_2 = \{A, B, C, D\}\).

Answer: \dfrac{4}{7}


Question: Find the local or global extrema of the function \( f(x_1, \ldots, x_n) = (1+x_1)\cdots(1+x_n) \) using the method of Lagrange multipliers, given the constraint \( x_1\cdots x_n = a^n \), where \( a > 0 \).

Answer: (1 + a)^n


Question: Determine whether the following statement is true or false: If the real part of a complex function is differentiable at a point, then the function is complex-differentiable at that point.

Answer: B


Question: In a convex quadrilateral $ABCD$, the diagonals intersect at point $E$. Given $AE=2$, $BE=5$, $CE=6$, $DE=10$, and $BC=5$, find the area of the quadrilateral.

Answer: 48


Question: Find a function of a natural number \( M \) that yields an odd square, such as 9 or 25, only when \( M \) is a triangular number.

Answer: 8M + 1


Question: Given a linear operator \( \psi : M \to M \) with minimal polynomial \( m_\psi (x) = (x-2)^2x^5 \), find the minimal polynomial of \( \psi^2 \).

Answer: (x - 4)^2 x^3


Question: Determine whether there exists a Galois extension $K$ of the field $\mathbb{Q}(i)$ such that the Galois group $\mathrm{Gal}(K/\mathbb{Q}(i))$ is isomorphic to the dihedral group $D_8$, which has 8 elements. Use the Fundamental Theorem of Galois Theory and consider the properties of the Galois group in your reasoning.

Answer: Yes


Question: If a function $f$ is continuous on the entire complex plane $\Bbb C$ and analytic on the complex plane excluding the boundary of the unit disk $\delta B_1(0)$, is $f$ necessarily analytic on $\Bbb C$?

Answer: Yes


Question: Evaluate the integral \( \int_{C_3} e^{X(z+iz-z^2/2)} dz \) as \( X \rightarrow \infty \) along the path \( z=\infty+iy, y\in [0,1] \).

Answer: 0


Question: Let $X_1, X_2, \ldots, X_J$ be independent, identically distributed random variables with finite expectation. If $\mathbb{P}(X_1 > a) \geq \delta$, does it imply that $\mathbb{P}\left (\frac{1}{J} \sum_{i=1}^J X_i > a \right ) \geq \delta$? Answer yes or no and justify your answer.

Answer: No


Question: Determine the condition under which the matrix expression \((P+Q-PQ-QP)\) is a projection matrix, given that \(P\) and \(Q\) are projection matrices.

Answer: PQ = QP


Question: Let $f(x) = x^4$ be approximated by a polynomial of degree at most 2 that interpolates $f(x)$ at $x = -1, 0, 1$. Determine the maximum absolute interpolation error over the interval $[-1, 1]$. Provide your answer as a numerical value.

Answer: \dfrac{1}{4}


Question: Let $\alpha$ be a root of the irreducible polynomial $X^3 + X^2 + 1$ over the finite field $\mathbb{F}_2$. Consider the field extension $K = \mathbb{F}_2(\alpha)$, which has 8 elements. Suppose $f \in K[X]$ is an irreducible polynomial of degree 4, and let $\beta$ be a root of $f$. Let $L$ be the splitting field of $f$ over $K$. Determine the number of elements in $L$ and the number of intermediate fields between $L$ and $K$. Provide your answer in the form $\boxed{(a, b)}$, where $a$ is the number of elements in $L$ and $b$ is the number of intermediate fields.

Answer: (4096, 3)


Question: Is there an element in the matrix algebra over the quaternions, $M(\mathbb{H})$, that cannot be expressed as a combination of a left multiplication map and a right multiplication map?

Answer: Yes


Question: Calculate the limit: $$ \lim_{x\to0}{\frac{x-\arcsin\left(x\right)}{\sin^{3}(x)}} $$ without using L'Hôpital's rule.

Answer: -\dfrac{1}{6}


Question: Let \( f(x) = \frac{1}{(1+x)(1+x^2)(1+x^4)} \). Find the coefficient of the term \( x^{50} \) in \( (f(x))^3 \).

Answer: 84


Question: Given an angle $AOB$ less than $180^\circ$ and an interior point $P$ of the angular region of $\angle AOB$, construct a line segment $CD$ using only a ruler and compass such that $C$ lies on ray $OA$, $D$ lies on ray $OB$, and $CP : PD = 1 : 2$. Provide the steps for the construction.

Answer: CD


Question: Given the transformation matrix \( M_B^B(\phi) = \begin{pmatrix} 1 & -1 & 3 & -1 & 1 \\ 0 & 0 & -2 & 9 & -4 \\ 0 & 0 & -1 & -3 & 18 \\ 0 & 0 & 0 & -2 & -4 \\ 0 & 0 & 0 & 0 & -3 \end{pmatrix} \) with respect to the basis \( B = (1, x, x^2, x^3, x^4) \), determine the basis for the kernel of the linear transformation \( \phi \).

Answer: \{1 + x\}


Question: Calculate the limit \(\lim_{n \to \infty} (n!e - \lfloor n!e \rfloor)\). Determine if the result is \(\infty\).

Answer: 0


Question: Evaluate \( \cos(z) \) where \( z = i \log(2+\sqrt{3}) \).

Answer: 2


Question: Determine whether every universal class of structures is closed under the formation of direct products. Answer true or false.

Answer: False


Question: Find the sum of the infinite series: $$ \frac{2}{1!}+\frac{2+4}{2!}+\frac{2+4+6}{3!}+\frac{2+4+6+8}{4!}+\cdots $$

Answer: 3e


Question: Calculate the residue of the function \( f(z)=\frac{1}{1+9z^2} \) at the pole \( z=\frac{i}{3} \).

Answer: -\dfrac{i}{6}


Question: Calculate the residue of the function \( \frac{1}{\cos^2 z} \) at the pole \( z = \frac{\pi}{2} + \pi k \), where \( k \) is an integer.

Answer: 0


Question: Consider a projective surface $X$ over $\mathbb{C}$ and a rank $2$ vector bundle $E$ on $X$. Assume that for a general global section $s \in H^0(X, E)$, the zero locus $Z(s)$ has codimension $2$, and $H^0(X,E)$ has dimension $\geq 2$. Is the vector bundle $E$ generically globally generated?

Answer: Yes


Question: Calculate the value of the summation \( \sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k} \).

Answer: H_n


Question: Find the greatest common factor (GCF) between the polynomials \((x+1)^{4n+3} + x^{2n}\) and \(x^3 - 1\), given that the roots of \(x^3 - 1\) are \(1\), \(\frac{1+i\sqrt{3}}{2}\), and \(\frac{1-i\sqrt{3}}{2}\).

Answer: x^2 + x + 1


Question: What is the probability that Sisyphus must labor forever, given that each rock he places on Mount Olympus is either vaporized with probability 10%, rolled back into the valley with probability 50%, or split into two rocks that roll back into the valley with probability 40%? Assume he starts with one rock in the valley and none on Mount Olympus.

Answer: \dfrac{3}{4}


Question: Evaluate the second-order partial derivative \( \frac{\partial^2 f}{\partial x \partial y}(0,0) \) for the function \( f(x,y)= \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} &\text {if }(x,y) \neq (0,0) \\ 0 & \text{otherwise}\end{cases} \).

Answer: 1


Question: Suppose that \(f: (0,\infty) \rightarrow (0,\infty)\) is uniformly continuous. Determine whether the following limit holds: \[\lim_{x \rightarrow \infty} \frac {f(x+\frac{1}{x})}{f(x)}=1\]

Answer: 1


Question: On the Argand plane, let \( a, b, c, \) and \( d \) represent the complex numbers corresponding to the points \( A, B, C, \) and \( D \) respectively, all of which lie on a circle with center at the origin. The chord \( AB \) is perpendicular to the chord \( CD \). Find the value of \( ab + cd \).

Answer: 0


Question: Find the minimum value of the function \( f(x) = \frac{9x^2(\sin^2x) + 4}{x\sin x} \) for \( x \in (0, \pi) \), and identify the condition under which this minimum occurs.

Answer: 12


Question: Determine the order of the group $\mathrm{Hom}(\mathbb{Z}_m,\mathbb{Z}_n)$ as a $\mathbb{Z}$-module.

Answer: \gcd(m, n)


Question:
Determine the maximum value of the sum of squared distances between ten points $P_1, P_2, \ldots, P_{10}$ on the unit circle, given by:
\[ \sum_{1 \le i < j \le 10} |P_i - P_j|^2. \]

Answer: 100


Question: Determine the number of real roots for the equation \( ax^2 + b|x| + c = 0 \), given that the quadratic equation \( ax^2 + bx + c = 0 \) has two real roots, \( m \) and \( n \), where \( m > 0 \), \( n < 0 \), and \( |m| < |n| \).

Answer: 2


Question: Find the volume of the standard $n$-simplex $S$ in $\mathbb{R}^n$ with base length $a > 0$, where $$S=\{(x_1, \ldots, x_n) \in \mathbb{R}^n\mid x_i \ge 0, \sum_{i=1}^n x_i \le a\}.$$

Answer: \dfrac{a^n}{n!}


Question: Determine whether it is possible to have an inequality of the form \( \|u-u_\epsilon\|_{L^2(\Omega)} \leq a_\epsilon \|u\|_{H^1(\Omega)} \) for a Sobolev function \( u \in H^1_0(\Omega) \) and its mollified version \( u_\epsilon \) in a bounded domain \( \Omega \subset \mathbb{R}^n \) with smooth boundary, where \( a_\epsilon \) converges to 0 as \( \epsilon \) approaches 0.

Answer: Yes


Question: Evaluate the limit: $$ \lim_{x\to\infty} e^{-x} \int_1^x \frac{e^t}{t} dt $$

Answer: 0


Question: Determine if the remainder is zero when the $S$-polynomial $S(f,g) = x_1^2x_2 - x_3$ is divided by the set of polynomials $\{f = x_3 - x_1^5, g = x_2 - x_1^3\}$ using Buchberger's algorithm with a lexicographic order where $x_1 > x_2 > x_3$. Note that $S(f,g)$ can be expressed as $-f + x_1^2g$. Does this imply the remainder is $0$?

Answer: 0


Question: Given a differentiable function $f: \mathbb{R} \to \mathbb{R}$ with $f(0) = 0$ and satisfying the inequality $(1+x^2)f'(x) \geq 1 + f^2(x)$, find the limit $\displaystyle \lim_{x\to \infty} f(x)$. \( \text{Provide your answer in the form of a number or expression.} \)

Answer: \infty


Question: Find the degree of the field extension \(\mathbb{Q}\Big(\sqrt[9]{3},\exp\left(\frac{2\pi i}5\right)\Big)\) over \(\mathbb{Q}\).

Answer: 36


Question: Find the derivative of the vector expression $A^Tx$ with respect to the matrix $A$, where $A \in \mathbb{R}^{n \times m}$ and $x \in \mathbb{R}^n$. Express your answer as $\frac{d(A^Tx)}{dA}$.

Answer: I_m \otimes x^T


Question: Evaluate the integral: $$\int_{0}^{\pi/2} \frac{\sin x \cos^5 x}{(1-2\sin^2x\cos^2x)^2}dx$$

Answer: \dfrac{\pi}{8}


Question: Let $S: U \rightarrow U$ be a linear transformation such that $S^3 = S$. Determine whether there exists a basis of $U$ consisting of eigenvectors of $S$.

Answer: Yes


Question: Let $X$ be a Banach space and $T:X \to X$ be a bounded bijective operator. Is $T^{-1}$ also bounded?

Answer: Yes


Question: Evaluate the limit \( \lim\limits_{x\to 0} \frac{e^x-1-x-\frac{1}{2}x^2}{x-\sin{x}} \) using Taylor's Theorem.

Answer: 1


Question: Determine the maximal number $k$ of linearly independent homogeneous quadratic forms $Q_1, \dots, Q_k$ in 5 variables such that the intersection $V(Q_1) \cap \dots \cap V(Q_k) \subseteq \mathbb{P}^4$ has a connected component that is positive dimensional and does not lie on a hyperplane.

Answer: 6


Question: Find the mass of the solid \( E \) that lies below the plane \( z=8 \) and above the cone \( z=\sqrt{x^2+y^2} \), given that the density function is \( \rho(x,y,z)=z \). Use triple integrals in cylindrical coordinates to solve the problem.

Answer: 1024\pi


Question: Calculate the residue of the function \( g(z) = \frac{1}{z^2 - \cosh(z)} \) at \( z = 0 \).

Answer: 0


Question:
Find the coefficient of $x^{15}$ in the expansion of the generating function:
$$G(x)=\frac{(x^3-x^{9})(1-x^{6})^2}{(1-x)^3}$$

Answer: 10


Question: Given the vectors $x_1=(1,-1,1,0)^T$, $x_2=(1,1,0,1)^T$, $x_3=(1,3,1,0)^T$, and $x_4=(1,1,1,0)^T$, determine the rank of the matrix $A = \sum_{i=1}^4 x_ix_i^T$ without explicitly computing $A$. Note that $x_1 = 2x_4 - x_3$. What is the rank of $A$?

Answer: 3


Question: Find the equation of the tangent line to the curve $y = g(u(x), v(x))$ at $x = 1$, given that $(u'(1), v'(1)) = (3, 1)$ and the tangent plane to the surface $z = g(u, v)$ at the point $(1, 2)$ is $4 - z - 2u + 5v = 0$. Assume $g(u, v)$ and $g(u(x), v(x))$ are related functions.

Answer: y = -x + 13


Question:
Which of the following interpretations correctly defines a "bounded process" in the context of a bounded, measurable, and \(\mathcal{F}_t\)-adapted process \(X\)?

1. There exists \(M>0\) such that for all \(t\) and almost every \(\omega \in \Omega\), \(\mathbb{P}\{|X_t(\omega )|\leq M\}=1\).
2. There exists \(M>0\) such that for all \(t\) and all \(\omega \in \Omega\), \(\mathbb{P}\{|X_t(\omega )|\leq M\}=1\).
3. For every \(\omega \in \Omega\), there exists an \(M(\omega )\) such that \(|X_t(\omega )|\leq M(\omega )\) for all \(t\).

Answer: 1


Question: Given an uncountable ordered set $\Theta$, determine if it is always possible to construct a sequence $(x_n)$ such that $x_1 < x_2 < \ldots < x_n < \ldots$ and for each $x \in \Theta$, there exists an $n$ such that $x_n > x$. Answer "yes" or "no".

Answer: no


Question:
Given a triangle with side lengths $a, b, c$ such that $a + b + c = 3$, find the minimum value of the expression:
\[a^8 + b^8 + c^8 + 2(a - 1)(b - 1)(c - 1)\]

Answer: 3


Question: Calculate the limit \( \lim_{m \to \infty} \left(\frac{2m^2 + m + 4}{2m^2 + 3m + 5}\right)^{3m-3} \) and express it in terms of \( e \).

Answer: e^{-3}


Question: Determine the number of terms, n, that need to be summed in the Maclaurin series for the arctangent function to ensure that \(|4P_n(1) − π| < 10^{-3}\), given that \(\tan \frac{π}{4} = 1\).

Answer: 2000


Question: In a category, if there is a unique morphism $f: A \to B$ and a unique morphism $g: B \to A$, does it follow that $f$ and $g$ are isomorphisms?

Answer: No


Question: A rubber band is initially 1 meter long, with one end fixed to a wall. The other end is pulled away from the wall at a constant speed of 1 m/s. A bug starts at the wall and crawls along the rubber band toward the moving end at a speed of 0.001 cm/s. Determine if the bug will ever reach the end of the rubber band, and if so, calculate the time it takes for the bug to reach the end.

Answer: e^{100000} - 1


Question:
Find a condition on the real numbers \(a, b, c\) such that the following equality holds:

\[ \alpha = \beta \]

where:

\[ \alpha = \sqrt{a} + \sqrt{2b + 2\sqrt{a}} \]
\[ \beta = \sqrt{b + 2\sqrt{c}} + \sqrt{a + b - 2\sqrt{c} + 2\sqrt{ab - 2a\sqrt{c}}} \]

and \(a = b^2 - 4c\). Assume all square roots are well-defined.

Answer: a = b^2 - 4c


Question: Given a real matrix $A$ of full rank, for which the matrix $A^TA$ is positive definite, determine if there exists an efficient algorithm to compute the Cholesky decomposition $A^TA = L^TL$ directly in terms of $A$ without recursion, and without explicitly forming $A^TA$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: For any connected graph $G$ with diameter less than or equal to two, there exist non-adjacent vertices $u$ and $v$ such that $\deg(u) + \deg(v) < n-1$, where $n$ is the order of $G$. Provide a proof or counterexample.

Answer: False


Question: Compute the limit: \(\lim\limits_{x\to0} \tan^{-1}\left(\frac{1}{x^2}\right)\).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0} \left( \frac{1}{x} - \frac{1}{e^x-1} \right) \] without using Taylor, Maclaurin series, or L'Hôpital's Rule.

Answer: \dfrac{1}{2}


Question: Calculate the probability of being in state $3$ after $500,000$ steps in a Markov chain, given that the chain starts in state $1$. Assume the transition probabilities are defined as in the original problem.

Answer: \dfrac{1}{3}


Question: Find the remainder when \(2^{1990}\) is divided by 1990.

Answer: 1024


Question: Evaluate the limit: $$\lim_{x\to 0}\frac{x-\sin(\sin(\overbrace {\cdot \ \cdot \ \cdot }^n(\sin(x))\overbrace {\cdot \ \cdot \ \cdot }^n))}{x^3}$$

Answer: \dfrac{n}{6}


Question:
Evaluate the limit:
\[ \lim_{n \to \infty} \left( \frac{\sqrt[n+1]{(n+1)! (2n+1)!!}}{n+1} - \frac{\sqrt[n]{n!(2n-1)!!}}{n} \right) \]
where \( n! \) is the factorial and \((2n-1)!!\) is the double factorial. Use Stirling's formula, \( n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \), to simplify the expression and determine the limit.

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


Question: Evaluate the limit using L'Hôpital's rule: $$\lim_{x\to0}\frac{e^{-\frac{1}{x^2}}}{x}$$

Answer: 0


Question: Calculate the remainder when \( \sum_{k=1}^{12} k \cdot k! + 3 \) is divided by \( 13! \).

Answer: 2


Question: Evaluate the integral \( I = \int_{0}^{\pi} \sin(x)\cos(x)\cos(2x)\cos(4x)\dots \cos(2^{n-1}x) \, \mathrm{d}x \) and determine if the application of the property \( \int_{a}^{b} f(x) \, \mathrm{d}x = \int_{a}^{b} f(a + b - x) \, \mathrm{d}x \) results in \( I = 0 \).

Answer: 0


Question: Find all positive integer solutions (x, y) to the equation \(1 + 2^x + 2^{2x+1} = y^2\).

Answer: (4, 23)


Question: Calculate the sum of the infinite series \( \sum\limits_{x=1}^\infty x^2\cdot\left(\frac{1}{2}\right)^{x+1} \).

Answer: 3


Question: Calculate the value of the definite integral \( \int_{0}^{1} \left(1 + x + x^2 + x^3 + \ldots + x^{n-1}\right) \left(1 + 3x + 5x^2 + \ldots + (2n-1)x^{n-1}\right) \, dx \).

Answer: n^2


Question: Given a function $h_0:\Bbb{N}\rightarrow\Bbb{N}$ and the recursive definition $h_{r+1}(m)=h_r(m)-h_r(m-1)$, if $h_k(m)$ is a polynomial of degree $d$ for some $k>0$, determine whether $h_0$ must be a polynomial of degree $d+k$. Answer true or false.

Answer: True


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac{\sum_{i=1}^n \log i}{n\log n}$$

Answer: 1


Question: Let $a$, $b \in \mathbb{N}$ with $3 < a < b$. Suppose all prime divisors of $a$ divide $b$ and all prime divisors of $b$ less than $a$ also divide $a$. Determine whether there always exists a prime $p$ such that $a < p < b$ and $p \nmid b$. Provide a justification for your answer.

Answer: Yes


Question: Let $Z_1$ and $Z_2$ be independent identically distributed standard Gaussian random variables with mean 0 and variance 1. Determine the distribution of the random variable \( X = \frac{(Z_1 + Z_2)^2}{(Z_1 - Z_2)^2} \).

Answer: F(1,1)


Question: Evaluate the double limit: \(\lim_{n\to\infty} \lim_{x\to 1^+} \frac{\ln(n!)}{n^x}\).

Answer: \infty


Question: Evaluate the contour integral \( \oint_C (4z - 1) \, dz \) along the quarter-circle \(|z| = 1\) from the point \(z_1 = 0\) to \(z_2 = 1\) in the first quadrant.

Answer: 1


Question: Calculate the limit of the function as \( n \) approaches infinity: \[ (1+n^{2})^{\frac{1}{\ln(n)}} \] Use the hint that the logarithm of the function can be written as \( \frac{\log(1+n^2)}{\log(n)} \).

Answer: e^2


Question: Determine whether there exists a function \( f: \mathbb{R} \to \mathbb{R} \) that is differentiable at exactly one point.

Answer: Yes


Question: Find the greatest value of \( n \) such that the equation \( 2\tan^2x - 5\sec x = 1 \) has exactly 7 distinct solutions for \( x \in [0, \frac{n\pi}{2}] \), where \( n \) is a natural number.

Answer: 15


Question: Evaluate the integral \( \int_{1}^{3} \frac{x^2+1}{\sqrt{x^3+3x}} \, dx \).

Answer: \dfrac{8}{3}


Question:
Evaluate the double integral of the function \( f(x,y) \) given by:

\[ f(x,y) = \begin{cases} \beta^2e^{-\beta y}, & 0 \leq x \leq y, \quad \text{with} \quad \beta > 0\\ 0, & \text{otherwise} \end{cases} \]

Find the value of:

\[ \int_0^\infty \int_0^\infty f(x,y) \, dx \, dy \]

Answer: 1


Question: Solve the equation: $$\log(x)+\log\left(\sqrt[3]{x}\right)+\log\left(\sqrt[9]{x}\right)+\log\left(\sqrt[27]{x}\right)+\ldots=6$$ Find the value of \(x\).

Answer: 10000


Question: Find the volume of the solid generated when the region bounded by the curve \( y = -\ln(x) \) and the x-axis on the interval \([0, 1]\) is rotated about the x-axis.

Answer: 2\pi


Question: Determine whether the following statement is true or false: For any non-empty subsets $S$ and $T$ of $\mathbb{R}^n$, if $S$ is compact, $T$ is closed, and $S \cap T = \emptyset$, then the distance between $S$ and $T$ is strictly greater than 0.

Answer: A


Question: Given the linear transformation $T:V\to V$ defined by $T(X) = AX + XA$, where $A = \begin{pmatrix}1&1&0\\0&2&0\\0&0&-1 \end{pmatrix}$, find the determinant $\det(T)$.

Answer: 0


Question: Given four positive integers \(a < b < c < d\) such that \(a + d = b + c = n\), determine if there exist integers \(p, q, r, s\) such that \(a = p + q\), \(b = p + r\), \(c = q + s\), and \(d = r + s\).

Answer: Yes


Question: Evaluate the limit \( \lim_{w\to 0} \frac{\cos (2m+1)w}{\cos w/2} \) as \( m \) approaches infinity.

Answer: 1


Question: Find the splitting field of $\alpha = \zeta_7 + \zeta_7^{-1}$ over $\mathbb{Q}$, given that the minimal polynomial of $\alpha$ over $\mathbb{Q}$ is $x^3 + x^2 - 2x - 1$. Determine the degree of the extension $\mathbb{Q}(\alpha)$ over $\mathbb{Q}$ and describe the splitting field.

Answer: 3


Question: If $A$ and $B$ are two normal operators such that $A + B = AB$, does it follow that $AB = BA$?

Answer: AB = BA


Question: Let \(\mathbf{A}\) be an \(s \times r\) block matrix and \(\mathbf{B}\) be an \(r \times q\) block matrix over a ring, where each entry \(A_{ij}\) and \(B_{ij}\) is a sub-matrix. If the block matrix multiplication \(\sum_{k=1}^{r} \left[\begin{array}{ccc} A_{1k}B_{k1} & A_{1k}B_{k2} & \dots & A_{1k}B_{kq} \\ A_{2k}B_{k1} & A_{2k}B_{k2} & \dots & A_{2k}B_{kq} \\ \vdots & \vdots & \ddots & \vdots \\ A_{sk}B_{k1} & A_{sk}B_{k2} & \dots & A_{sk}B_{kq} \end{array}\right]\) is valid, is it equal to \(\mathbf{AB}\)?

Answer: Yes


Question: Provide an example of a non-closed subspace of the Banach space $l^1$. Express your answer clearly and justify why the subspace is not closed.

Answer: c_{00}


Question: Evaluate the limit: $$\lim_{h\rightarrow0}\frac{1}{2h}\int_{t-h}^{t+h}f(s+h)ds$$ where $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function and $t\in\mathbb{R}$. Provide your answer in terms of $f$ and $t$.

Answer: f(t)


Question: Determine the Galois group of the polynomial \( f(x) = x^3 + 2x + 2 \) over the field \( \mathbb{Q}(i\sqrt{19}) \).

Answer: S_3


Question: Determine the values of $r$ for which the set $S_r = \{ \alpha \in \mathbb{C}^* : |\alpha| = r \}$ forms a subgroup of the multiplicative group of non-zero complex numbers $\mathbb{C}^*$. \(r\) is a positive real number.

Answer: 1


Question: If $f$ is uniformly continuous on $\mathbb{R}$, does there exist a constant $C$ such that \(\lim_{|x-y|\to\infty} (f(y)-f(x)-C|x-y|)=0\)? Answer yes or no.

Answer: No


Question: Determine whether the sequence $(Y_n)_n$ defined by $Y_{n+1} = \alpha + \beta Y_n$ or $Y_{n+1} = \beta Y_n$ with probabilities $Y_n$ and $1-Y_n$ respectively, where $Y_0 = \gamma$, $\alpha$, $\beta > 0$ and $\alpha + \beta = 1$, converges in $L_1$.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to\infty} n\int_{0}^{\pi/4} \tan^n(x) \,{\rm d}x$$

Answer: \dfrac{1}{2}


Question: In triangle $ABC$, where $AC = 3$, $BC = 4$, and $\angle C = 90^\circ$, point $P$ moves in the plane of triangle $ABC$ such that $PC = 1$. Determine the range of values for the dot product $\vec{PA} \cdot \vec{PB}$. \(\text{Express your answer as an interval.}\)

Answer: [-4, 6]


Question: Find the center of mass of a thin sheet in the shape of the cone \(z = \sqrt{x^2 + y^2}, z \le 4\), with density function \(\rho(x, y, z) = z\).

Answer: (0, 0, 3)


Question: If a square matrix $B$ has integer elements and an inverse $B^{-1}$ with integer elements, what possible values can the determinant of $B^2$ have?

Answer: 1


Question: Evaluate the limit using L'Hôpital's rule: $$\lim_{x \to 0^+} x^{x^x - 1}$$

Answer: 1


Question: Consider a non-increasing sequence of convex sets \(\{S_k\}_{k=1}^\infty\) in a Euclidean space, where \(S_{k+1} \subset S_k\) for each \(k \in \mathbb{N}\). If the limit of this sequence as \(k\) approaches infinity is denoted by \(S\), is \(S\) a convex set?

Answer: Yes


Question: Can every real, symmetric matrix be orthogonally diagonalized as \( A = QDQ^T \), where \( Q \) is an orthogonal matrix?

Answer: Yes


Question: Find a possible expression for the differentiable function $g(x)$ given that \( \int g(x)e^{\frac{x}{4}}dx=4g(x)e^{\frac{x}{4}}-\int 8x^2e^{\frac{x}{4}}dx \).

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


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

Answer: -\infty


Question: Determine whether the series \( \lim\limits_{n \to \infty} \frac{\sum_{k=1}^{n} \left(\frac{1}{k+1 }\right)}{\log(n+1)} \) converges.

Answer: 1


Question: Given the function \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that \( f(f(x+y)) = f(x^2) + f(y^2) \), and \( f(f(2020)) = 1010 \), determine the value of \( f(2025) \).

Answer: 505


Question: Determine whether the limit \( \lim_{a \rightarrow 0^+}a \int_{b}^{-\log a}\frac{e^y}{y} \, dy \) exists. If it exists, find its value.

Answer: 0


Question: Evaluate the limit of the function \( f(x,y) = \frac{e^{-\frac{1}{x^2+y^2}}}{x^6+y^6} \) as \((x,y)\) approaches \((0,0)\).

Answer: 0


Question: If for every sentence $\varphi$, $M \models \varphi$ if and only if $N \models \varphi$, are the models $M$ and $N$ necessarily isomorphic?

Answer: No


Question: In metric spaces $(X_1,d_1)$ and $(X_2,d_2)$, consider the product space $X_1 \times X_2$ equipped with a metric naturally related to $d_1$ and $d_2$. Does a Cauchy sequence in this product space imply that the sequences in each individual space are also Cauchy?

Answer: Yes


Question: Given that $f(z)$ is a holomorphic function in a disk centered at $0$ with radius $r$, and $b$ is a point inside this disk, find the relationship between the integral $\int_{\gamma} \frac{f(z)}{z-b}dz$ and $f(b)$, where $\gamma(t) = re^{it}$ for $0 \leq t \leq 2\pi$. Use the Cauchy Integral Formula to express the integral in terms of $f(b)$.

Answer: 2\pi i f(b)


Question:
Given that $a + b + c = 0$, find the value of the expression:

\[ L = \frac{a^2}{2a^2 + bc} + \frac{b^2}{2b^2 + ac} + \frac{c^2}{2c^2 + ab} \]

Answer: 1


Question: Prove that \(\lim_{x \to \infty} \frac{e^{e^{x}}}{x!} = \infty\).

Answer: \infty


Question: Find the correct value of the expression \(\sqrt{i} + \sqrt{-i}\).

Answer: \sqrt{2}


Question: Evaluate the expression \( \frac{\int_0^\pi x^3\ln(\sin x)\,dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x))\,dx} \).

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


Question: Given a non-constant, thrice-differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f\left(1 + \frac{1}{n}\right) = 1$ for all integers $n$, and $f(1) = 1$, find $f''(1)$.

Answer: 0


Question: Calculate \( \iint_{S} \text{curl} \vec{F} \cdot d\vec{S} \) for \( \vec{F} = \begin{pmatrix} y^2z^3 \\ 2xy \\ x^2z \end{pmatrix} \), where \( S \) is the surface of the paraboloid \( z = x^2 + y^2 \) that lies inside the cylinder \( x^2 + y^2 = 4 \), oriented with an upward-facing normal.

Answer: 0


Question: Evaluate the limit: \[ \lim_{z\rightarrow 0}\left(\frac{e^z-1}{z}\right)^{1/z} \]

Answer: \sqrt{e}


Question: Evaluate the double integral \( \int_0^1 \int_0^1 f(x,y) \, dx \, dy \) for the function \( f(x,y) = \sum_{n=1}^\infty [g_n(x) - g_{n+1}(x)]g_n(y) \), where \( g_n \) are continuous functions with support in \((\delta_n, \delta_{n+1})\) and satisfy \( \int_0^1 g_n(t) \, dt = 1 \). Explain why the computation might incorrectly yield 0 and provide the correct evaluation.

Answer: 1


Question: Calculate the limit of the expression \(\frac{\ln{2^{\frac{1}{x}}}-\ln{x^2}}{1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{x}}\) as \(x\) approaches infinity.

Answer: -2


Question: Determine the order of a finite group $G$ where every non-identity element is a generator of $G$.

Answer: p


Question: Consider a bipartite graph that has a Hamiltonian path but does not have a Hamiltonian cycle. Is it possible for the partitions X and Y of this graph to have different sizes?

Answer: Yes


Question: Find the coefficient of $x^{20}$ in the expansion of $(x^3+2x^2+5x+3)^5(x^2+4x+5)^4$. Provide your answer as a single integer.

Answer: 2991


Question: Let $V$ be an $n$-dimensional vector space over the field $K$, and let $T: V \rightarrow V$ be a linear operator. Suppose there exists a vector $v_0 \in V$ that does not belong to any proper subspace of $V$ invariant under $T$. Can we conclude that the minimal polynomial of $T$ over $K$ has degree $n$?

Answer: Yes


Question: Find a polynomial $f(x)$ with real coefficients that satisfies the equation $$(x+1)f(x-1) = (x-2)f(x)$$ given that $f(-1) = 0$ and $f(1) = 0$. What is the polynomial $f(x)$?

Answer: x^3 - x


Question: Evaluate the surface integral \( \iint_S (y^2z^2 \textbf{i} \, +z^2x^2\textbf{j}+z^2y^2\textbf{k})\cdot\textbf{n} \,\mathrm{d}S \) where \( S \) is the part of the sphere \( x^2+y^2+z^2=1 \) that lies above the \( xy \) plane.

Answer: \dfrac{\pi}{12}


Question: Determine if all nontrivial elements of a nonabelian semidirect product of $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p$ have order $p$, where $p$ is a prime.

Answer: No


Question: Let $A$ be an $n \times n$ complex matrix whose characteristic polynomial has no repeated roots. Determine the number of $n \times n$ matrices over $\mathbb{C}$ that are both similar to and commute with $A$. Provide your answer as a single number.

Answer: n!


Question: Find the set of values of $b$ for which $y(x) = e^{bx}$ is a solution to the differential equation: $$x^2\frac{d^2y}{dx^2} + 2x\frac{dy}{dx} - 6y = 0, \quad x > 0.$$

Answer: \emptyset


Question: Find the value(s) of $x$ that satisfy the equation: $$\frac{x^{3} - 2x^{2} + x + 4}{x+1} = \frac{7}{4}\sin\left(\frac{\pi x}{3}\right)$$

Answer: \dfrac{3}{2}


Question: If a function $f$ from a planar domain $D$ to a Banach space $A$ is strongly continuous, does it imply that $f$ is weakly continuous? Answer yes or no.

Answer: Yes


Question: Find the limit of the function \( g(x) = \int_x^{x^2} \frac{1}{\ln t} \, dt \) as \( x \) approaches 1, given the inequalities \( t - \frac{t^2}{2} \leq \ln(1+t) \leq t \) and \( \frac{1}{t - \frac{t^2}{2}} = \frac{1}{t} + \frac{\frac{1}{2}}{1 - \frac{t}{2}} \) for \( x > 0 \) and \( x \neq 1 \).

Answer: \ln 2


Question: Given positive integers $a$ and $b$ with $\gcd(a,b)$ divisible by exactly 7 distinct primes and $\mathop{\text{lcm}}[a,b]$ divisible by exactly 28 distinct primes, where $a$ has fewer distinct prime factors than $b$, find the maximum number of distinct prime factors that $a$ can have.

Answer: 17


Question: Given a $4 \times 4$ real matrix $A$ with eigenvalues $1, -1, 2, -2$ that satisfies the equation $A^4 - 5A^2 + 4I = 0$, find the determinant of the matrix $C = 3A - A^{-1} + 2I$. \(A^{-1}\) denotes the inverse of matrix \(A\), and \(I\) is the identity matrix.

Answer: 0


Question: An image is divided into two regions: one white and one black. A reading from a point in the white region follows a normal distribution with mean $\mu_w = 4$ and variance $\sigma_w^2 = 4$. A reading from a point in the black region follows a normal distribution with mean $\mu_b = 6$ and variance $\sigma_b^2 = 9$. Determine the value of $x$ such that the probability of error is the same whether the point is concluded to be in the black region or the white region.

Answer: 4.8


Question: Find the value of the limit \( \lim_{x\to\infty}e^{-\left((\log(\frac{x+a}{\sigma}))^\nu-(\log(\frac{x}{\sigma}))^\nu\right)} \), where \(\nu\) and \(\sigma\) are positive constants.

Answer: 1


Question: Find the roots of the equation $e^{\tan(x)} + 3^{\sin(x)} = 2 - 2010x - \arctan(x)$ for $x \neq \frac{\pi}{2}(2k + 1)$, where $k$ is an integer.

Answer: 0


Question: How many $k$-simplices in a triangulation of the simplex $\sigma_k:=\{(x_1,...,x_{k+1}):x_1+\cdots+x_{k+1}=1, x_i\ge 0\}$ does a non-boundary $(k-1)$-simplex lie in?

Answer: 2


Question: Calculate the surface area of the solid formed by rotating the astroid given by the equation \(x^{2/3} + y^{2/3} = 1\) around the \(y\)-axis, for \(y\) in the interval \([0, 1]\).

Answer: \dfrac{6\pi}{5}


Question: Determine whether the following statement is true or false: The Henstock-Kurzweil integral extends the concept of integrability to include all differentiable functions.

Answer: B


Question:
Find all continuously differentiable functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), the following equation holds:

\[ f(y^2f(x) + x^2 + y) = x(f(y))^2 + f(y) + x^2 \]

Assume that the domain of \( f \) is connected and contains 0.

Answer: f(x) = x


Question: Suppose \( A \) is a Dedekind domain, \( K \) its field of fractions, \( L \) a finite, separable extension of \( K \), and \( B \) the integral closure of \( A \) in \( L \). Let \( J \) be any integral ideal of \( B \). Determine whether there exists a \( \theta \in B \) such that \( \theta \) is a primitive element (i.e., \( L=K(\theta) \)) and \( J \) is relatively prime to the conductor \( \mathfrak{f} = \{\alpha \in B|\alpha B \subseteq A[\theta] \} \) of \( A[\theta] \).

Answer: Yes


Question:
Find the equation form of a surface $z(x,y)$ that satisfies the following conditions:
- For any fixed $x$, $z$ can be expressed as $z = px + q$.
- For any fixed $y$, $z$ can be expressed as $z = r \ln(y) + s$, where $p, q, r, s$ are constants.

Answer: z = C


Question: Evaluate the limit: $$ \lim_{x \to \infty} \frac{\sqrt[5]{x^7+3} - \sqrt[4]{2x^3 - 1}}{\sqrt[8]{x^7 + x^2 + 1} - x} $$

Answer: -\infty


Question: Evaluate the limit: \[ \lim_{x\to\infty} \frac{(x-1)^{15}+(x-2)^{15}+\cdots+(x-100)^{15}}{x^{15}-10^{15}} \]

Answer: 100


Question: Is a homeomorphism from a topological space to its image always bijective? Provide a justification for your answer.

Answer: Yes


Question: Find 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: In a nonabelian group $G$ of order $pqr$, where $p$, $q$, and $r$ are distinct primes with $p < q < r$, determine the number of elements of order $r$.

Answer: r - 1


Question: Determine whether the set of elements in a division ring that commute with a given element forms a division ring. Answer with true or false.

Answer: True


Question: Find the value of $x$ such that $x^{17} \equiv 20 \pmod{1001}$. Provide your answer as the smallest non-negative integer solution.

Answer: 531


Question: Determine whether a complete metric can be defined on the open interval (0,1) that induces the standard subspace topology.

Answer: Yes


Question: Determine the form of the function $f(x)$ that satisfies the equation $f(x^2 + yf(x)) = f(0)$ for all real numbers $x$ and $y$.

Answer: f(x) = c


Question: Determine whether the minimal polynomial $P$ of a square matrix over a field $F$ can be factored as $P=Q(X-\lambda)^2$ for some $\lambda \in F$, where $Q$ is a non-zero polynomial.

Answer: Yes


Question: Calculate the sum of the infinite series: \[ \sum_{n=0}^\infty \frac{n^3}{2^n}. \]

Answer: 26


Question: If a sequence of functions converges uniformly to a discontinuous function, must all functions in the sequence be discontinuous?

Answer: No


Question: Let $f:[0,1]\to\mathbb{R}$ be a function in $L^1(0,1)$ such that $\int_0^t f(s) \, ds = 0$ for all $t \in [0,1]$. Does this imply that $f(t) = 0$ almost everywhere on $[0,1]$?

Answer: Yes


Question: Determine whether the ring \(\mathbb{Z}[x]/(23,x-2)\) is an integral domain or isomorphic to another ring.

Answer: \mathbb{Z}_{23}


Question:
Evaluate the limit:

\[ \lim_{n\to +\infty}\left[\frac{\frac{a_n}{s_n}}{-\ln(1-\frac{a_n}{s_n})}\right] \] where \( a_n = \sqrt{n} \) and \( s_n = a_1 + a_2 + \ldots + a_n \).

Answer: 1


Question: Does there exist an infinite $\sigma$-algebra in which every nonempty member has at least one nonempty proper measurable subset?

Answer: Yes


Question:
Find the real solutions of the following system of equations:
\[ x^3 - 6z^2 + 12z - 8 = 0 \]
\[ y^3 - 6x^2 + 12x - 8 = 0 \]
\[ z^3 - 6y^2 + 12y - 8 = 0 \]

Answer: (2, 2, 2)


Question: Evaluate the double integral \( \displaystyle \iint_{x^2+y^2\leq 1} (\sin x + y + 3)\,dA \) over a disk of radius 1.

Answer: 3\pi


Question: Find a simplified formula for the integral \( \int_a^\infty \Phi(a - x) f(x) \, dx \), where \( f(x) \) is the density of a normal distribution with mean \( a \) and variance \( 1 \), and \( \Phi \) is the standard normal cumulative distribution function.

Answer: \dfrac{1}{8}


Question: Calculate the limit of the sequence \( S_n \) defined by \[ S_n = \frac{1}{\sqrt{n}}\left(1+\frac{1}{\sqrt{2}}+\cdots +\frac{1}{\sqrt{n}}\right). \]

Answer: 2


Question: Determine whether the sequence of random variables $X_n$ with distribution $P(X_n = n) = \frac{1}{n}$ and $P(X_n = 0)= 1-\frac{1}{n}$ for $n \geq 1$ converges to 1 in $L^1$ as $n \rightarrow \infty$.

Answer: No


Question: Evaluate the limit: \[ \lim_{q\to\infty}\int_{0}^{\pi}\cos^{2q}{x}\,dx \] using the Laplace method.

Answer: 0


Question: Evaluate the integral \( \frac{1}{2\pi i} \oint_{|z|=1} \frac{\cos \left(e^{-z}\right)}{z^2} \, dz \) using the residue theorem.

Answer: \sin 1


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

Answer: \sqrt[4]{e^5}


Question: Is it possible for a generating set of a group to have fewer subsets than the number of subgroups of the group?

Answer: Yes


Question: Determine whether it is possible to color the edges of an $n$-dimensional hypercube using 3 colors such that every 2-dimensional face contains edges of all 3 colors.

Answer: Yes


Question: Find the number of connected components of the set \( Z(q) \), where \( q(z,w) = z^2 + w^2 - zw + 1 \).

Answer: 1


Question: For any subset $T \subseteq \{1, 2, \ldots, 20\}$, a number $m$ is called a median of $T$ if both $m$ and $m + |T|$ are elements of $T$. If $T$ is randomly chosen from all $2^{20}$ subsets of $\{1, 2, \ldots, 20\}$, what is the expected value of the number of medians of $T$?

Answer: \dfrac{9}{4}


Question:
Determine whether the following limit exists:
\[\lim _{ (x, y) \to (0,0) } \frac{ xy \sin y }{ x^2 + y^2 }\]

Answer: 0


Question:
Factorize the expression \(a(b^2-c^2) + b(c^2-a^2) + c(a^2-b^2)\) and find the coefficient \(m\) such that:
\[a(b^2-c^2) + b(c^2-a^2) + c(a^2-b^2) = m(a-b)(b-c)(c-a)\]

Answer: 1


Question: Determine the set of values of $a$ for which the improper integral \( \int_{0}^{\infty}\frac{t^a}{1+t}\,\mathrm{d}t \) converges.

Answer: (-1, 0)


Question: Compute the integral \( \iiint_V \nabla \cdot \vec{G} \, dV \) using Gauss's divergence theorem, where \( \vec{G} = (y^2z, x − yz^2, x^2) \) and \( V \) is the region bounded by the paraboloid \( z = x^2 + y^2 \) and the plane \( z = 4 \).

Answer: -64\pi


Question: Determine the number of subgroups of the group $C_3 \times C_3 \times C_3$ that are isomorphic to $C_9$. Provide your answer as a single integer.

Answer: 0


Question: Evaluate the limit: $$ \lim_{(x,y)\to(1,1)} \frac{(x-1)^{4/3}-(y-1)^{4/3}}{(x-1)^{2/3}+(y-1)^{2/3}} $$

Answer: 0


Question: Determine the number of natural numbers \( x \) for which the expression \( \left( x(x + 3) + 1 \right)^2 - 1 \) has exactly three different prime factors.

Answer: 3


Question: In a coin-tossing game, the game ends when the difference between the number of heads and tails is 3. What is the expected length of this game?

Answer: 9


Question:
Let \( f:\mathbb{R}\to\mathbb{R}^n \) be an absolutely continuous function such that \( |f(t)| \) is nonincreasing as \( t \to \infty \). Compute the limit:
\[ \lim_{t\to\infty}\dfrac{1}{t}\sup_{r\in[0,t]}|f(r)|^2 \]

Answer: 0


Question: Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain. Let $f\in L^{1}\left(\Omega\right)$ and a sequence $\left\{ f_{k}\right\} $ such that $f_{k}\rightarrow f$ uniformly on each compact subset of $\Omega.$ Assume $\intop_{\Omega}f_{k}=1$ for any $k.$ Can we conclude that $\intop_{\Omega}f=1$?

Answer: No


Question: Determine whether it is possible to define a group operation on the set $G = \{0, 1, 2, 3, 4, 5, 6, 7\}$ such that $a \ast b \leq a + b$ for all $a, b \in G$ and $a \ast a = 0$ for all $a \in G$. Provide a justification for your answer.

Answer: Yes


Question: Given that \( \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = -4 \), find the determinant of the matrix \( \begin{vmatrix} 2a_3 & 2a_2 & 2a_1 \\ b_3-a_3 & b_2-a_2 & b_1-a_1 \\ c_3+3b_3 & c_2+3b_2 & c_1+3b_1 \end{vmatrix} \).

Answer: 8


Question: Given points $A(0,0,2)$ and $B(3,4,1)$ in the coordinate system $Oxyz$, find the minimum value of $AX + BY$ where $X$ and $Y$ are points in the $Oxy$ plane such that the distance $XY = 1$. Provide your answer as a single number.

Answer: 5


Question:
Derive the maximum likelihood estimator for the uniform distribution $\mathcal{U}(\theta, 0)$, where $\theta>0$, using the likelihood function:
$$
L(\theta) = \begin{cases} \dfrac 1 {\theta^n} & \text{if }\theta\ge Y_n, \\
0 & \text{if }\theta < Y_n. \end{cases}
$$
Here, $Y_n=\max_{1 \leq k \leq n}\{x_k \}$. What is the maximum likelihood estimator for $\theta$?

Answer: Y_n


Question: Evaluate the integral: $$ \int\limits_{-\infty}^\infty \left[ e^{-at^3} \delta(t-10) + \sin (5 \pi t) \delta(t) \right] \mathrm{d}t $$

Answer: e^{-1000a}


Question: Evaluate the triple integral \( \iiint_S x^2 \, dx\, dy\, dz \), where \( S \) is the region bounded by \( 4x^2 + y^2 = 4 \), \( z + x = 2 \), and \( z = 0 \).

Answer: \pi


Question: Evaluate the limit: \[ L = \lim_{n\to\infty} e^{\frac{n}{4}}n^{-\frac{n+1}{2}}\left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n}} \]

Answer: \sqrt{2\pi}


Question: Evaluate the integral \( \int u^{p-1} e^{-u} \, du \) and determine if the result is \( \Gamma(p) \) or \( -\Gamma(p) \).

Answer: \Gamma(p)


Question: Compute the determinant of the matrix \( M := \text{diag} (1, 2, \dots, n) - n \, I_n + n \,1_n 1_n^T \).

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


Question: Evaluate the limit: $$\lim_{x\to-\infty} \frac{\int_0^x e^{t^2} dt}{\sqrt{-\pi x}}$$

Answer: -\infty


Question: Compute the limit: \(\lim\limits_{x\to \infty} \left(|x|^\beta-|x-c|^\beta\right)\), where \(c\in\mathbb{R}\) and \(\beta\in(0,1)\).

Answer: 0


Question: A biased coin lands on heads with an unknown probability $p$, where the cumulative distribution function of $p$ is given by $P(p \leq x) = x^4$. If you flip the coin 5 times and it lands on heads each time, calculate the probability that the next flip will also result in heads.

Answer: \dfrac{9}{10}


Question: Given three cell towers and a vehicle moving along a straight line, determine if the vehicle's trajectory can be calculated using only two pings from each tower, assuming the vehicle's speed is known.

Answer: Yes


Question: In a simple group with 180 elements, the number of Sylow 5-subgroups is 6. Determine the index of the normalizer of a Sylow 5-subgroup in the group.

Answer: 6


Question: Find the values of \( p \in \mathbb{R} \) such that the series \( \sum^{\infty}_{n=4}{\frac{1}{n\log(n)\log(\log(n))^p}} \) converges.

Answer: p > 1


Question: Find the conditions on the real exponent $p$ for which the series \( \sum_{n=2}^{\infty} \frac{1}{n(\log(n))^p} \) converges.

Answer: p > 1


Question: Given that $f$ and $g$ are monic polynomials in $\mathbb{Z}[x]$, and there exists a polynomial $h$ in $\mathbb{R}[x]$ such that $f(x) = g(x)h(x)$ for all real $x$, must $h$ also be in $\mathbb{Z}[x]$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Evaluate the integral \( \int_{|z|=2} z^2 e^{\frac{3}{z}} \, dz \) using the residue theorem.

Answer: 9\pi i


Question: Determine the number of ways to tile a $3 \times 8$ board using tiles of sizes $3 \times 1$, $1 \times 3$, and $3 \times 3$.

Answer: 37


Question: Does Cauchy's estimate imply that if $K$ is a compact subset of the unit disc $\mathbb{D}$ and $\{f_n\}_{n=1}^{\infty}$ is a sequence of functions converging uniformly to $0$ on $K$ as $n \to \infty$, then the sequence of derivatives $\{f_n'\}_{n=1}^{\infty}$ also converges uniformly to $0$ on $K$? Answer yes or no.

Answer: No


Question: Evaluate the limit using Stirling's approximation: $$\lim_{x \to \infty} \frac{5^{-x} x!}{m! (x-m)!}$$ where $m$ is a positive integer less than $x$. Provide your answer in terms of $m$.

Answer: 0


Question: Suppose $E$ is a Lebesgue measurable set with positive measure. For any $0 < \alpha < 1$, determine if there exists an open interval $I$ such that the measure of $E \cap I$ is greater than $\alpha$ times the measure of $I$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^{\infty} \frac{\sqrt{x}}{(x+1)^2}{\rm d}x \) using the Residue theorem.

Answer: \dfrac{\pi}{2}


Question: Consider the distribution \( f_a(x)=\frac{H(x+a)-H(x-a)}{2a}\). Determine the \(a\)-derivative of this distribution \( \left < \frac{\partial f_a}{\partial a},\phi \right> = \lim_{h\to0} \frac{ \left <f_{a+h} ,\phi \right > - \left <f_{a} ,\phi \right > }{h}\) and evaluate the limit \( \lim_{a\to 0} \frac{\partial f_a}{\partial a}\).

Answer: 0


Question: Determine the number of monic irreducible quadratic polynomials over the finite field $\Bbb{F}_{11}$.

Answer: 55


Question: How many homomorphisms exist from the dihedral group $D_5$ onto the group $\mathbb{Z}_2 \oplus \mathbb{Z}_2$?

Answer: 0


Question: Find the dimension of the algebraic set $X = V(x^2 - y^2, x^3 - z^3)$ over an algebraically closed field $F$. Express your answer as a single integer.

Answer: 1


Question: Determine the number of invertible matrices of size $3 \times 3$ over the finite field $\mathbb{F}_5$. Provide your answer as a single integer.

Answer: 1488000


Question: Determine an asymptotic upper bound for the recurrence relation \( T(n) = (T(\sqrt{n}))^2 \), with the base case \( T(n) = n \) for \( n \leq 2 \).

Answer: O(n)


Question: Is every subset of $\mathbb{R}$ a $G_\delta$ set?

Answer: No


Question: Evaluate the limit without using L'Hopital's Rule: $$\lim_{x\to 0} \left( \frac{2+x^{2}}{2-x^{2}} \right)^{\frac{1}{x^2}}$$

Answer: e


Question: Evaluate the limit using Taylor series expansions: $$\lim_{x\rightarrow 0}\frac{\ln(1+\alpha x)}{\ln(\ln(1+\text{e}^{\beta x}))}$$ where \(\alpha, \ \beta \in \mathbb{R}^+\).

Answer: 0


Question: Determine the number of rotations in \( \mathbb{R}^3 \) that map the cube \( [-1,1]^3 \) onto itself.

Answer: 24


Question: In a linear programming problem, consider a variable $x_k$ without a non-negativity constraint, expressed as $x_k = u_k - v_k$ where $u_k \ge 0$ and $v_k \ge 0$. Can a basic feasible solution exist where both $u_k$ and $v_k$ are basic variables? Answer yes or no.

Answer: No


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

Answer: 1


Question: Let $x$, $y$, and $z$ be the lengths of perpendiculars drawn from a point inside an equilateral triangle with side length 2 to each of its sides. Given that $x + y + z = \sqrt{3}$, find the minimum value of the expression \[ I = x^2 + y^2 + z^2 - xy - yz - zx. \]

Answer: 0


Question: Suppose $A$ is a self-adjoint linear map with respect to a nondegenerate symmetric bilinear form $q$. If $A$ has an orthonormal basis of eigenvectors, do all the eigenspaces of $A$ have orthonormal bases? Answer yes or no.

Answer: Yes


Question: Determine the greatest common divisor of 4 and 6 in the ring \( \mathbb{Z}[\sqrt{2}] \).

Answer: 2


Question: Consider a differentiable function $g$. If for sufficiently large $x$ we have \(g'(x)=x^{-1}+O(x^{-3})\), does this imply that $g(x)$ is monotone increasing for sufficiently large $x$? Provide a justification for your answer.

Answer: Yes


Question: Given the equation $8iz^3 + 12z^2 - 18z + 27i = 0$, where $z$ is a complex number, find the value of $4|z|^2$.

Answer: 9


Question: Evaluate the integral \( \int_{\gamma} \sin(z) \, dz \) where \( \gamma \) is a circle of radius 5 centered at the point \( i \).

Answer: 0


Question: Consider a function \( f(z) \) with an expansion of the form \( f(z) = \sum_{n\geq 1} a_n {\rm e}^{2\pi i n z} \). If \( f(z) \) is bounded on the upper half-plane, determine the decay rate of the terms \( a_n {\rm e}^{2\pi i n z} \) in the series as the imaginary part of \( z \) tends to infinity.

Answer: e^{-2\pi n y}


Question: Determine whether the following asymptotic relation is true for some constant \( c \): \[ \prod_{p < x} \left(1 - \frac{n}{p} \right) \sim c(\log(x))^{-n}. \]

Answer: Yes


Question: Determine the limit \(\lim_{(x,y)\to (0,0)} \frac{ x^2 +[g(y)]^2}{[g(x)]^2+y^2}\) if it exists, given that \(g:\mathbb{R}\to \mathbb{R}\) and \(\lim_{t \to 0}\frac{g(t)}{t}= 1\) with \(g(0)= 0\).

Answer: 1


Question: Determine whether a $C_0$ semigroup can be strongly continuous at $0$ but not uniformly continuous at $0$. Provide a justification for your answer.

Answer: Yes


Question: Let $X$ be a topological space homeomorphic to a 3-sphere, and $X\hookrightarrow S^2\times S^2$ a smooth embedding. Determine if $[X]$ is a generator of $H_3(S^2\times S^2;\mathbb{Z})$. Provide your answer as either "Yes" or "No".

Answer: No


Question: Consider all natural numbers whose decimal expansion consists only of the even digits $0, 2, 4, 6, 8$. These numbers are arranged in increasing order. Let $a_n$ denote the $n$-th number in this sequence. Determine the value of the limit: \[ \lim_{n\to \infty}\frac{\log a_n}{\log n} \] (a) $0$ (b) $\log_5 10$ (c) $\log_2 10$ (d) $2$

Answer: B


Question: Given coprime integers $p$ and $q$ with $p < q < 2p$, consider four unknowns $w, x, y, z$ satisfying $w,x,y,z < \min(p,q)$. If the following congruences are known: $(wy+xz) \bmod (p+q)$, $(wz+xy) \bmod (p+q)$, $wy \bmod q$, and $xz \bmod p$, are $w, x, y, z$ uniquely determined up to a symmetry?

Answer: Yes


Question: Determine the dimension of the vector space $V_r$ over the complex numbers $\mathbb{C}$, where $V_r$ consists of all holomorphic functions $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $|f(z)| < \infty$ for all $z \in \mathbb{C}$. Provide your answer as a single number.

Answer: \infty


Question: Let \(f: \mathbb{N}^+ \to \mathbb{Z}\) be a monotonic nondecreasing function such that \(f(m)f(n) = f(mn) + f(m+n-1)\) for all positive integers \(m\) and \(n\), and \(f(4) = 5\). Find all possible values of \(f(n)\) for positive integers \(n\).

Answer: n + 1


Question:
Find the integer solution to the equation \(975a + 880b + 790c + 585d + 487e + 440f + 292g + 260h + 530i + 195j + 125k = 1002\) that minimizes the sum \(a + b + c + d + e + f + g + h + i + j + k\), given the constraints:

\[0 \leq a \leq 27\]
\[0 \leq b \leq 37\]
\[0 \leq c \leq 14\]
\[0 \leq d \leq 15\]
\[0 \leq e \leq 1\]
\[0 \leq f \leq 2\]
\[0 \leq g \leq 1\]
\[0 \leq h \leq 6\]
\[0 \leq i \leq 1\]
\[j \in \mathbb{N}\]
\[k \in \mathbb{N}\]

Answer: 3


Question: In a game involving the polynomial equation $x^3 + Ax^2 + Bx + C = 0$, two players, Boy and Girl, take turns as follows: Boy chooses a real number, and Girl assigns it to one of the coefficients $A$, $B$, or $C$. This process is repeated until all coefficients are determined. If the resulting polynomial has three distinct integer roots, Boy wins; otherwise, Girl wins. Determine if either player has a winning strategy, and if so, identify the player.

Answer: Girl


Question: In a party with \( n + 1 \) couples, where handshakes between spouses are forbidden, a total of \( 2n + 1 \) guests (excluding the host and hostess) report \( 2n + 1 \) unique numbers of handshakes. Determine the number of handshakes the host made.

Answer: n


Question: Consider the sequence defined by the recurrence relation \( a_{n+1} = -ta_n^2 + (t+1)a_n \) with initial condition \( a_0 \in \left(0, \frac{1}{t} + 1\right) \) and parameter \( t \in (0, 1] \). Determine whether the limit \( \lim_{n \to \infty} a_n \) exists for all such \( t \) and \( a_0 \). If the limit exists, find its value.

Answer: 1


Question:
Let \( f \colon [0,1]\rightarrow \mathbb{R} \) be a continuous function such that \( f(1)=1 \). Evaluate the limit:
\[ \lim_{y\rightarrow \infty}\int_0^1 yx^y f(x) \, dx \]

Answer: 1


Question: Evaluate the limit \( \lim_{n\rightarrow \infty}\frac{I_{n+1}}{I_{n}} \), where \( I_{n}=\int^{1}_{0}x^2(1-x^2)^n\,dx \).

Answer: 1


Question: Determine whether there exists a second-degree polynomial in two variables whose values form an infinite arithmetic progression. Provide a justification for your answer.

Answer: Yes


Question: Find the limit of the sequence \( \frac{1}{n}\sum_{k=1}^n k^{\frac{1}{k}} \) as \( n \to \infty \).

Answer: 1


Question: Evaluate the contour integral \( \oint_{|z-1|=R} \frac{1}{z^4+1} \, dz \) where \( R > 2 \).

Answer: 0


Question: Calculate the limit \(\lim_{x \to 0} \frac{\left(1+o\left(x^2\right)\right)^{x + o\left(x^2\right)}-1}{x^3}\) using the generalized binomial theorem.

Answer: 0


Question: Find the value of the Eisenstein series $G_2(\mathbb{Z}[i])$, where $\mathbb{Z}[i]$ is the lattice of Gaussian integers and $k=1$. Express your answer in a simplified form.

Answer: 0


Question: In an irreducible topological space, can the intersection of an infinite number of open sets be a closed set?

Answer: Yes


Question: Evaluate the limit: $$\lim \limits_{x \to 0} \frac{(\sin{x})^{x-1}}{\frac{1}{x}}$$

Answer: 1


Question:
Find the coefficient of \(x^6\) in the expansion of the generating function:

\[f(x) = \frac{1}{x(2x-1)^2}\]

Answer: 1024


Question: Given three numbers such that the sum of the numbers is 6, the sum of their squares is 8, and the sum of their cubes is 5, find the sum of their fifth powers.

Answer: \dfrac{118}{3}


Question: Consider a Markov chain with a finite state space of size $k$ and a transition matrix where all non-diagonal entries are equal to $1/k$. Determine the steady-state probability distribution for this Markov chain.

Answer: \dfrac{1}{k}


Question: Let $g$ be a differentiable function satisfying $$g(x - y) = e^xg(y) - e^yg(x)$$ for all $x, y \in \mathbb{R}$. Determine $g'(0)$. \( \boxed{} \)

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{(\cos 3x-1)(\cot 4x+\cot(\frac{\pi}{4}-3x)-\cot(\frac{\pi}{4}+2x))}{x\cos x \cot x\cos 3x}$$ without using L'Hopital's rule or Taylor series.

Answer: 0


Question: Determine the automorphism group of the complete bipartite graph $\mathcal{V}_{m,n}$ where $m \ne n$. Provide your answer in terms of group structure.

Answer: S_m \times S_n


Question: For any pair of matrices $A$ and $B$ with complex entries, is it true that the inequality \(|A+B| \prec |A| + |B|\) holds in the Loewner order, where the modulus of a matrix $A$ is defined as \(|A| = \sqrt{A^{\dagger}A}\)?

Answer: No


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \) where \( \mathbf{F}(x,y,z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), \( S \) is the plane surface whose boundary is the triangle with vertices at \((1,0,0), (0,1,0),\) and \((0,0,1)\), and \( \mathbf{n} \) is the unit normal to \( S \) having a nonnegative \( z \)-component. Use the vector representation \( \mathbf{r}(u,v) = (u+v)\mathbf{i} + (u-v)\mathbf{j} + (1-2u)\mathbf{k} \).

Answer: \dfrac{1}{2}


Question: Find the least positive integer \(x\) that satisfies the inequality \(x^2 - 4x > \cot^{-1}(x)\).

Answer: 5


Question: Determine the convergence and find the value of the series \( \sum_{n=1}^{\infty} (e^{-4n}-e^{-4(n-1)}) \).

Answer: -1


Question: Find the minimum value of \( \frac{\tan x - x}{x - \sin x} \) on the interval \((0, \pi/2)\).

Answer: 2


Question: Find the limit of the expression as \( n \) approaches infinity: \[ \lim_{n\to\infty} \left(\frac{40}{40-41n^2}\right)^{41n^2-n^2} \]

Answer: 0


Question: Provide an example of a compact operator in a Banach space such that the spectrum of the operator is equal to the set of continuous spectrum, and both are equal to \( \{0\} \).

Answer: V


Question: Evaluate the integral \( \int_0^{+\infty} \frac{\sin^4{x}}{x^4}dx \) given that \( \int_0^{+\infty} \frac{\sin{x}}{x}dx = \frac{\pi}{2} \).

Answer: \dfrac{\pi}{3}


Question: Determine the number of elements $\bar{x} \in \mathbb{Z}^*_n$ such that $\bar{x}^2 = \bar{1}$, where $n$ is the product of three distinct prime numbers.

Answer: 8


Question: Evaluate the limit: $$\lim_{k \to \infty} \int_{0}^{1} \frac{kx^k}{1+x} \, dx$$

Answer: \dfrac{1}{2}


Question: Evaluate the limit: $$ \lim_{x\to\infty} e^{-x} \int_1^x \frac{e^t}{t} dt $$

Answer: 0


Question: In a row of eight seats, four married couples are randomly seated. What is the probability that no husband sits next to his wife?

Answer: \dfrac{12}{35}


Question: Find the limit of the sequence \( P_n \) defined by \( a_1=1 \) and \( a_{n+1}=(n+1)(1+a_n) \), where \( P_n=\prod_{i=1}^n(1+a_i^{-1}) \), as \( n \) approaches infinity.

Answer: e


Question: Evaluate the limit of the expression \((1+3x^2)^{5\cot x + \frac{2\csc x}{x}}\) as \(x\) approaches 0.

Answer: e^6


Question: Determine whether the following statement is true or false: If $H(X|Y)=0$, then $H(Y|X)=0$, where $H$ denotes the Shannon entropy.

Answer: False


Question: Find the sum of the series \( \frac{1}{1!} + \frac{1+2}{2!} + \frac{1+2+3}{3!} + \ldots \). Use the properties of exponential functions to derive your answer.

Answer: \dfrac{3e}{2}


Question: Calculate the Morley rank of the type $x = x$ in the theory $ACF_{0}$.

Answer: 1


Question: Assuming a Poisson distribution with an unknown rate parameter \(\lambda\) for software failures, where two failures occurred in one week, determine the waiting time without any additional failures needed to declare with 95% confidence that a workaround is successful. A successful workaround will permanently eliminate the problem. Calculate this waiting time.

Answer: 1.5


Question: Find the extremal for the functional \( J[y]=\int_1^2(y'^2+2yy'+y^2)\, dx \) subject to the boundary conditions \( y(1) = 1 \) and \( y(2) \) is arbitrary.

Answer: e^{1 - x}


Question: Evaluate the triple integral \( \iiint e^{-2r} \cos^2\theta \, dV \) over all space using spherical coordinates.

Answer: \dfrac{\pi}{3}


Question: Is the algebra generated by the set \( \{1,x^3\} \) dense in \( C[-1,1] \) with respect to the supremum norm?

Answer: Yes


Question: Evaluate \( \lim_{n \to 0} \int_1^x t^{n-1} \, dt \) without using L'Hospital's rule.

Answer: \ln x


Question: Let $f(x)$ be a real-valued function defined on the interval $(-5,5)$ such that $e^{-x}f(x)=2+\int_0^x\sqrt{t^4+1}\,dt$ for all $x\in(-5,5)$. Find the derivative of the inverse function of $f(x)$ at $x=2$. \( \text{Express your answer as a single number.} \)

Answer: \dfrac{1}{3}


Question: Consider a population of people where each person's height is independently and identically distributed (IID). You randomly choose a person and observe that their height is $X$. Let $N$ be the number of additional samples needed to randomly select a person whose height is greater than $X$. Calculate the expected value $E[N]$.

Answer: \infty


Question: Find the value of $15^{100!} \bmod 5000$ using basic number theory concepts, such as Fermat's Little Theorem, Euler's theorem, and the Chinese Remainder Theorem.

Answer: 625


Question: Determine the number of positive roots of the equation \( bP + \frac{b}{8} P^3 = \frac{3}{2} \) using the parameterization of a parabola.

Answer: 1


Question: Let $G$ be a free abelian group of rank $k$, and let $S$ be a subset of $G$ that is linearly independent over $\mathbb{Z}$. Determine whether it is true that $|S| \leq k$. Provide a justification for your answer.

Answer: Yes


Question: Given a continuous, smooth curve $\mathcal{C}$ of length $\ell$ in $\mathbb{R}^2$, at each point on the curve, draw a line segment of length $d$ normal to the curve and centered at the point. Assuming no two line segments intersect, is the area covered by all the line segments equal to $d \cdot \ell$?

Answer: d \cdot \ell


Question: Solve the equation \( \frac{x^{3} - 2x^{2} + x + 4}{x+1} = \frac{7}{4}\sin\frac{\pi x}{3} \).

Answer: \dfrac{3}{2}


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

Answer: \infty


Question: Determine whether the set of all closed subsets in \(\mathbb{R}\) can serve as a basis for some topology on \(\mathbb{R}\).

Answer: Yes


Question: Find the limit of the sequence \(a_n = \sqrt[n]{4^{2n} - 5^{n+2} - \sin(\sqrt{n})}\) as \(n\) approaches infinity.

Answer: 16


Question: Consider the polynomial \( P(x) = ax_1 + bx_2^2 + cx_3^2 \) where \( a \ge 0, b \ge 0, \) and \( c \le 0 \). Determine whether the following equivalence is true: \( P \) does not have any local minimum if and only if \( a - c \neq 0 \).

Answer: True


Question: Determine all integer values of \(n\) for which \(\frac{n^3-1}{5}\) is a prime number.

Answer: 6


Question: Evaluate the limit \( I = \lim_{x \to 0} \frac{(1+x)^{\frac{1}{x}} - (1+2x)^{\frac{1}{2x}}}{x} \) using L'Hôpital's rule and Taylor's expansion.

Answer: \dfrac{e}{2}


Question:
Consider a sequence $(g_n)$ defined recursively as:

$$ g_{0}(x)=\frac{1}{(1+x)^3}, \quad g_{n}(x)=\int^{x}_{0}g_{n-1}(y)\;\text{dy},\ n\ge1.$$

Evaluate the limit: $$\lim_{n\to \infty}\frac{g_{n}(1)}{n!}. $$

Answer: 0


Question: Find the number of $2 \times 2$ nilpotent matrices over the finite field $\mathbb{F}_q$, where $q$ is a power of a prime.

Answer: q^2


Question:
Evaluate the limit:

$$\lim_{n \to \infty} \int _{2n}^{3n}\frac{dx}{(f^{-1}(x))^2+2f^{-1}(x)}$$

where $f(x)=x^4+2x^2+1$. Use the concept of inverse functions in your solution.

Answer: \infty


Question: Is it possible to find the maximum value of a discrete quasiconcave function $p(h)$, defined on a finite set of integers $\{1, \dots, n\}$, in $O(\log n)$ time?

Answer: Yes


Question: Evaluate the integral \( \iint_R \left[5+\frac{1}{4}\left( \frac{x^2}{c^2}-\frac{y^2}{d^2}\right)\right] \, dx \, dy \) where \( R \) is the region defined by \( \frac{x^2}{c^2}-\frac{y^2}{d^2} \ge 1 \). Use a suitable transformation to simplify the integral.

Answer: \infty


Question: Evaluate the integral \( \int_{0}^{2\pi} \frac{x(t)x'(t) + y(t)y'(t)}{x^2(t) + y^2(t)}dt \) for a closed continuously differentiable curve \( F(t) = (x(t), y(t)) \) in \( \mathbb{R}^2 \setminus \{(0,0)\} \).

Answer: 0


Question: Determine whether the condition \( \int_{\mathbb{R}} |f(t)g(t)|\,dt < K\| g\| \) for all \( g\in L^2 \) and some constant \( K \) implies that \( f\in L^2 \).

Answer: Yes


Question:
Let $A = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)$ where $a, b, c, d$ are positive real numbers satisfying $a + b < \frac{1}{2}$ and $c + d < \frac{1}{2}$. Determine which of the following statements is FALSE:

A. Matrix $I + A$ is invertible.
B. Matrix $I - A$ is invertible.
C. Matrix $I + A^2$ is invertible.
D. There exists a natural number $m$ such that matrix $I + A^m$ is not invertible.

Answer: D


Question: Let $G = C_6 = \langle g \mid g^6 = 1 \rangle$ and let $V_1, V_2$ be two $\mathbb{C}G$-modules with bases $\mathcal{B}_1 = \{v_1, v_2, v_3\}$ and $\mathcal{B}_2 = \{w_1, w_2\}$ respectively. The action of $G$ on $V_1$ and $V_2$ is given by $[g]_{V_1} = \begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{pmatrix}$ and $[g]_{V_2} = \begin{pmatrix}2 & 5\\ -1 & 1\end{pmatrix}$. Find a basis for $\operatorname{Hom}_{\mathbb{C}G}(V_1, V_2)$. \(\operatorname{Hom}_{\mathbb{C}G}(V_1, V_2)\) denotes the space of $\mathbb{C}G$-module homomorphisms from $V_1$ to $V_2$. \(\mathbb{C}G\) is the group algebra of $G$ over the complex numbers.


Question:
Consider two matrices $A$ of order $n \times n$ and $C$ of order $m \times n$. Define the block matrix $D$ as follows:
$$
D = \left( \begin{matrix}
C\\
CA\\
\vdots\\
CA^{q-1}
\end{matrix} \right)
$$
where $q \in \mathbb{N}^*$. Determine the smallest value of $q$ such that $\text{rank}(D) = n$. Provide a justification for your answer.

Answer: n


Question: In triangle $ABC$, the orthocenter is $H$. Given $AB = 84$, $AC = 32\sqrt{3}$, and $\angle BAC = 60^\circ$, points $D, E, F$ are on $AH, BH, CH$ respectively such that $7AD = AH$, $7BE = BH$, and $7CF = CH$. Calculate the value of $AB \cdot CF + BC \cdot AD + AC \cdot BE$. \(\boxed{\text{Your answer here}}\)

Answer: 1152


Question:
Consider the set of integers \(\{1, 2, 3, \ldots, 100\}\). Let \(\{x_1, x_2, x_3, \ldots, x_{100}\}\) be a permutation of this set, where each \(x_i\) is distinct. Find the minimum value of the sum:
\[ S = |x_2 - x_1| + |x_3 - x_2| + \cdots + |x_{100} - x_{99}| + |x_{100} - x_1|. \]

Answer: 198


Question: Find the total derivative of \( u(x,z) = \ln\left(\frac{x^\alpha + z^\alpha}{z^\alpha}\right) \) with respect to \( x \), given that \( z = h(x) = (ax^4 + b)^{1/3} \) and \( u'(x,z) = 0 \).

Answer: 0


Question:
Consider the setting where $X=\mathbb{R}^m$, $Y=\mathbb{R}^n$, $\alpha:X\to X$ is a $C^1$ diffeomorphism, and $\beta:X\times Y\to Y$ is such that for all $x\in X$, $\beta_x:Y\to Y$ is a $C^1$ diffeomorphism. Suppose $\pi(x,y)=\pi_2(y\mid x)\pi_1(x)$ is a continuous joint probability density function. If $\beta(x,y)$ is not differentiable in $x$, determine whether the following proposition is true:

Proposition: The pushforward density of $\pi$ by the map $f(x,y)=(\alpha(x),\beta_x(y))$ can be expressed as:
$$ f_*\pi(x,y) = (\beta_{\alpha^{-1}(x)})_*\pi_2(y\mid\alpha^{-1}(x))\alpha_*\pi_1(x) $$

Answer: True


Question: For an integrable function $f:\mathbb{R}\to\mathbb{R^+}$ with Fourier transform $\widehat f(\lambda)\ne 0$ for almost every $\lambda\in\mathbb{R}$, determine if there exist sequences $R_n\to\infty$ and $r_n\to 0$ such that the Lebesgue measure of the sets $A_{R_n,r_n} = \left\{ \lambda:\left| \lambda \right| \leqslant R_n,\left| \hat f\left( \lambda \right) \right| \leqslant r_n \right\}$ approaches zero.

Answer: Yes


Question: Let $\mathbb{Q}\cap [0,1]=\{x_1,x_2,x_3,\dots\}$ and define the open interval $$G_n=\left(x_n-\frac{1}{2^{n+2}},x_n+\frac{1}{2^{n+2}}\right),\,n\in\mathbb{N}.$$ Define $G=\bigcup_{n=1}^{\infty}G_n$ and the Borel set $B$ as $$B=[0,1]\cap G^c.$$ Consider the Lebesgue measure space $(\mathbb{R},\Lambda,\lambda).$ Prove that if $I\subset B\cup E$, where $I$ is an interval and $\lambda(E)=0$, then $\lambda(I)=0$. Use the result: Let $G\subset \mathbb{R}$ be an open set and $E\subset G$ such that $\lambda(E)=0$. Then $G\backslash E$ is dense in $G$. \(\text{Provide a proof or justification for this statement.}\)

Answer: \lambda(I) = 0


Question: Determine the value of $p$ such that the function \( f(x,y) = \frac{1}{(x^2+y^3-2)^p} \) is integrable over the region \( 1 \leq x \leq 2, 1 \leq y \leq 3 \).

Answer: p < 2


Question: Determine whether every limit ordinal is an infinite cardinal. Provide a yes or no answer.

Answer: No


Question: Determine whether the following statement is generally true: \(\frac{\mathrm d}{\mathrm da} \int_{-\infty}^{a-y} f(x)\, \mathrm dx = f(a-y)\), where \(a\) and \(y\) are constants.

Answer: True


Question:
Determine which of the following claims about exponential distributions is correct and explain why:

1. The PDF for the sum of two independent exponential random variables with parameter \( \lambda \) is given by \( \rho(x_1,x_2) = \lambda^2 e^{-\lambda(x_1+x_2)} \).

2. The combined PDF for independent events is \( \lambda e^{-\lambda x_1} \lambda e^{-\lambda x_2} = \lambda^2 e^{-\lambda(x_1+x_2)} \).

Answer: 2


Question: Let $G$ be a finite group, $V$ an irreducible complex representation, and $\mathbb{1}$ the trivial representation. Determine whether there exists a positive integer $n$ such that $\mathbb{1} \le V^{\otimes n}$.

Answer: Yes


Question: As the area of a hyperbolic triangle in the open upper half complex plane becomes arbitrarily large, what happens to the sum of its angles?

Answer: 0


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

Answer: \dfrac{1}{6}


Question: Find the number of elements in the kernel of the trace map for the finite field $\mathbb{F}_{p^f}$, where $p$ is a prime and $f$ is a positive integer.

Answer: p^{f - 1}


Question: Determine the Galois group of the polynomial \(x^{4}+2x^{2}+2\) over \(\mathbb{Q}\).

Answer: D_4


Question: Does a ring homomorphism from a ring \( R \) with unity to the integers \( \mathbb{Z} \) always map the unity of \( R \) to the unity of \( \mathbb{Z} \)?

Answer: No


Question: Evaluate the limit \( \lim_{x\to\infty} \frac{1}{x \sqrt{x}} \int_{0}^{x} \ln \left( \frac{t^3 + 2}{t^2 + 1} \right) dt \) using L'Hopital's Rule. Justify the application of the rule and find the value of the limit.

Answer: 0


Question: Evaluate the integral \(\int_{\gamma}f(z)dz\), where \(f(z)=\frac{z^2-1}{(z+1)^2(z-2)}\), and \(\gamma\) is the circle centered at the origin with radius \(R > 2\).

Answer: 2\pi i


Question: Suppose $f(x)$ is a function with a singularity at $x = a$ that can be expressed as $f(x) = \frac{g(x)}{x-a}$, where $g(x)$ is nonsingular everywhere. Determine the conditions under which the integral $\int_{-\infty}^{\infty}f(x)\delta(x-a)dx$ exists and find its value.

Answer: g'(a)


Question: Find the cardinality of the set of all functions from \( \mathbb{Q} \) to \( \{0,1\} \) that are zero outside a countable subset of \( \mathbb{Q} \).

Answer: 2^{\aleph_0}


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

Answer: f(x) = c


Question: Compute the residue of the function \( g(z) = \frac{e^z - 1}{z^3} \) at \( z = 0 \).

Answer: \dfrac{1}{2}


Question: Find the smallest integer \( n \) such that the symmetric group \( S_n \) contains an even permutation of order 312, given that there exists an 8-cycle, a 3-cycle, and a 13-cycle in the group.

Answer: 26


Question: Find the elements of $U(L)$ that satisfy the grouplike element property for a Lie algebra $L$.

Answer: 1


Question:
Let \(f:[0,+\infty) \to \mathbb{R}\) be a function that satisfies the following properties:

1. \(f\) is uniformly continuous on \([0,+\infty)\).
2. For any \(x_0 \in [0,1]\), we have \(\lim_{n \to +\infty} f(x_0 + n) = 0\).

Determine \(\lim_{x \to +\infty} f(x)\).

Answer: 0


Question: Find the smallest integer \( n \) such that \( f_3(n) > T(n+3) \), using the bounds for \( f_3(n) \) provided.

Answer: 4


Question: Evaluate the surface integral \(\iint_S(z+x^2y)\,dS\) where \(S\) is the part of the cylinder \(y^2 + z^2=1\) that lies between the planes \(x=0\) and \(x=3\) in the first octant \(x, y, z \ge 0\).

Answer: 12


Question: Determine the number of real roots of the polynomial \( P_n(x) = x^{2n+1} - \left(2 + \frac{1}{n}\right)x^{2n} + \left(1 - \frac{1}{n}\right)x + 1 \), where \( n \geq 1 \) is an integer.

Answer: 3


Question: Evaluate the limit \( \lim_{n \to \infty} \sum_{i=1}^{n} \frac{i}{n^2 + i} \).

Answer: \dfrac{1}{2}


Question: Find the Laurent series of the function \( f(z) = \frac{2}{z} - \frac{3}{z-2} + \frac{1}{z+4} \) in all possible regions of convergence about \( z=0 \). Determine the residue of \( f(z) \) at \( z=0 \).

Answer: 2


Question: Determine \( \lim_{t \to \infty} u(1,t) \), where \( u(x,t) \) is the solution to the heat equation \( u_t = u_{xx} \) with the initial condition \( u(x,0)=\frac{e^{2x}-1}{e^{2x}+1} \).

Answer: 0


Question: Determine if there are any positive integer solutions to the equation \(2x^3 - 1 = z^3\).

Answer: (1, 1)


Question: In the context of knot theory, consider a free product with amalgamation $G \ast_K H = \mathbb{Z} \oplus \mathbb{Z}$, where $K = \mathbb{Z} \oplus \mathbb{Z}$. Given that $G$ is a free group, determine if $H$ must also be a free group.

Answer: No


Question: Evaluate the limit: $$ \lim_{n\rightarrow+\infty}n\int_{0}^{1}\frac{x^{n}}{x^{n}+x+1}\,dx $$

Answer: \ln \frac{3}{2}


Question: Determine the values of $b$ for which the set \( N_b := \{ (x,y,z) \in \mathbb{R}^{3} : x^2 + y^2 - z^2 = b\} \) is a submanifold of \( \mathbb{R}^{3} \) using the regular value theorem.

Answer: b \neq 0


Question: In a game, player A selects a number n from the set S = \{1, 2, \ldots, 1001\}. Player B must guess the value of n by choosing subsets of S. After each guess, player A reveals the number of subsets that contain n. Player B makes three guesses, with the number of subsets in each guess denoted by k_1, k_2, and k_3, respectively. Determine the minimum possible value of k_1 + k_2 + k_3 such that player B can always correctly identify n.

Answer: 28


Question: Determine whether there exists a function $f \in \bigcap_{p \geq 1} L^p(\mathbb{R})$ such that $\|f\|_p = 1$ for every $p \geq 1$, where $\mathbb{R}$ is equipped with the Borel $\sigma$-algebra and the Lebesgue measure.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to \infty} \int_{0}^{\pi/3} \frac{\sin^n x}{\sin^n x + \cos^n x} \, dx$$

Answer: \dfrac{\pi}{12}


Question: Determine the number of regions formed by the intersection of $n$ unit spheres in $\mathbb{R}^3$. Provide your answer as a single number.

Answer: \dfrac{n^3 - 3n^2 + 8n}{3}


Question: Determine whether the following statement is true or false: If the localization of a ring $R$ at every prime ideal is a regular local ring, then $R$ is also a regular local ring.

Answer: B


Question: Let $(x_n)_{n\in \mathbb{N}} \subset \ell^2$ converge component-wise to $x \in \ell^2$. If $f: \ell^2 \to \ell^2$ is sequentially continuous, does it follow that $\lim_{n \to \infty} f(x_n) = f(x)$? Answer yes or no and justify your answer.

Answer: No


Question: Determine the range of values for $a$ such that the polynomial \( p(x) = x^4 + ax^3 + bx^2 + ax + 1 \) has complex (non-real) roots, given that \( x = 1 \) is a root and \( b = -2(a+1) \).

Answer: (-4, 0)


Question: Calculate the limit: $$\lim\limits_{(x, y) \rightarrow (+\infty, +\infty)} \left(1 + \frac{1}{xy}\right)^{x\sin y}$$

Answer: 1


Question: Find a function $h(x)$ such that the power series expansion of $h(x)$ around $x=0$ has coefficients $b_k$ satisfying \( \sum_{k\geq 0} (k+1) b_k = 0 \).

Answer: e^{-x}


Question: Determine all positive integers \( n > 1 \) such that \( n^2 \mid 2^n + 1 \).

Answer: 3


Question: Evaluate the limit: $$\lim_{(x,y) \to (0,0)} \exp\left( -\frac{\ln  (1+x^4y^4)}{x^2 + y^2}  \right)$$

Answer: 1


Question:
As \( n \rightarrow \infty \), to which distribution does \( \sqrt{n} \sum_{i=1}^{n} (X_i - p) \) converge? Choose from the following options:
1. \( N(p, \frac{1}{4n}) \)
2. \( N(0, \frac{1}{4}) \)
3. \( N(0, \frac{1}{2}) \)
4. \( N(p, \frac{1}{4n}) \)
5. \( N(p, \frac{1}{2n^{1/2}}) \)

Where \( N(m, r) \) denotes a normal distribution with mean \( m \) and variance \( r \).

Answer: 2


Question: Determine the correct limit for \( r \) when transforming the integral \( \iiint_{E}z\sqrt{x^2+y^2}dV \) to cylindrical coordinates, where \( E \) is the region bounded by the cylinder \( x^2+y^2=2x \) and the planes \( y=0 \), \( z=0 \), and \( z=a \).

Answer: 0 \leq r \leq 2\cos\theta


Question:
Let $C[0,1]$ be the space of continuous functions on $[0,1]$ with the supremum norm. Consider the subset $$S = \{f \in C[0,1] : \lim_{n\to\infty} f(1/n) = 0\}.$$ Determine which of the following statements is correct:

a) $S$ is a vector space but not closed in $C[0,1]$.
b) $S$ is closed but does not form a vector space.
c) $S$ is a closed vector space but not an algebra.
d) $S$ is a closed algebra.

Answer: d


Question: Find all continuous functions $f: \mathbb{R} \to \mathbb{C}$ such that $f(x) = f(x+1)$ for all $x \in \mathbb{R}$ and $f(x+y) = f(x) f(y)$ for all $x, y \in \mathbb{R}$. Provide a complete characterization of such functions.

Answer: f(x) = e^{2\pi i n x}


Question: Given a topological space $X$ and a connected subset $C \subset X$, consider a partition of $X$ into disjoint open sets $\{X_\alpha\}$. Must there exist an $\alpha$ such that $C \subset X_\alpha$? Provide a justification for your answer.

Answer: Yes


Question: Suppose $g$ is a differentiable and positive function on $[1,+\infty)$ with $g(1)=1$. If $\lim_{x \to +\infty} \frac{x g'(x)}{g(x)} = \beta$ for some real number $\beta$, find the limit of $\frac{\ln(g(x))}{\ln(x)}$ as $x \to +\infty$. \( \text{Express your answer in terms of } \beta. \)

Answer: \beta


Question: Can the symmetric group $\Sigma_n$ act non-trivially on a set of $k$ elements, where $k$ is an odd number less than $n$?

Answer: Yes


Question: Determine whether there exists a continuous and bounded function between normed spaces that is not linear. Provide a justification for your answer.

Answer: Yes


Question: Suppose $f$ is a non-constant polynomial over a field $K$ and $a \neq b \in K$ such that $f(a) = f(b) = 0$. Let $n \in \mathbb{N}$ be the maximum $k$ such that $(X-b)^k \mid f$. Let $h$ be the polynomial such that $f = (X-b)^n h$. Determine whether $a$ is a zero of $h$. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of standard ways to number the faces of a regular polyhedron die such that the sum of the numbers on opposite faces is constant.

Answer: 2


Question:
Let $T:M_2(\mathbb{R})\to M_2(\mathbb{R})$ be a linear map defined by $T(B) = AB$, where $A$ is a $2\times 2$ real matrix with trace $2$ and determinant $-3$. If the matrix of $T$ is given by

$$\begin{pmatrix}a_{11}&a_{12}&0&0\\a_{21}&a_{22}&0&0\\0&0&a_{11}&a_{12}\\0&0&a_{21}&a_{22}\end{pmatrix},$$

determine whether the eigenvalues of $T$ are the same as the eigenvalues of $A$. State whether the statement is true or false.

Answer: B


Question:
Evaluate the curl of the vector field \( \vec{F}(x,y,z) = (z,x,y) \) at the point \( P = (0,0,0) \) using the formula:

$$\left(\vec\nabla\times\vec{F}\right)\Biggr|_{P}\cdot \hat{n} = \lim_{A\to0}\frac{1}{|A|}\oint\limits_{\partial A}\vec{F}\cdot d\vec{r}$$

where \( A \) is the disk \( \{(x,y,z) \mid x^2+y^2\leq r^2, z=0\} \) and \( \hat{n} \) is the unit normal vector to \( A \) at \( P \).

Answer: 1


Question: Let \(T\) be a linear transformation from \(\mathbb{R}^4\) to \(\mathbb{R}^4\) such that the null space of \(T\) is \(\{(x,y,z,w) \in \mathbb{R}^4 : x+y+z+w=0\}\). If the minimal polynomial of \(T\) is \(x(x-4)^a\), find the value of \(a\).

Answer: 1


Question: Let \(g(x) = px^2 + qx + r\), where \(p, q, r \in \mathbb{R}\). Suppose \(|g(x)| \leq 2\) for all \(x \in [-1, 2]\). Find the maximum possible value of \(|p| + |q| + |r|\).

Answer: \dfrac{46}{9}


Question:
Determine the order of a $p$-Sylow subgroup of the group $GL_{50}(F_p)$, where $p$ is a prime number. The group consists of invertible $50 \times 50$ matrices with entries from the finite field $F_p$. Given that the order of $GL_{50}(F_p)$ is $p^{2500}$, find the order of the $p$-Sylow subgroup from the following options:

1. $p^{50}$
2. $p^{125}$
3. $p^{1250}$
4. $p^{1225}$

Provide your answer as the correct option number.

Answer: 4


Question: Determine whether a function can be weakly convex without being convex. Provide a justification for your answer.

Answer: Yes


Question: A rectangle ABCD has dimensions 9 ft by 12 ft. It is folded once perpendicular to diagonal AC so that vertices A and C coincide. Determine the length of the fold.

Answer: \dfrac{45}{4}


Question: For random variables $X$ and $Y$ in $\mathbb{R}^d$, if the characteristic function of $(X,Y)$ equals the product of the characteristic functions of $X$ and $Y$ for all $\xi,\eta \in \mathbb{R}^d$, are $X$ and $Y$ necessarily independent? Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: $$\lim_{n\to\infty} \frac{1}{n+1}\sum_{k=1}^n \frac{k}{1 + \sqrt{2} +\sqrt{3} +\cdots+\sqrt{k+1}}$$

Answer: 0


Question: Find the coefficient of \(x^{20}\) in the expansion of \((x^{3} + x^{4} + x^{5} + \cdots)^{3}\).

Answer: 78


Question: Determine whether all torsion-free solvable groups are left-orderable.

Answer: Yes


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

Answer: \infty


Question: Calculate the value of the infinite sum \( \sum_{j=0}^\infty \frac{\binom{2j}{j}}{2^{2j}(1-4j^2)} \) without using Riemann sums or integrals.

Answer: \dfrac{\pi}{4}


Question: Determine the number of different ring homomorphisms from \(\mathbb{Z} \times \mathbb{Z}\) to \(\mathbb{Z} \times \mathbb{Z}\), given that a ring homomorphism must map identities to identities.

Answer: 4


Question: Evaluate the double integral \( \displaystyle\iint dydx \) over the region bounded by \( \frac{\pi}{4} \leq \theta \leq \frac{3\pi}{4} \) and \( 0 \leq r \leq 2 \) without using polar coordinates.

Answer: \pi


Question: Find the Lebesgue measure of the set of numbers in the interval $[0, 10]$ that do not contain the digit $5$. Express your answer as a single number.

Answer: 0


Question: Evaluate the integral \( \int_0^{\frac{1}{4}} \frac{x-1}{\sqrt{x}-1} \, dx \) without using a calculator.

Answer: \dfrac{1}{3}


Question: Find the limit of the sequence of functions \( f_n(x) = \frac{x^2}{x^2+(1-nx)^2} \) as \( n \to \infty \).

Answer: 0


Question: Evaluate the limit \( \lim_{y\rightarrow \infty}\int_0^1 yx^yf(x)dx \) where \( f \colon [0,1]\rightarrow \mathbb{R} \) is a function with a continuous derivative, and it is given that \( f(1)=1 \).

Answer: 1


Question:
Simplify the expression:

$$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$

Answer: 373


Question: Evaluate the contour integral $$\oint_C \frac{-y}{x^2 + y^2} \,dx + \frac{x}{x^2 + y^2} \,dy$$ where $C$ is the unit circle $x^2 + y^2 = 1$.

Answer: 2\pi


Question: Let $T$ be an invertible linear operator on a Hilbert space $H$ with $\|T\| = a$ and $\|T^{-1}\| = b$. Determine a lower bound for the norm $\|(T^*T)^\frac{1}{2}\|$ in terms of $a$ and $b$. Provide your answer as a function of $a$ and $b$.

Answer: a


Question: Evaluate the limit \( \lim_{z\to0} \left(\frac{1}{e^{z}-1}-\frac{1}{z}\right) \) using the Taylor series expansion of \( e^z \).

Answer: -\dfrac{1}{2}


Question:
Evaluate the expression:
\[10^{10^{10}} \sin\left( \frac{109}{10^{10^{10}}} \right) - 9^{9^{9}} \sin\left( \frac{101}{9^{9^{9}}} \right) - 8^{8^{8}} \sin\left( \frac{17}{8^{8^{8}}} \right) + 7^{7^{7}} \sin\left( \frac{76}{7^{7^{7}}} \right) + 6^{6^{6}} \sin\left( \frac{113}{6^{6^{6}}} \right)\]

Answer: 180


Question:
Solve for the positive real number $x$ in the equation:

$$2^x+4^x+2^{\lfloor x \rfloor}+4^{\lfloor x \rfloor}+2^{x- \lfloor x \rfloor}-4^{x-\lfloor x \rfloor}=50+\sqrt{50}.$$

Answer: \dfrac{5}{2}


Question: Find the index of the module $\mathbb{Z}[\sqrt{7}]$ as a submodule of $\mathbb{Z}\left[\frac{1+\sqrt{7}}{2}\right]$. Provide your answer as a single integer.

Answer: 2


Question: Identify a subset $B$ of the real numbers $\mathbb{R}$ such that every point in $B$ is an accumulation point, but no point in $B$ is a limit point.

Answer: \emptyset


Question: Does Frobenius' theorem guarantee the existence of a submanifold tangent to two vectors $A_{1}$ and $A_{2}$ if their bracket is zero?

Answer: Yes


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

Answer: \dfrac{\pi}{24}


Question: Suppose $X_{1}, X_{2}, \ldots, X_{20}$ are twenty sets each with seven elements and $Y_{1}, Y_{2}, \ldots, Y_{m}$ are $m$ sets each with five elements. Let $\bigcup_{i=1}^{20} X_{i}=\bigcup_{j=1}^{m} Y_{j}=T$. Assume that each element of $T$ belongs to exactly $8$ of the sets $X_{i}$ and to exactly $6$ of the sets $Y_{j}$. Find the value of $m$. \(\boxed{\phantom{m}}\)

Answer: 21


Question: Using a beam balance, determine the minimum number of weighings required to identify the odd ball among 200 balls, where one ball is either heavier or lighter than the others.

Answer: 6


Question: Consider a closed set $S$ and an open set $A$. Is it possible for the set $S \setminus A$ to be compact, even if neither $A$ nor $S$ is compact?

Answer: Yes


Question: Given a set of points $X$ in $\mathbb{R}^2$, define $g(X)$ as the geometric median of $X$, where $g(X) = \arg \min_{y \in \mathbb{R}^2} \sum_{x_i \in X} ||x_i - y||_2$. For two neighboring sets $X$ and $X'$ that differ by only one element ($X \sim X'$), determine the supremum of the Euclidean distance between their geometric medians, $\sup_{X':X \sim X'} ||g(X) - g(X')||_2$. Find this supremum.

Answer: \infty


Question: Determine if the function \( f(x, y) = |xy| \) is differentiable at the origin \((0,0)\). If it is differentiable, find the derivative at that point.

Answer: 0


Question: If the third derivative $f'''(x)$ of a function $f(x)$ exists, does this imply that $f(x)$ is continuous and that both its first and second derivatives are also continuous?

Answer: Yes


Question:
Determine whether the following inequality is true for $u \in W^{s,p}_0(\Omega)$:
\[ \|u\|_{L^p(\Omega)} \leq c \left(\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}\,dx\,dy\right)^{1/p} \]
where $W^{s,p}_0(\Omega)$ is the closure of $C^{\infty}_c(\Omega)$ in $(W^{s,p}(\Omega),\|\cdot\|_{W^{s,p}(\Omega)})$ under the norm
\[ \|u\|_{W^{s,p}(\Omega)}=\|u\|_{L^p(\Omega)}+\left(\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}\,dx\,dy\right)^{1/p}. \]

Answer: True


Question: Determine the values of \( p \) for which the integral \( \int _0 ^\infty \frac {\sin x}{x^p} \, dx \) converges.

Answer: (0, 2)


Question: Find the value of $n$ for which the integral \( I = \int_{n}^{n^2} \frac{1}{x+\sqrt{x}} \, dx \) attains its minimum.

Answer: 3 - 2\sqrt{2}


Question: If \( \frac{1}{N}\sum_{i=1}^N |a_i| \to 0 \) as \( N \to \infty \), does it necessarily follow that \( \frac{1}{N}\sum_{i=1}^N a_i \to 0 \)? Justify your answer.

Answer: Yes


Question: Determine the number of ways to distribute 26 different chocolates to 5 people such that each person receives an odd number of chocolates and no two people receive the same number of chocolates.

Answer: 0


Question: Evaluate the integral \( \int_{|z| = 3} \frac{1}{z^{800} + 1} \, dz. \)

Answer: 0


Question:
Determine whether the following equality holds:
\[ \int_{B(0,2)} (xy^2+y^2z) \, d\mu(x,y,z) = \int_{B(0,1)} (xy^2+y^2z) \sqrt{x^2+y^2+z^2} \, d\lambda_3(x,y,z) \] where \( \mu \) is defined by \( \mu(E) = \int_{E \cap B(0,1)} \sqrt{x^2+y^2+z^2} \, d\lambda_3(x,y,z) \) for every measurable set \( E \subset \mathbb{R}^3 \).

Answer: Yes


Question: Determine the values of the complex constant $c = c_1 + ic_2$ for which the function $f(z) = (x^2 - y^2 + 2c_1xy) + i(2c_2xy)$, where $z = x + iy$, is analytic in the complex plane.

Answer: i


Question: Given positive real numbers $x$, $y$, and $z$ such that $\frac{1}{3} \leq xy + yz + zx \leq 3$, find the range of possible values for $x + y + z$. Express your answer as an interval.

Answer: [1, \infty)


Question: Given two disjoint and compact sets \( S \) and \( T \) in a normed vector space \( W \), does there exist elements \( s \in S \) and \( t \in T \) such that \( f(S,T) = ||s-t|| \), where \( f(S,T) = \inf\{||s-t||:s \in S, t \in T\} \)?

Answer: Yes


Question: Consider a plane $P$ passing through the points $A(\lambda,3,\mu)$, $B(-1,3,2)$, and $C(7,5,10)$. A line $L$ with positive direction cosines passes through $A$, bisects $BC$, and makes equal angles with the coordinate axes. Let $L_1$ be a line parallel to $L$ and passing through the origin. Determine the equation of the plane that is perpendicular to plane $P$ and contains the line $L_1$. Provide your answer in the form of a plane equation.

Answer: x - 2y + z = 0


Question: Evaluate the sum \( \displaystyle \sum\limits_{k=1}^{6}\tan^2 \frac{k\pi}{7} \).

Answer: 42


Question: Find all solutions to the congruence \(x^{2023} + x^{2022} + \dots + x + 1 \equiv 0 \pmod{103}\) that are not congruent to \(1\) or \(0\) modulo \(103\).

Answer: 102


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

Answer: 2


Question: Consider a bounded continuous function \(f(x)\) on \([c,d]\). Define \[f_{n}(x)=\sum_{i=1}^{n}f(x_{i})X_{[x_{i},x_{i+1}]}(x)\] where \(x_{i}^{n}\) is a partition of \([c,d]\) with \(\max_{i=1...n}|x_{i+1}-x_{i}|\to 0\) as \(n\to \infty\). Does \(f_{n}\) converge uniformly to \(f\) on \([c,d]\)?

Answer: Yes


Question: Find the largest positive integer \( n \) such that the inequality \[ \frac{a^2}{\frac{b}{29}+\frac{c}{31}} + \frac{b^2}{\frac{c}{29}+\frac{a}{31}} + \frac{c^2}{\frac{a}{29}+\frac{b}{31}} \ge n\,(a+b+c) \] holds for all positive real numbers \( a, b, c \).

Answer: 14


Question: If $h$ is a real differentiable function defined on $(0,1)$ with a one-to-one derivative, determine if the derivative is continuous on $(0,1)$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{n \to \infty}\frac{1}{\ln{n}}\sum_{k=1}^{\lfloor Bn \rfloor}\frac{1}{k}$$ where $B$ is a positive constant greater than or equal to 1.

Answer: 1


Question: Determine whether the following statement is true: If two random variables $X$ and $Y$ have the same probability distribution, then $E\left(\frac{X}{X+Y}\right) = \frac{1}{2}$. Provide a justification for your answer.

Answer: False


Question: Determine the smallest cardinal \( \kappa \) such that \( \kappa^{<\kappa} = \kappa \) in models of ZFC without large cardinal assumptions.

Answer: \aleph_0


Question: Evaluate the limit: $$\lim _{x\to +\infty } \left(\frac{2+x^{\beta}}{3+x}\right)^{x}$$ for $\beta < 1$. Provide your answer as a single value.

Answer: 0


Question: Calculate the gradient with respect to the matrix $A$ of the function $f(A) = \log(\det(AX))$, where $X$ and $A$ are positive definite matrices.

Answer: A^{-1}


Question: Given any 10 consecutive positive integers, determine if there exists at least one integer that is relatively prime to the product of the other nine integers.

Answer: Yes


Question: Determine if the projection operator $P: W \rightarrow W$ on a subspace $W$ of a real vector space $V$ is diagonalizable.

Answer: Yes


Question: In the context of the equation $A^{-1} A = I$ for the multiplicative inverse of a multivector, is the $I$ on the right-hand side a scalar?

Answer: Yes


Question: Determine the number of free indices in the result when taking the derivative of a matrix-valued function with respect to a matrix.

Answer: 4


Question: Find the sum of the infinite series: $$\sum_{n=0}^\infty \frac{(-1)^n \pi^{2n}}{9^n(2n)!}$$

Answer: \dfrac{1}{2}


Question: Determine the minimum number of iterations of the logarithm function with base TREE(4) that must be applied to TREE(3) to obtain a negative result.

Answer: 2


Question: Let $G$ be a group with subgroups $H$ and $K$ such that $H \cap K = \{1\}$. Given that $H$ centralizes $K$, can we conclude that $K = \{1\}$?

Answer: No


Question: Evaluate the line integral \( \oint\limits_{C} xy^2dx +x^2ydy+z^3dz \), where \( C \) is the curve of intersection of the plane \( x+y+z=1 \) and the cylinder \( x^2 +y^2=4 \), oriented counterclockwise when viewed from above.

Answer: 0


Question: Determine whether a stochastic process $X$ with independent increments is necessarily a Gaussian process. Answer with True or False.

Answer: False


Question: Find all continuous functions \( f: [a, b] \to \mathbb{R} \) that satisfy the condition \( \int_{a}^{x} f(t) \, dt = \int_{x}^{b} f(t) \, dt \) for every \( x \in (a, b) \).

Answer: 0


Question: Given that $X$ and $Y$ are independent standard normal random variables, find the probability that both $X$ and $X+Y$ are positive.

Answer: \dfrac{3}{8}


Question: Determine the minimum number of input/output pairs required to uniquely identify the fixed matrices \( X_{3 \times 3} \) and \( Y_{3 \times 1} \) in the equation \( O_{3 \times 1} = X_{3 \times 3} \times (I_{3 \times 1} - Y_{3 \times 1}) + Y_{3 \times 1} \), given the input/output data \( (I_{3 \times 1}, O_{3 \times 1}) \).

Answer: 4


Question: Find all functions $f(x)$ that satisfy the functional equation $f(x)f(y) = f(2xy + 3) + 3f(x + y) - 3f(x) + 6x$. Then, determine the value of $f(8)$. Provide your answer as a single value for $f(8)$. \( \boxed{\text{value}} \)

Answer: 19


Question:
Evaluate the limit as \( n \to \infty \):
\[ \lim_{n \to \infty} \sqrt[n]{n^{(4n)}+(4n)^n}\left[\left(2+\frac{1}{n^2}\right)^{18}-\left(4+\frac{4}{n^2}\right)^9\right] \]

Answer: 589824


Question: A bracelet is made using 7 identical green beads, 4 identical yellow beads, and 6 identical purple beads. Determine the number of distinct bracelets that can be formed if the beads can move freely around the bracelet and reflections are considered distinct.

Answer: 240240


Question: Determine the order of the subgroup \( H \) of the group of \( 2 \times 2 \) integer matrices modulo a prime \( p \), where each matrix in \( H \) has a determinant of 1.

Answer: p(p^2 - 1)


Question: For primes $p \geq 5$, is the number of generators of the multiplicative group $\mathbb{F}_p^{\times}$ always greater than 1?

Answer: Yes


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

Answer: \dfrac{3}{2}


Question: Find the coefficient of \(x^{57}\) in the expansion of \((x^2 + x^7 + x^9)^{20}\) without using the straightforward binomial expansion.

Answer: 3420


Question: Determine whether there exist any polygons, other than regular polygons, where the length of each diagonal is greater than the length of each side.

Answer: Yes