Question: Evaluate the integral \( \int_0^\infty e^{-\sqrt[4]{x}} \, dx \) using the relation between the factorial and the gamma function.

Answer: 24


Question: Determine the dual number that corresponds to the matrix \( \begin{pmatrix}0 & 0 \\1 & 0 \end{pmatrix} \) in the algebra of dual numbers.

Answer: \epsilon


Question: Find the sum of the series \(\sum\limits_{n=1}^\infty \frac{n^3}{n!}\) by demonstrating the convergence of the series using the definition that a series converges if its sequence of partial sums converges.

Answer: 5e


Question: Find the 10th derivative of \( f(x) = \ln(2+x^2) \) at \( x=0 \) using a Taylor series expansion. Use the properties of the Taylor series and the relation \( f^{(n)}(0) = a_n \cdot n! \) to simplify the process.

Answer: 22680


Question: Evaluate the limit \( \lim_{\{x,y\} \to \{0,0\}} \frac{-\frac{x y}{2}+\sqrt{x y+1}-1}{y \sqrt{x^2+y^2}} \). Determine whether the limit exists, and if so, find its value.

Answer: 0


Question: Determine whether a countable set can have uncountably many infinite subsets such that the intersection of any two distinct subsets is finite. Answer 'yes' or 'no'.

Answer: yes


Question: Evaluate the limit \( \lim_{n\to\infty}\int_{0}^{\infty}\frac{x}{\sqrt[n]{1+x^{3n}}}dx \). Determine whether the limit converges or diverges.

Answer: \dfrac{3}{2}


Question: In a self-complementary graph G with n vertices, suppose there are m vertices with degree k. Determine the number of vertices in G that have a degree of n - k - 1.

Answer: m


Question: Determine the number of elements of order $n$ in the group $\mathbb{Z}/(n,p-1)\mathbb{Z}$, where $p$ is a prime number. Assume that for any divisor $d$ of $n$, there are exactly $\varphi(d)$ elements of order $d$ in $\mathbb{Z}/n\mathbb{Z}$, and $n$ divides $(n,p-1)$. Provide your answer as a single number.

Answer: \varphi(n)


Question: Determine the number of distinct principal ideals in the ring \( \mathbb{Z}_4 \times \mathbb{Z}_6 \).

Answer: 12


Question:
Determine whether the following limit is true:
\[ \lim_{y\rightarrow\infty}\zeta\left(\frac{1}{2}+iy\right)+\frac{1}{2}\frac{1}{[y/\pi]^{1/2+iy}}-\sum_{n=1}^{[y/\pi]}\frac{1}{n^{1/2+iy}}=0. \]

Answer: 0


Question: Determine whether the equation $2^n - 3^m = C$ has only finitely many solutions for any fixed integer $C$, given that $(m, n) \neq (2, 2)$.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\rightarrow 0} \frac{\{x\}}{\tan\{x\}}$$ where $\{x\}$ denotes the fractional part of $x$. Provide your answer as a single number.

Answer: 1


Question: Let \(a, b \in \mathbb{N}\) be two coprime numbers. Determine the number of ordered pairs \((a, b)\) such that \(\frac{b}{a} + \frac{101b}{5050a} \in \mathbb{N}\), given that \(5050=2\cdot5\cdot505\).

Answer: 4


Question: Does there exist a Riemannian manifold with a point $p \in M$ such that there are infinitely many points $q \in M$ where more than one minimizing geodesic exists from $p$ to $q$?

Answer: Yes


Question: Determine the cardinality of the set of automorphisms of the group of rational numbers under addition, excluding those of the form \(x \mapsto qx\) for \(q \in \mathbb{Q}\).

Answer: 0


Question: Determine the range of positive values of \( x \) for which the series \( \sum_{n=0}^{\infty}\frac{1}{1+x^{n}} \) is both convergent and uniformly convergent.

Answer: (1, \infty)


Question: Find the limit of \( \lim\limits_{n\to \infty}\left\{\left(\frac{3}{2}\right) \left(\frac{4}{3}\right)^{2} \left(\frac{5}{4}\right)^{3} \cdots \left(\frac{n+2}{n+1}\right)^{n+1} \right\}^{\frac{1}{n+1}} \).

Answer: e


Question: Determine the number of solutions to the equation \(\cos^{-1}[x^2-\frac{1}{2}]+\sin^{-1}[x^2+\frac{1}{2}]=x^2\), where \(x\in[-1,1]\) and \([x]\) is the greatest integer function of \(x\).

Answer: 0


Question: Solve the equation \( \frac{10^4}{x^2} = \frac{x^{(8 - 2\log x)}}{10^4} \) for \( x \). Given that one solution is \( x=10 \), find any remaining solutions.

Answer: 10000


Question: Use Stokes' Theorem to evaluate \(\displaystyle\iint_S\mathrm{curl}~\mathbf{F}\cdot d\mathbf{S}\) where \(\mathbf{F}(x,y,z) = (x+y)z~\mathbf{i}+ (x-z)y~\mathbf{j}+ (z-y)x~\mathbf{k}\) and \(S\) is the part of the sphere \(x^2+y^2+z^2 = 16\) that lies above the plane \(z = 2\), oriented upward.

Answer: -24\pi


Question: Find a function \( w: \mathbb{R}^2 \to \mathbb{R} \) such that \( iv + w \) is holomorphic in \( \mathbb{C} \), given that \( v(x,y) = \frac{y^2}{2} - \frac{x^2}{2} \).

Answer: xy


Question: Let $I$ and $J$ be ideals of a Noetherian ring $A$. If $IA_P \subseteq JA_P$ for every $P \in \operatorname{Ass}_A(A/J)$, does it follow that $I \subseteq J$? Provide a proof or a counterexample.

Answer: I \subseteq J


Question: Calculate the volume of the region bounded by the coordinate planes, the plane \(x + 2y = 6\), and the elliptic paraboloid \(z = x^2 + 2y^2\).

Answer: 81


Question: Calculate the residues of \(\cot^2(z)\) at its poles.

Answer: 0


Question: Let $X \sim \mathrm{Exp}(1)$ and $Y \sim \mathrm{Exp}(2)$ be independent random variables. Calculate the expected value of $Z = \max(X, Y)$. \( \boxed{?} \)

Answer: \dfrac{7}{6}


Question: Evaluate the integral \( \int_0^2\int_1^3\min(x,y)\,dx\,dy \) given that \( x < y \).

Answer: \dfrac{2}{3}


Question: Evaluate the integral: $$\int_{0}^{1} \left( 1 + \sin^2 \left( \frac{1}{x} \right) + \frac{1}{x^2} \cos^2 \left( \frac{1}{x} \right) - \frac{1}{x} \sin \left( \frac{2}{x} \right) \right) dx$$

Answer: 1


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

Answer: 1


Question: Calculate the surface area of the cylinder defined by the equation $x^2 + y^2 = 2x$, which is bounded by the cone $z = \sqrt{x^2 + y^2}$ and the plane $z = 0$. The center of the cylinder is at $(1, 0)$. Provide a parameterization of this surface to facilitate the calculation of the area.

Answer: 8


Question:
Given that $A$ is a subset of the real numbers $\mathbb{R}$ with $|A| > \aleph_0$, which of the following statements is correct?

A. The cardinality of the union $A \cup \mathbb{Q}$ is greater than the cardinality of the difference $A \setminus \mathbb{Q}$.
B. The cardinality of the intersection $A \cap (\mathbb{R} \setminus \mathbb{Q})$ is greater than $\aleph_0$.
C. The cardinality of the complement $\mathbb{R} \setminus A$ is less than or equal to $\aleph_0$.
D. None of the options A, B, or C is correct.

Determine the correct statement.

Answer: B


Question: Determine whether a holomorphic function $f: \mathbb{C} \to \mathbb{C}$ that is injective and has finite fibers must necessarily be a polynomial.

Answer: Yes


Question:
Determine whether the following matrix equation is true or false:

$$I-X(X^TX)^{-1}X^T = (I-X(X^TX)^{-1}X^T)^2$$

Answer: True


Question: Given a triangle with specified vertices, calculate the discrepancy between its surface area obtained using the surface integral method and the geometric method. Provide the numerical difference between the two results.

Answer: 0


Question: Calculate the limit: $$\lim_{n\rightarrow \infty}\int_0^\infty \frac{e^{\sin(\ln(x))}}{1+\sqrt{n}x^{42}}\,dx.$$

Answer: 0


Question: Calculate the gradient with respect to a positive definite matrix $X$ of the function $f(X) = \log \det(X^{-1})$. What is the gradient in this case?

Answer: -X^{-1}


Question: Given the equations \( x^2 - y^2 - u^3 + v^2 + 4 = 0 \) and \( 2xy + y^2 - 2u^2 + 3v^4 + 8 = 0 \), find \( \frac{\partial v}{\partial y} \) at \((x,y) = (2,-1)\). Assume that these equations determine functions \(u(x,y)\) and \(v(x,y)\) near the point \((x,y,u,v) = (2, -1, 2, 1)\).

Answer: -\dfrac{1}{16}


Question: Determine the pullback of the 2-form $\omega = ze^{xy} \, dx \wedge dy$ on $\mathbb{R}^3$ under the map $\varphi: \mathbb{R}^2_{\ne 0} \to \mathbb{R}^3$ defined by $\varphi(u,v) = \left( \frac{u}{v}, \frac{v}{u}, uv \right)$. Express $\varphi^*\omega$ in terms of $u$ and $v$.

Answer: 0


Question: Determine if the equation \(x^{17^{-x}}=17^{17^{17^{-x}}}\) has a solution. If it does, find the value of \(x^{x^{-1}}\).

Answer: 17


Question:
Let $a, b, c$ be positive real numbers, and let $\Gamma$ denote the Gamma function. Evaluate the sum:
\[ \sum_{n=0}^\infty \frac{a^n}{\Gamma(b+nc)} \]

Answer: E_{c,b}(a)


Question: Evaluate the contour integral \( \int_{C(0,2)} \frac{e^z}{z-1} \, dz \) using Cauchy's Integral Theorem, where \( C(0,2) \) is the circle centered at the origin with radius 2.

Answer: 2\pi i e


Question: Given that $f(x)$ and $f(x)g(x)$ belong to the Sobolev space $\mathcal{W}^{s+1}$ with $s \ge 1$, and $g(x)$ is a bounded function on $[0,\infty)$, with $\int_{0}^{\infty} x^k f(x) \, dx < \infty$ for all $k \ge 0$, and $0 < \int_{0}^{\infty} f(x)g(x) \, dx < \infty$, determine whether $\lim_{x\rightarrow \infty} f(x) = 0$. Provide a justification for your conclusion.

Answer: 0


Question: Determine the domain of the function \( \frac{1}{\sqrt{g(x)}} \), given that \( g: \mathbb{R} \rightarrow \mathbb{R} \) satisfies the functional equation \( g(x-y) = g(x) + g(y) \).

Answer: \emptyset


Question: Let $f(x+y) = f(x)f(y)$ for all $x$ and $y$, with $f(5) = -2$ and $f'(0) = 3$. Determine the value of $f'(5)$. \( \boxed{} \)

Answer: -6


Question: Let $f:\mathbb{R} \to \mathbb{R}$ be a convex function. Determine whether there exists a linear map $g(y) = ay + b$ such that $f(y) \geq g(y)$ for all $y \in \mathbb{R}$ and $f(x) = g(x)$ for some $x \in \mathbb{R}$. Provide a justification for your answer.

Answer: Yes


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

Answer: 0


Question: Evaluate the limit: \[ \lim_{{(x,y) \to (1,0)}} \frac{{(x-1)^2 \ln x}}{{(x-1)^2 + y^2}} \] Prove that the limit exists for all paths approaching \((1, 0)\).

Answer: 0


Question: Calculate the operator norm of the operator \( S: C[0,1] \rightarrow C[0,1] \) defined by \( Sx(t) = t^2 \int_0^t x(s)\,ds \).

Answer: 1


Question: Suppose $T: V \rightarrow W$ is a bijective linear function. If vector addition and scalar multiplication on $W$ are defined using the inverse of $T$, does $W$ become a vector space?

Answer: Yes


Question: Determine the values of $z$ for which the alternating series \( \sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt n}z^n \) converges conditionally.

Answer: 1


Question: Find the general solution for the differential equation \( f'(x) = f(\cos(x)) \), given that \( f(x) \) is bounded.

Answer: 0


Question: Determine whether the following statement is true or false: If a function $f$ is continuous on the entire complex plane, then the function $G(w) = \int_0^w f(z)dz$ (where the integral is taken over the circle of radius $R$ centered at 0) is holomorphic on the closed disk of radius $R$ centered at 0.

Answer: B


Question: Determine whether the expression $27195^8 - 10887^8 + 10152^8$ is divisible by $26460$, where $26460 = 2^2 \times 3^3 \times 5 \times 7^2$. Use the given factorizations: $10152 = 2^3 \times 3^3 \times 47$ and $27195 - 10887 = 16308 = 2^2 \times 3^3 \times 151$. Justify your answer.

Answer: Yes


Question: Determine whether the function \( f = \sum_{n=0}^{\infty} 2^{-n}\chi_{[n,n+1)} \) is Lebesgue integrable on \( \mathbb{R} \). Justify your answer.

Answer: Yes


Question: Calculate the average value of \( S \), where \( S \) represents the number of adjacent pairs of a boy and a girl in a lineup of 7 boys and 13 girls arranged in all possible orders.

Answer: \dfrac{91}{10}


Question: Calculate the expected value of \(\frac{X}{Y}\) given that \(X \sim \operatorname{Exp}(\lambda =1)\) and \(Y\sim U(1,2)\) are independent continuous random variables.

Answer: \ln 2


Question: Suppose $U$ is a random variable uniformly distributed on $[0,1]$, and $X_1$, $X_2$ are independent, identically distributed, non-negative random variables. If $U$, $X_1$, and $X_2$ are independent, find the conditional probability $P(U<X_1|X_1+X_2=1)$. Provide your answer as a single value.

Answer: \dfrac{1}{2}


Question: Given the differential equation \( xy'' + y' = 0 \) and a known solution \( y_1 = \ln(x) \), find the second linearly independent solution.

Answer: 1


Question: Determine the radius of convergence of the series $$ \sum_{n=0}^{\infty} z^{a^n} $$ where $a>1$ is an integer.

Answer: 1


Question: Evaluate the limit: \[ \lim_{x \to 0^{+}} \frac{x}{a} \left\lfloor \frac{b}{x} \right\rfloor \] where \( a, b > 0 \).

Answer: \dfrac{b}{a}


Question: In the measure space $(X, M, \mu)$, does convergence in mean imply convergence in measure?

Answer: Yes


Question: Determine whether the function \( f(t) = \int_0^t \sin(t\sqrt{1+x^2}) \, dx \) has a local minimum at \( t = 0 \).

Answer: Yes


Question: Consider the operator $A = (-\Delta)^{-1}(u\cdot\nabla)$, where $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is defined such that $u(x,y)=(u_1(x,y),u_2(x,y))$. What is $A^0$?

Answer: I


Question: Let $f$ be a continuous function on $[0,+\infty)$ such that $\int_0^{\infty}f(x)\ dx=2$. Determine the limit \( \lim \limits_{x\to \infty}\frac{\int_0^x tf(t)\ dt}{x} \).

Answer: 0


Question: Find the lower bound for the sum: \[\sum_{n=N^\delta}^N\sqrt{\ln(n)}\] where \(\delta\) is a positive constant less than 1.

Answer: \Omega(N \sqrt{\ln N})


Question: Determine all possible values of $f(36)$ for the function $f: \mathbb{R} \to \mathbb{R}$ satisfying the equation \[ \frac{f(x)f(y) - f(xy)}{3} = x + y + 2 \] for all real numbers $x$ and $y$. Provide your answer as a set of values.

Answer: \{39\}


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty}{\left(\sum_{k=1}^{n}{\frac{1}{k}}\right)}z^n \).

Answer: 1


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(\frac{\sqrt{1\times2}}{n^2+1}+\frac{\sqrt{2\times3}}{n^2+2}+\cdots+\frac{\sqrt{n\times(n+1)}}{n^2+n}\right)$$

Answer: \dfrac{1}{2}


Question: Determine whether a star compact subset $A$ of the real numbers $\mathbb{R}$ is necessarily closed. A set $X$ is star compact if every open cover of $X$ admits a finite subset $A\subseteq X$ such that $X=\operatorname{st}(A,\mathcal{U})$, where $\operatorname{st}(A,\mathcal{U}) = \bigcup \{ U \in \mathcal{U} : A \cap U \neq \emptyset \}$. A subset of $\mathbb{R}$ is closed if and only if every convergent sequence of points in it converges to a point within the set.

Answer: Yes


Question: Evaluate the integral: $$ \int_{-\infty}^{+\infty} \frac{x^2}{(x^2 - 8x + 20)^2} \, dx $$

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


Question: Determine the number of zeros of the polynomial \(3z^4 - 8z^2 + z - 2\) inside the annulus \(\{z : 2 < |z| < 3\}\) using Rouché's theorem.

Answer: 0


Question: Determine whether the $j$-invariant \( j(\tau) = 1728 \frac{g_2^3(\tau)}{\Delta(\tau)} \) for \( \tau \in \mathbb{H} \) attains every complex value.

Answer: Yes


Question: Calculate the limit: $$ \lim_{x \to \infty} \frac{1}{x^2}\log\left(\left(1+(\exp(2x)-1)^2\right)\right)^2. $$

Answer: 16


Question: Find the maximum value of \( P(X=1) \) for a random variable \( X \) that takes non-negative natural number values, given that \( E[X]=1 \), \( E[X^2]=2 \), and \( E[X^3]=5 \). Use the Taylor expansion of the probability generating function at \( z=1 \).

Answer: \dfrac{1}{2}


Question: The base of a solid S is an elliptical region with the boundary curve given by the equation \(25x^2 + 16y^2 = 400\). Cross-sections of the solid perpendicular to the x-axis are isosceles right triangles with their hypotenuse lying along the base. Calculate the volume of the solid S.

Answer: \dfrac{400}{3}


Question: Find the value of $x$ that maximizes the area of an isosceles triangle inscribed inside the ellipse given by the equation $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, with one vertex at $(a,0)$ and the other vertices at $(x,y)$ and $(x, -y)$. Express the area $A$ using Heron's formula.

Answer: -\dfrac{a}{2}


Question: Find the limit of the function \( g(x)=\left[\frac{cx+3}{dx-1}\right]^{x} \) as \( x \) approaches infinity, where \( c,d >0 \) and \( c>d \).

Answer: \infty


Question: Evaluate the infinite series \( \sum_{n = 1}^\infty \left( \frac{1}{n} - \frac{1}{n + x} \right) \) for \( x = 0.500 \). Round your answer to 3 decimal places and ensure the precision is at least 12 significant digits.

Answer: 0.614


Question: Solve the differential equation \( v'v = 8yv \) with the initial condition \( v(0) = 1 \).

Answer: 4y^2 + 1


Question: Given a function \( f(x,y) \) with continuous partial derivatives everywhere, and \( \nabla f(0,-18) = -2i + 3j \), define a new function \( g(x,y) = f(xy + x^2, xy - y^2) \). Calculate \( \nabla g(3,-3) \).

Answer: -15\mathbf{i} + 21\mathbf{j}


Question: Evaluate the integral: $$\int_0^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx$$

Answer: 0


Question: Find the value of $K$ such that the expression \( R = \sin^6\alpha + \cos^6\alpha + K(\sin^4\alpha + \cos^4\alpha) \) is independent of \( \alpha \).

Answer: -\dfrac{3}{2}


Question: Evaluate the limit \( \lim_{n\to\infty} \int_{-1/n}^{1/n} \frac{f(x)}{1+n^2x^2} \, dx \), where \( f(x) \) is a continuous function on \([-1,1]\).

Answer: 0


Question: Evaluate the limit: $$\lim _{x\rightarrow \frac{\pi}{4}}\left( \tan x\right) ^{\sec ^{2}x}$$

Answer: 1


Question: Evaluate the limit: $$\lim_{x\rightarrow\infty}\left(\Gamma\left(\frac{1}{x}\right)\right)^{-1}\int_{0}^{x}\frac{|\sin\left(t\right)|}{t}\:dt$$

Answer: 0


Question: Provide an example of a group \( G \) such that the quotient group \( G/Z(G) \) is not abelian.

Answer: S_3


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty}\frac{\log((n+1)!)}{n+1}$$

Answer: \infty


Question: Evaluate the integral: $$\int_{-\infty}^{\infty} \frac{1}{x^6+1}dx$$

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


Question: In the figure, quarter arc $AD$ of the circle $x^2+y^2=16$ is given. Points $B$ and $C$ are arbitrary on the arc $AD$ such that $C$ is between $D$ and $B$. Determine the maximum possible area of the pentagon $OABCD$, where $O$ is the origin.

Answer: 12


Question: Determine the smallest prime factor of the expression \(4^{52} + 52^{2013} + 2013^{52}\) without using a calculator.

Answer: 7


Question: Find the intersection number at the point \((0,0)\) of the curves given by the equations \(y^3 = x^2\) and \(y = x^2\).

Answer: 2


Question: In triangle $\triangle ABC$, $\angle B$ is obtuse. The interior bisector $BM$ is drawn from point $B$, along with the altitudes $AN$ and $CQ$. Given that $AN = 8$ cm and $CQ = 12$ cm, determine the length of the perpendicular drawn from $M$ to line $\overline{AC}$.

Answer: \dfrac{24}{5}


Question: Determine the number of 3x3 matrices over the field of three elements that are conjugate to a diagonal matrix.

Answer: 2109


Question: Evaluate the integral: $$\int_{0}^{\frac{\pi}{2}} \sqrt{\frac{1+\sin x}{2}} \, dx$$

Answer: \sqrt{2}


Question: Consider a matrix with a characteristic polynomial of $x^4$. Determine if its Jordan normal form can consist of two Jordan blocks of size $2 \times 2$. Justify your answer.

Answer: Yes


Question: Verify the integral equality: \( \int_{0}^{1}x\ln[\ln{x}\ln(1-x)]\,dx=-\gamma \).

Answer: -\gamma


Question: Determine the maximum possible rank of a set such that the ranks of all its finite subsets are bounded by a given ordinal \( \beta \).

Answer: \beta


Question: In the ordered square topology, determine whether all basis elements of the form $(a \times b, c \times d)$ are open sets.

Answer: Yes


Question: Determine whether there exists a non-commutative ring $R$ such that $R^m$ and $R^n$ are isomorphic as left $R$-modules, where $m$ and $n$ are distinct.

Answer: Yes


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

Answer: 0


Question: Determine if the cardinality of computable numbers in the interval \((0, 1)\) is equal to the cardinality of all computable numbers, given that the tangent function preserves computability in both directions.

Answer: Yes


Question: Find the residue of the function \( f(z) = \frac{1}{(z-2i)^5} + \frac{2}{z+3i} \) at the pole \( z = 2i \).

Answer: 0


Question: Find the standard expression for the sum \( \sum_{r=0}^{n}\binom{n}{r}^2 \).

Answer: \dbinom{2n}{n}


Question: In triangle $\triangle ABC$, point $P$ is an interior point such that $Q$ and $R$ are the reflections of $P$ across sides $AB$ and $AC$, respectively. If points $Q, A,$ and $R$ are collinear, find the measure of $\angle A$. \(\text{Express your answer in degrees.}\)

Answer: 90


Question: For a bijective map \( f: \mathbb{Z}_{2n} \rightarrow \mathbb{Z}_{2n} \), where \( n \geq 2 \), must the set \( \{f(t) - t: t \in \mathbb{Z}_{2n}\} \) contain at least two distinct elements that are congruent modulo \( 2n \)?

Answer: Yes


Question: Consider the sequence \( \{y_n\} \) defined by \( y_n = \frac{n}{\lfloor nx \rfloor} \) for \( x \in \mathbb{R} \) and \( x \neq 0 \), where \( \lfloor \cdot \rfloor \) denotes the integer part. Determine whether the sequence \( \{y_n\} \) converges, and if so, find its limit.

Answer: \dfrac{1}{x}


Question: Determine the maximum value of the function \( f(x) = \int_{1/2}^1 e^{|t-x|} \, dt \) for \( \frac{1}{2} \leq x \leq 1 \).

Answer: \sqrt{e} - 1


Question: Determine the point on the surface \(x^2+2y^2-z^2=4\) that is farthest from the point \((1,0,0)\).

Answer: (-2, 0, 0)


Question: Let $\{W_t\}_{t\ge 0}$ be a standard 2D Brownian motion starting at $(1,1)$ in $\mathbb{R}^2$. Determine the probability that $W_t$ hits the positive x-axis before it hits the negative x-axis.

Answer: \dfrac{3}{4}


Question: Compute the limit \( \displaystyle \lim_{x\to\frac{2}{3}}\frac{x^2 \cos(\pi/x)}{x - \frac{2}{3}} \) without using L'Hôpital's rule.

Answer: -\pi


Question: Evaluate the limit: $$ \lim_{x \rightarrow 0} \frac{(\ln(1+x))^x}{(1+x)^{\ln(1+x)}} $$.

Answer: 1


Question: Determine the number of possible sets $A_1, A_2, A_3, A_4 \subseteq [n]$ such that $A_1 \cap A_2 \cap A_3 \cap A_4 = \emptyset$ and $A_1 \cup A_2 \cup A_3 \cup A_4 = [n]$. Provide your answer as a single integer.

Answer: 14^n


Question: Find the limit of \( \left\lfloor \frac{1}{\int (\frac{rn}{n^3 + r}) \, dn} \right\rfloor \) as \( n \to \infty \).

Answer: 0


Question: Find the multiplicative inverse of the polynomial $4x+3$ in the quotient ring $\mathbb{Z}_{11}[x]/\langle x^2+1 \rangle$. If no inverse exists, state so.

Answer: 6x + 1


Question: Determine the limit of the solution to the initial value problem $x'(t)=x^2-x$, $x(0)=\frac{1}{3}$ as $t\rightarrow\infty$, if it exists.

Answer: 0


Question: Given real positive numbers \( a, b, c \), find the value of \( \tan \theta \) where \( \theta = \tan ^{-1}\left[\left(\frac{a(a+b+c)}{bc}\right)^{\frac{1}{2}}\right] + \tan ^{-1}\left[\left(\frac{b(a+b+c)}{ca}\right)^{\frac{1}{2}}\right] + \tan ^{-1}\left[\left(\frac{c(a+b+c)}{ab}\right)^{\frac{1}{2}}\right].\)

Answer: 0


Question: Find the quotient group of the multiplicative group of non-zero complex numbers \( \mathbb{C}^* \) modulo the subgroup of complex numbers with modulus 1, \( S^1 \).

Answer: \mathbb{R}^+


Question: Minimize \( f(x_1) + f(x_2) + f(x_3) \) subject to the constraints \( x_1 + x_2 + x_3 = 3 \) and \( x_1, x_2, x_3 \in [0,3] \), where \( f(x) = \frac{2x+1}{x^2+3} \).

Answer: \dfrac{5}{4}


Question: Given the function \(f(z) = (4 + i)z^2 + az + b\), where \(z\), \(a\), and \(b\) are complex numbers, find the minimum value of \(|a| + |b|\) if both \(f(1)\) and \(f(i)\) are purely real.

Answer: \sqrt{2}


Question: Determine whether the following statement is true or false: For positive integers $i$ and $j$, and a natural number $n > 1$, if $a_{0}^{n} + a_{1}^{n} + \ldots + a_{i}^{n} < b_{0}^{n} + b_{1}^{n} + \ldots + b_{j}^{n}$, then $a_{0}^{n + 1} + a_{1}^{n + 1} + \ldots + a_{i}^{n + 1} < b_{0}^{n + 1} + b_{1}^{n + 1} + \ldots + b_{j}^{n + 1}$ for all $a_{k} \geq 0$ (where $0 \leq k \leq i$) and $b_{k} \geq 0$ (where $0 \leq k \leq j$).

Answer: False


Question: Find the value of $m \in \mathbb{R}$ such that the polynomial $P(X) = (X+1)^{2n+1} + (m-1)X^{n+2}$ is divisible by $X^2 + X + 1$ for all $n \in \mathbb{N}$. Provide your answer as a single real number.

Answer: 2


Question: Compute the minimal number of generators, \( \mu(mA_m) \), of the ideal \( mA_m \), where \( A=\mathbb{C}[x,y]/(x^3-y^2) \) and \( m=(\bar{x} -1 , \bar{y} -1) \) is a maximal ideal of \( A \).

Answer: 1


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

Answer: 0


Question: Determine whether there exist an integer $k$ and two infinite sequences of natural numbers $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$, with $0\leq b_n<a_n$ for all $n$, such that $k\equiv b_n \pmod{a_n}$ for all $n$. Provide a justification for your answer.

Answer: Yes


Question:
Let $f$ be an entire function with a zero at the origin of multiplicity $7$. Determine which statement is false:

(A) There exists a neighborhood of the origin where $f(z)$ takes on every value close to zero at least once.
(B) The equation $f(z) = a$ has exactly $7$ solutions in a small neighborhood of the origin for $a$ close to zero.
(C) The function $f(z)$ has a critical point at the origin.
(D) The image of a small neighborhood of the origin under $f(z)$ covers a neighborhood of zero.

Answer: B


Question: Provide an example of an infinite group where every nontrivial subgroup is also infinite.

Answer: \mathbb{Z}


Question: Determine whether the polynomial \( f(x) = x^4 + x^3 + x^2 + x + 1 \) can be factored into a product of two quadratic polynomials in the ring \( R = (\mathbb{Z}/11\mathbb{Z})[x] \).

Answer: Yes


Question: Determine the asymptotic probability that a randomly chosen $n \times n$ matrix $A$ over $\mathbb{F}_2$ satisfies $A^2 = 0$ as $n$ approaches infinity, where each entry of the matrix is independently $0$ or $1$ with equal probability.

Answer: 0


Question:
Find the limit of the sequence \((U_n)\) defined by the recursion:
\[ x_1 = \sqrt[2022]{\frac{2022}{2021}} \]
\[ x_{n+1} = \frac{n+2}{3n+11}\left(\sqrt{x_n} + \sqrt[3]{7 + x_n}\right), \quad \forall n \in \mathbb{N^*} \]

Answer: 1


Question:
Find all functions \( g: \mathbb{R} \to \mathbb{R} \) that satisfy the following conditions:

1. \( g \) has the intermediate value property (IVP).
2. \( g(x-y) = g(x) - g(y) \) for all \( x, y \in \mathbb{R} \).

Answer: g(x) = kx


Question: Evaluate the limit \(\lim_{x\rightarrow0^+}x^{\tan(x^2)}\).

Answer: 1


Question: Evaluate the integral \( I_{k} = \int_{t = 0}^{1}\int_{y = y_{k}}^{y_{k+1}}\int_{x = x_{k}}^{x_{k+1}} \left\{\dfrac{y}{t}\right\}\left\{\dfrac{t}{x}\right\} dt dy dx \) where \( x_{k} = y_{k} = \frac{k}{n} \), \( k = 0,\dots,n-1 \) and \( n \geqslant 2 \). Express the result in terms of \( h = \frac{1}{n} \) and \( k \). Here, \( \{z\} \) denotes the fractional part of \( z \) (\( z > 0 \)), i.e., \( \{z\} = z - \lfloor z\rfloor \) where \( \lfloor z \rfloor \) is the greatest integer less than or equal to \( z \).

Answer: \dfrac{h^3}{12}


Question: Given the equations $ac + bd + bc + ad = 42$ and $c^2 - d^2 = 12$, where $a$, $b$, $c$, and $d$ are non-negative integers, determine all possible values of $a + b + c + d$.

Answer: 13


Question: Evaluate the limit: $$\lim_{x\rightarrow\infty}\left(\frac{p\left(x\right)}{q\left(x\right)}\right)^{\frac{1}{\ln x}}$$ where $q(x)$ is a polynomial of degree $m$ with leading coefficient $b_m$.

Answer: e^{n - m}


Question: Starting at 6 a.m., cars, buses, and motorcycles arrive at a highway toll booth according to independent Poisson processes. Cars arrive about once every 5 minutes. Buses arrive about once every 10 minutes. Motorcycles arrive about once every 30 minutes. Calculate the probability that the seventh motorcycle arrives within 45 minutes of the third motorcycle.

Answer: 0.0656


Question: Find the minimum value of the expression \( \frac{18}{a+b} + \frac{12}{ab} + 8a + 5b \), given that \( a \) and \( b \) are positive real numbers.

Answer: 30


Question: Under the assumption of the Riemann Hypothesis, what percentage of Riemann zeros are known to be simple?

Answer: 70.37\%


Question: Evaluate the integral: $$\int_{0}^{\infty}\frac{\ln x}{(x+1)^{3}}\mathrm{d}x$$

Answer: -\dfrac{1}{2}


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty }\int_{0}^{\pi /2}\sqrt[n]{\sin^n x + \cos^n x}\,dx$$

Answer: \sqrt{2}


Question: Suppose $X$ is a complete metric space and $f, g$ are self-maps on $X$. If $f$ is a contraction and $f(g(x)) = g(f(x))$ for all $x \in X$, does $g$ have a fixed point? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty }^{\infty }\dfrac{dx}{(x^{2}+1)^{3}} \).

Answer: \dfrac{3\pi}{8}


Question: Find the coefficient of $x^{16}$ in the expansion of $(x+3)^7(x+2)^7(x+1)^3$. Use the factored form to determine the coefficient.

Answer: 38


Question: Let $p$ be a prime number such that $p \equiv 1 \pmod{4}$. Show that the Legendre symbol $\left(\frac{-1}{p}\right) = 1$. Provide a brief explanation.

Answer: 1


Question: Determine all functions \( f: \mathbb{N} \to \mathbb{N} \) that satisfy the equation \( f(m^2 + n^2) = f(m)^2 + f(n)^2 \) for all \( m, n \in \mathbb{N} \).

Answer: f(n) = n


Question: Determine the number of elements \( \alpha \in F_{83} \) such that the polynomial \( t^2 + 5t + \alpha \) is irreducible over the field of 83 elements.

Answer: 41


Question: Evaluate the double integral \( \int \int_R |xy| \, dx \, dy \), where \( R \) is the region enclosed by \( x^2+4y^2 \geq 1 \) and \( x^2+y^2 \leq 1 \).

Answer: \dfrac{3}{8}


Question: Determine the time complexity of finding a perfect matching in a graph with $n$ vertices and $m$ edges using the Edmonds matching algorithm.

Answer: O(n^2 m)


Question: Determine the number of positive roots for the equation \(e^{2x} - \ln(2x) - 2^{2015} = 0\) using the concept of concavity.

Answer: 2


Question: Determine whether the function \( u(x, y) = \ln(x^2 + y^2) \) can be the real part of a holomorphic function \( f(z) \) on a domain that does not include the origin. If it can, find the function \( f(z) \).

Answer: 2 \log z


Question: Evaluate the limit \( \lim_{(x,y)\rightarrow (0,0)}\frac{e^{-\frac{1}{x^2+y^2}}}{x^6+y^6} \).

Answer: 0


Question: Evaluate the limit \( \lim_{n \to \infty} \sin^n\left(\frac{2\pi n}{3n+1}\right) \).

Answer: 0


Question: Evaluate the integral \( K = \int_0^4 \frac{\ln (4x-x^2)}{\sqrt{4x-x^2}} \, dx \).

Answer: 0


Question: Determine the values of $z_0$ for which the function $f(z) = (\bar{z})^2$ is complex differentiable or holomorphic.

Answer: 0


Question: Consider the sequence defined by $x_1 = 1$ and $x_{n+1} = 4 - \frac{1}{x_n}$ for $n \geq 1$. Determine the limit of this sequence as $n$ approaches infinity. Is it $2 + \sqrt{3}$ or $2 - \sqrt{3}$?

Answer: 2 + \sqrt{3}


Question: Solve the heat equation \( u_t = u_{xx} \) for \( x \in \mathbb{R}, t > 0 \) with the initial condition \( u(x,0) = \begin{cases} 0, & 0 \leq x \leq 1 \\ 1, & \text{otherwise.} \end{cases} \). Find \( \lim_{t \to 0^+} u(1,t) \).

Answer: \dfrac{1}{2}


Question: Determine whether the equation $PP^* = PP^*P$ holds for any complex matrix $P$. Provide a proof or counterexample.

Answer: PP^* \neq PP^*P


Question: Given i.i.d. random variables $X_1, \dots, X_n$ from a normal distribution with unknown mean $\theta$ and variance 1, find the maximum likelihood estimator for the probability that $X_i \leq 0$. Express your answer in terms of $\theta$. \( \text{MLE} = \boxed{\Phi(-\bar{X})} \) where $\bar{X}$ is the sample mean.

Answer: \Phi(-\bar{X})


Question: For which real numbers \(\alpha\) does the sequence \(a_n = \sin(\alpha n)\) converge to a limit?

Answer: \alpha = k\pi


Question: Find the general solution to the recurrence relation \(a_n=1+\frac{1}{2}a_{n+1}+\frac{1}{2}a_{n-1}\) for \(n=0,1,\ldots,T\) with boundary conditions \(a_0=a_T=0\).

Answer: a_n = n(T - n)


Question: Evaluate the sum \(\sum_{0\le k\le n/2}\binom{n-k}{k}\).

Answer: F_{n+1}


Question: Find the number of permutations in the symmetric group $S_{12}$ that have the cycle structure $(2,2,0,1,2,0,0,0,0,0)$. Express your answer as a single integer.

Answer: 0


Question: Verify the equality: \[ \sum_{j=1}^{\infty} \frac{e^{-j}j^{j-1}}{j!}=1. \]

Answer: 1


Question: Determine whether the random variable \(X = U_{1}^{2} + U_{2}^{2} \mid U_{1}^{2} + U_{2}^{2} \leq 1\) is uniformly distributed on the interval \([0, 1]\), where \(U_{1}\) and \(U_{2}\) are independent and identically distributed uniform random variables on the interval \([-1, 1]\).

Answer: Yes


Question:
Given three points \((a, b), (c, d), (x, y)\) on a circle with radius \(r\) in a rectangular coordinate plane, determine the minimum possible value of the expression:
\[
(ax + by - c)^2 + (bx - ay + d)^2 + (cx + dy + a)^2 + (dx - cy - b)^2.
\]

Answer: 2r^4 + 2r^2


Question: Determine whether the following statement is true or false: For data sets containing only rational numbers, the square of the correlation coefficient will always be rational.

Answer: True


Question: Determine all values of \( M \) for which the bilinear form \( \langle u, v \rangle = ac - 3ad - 3bc + Mbd \) represents an inner product on \( \mathbb{R}^2 \), where \( u = \langle a, b \rangle \) and \( v = \langle c, d \rangle \).

Answer: M > 9


Question: Simplify the expression \(x=(1-2\cdot 10^{-23})^{(2\cdot 10^{22})}\) analytically and show that it converges to \(x = \exp(-0.4)\).

Answer: \exp(-0.4)


Question: Determine whether the following statement is true or false: If $C$ is a square matrix of size $n$ and $\det(C) = 0,$ then $C^n = O_n$ (the zero matrix). If the statement is false, provide a counterexample.

Answer: False


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

Answer: 43


Question: Given the sequence of sets $E_n = (n, \infty)$, where $E = \cap_{n=1}^{\infty}E_n$ is empty, determine the value of $\lim_{n \to \infty} m(E_n)$.

Answer: \infty


Question: Given any uncountable subset $S$ of the unit interval $[0, 1]$, does $S$ have an accumulation point that also lies in $S$? Provide a proof or counterexample.

Answer: Yes


Question:
Consider an interval $I$ and functions $g$, $f$, and $h$ defined on $I$ except possibly at $x_0$. For every $\varepsilon > 0$ and all $x \in I$, there exists a constant $L$ such that
$$ (L - \varepsilon)g(x) < f(x) \leq Lh(x). $$
Moreover, as $x$ approaches $x_0$, the limits of $g(x)$ and $h(x)$ are equal to $1$, i.e.,
$$ \lim_{x \to x_0} g(x) = \lim_{x \to x_0} h(x) = 1. $$
Does this imply that
$$ \lim_{x \to x_0} f(x) = L? $$

Answer: Yes


Question: Suppose $G_1$ and $G_2$ are groups with orders $|G_1| = 2p$ and $|G_2| = 2q$, where $p$ and $q$ are distinct prime numbers. Consider a non-trivial homomorphism $\phi: G_1 \rightarrow G_2$. Determine the order of the kernel, $|\ker(\phi)|$. \(\boxed{\text{Your answer here}}\)

Answer: p


Question: Calculate the area of the region enclosed by the polar curve \( r = \arctan(r) \) from \( \phi = 0 \) to \( \phi = \frac{\pi}{\sqrt{3}} \).

Answer: 0


Question: Determine whether it is possible to construct a holomorphic function on the unit disk that approaches infinity at multiple distinct points on the boundary.

Answer: Yes


Question: Determine whether a matrix in $GL_n(\mathbb{Z})$ can be constructed such that its first row is $(a_1, \ldots, a_n)$, where $a_1, \ldots, a_n$ are integers with GCD 1.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0}\left[1^{\frac{1}{\sin^2x}}+2^{\frac{1}{\sin^2x}}+3^{\frac{1}{\sin^2x}}+\cdots + n^{\frac{1}{\sin^2x}}\right]^{\sin^2x} \]

Answer: n


Question: Determine the maximum possible length of a path that includes all vertices in an Eulerian semi-graceful graph with $n$ vertices.

Answer: n-1


Question: Evaluate the integral \( \int_{-\infty}^\infty x^3 e^{-ax^2} \, dx \) using integration by parts, assuming \( a \) is a non-zero constant.

Answer: 0


Question:
Given a discrete time Markov chain with state space \( \{A, B, C\} \) and the transition matrix:

\[\begin{bmatrix}
1/4 & 1/2 & 1/4 \\
1/3 & 1/3 & 1/3 \\
1/2 & 1/4 & 1/4
\end{bmatrix}\]

Find the expected value of the time it takes to move from state A to state C.

Answer: \dfrac{7}{2}


Question: Let \( N \) be the set of all positive integers and \( S = \{(a,b,c,d) \in N^4 : a^2 + b^2 + c^2 = d^2 \} \). Determine the largest positive integer \( m \) such that \( m \) divides \( abcd \) for all \((a,b,c,d) \in S\).

Answer: 12


Question: Let $M$ be a connected smooth manifold and $N$ a closed embedded submanifold of $M$ with the same dimension as $M$. Are $M$ and $N$ necessarily equal?

Answer: Yes


Question: Determine whether there exists a real number $r$ that cannot be expressed as a finite sum of integral powers of two given transcendental numbers $a$ and $b$ with rational coefficients. Specifically, is there an $r \in \mathbb{R}$ such that \( r \notin \left\{ \sum_{i=0}^n (c_i a^i + d_i b^i) \mid c_i, d_i \in \mathbb{Q}, \; 0 \le n \in \mathbb{Z} \right\} \) for any finite $n$?

Answer: Yes


Question: Let $X$ be a measure space with finite measure. Suppose $\{f_n\}$ is a sequence in $L^p(X)$ that converges to $f$ in $L^p(X)$, where $1\le p<\infty$. If there exists a constant $K$ such that $|f_n(x)|\le K$ for all $n\in \mathbb{N}$ and almost every $x\in X$, does $\{f_n\}$ converge to $f$ in $L^r(X)$ for all $1\le r<p$?

Answer: Yes


Question: Let $R$ be a commutative ring with unity, and consider the polynomial ring $R[x]$. For a non-negative integer $n$, define $\mathcal{P}_n$ as the set of all polynomials in $R[x]$ that are either zero or have a degree of at most $n$. If $a \in R$ is a unit (invertible) and $p_1(x), \ldots, p_n(x) \in R[x]$ are polynomials with invertible leading coefficients, such that $\deg p_i(x) = i$ for all $i = 1, \ldots, n$, does the set $\{a, p_1(x), \ldots, p_n(x)\}$ generate $\mathcal{P}_n$ as an $R$-module?

Answer: Yes


Question: Find the \(3 \times 3\) matrix that converts a vector from basis \(A[a,b,c]\) to basis \(B[d,e,f]\) in \(\mathbb{R}^3\).

Answer: B^{-1}A


Question: Determine whether the function \( f(x) = \frac{1}{x} - \frac{1}{x+1} + \frac{1}{x+2} - \frac{1}{x+3} + \cdots \) is continuous on the interval \((0, \infty)\).

Answer: Yes


Question: Suppose \(a_n\) for \(n\geq 0\) is a complex number such that \[\sum_{n\geq 0}a_n(z-2i)^n=\sum_{n\geq 0}(n+1)z^n\] holds for each \(z\in \mathbb{C}\) such that \(|z-2i|<1\). Find the value of \(|a_2|\).

Answer: \dfrac{3}{25}


Question: Verify Stokes' theorem for the vector field \( F = (z, x, y) \) over the surface of the hemisphere \( x^2 + y^2 + z^2 = 4 \), \( z \geq 0 \), oriented in the direction of the positive z-axis.

Answer: 4\pi


Question: Evaluate the integral \( \int_{0}^{1}{x^{a - 1} \over \left(1 - x\right)^{a + 1}} \,\exp\left(x \over x - 1\right)\,\mathrm{d}x \) where \( a \in \mathbb{R} \) is a constant.

Answer: \Gamma(a)


Question: Determine the smallest integer \( n \) such that \( n^n \) does not divide \( 2016! \). Prove that \( n = 47 \) is indeed the smallest integer for which this condition holds.

Answer: 47


Question: Calculate the limit \( \lim _{x \rightarrow \infty} \int _1^{x} \frac{f(u/x)-g(0+)}{u^2} {\rm d u} \), where \( f \) and \( g \) are integrable functions on \([0,1]\) such that \( \lim_{x \to 0^+}f(x) = f(0+) \) and \( \lim_{x \to 0^+}g(x) = g(0+) \).

Answer: f(0+) - g(0+)


Question: Evaluate the contour integral \(\oint_{C} \frac{z^2}{(z+2)(z-2)^3}dz\) where \(C\) is the ellipse \(\frac{x^2}{4^2}+\frac{y^2}{3^2}=1\).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x \to \infty} \left(\frac{\pi}{2}\arctan x\right)^x \]

Answer: \infty


Question: Determine the number of solutions of the equation \(f(x) = x^{19} - 3x + 2 = 0\) over the ring of \(19\)-adic integers, \(\mathbb{Z}_{19}\). Use the strategy of checking for all integers \(n\) in the set \(\{0, 1, \ldots, 18\}\) whether \(f(n) \equiv 0 \pmod{19}\) and \(f'(n) \not\equiv 0 \pmod{19}\) both hold. Apply Hensel's lemma to determine the number of solutions.

Answer: 1


Question:
Evaluate the line integral \( \oint_C \textbf{f} \cdot d\textbf{r} \) using Green's Theorem, where \( \textbf{f}(x,y) = (ye^{xy} + y^2\sqrt{x})\textbf{i} + (xe^{xy} + \frac{4}{3}yx^{\frac{3}{2}})\textbf{j} \) and \( C \) is the ellipse defined by \((x-1)^2 + \frac{y^2}{9} = 1\), oriented counterclockwise. Compute the double integral:

\[ \iint_R \left(2y\sqrt{x} - 2\sqrt{x}y\right) \, dx \, dy \]

using the bounds \(0 \leq x \leq 2\) and \(-\sqrt{9(1 - (x - 1)^2)} \leq y \leq +\sqrt{9(1 - (x - 1)^2)}\).

Answer: 0


Question: Given an $m \times n$ matrix $A$, where $m < n$, simplify the expression $(AA^T)^{-1}A$ if possible.

Answer: (A^T)^+


Question: Find the image of the circle with center \(A(1/2,1/2)\) and radius \(\frac{1}{2} \sqrt{2}\) under inversion with respect to the unit circle centered at the origin \((0,0)\) in the Euclidean plane \(\mathbb{E}^2\).

Answer: x + y = 1


Question: For a grid graph \(M_{n,n}\), determine the smallest value of \(\alpha > 0\) such that an \(\alpha\)-Spanning Tree exists. An \(\alpha\)-Spanning Tree \(H = (V_H, E_H)\) of \(G\) is defined as a subgraph where \(V_H = V\), \(E_H \subseteq E\), and the shortest path distance in \(H\), denoted by \(d_H(u,v)\), satisfies \(d_H(u,v) \leq \alpha \cdot d_G(u,v)\) for all vertices \(u\) and \(v\), where \(d_G(u,v)\) is the shortest path distance in \(G\).

Answer: 3


Question: Determine whether a finite quotient Dedekind domain can have infinitely many primes.

Answer: Yes


Question: Evaluate the improper integral \( \int_{0}^{1} \frac{1}{\sqrt{x(1-x)}}\,dx \) using a series expansion.

Answer: \pi


Question: Determine if a parity check matrix can be derived from a generator matrix that is not in standard form. Provide a justification for your answer.

Answer: Yes


Question: Determine the number of solutions to the equation \(2^{\sin^2(x)} - 2^{\cos^2(x)} + 2\sin^2(x) = 1\) within the interval \(x \in \left[0, \frac{\pi}{2}\right]\).

Answer: 1


Question: Determine whether the following statement is true or false: If $X$ and $Y$ are topological vector spaces with $Y$ compact, and $T:X\to Y$ is a linear surjection, then for any closed subset $W\subset Y$, the inverse image $T^{-1}(W)\subset X$ is also closed.

Answer: A


Question: Determine the number of roots of the polynomial \( f(z) = 1 + 6z^3 + 3z^{10} + z^{11} \) that lie within the annulus defined by \( 1 < |z| < 2 \).

Answer: 7


Question: Determine if the real number \( \alpha = 3 - \sqrt[5]{5} - \sqrt[5]{25} \) is algebraic. If it is, find the degree of the extension \([\mathbb{Q}(\alpha):\mathbb{Q}]\).

Answer: 5


Question: Suppose we have two sequences $(c_k)_k$ and $(d_k)_k$ in $\mathbb{R}^3$ that converge to $0$ and $d\neq0$, respectively. Determine whether the sequence $$\frac{\|c_k\times d_k\|}{\|d_k\|}$$ converges, where $c_k\times d_k$ denotes the cross product.

Answer: 0


Question:
Find the value of $k$ in the expression $$(20)^{19}+2(21)(20)^{18}+3(21)^2(20)^{17}+
cdots+20(21)^{19}=k(20)^{19}$$ using a geometric series.

Answer: 400


Question: Determine the degree of the map \( \phi: S^3 \rightarrow S^3 \) given by \( \phi(x_1,x_2,x_3,x_4) = (x_1,-x_3,-x_2,x_4) \). Does \( \phi \) preserve or reverse orientation?

Answer: -1


Question: Evaluate the integral \( \frac{1}{2 \pi i} \int_{|z|=2} \frac{4z^{50}}{15z^{51}-3z^{30}+z^2-9} \, dz \).

Answer: \dfrac{4}{15}


Question: Evaluate the limit: $$\lim _{x \rightarrow \infty} e^{-x^{2}}\int_{0}^{x} e^{t^{2}} d t$$

Answer: 0


Question: Determine whether the following statement is true or false: If a group $G$ has infinitely many ends, then it is not boundedly generated.

Answer: True


Question: Determine whether the set \( G_n(p) = \{x + a_2x^2 + \cdots + a_nx^n \mid a_i \in \mathbb{F}_p \} \) forms a group under the operation of function composition modulo \(x^{n+1}\).

Answer: Yes


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

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to\infty}\left(\frac {x^5+\pi x^4+e}{x^5+ex^4+\pi}\right)^x$$

Answer: e^{\pi - e}


Question: Determine the degree of the field extension $[\Bbb Q(\sqrt{42}, \sqrt{-42}):\Bbb Q]$. Provide your answer as a single integer.

Answer: 4


Question:
Evaluate the sum of the series for \(p > 1\):
\[ \sum_{n=1}^\infty \sum_{m=n}^\infty m^{-p} \]

Answer: \zeta(p-1)


Question: Determine whether the following statement is true or false: If $A \sim B$ denotes the existence of a bijection between Dedekind-infinite sets $A$ and $B$, then $A \sim B \cup \{x\}$. If the statement is false, provide a counterexample.

Answer: True


Question: Let $X_1, X_2, \ldots$ be independent random variables. If $P( X_n > n \ \text{i.o.} ) = 0$, does it follow that $\limsup_{n \to \infty} X_n < \infty$ almost surely? Answer with "Yes" or "No".

Answer: No


Question: An investor buys a stock worth $100. Each month, the stock has a 50% chance of increasing by $1.10 and a 50% chance of decreasing by $0.90, with changes being mutually independent. What is the probability that, after 100 months, the stock's value exceeds $91?

Answer: 0.9713


Question: Given distinct real numbers $a, b, c, d$ such that $a$ and $b$ are roots of the quadratic equation $x^2 - 5cx - 6d = 0$, and $c$ and $d$ are roots of the quadratic equation $x^2 - 5ax - 6b = 0$, determine the value of $b + d$. Express your answer as a single number.

Answer: 144


Question:
Evaluate the integral for $N > 0$:
$$ \frac{\exp\left(-\frac N2\right)}{2\pi N} \iint \exp\left(-\left( \frac {x^2-2xy-2Ny+2y^2}{2N}\right)\right) \,dx\,dy $$

Answer: 1


Question: Determine whether the statement "Every saddle point is an isolated critical point" is true or false.

Answer: B


Question: Find the last three digits of the sum \( T = \sum_{i=10^5}^{10^6} (i^3 + i^2) \).

Answer: 000


Question: Given a commutative ring $R$ and a non-zero ideal $I$ with the prime factorization $I = P_1^{e_1} P_2^{e_2} P_3^{e_3}$, determine the cardinality of $R/I$ if the cardinalities of $R/P_i^{e_i}$ for $i = 1, 2, 3$ are known.

Answer: |R/P_1^{e_1}| \cdot |R/P_2^{e_2}| \cdot |R/P_3^{e_3}|


Question:
Consider a Markov chain $\{X_n\}_{n\in\mathbb{N}}$ on the state space $(S,\mathcal{F})$ that is aperiodic, irreducible, and positive recurrent, with a unique stationary distribution $\pi$. Determine whether the following statement is true for any $x\in S$:
\[\lim_{n\rightarrow\infty}\sup_{\text{A}\in\mathcal{F}^{\mathbb{Z}_+}}|P_x((X_{n},X_{n+1},\dots)\in\text{A})-P_\pi(X\in A)| =0.\]

Answer: True


Question: Determine the cardinality of the set of all polynomials with complex coefficients, \(\mathbb{C}[x]\), using an expression involving cardinal operations, 2, and \(\omega\).

Answer: 2^{\omega}


Question: Determine the smallest cardinal number \(\kappa\) such that the ordered interval \([0, \omega_1)\) can be embedded into \(\mathbb{R}^\kappa\). Assume \(\aleph_1 \leq \kappa \leq \mathfrak{c}\), given that \(\mathbb{R}^{\aleph_0}\) is metrizable and the cardinality of the space of continuous functions on \([0, \omega_1)\), denoted by \(C([0, \omega_1))\), is \(\mathfrak{c}\).

Answer: \aleph_1


Question: Evaluate the integral \( \frac{1}{2\pi i}\oint_C \frac{z^2}{z^2+4}dz \), where \( C \) is the square with vertices \( \pm2 \) and \( \pm2+4i \).

Answer: i


Question: Calculate the arc length of the curve defined by the parametric equations $x(t) = \frac{\cos(3t)}{t^3}$ and $y(t) = \frac{\sin(3t)}{t^3}$ for $t \geq \frac{1}{\sqrt{3}}$. Use the arc length formula $l = \int_{t_1}^{t_2} \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$ to find the total arc length. Determine the appropriate limits of integration.

Answer: 7


Question: Find the interval of convergence for the series \( \sum_{n=1}^{\infty} \frac{x^n}{n^x} \).

Answer: (-1, 1)


Question: Given a matrix $A$ and a unitary matrix $U$, find a unitary matrix $V$ such that $V^H A U = \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$.

Answer: V


Question: Given the rate of population growth \( \frac{dP}{dt} = \frac{0.2P}{4(4-\frac{P}{400})} \), and an initial population of 3 at time \( t = 0 \), find the time \( t \) when the population \( P \) will reach 1200.

Answer: 419.47


Question: Determine whether there exist positive integers $a$ and $b$ such that $a \bmod p > b \bmod p$ for all primes $p$ where $(p,a)=1$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the sequence \( a_n = \frac{\sqrt{b^n}}{\sqrt{n!}} \) converges or diverges, where \( b > 1 \) is a real number.

Answer: 0


Question: Find the volume of the region obtained by applying the transformation \( T:(x, y, z)\to(x, y+\tan(\alpha z), z) \) to the region \( \Omega:\{ (x, y, z): x^2+y^2\leq 1, 0\leq z\leq 2 \} \), where \( \alpha\in (0, \pi) \).

Answer: 2\pi


Question: In an acute triangle $\triangle ABC$ with $\angle C = 60^{\circ}$, perpendiculars $AA_1$ and $BB_1$ are drawn from points $A$ and $B$ to the sides $BC$ and $AC$ respectively. Let $M$ be the midpoint of $AB$. Calculate the ratio $\frac{\angle A_1MB_1}{\angle A_1CB_1}$.

Answer: 1


Question: Let \( f:\mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \( f(2x) = f(x) \) for all \( x \in \mathbb{R} \). Given that \( f(1) = 3 \), find the value of \( \int_{-1}^1 f(f(f(x))) \, \mathrm{d}x \).

Answer: 6


Question: Find a modulus $m$ such that the equation $2^{a_1} + 2^{a_2} + 2^{a_3} \equiv 0 \pmod{m}$ has no solution for integers $a_1, a_2, a_3 \geq 0$. What is the smallest such modulus $m$?

Answer: 15


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

Answer: 0


Question: Calculate the limit of the sequence: \[ \lim_{n \to \infty} \frac{\left(2\,\sqrt[\Large n]{\, n\,}\, - \,\sqrt[\Large n]{\, 2\,}\right)^{n}}{n^2} \]

Answer: \dfrac{1}{2}


Question: Consider a linear programming problem stated as follows: minimize $c^Tx$, subject to $Ax = b$ and $x \geq 0$. Its dual is given by maximize $b^Ty$, subject to $A^T y \leq c$, assuming both the primal and dual are feasible and bounded. Let $y^*$ denote the optimal solution of the dual program. What is the effect on the dual solution when we multiply the objective function of the primal problem by a constant $k$? Specifically, is the new dual solution simply $k \cdot y^*$, or does this relationship have more complexity?

Answer: k \cdot y^*


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_{1385}^{2006} f(nx) \, dx$$ where $f(x)$ is a continuous function satisfying $\lim_{x \to \infty} f(x) = 1$. What is the value of this limit?

Answer: 621


Question: Consider a three times continuously differentiable real-valued function $f(x)$ defined on the interval $(0,5)$. If the third derivative of $f(x)$, denoted as $f'''(x)$, is equal to $\frac{1}{100}$ for all $x \in (0,5)$, find the absolute value of the difference $|f(4) - P(4)|$, where $P(x)$ is a quadratic polynomial satisfying $P(1) = f(1)$, $P(2) = f(2)$, and $P(3) = f(3)$. \( \boxed{} \)

Answer: \dfrac{1}{100}


Question: Find the number of distinct points in the projective plane \( \mathbb{P}^{2}(k) \) over a finite field \( k \) with \( q \) elements.

Answer: q^2 + q + 1


Question: Determine whether the series \( \sum_{n=0}^\infty \frac{(2n)!}{n!^2} x^n \) converges or diverges, and find its radius of convergence.

Answer: \dfrac{1}{4}


Question: Determine the number of homomorphisms from the additive group \( \mathbb{Z}_{15} \) to the additive group \( \mathbb{Z}_{10} \).

Answer: 5


Question: Is it always possible to diagonalize the covariance matrix of a multivariate Gaussian distribution to obtain independent variables?

Answer: Yes


Question: Calculate \( \int_C \bar{z}^2 \, dz \) where \( C \) is the circle defined by \( |z| = 1 \).

Answer: 0


Question: Find the maximum value of \( \sum_{i=1}^n k_i \) given that \( \sum_{i=1}^n\left(\frac{1}{i^2}k_i\right) \leq \sum_{i=1}^n \frac{1}{i^2} \) for positive integers \( k_i \) and \( n \).

Answer: n


Question: Is there a set \( A \) such that the statement \( |\mathbb{Z}| < |A| < |\mathbb{R}| \) is undecidable without the Continuum Hypothesis (CH) or its negation?

Answer: Yes


Question:
Determine if the following determinant vanishes identically for a $C^3$ function $\phi: \mathbb{R}^n \rightarrow \mathbb{R}^{n-2}$:

$$\begin{bmatrix} \frac{\partial }{\partial x_1} & \frac{\partial\phi_2 }{\partial x_1} & \dots & \frac{\partial \phi_{n-1} }{\partial x_1} \\
\\ \frac{\partial }{\partial x_n} & \frac{\partial\phi_2 }{\partial x_n} & \dots & \frac{\partial \phi_{n-1} }{\partial x_n} \ \end{bmatrix}$$

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to 0} \int_{x}^{\sin x} \frac{1}{t^3(1+t^{1/2})} \, dt. \]

Answer: -\dfrac{1}{6}


Question: For a group with prime order $p$, determine the conditions under which the function $f(k) = k^n$ is invertible.

Answer: n \not\equiv 0 \pmod{p}


Question: Determine the dimension of the vector space \( \mathbb{C}[x,y]/I \) over \( \mathbb{C} \), where \( I \) is the ideal generated by the polynomials \( y^2 - x^2 \) and \( y^2 + x^2 \) in \( \mathbb{C}[x,y] \).

Answer: 4


Question: Evaluate the integral \( \iint_D x \, dA \) using polar coordinates, where \( D \) is the region inside the circle \( x^2 +(y-1)^2 = 1 \) but outside the circle \( x^2 +y^2 = 1 \).

Answer: 0


Question: Evaluate the limit: $$\lim_{x\to\infty}\left((x+1)!^{\frac{1}{x+1}}-\frac{x+1}{e}\right)$$ Provide justification for your answer.

Answer: \infty


Question: Evaluate the limit \(\lim_{x \to 0}\frac{\left(1-\cos{x}\right)\left(1-\cos{2x}\right)...\left(1-\cos{nx}\right)}{\left(1-x^2\right)^{\left(n-1\right)}}\) without using Taylor series.

Answer: 0


Question: Evaluate the double sum: $$\sum_{n=2}^{\infty} \sum_{m=2}^{\infty} m^{-n}$$

Answer: 1


Question: Given two matrices A and B, each with dimensions N x N, where rank(A) = N-1 and rank(B) = 2, determine the upper bound of rank(A-B).

Answer: N


Question: Determine whether the following statement is true or false: For a normed space \((X, \|\cdot\|_X)\) and a closed subspace \(M\), the quotient space \(X/M\) is always isomorphic to the orthogonal of \(M\), denoted as \(M^\perp\).

Answer: False


Question: Find the residue of the function \( f(z) = \frac{1}{(z+i)^2(z-i)^2} \) at the double pole \( z = -i \) using the method of change of variables.

Answer: \dfrac{i}{4}


Question: Consider a square $ABCD$ whose vertices lie on the curve $xy = \pm 1$. If the midpoints of its sides also lie on the same curve, determine the area of square $ABCD$. \( \boxed{?} \)

Answer: 4\sqrt{5}


Question: Given a positive real random variable $X$ with a finite first moment ($E[X] < \infty$), determine whether the second moment $E[X^2]$ is necessarily finite. Answer with 'True' if it is always finite, or 'False' if it is not necessarily finite.

Answer: False


Question:
Find the maximum value of the expression:
\[ \frac{p + q + r}{3} + \sqrt{p(1 - p) + q(1 - q) + r(1 - r) + 2pqr} \]
where \( p, q, r \in \mathbb{R} \) and \( 0 \leq p, q, r \leq 1 \).

Answer: 1 + \sqrt{2}


Question: Evaluate the limit: $$\lim_{n \rightarrow \infty} \sum_{i=1}^{n}\sum_{j=1}^{i} \frac{j}{n^3}.$$

Answer: \dfrac{1}{6}


Question:
Let $A$, $B$, and $C$ be finite subsets of the plane such that $A \cap B$, $B \cap C$, and $C \cap A$ are all empty. Let $S = A \cup B \cup C$. Assume that no three points of $S$ are collinear and that each of $A$, $B$, and $C$ has at least 3 points. Which of the following statements is always true?

(a) There exists a triangle having a vertex from each of $A$, $B$, and $C$ that doesn't contain any point of $S$ in its interior.
(b) There exists a triangle having a vertex from each of $A$, $B$, and $C$ that contains all the remaining points of $S$ in its interior.
(c) Any triangle having a vertex from each $A$, $B$, and $C$ must contain a point of $S$ in its interior.
(d) There exist 2 triangles both having a vertex from each of $A$, $B$, and $C$ such that the two triangles do not intersect.

Answer: a


Question: Calculate the variance of the expression \((W_1W_2 - Z_1Z_2)^2\), where \(W_1, W_2, Z_1,\) and \(Z_2\) are independent random variables.

Answer: 20


Question: Let $\Omega$ be a bounded domain in $\mathbb{R}^N$, $N\geq 2$. Let $u\in W^{1,p}(\Omega)$, the class of all measurable functions $u:\Omega\to\mathbb{R}$ such that \( \|u\|_{W^{1,p}(\Omega)}:=\|u\|_{L^p(\Omega)}+\|\nabla u\|_{L^p(\Omega)}<\infty. \) Let $1<q<p^*=\frac{Np}{N-p}$ if $1<p<N$ and $1<q<\infty$ if $p\geq N$. Assume that \( \int_{\Omega}|u|^{q-2}u\,dx=0. \) Determine whether this implies \( \int_{\Omega}u\,dx=0. \)

Answer: No


Question: Determine whether the statement $A^3 = A$ is true for any square matrix $A$, given that $\text{Image}(A^2) = \text{Image}(A)$ and $\dim(\text{Image}(A^2)) = \dim(\text{Image}(A))$. Provide a proof if the statement is true, or a counter-example if it is false.

Answer: A^3 \neq A


Question: Determine the rate of convergence of the sequence \( \{x_n\} \), where \( x_n = \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} \). Find the value of \( p \) such that \( \displaystyle \lim_{n\to \infty} \frac{|x_{n+1}-l|}{|x_n-l|^p} \) exists and is non-zero finite.

Answer: 1


Question: Find the limit of the sequence defined by \( a_n = \sqrt{1+\sqrt{5+\sqrt{11+\ldots+\sqrt{t_n}}}} \), where \( t_n = n^2 - n - 1 \), as \( n \) approaches infinity.

Answer: 2


Question: Let $f(x) = [n + p \sin x]$, where $x \in (0, \pi)$, $n \in \mathbb{Z}$, $p$ is a prime number, and $[x]$ denotes the greatest integer less than or equal to $x$. Determine the number of points at which $f(x)$ is not differentiable.

Answer: 2p - 1


Question: Prove the identity \( S_n = (n+1)H_n - n \), where \( S_n = \sum_{k=1}^{n}H_k \) and \( H_k \) is the \( k \)th harmonic number. Start with the expression \( S_n = \sum_{k=1}^{n} \sum_{j=1}^{k}\frac{1}{j} \) and manipulate it to reach the desired form.

Answer: S_n = (n+1)H_n - n


Question: Derive an explicit formula for the sequence defined by the recursive equation \( F(n+1) = F(n) + \frac{n}{F(n)} \), with the initial condition \( F(1) = 1 \).

Answer: n


Question: Let $f$ be a real-valued function defined on the interval $[1, \infty)$ with $f(1) = 3$. For all $x \geq 1$, the function satisfies the equation \( 2 \int_{2}^x f(t)\,dt = xf(x) + x^3 \). Determine the value of $f(2)$. \( \boxed{} \)

Answer: -4


Question: Find the minimum value of the function \( y(x) = x^x \) using calculus.

Answer: e^{-1/e}


Question: Given that \(\alpha\) is a zero of \(f(x)=x^3+x+1\) in \(\mathbb{F_2}[x]\), determine the number of distinct elements in the splitting field \(\mathbb{F_2}(\alpha)\) over \(\mathbb{F_2}\).

Answer: 8


Question: Let $h(x)$ be a polynomial of degree 4 such that $h(-3) = 10$, $h(-1) = -2$, $h(1) = 2$, and $h(3) = 10$. Let $k(x) = h(x)^{20}$. Find the sum of the coefficients of the terms in $k(x)$ with odd exponents.

Answer: 0


Question: Evaluate the limit \[ I = \lim_{n \to \infty} \left( \frac{\sin\frac{1}{n^2} + \binom{n}{1}\sin\frac{2}{n^2} + \binom{n}{2}\sin\frac{3}{n^2} + \cdots + \binom{n}{n}\sin\frac{n+1}{n^2}}{\cos\frac{1}{n^2} + \binom{n}{1}\cos\frac{2}{n^2} + \binom{n}{2}\cos\frac{3}{n^2} + \cdots + \binom{n}{n}\cos\frac{n+1}{n^2}} + 1 \right)^n, \] where \( 0 \leq \frac{n+1}{n^2} \leq \frac{\pi}{2} \). Determine the value of \( I \) as \( n \to \infty \).

Answer: \sqrt{e}


Question: Determine the values of $r$ and $s$ for which the inequality $\|f\|_s \le \|f\|_r$ holds for any complex measurable function $f$ on a measure space $X$ with $\mu(X) = 1$.

Answer: s \leq r


Question:
Calculate the determinant of the following tridiagonal matrix:

\[
\begin{vmatrix}
3 & 2 & 0 & 0 & \cdots & 0 & 0 \\
1 & 3 & 2 & 0 & \cdots & 0 & 0 \\
0 & 1 & 3 & 2 & \cdots & 0 & 0 \\
0 & 0 & 1 & 3 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \cdots & 1 & 3
\end{vmatrix}
\]

Answer: 2^{n+1} - 1


Question: Determine the proportion of prime numbers among all natural numbers as the number of natural numbers increases indefinitely.

Answer: 0


Question: Let \( A, B \in M_n(\mathbb{R}) \) be two matrices, and let \( \lVert \cdot \rVert \) be a norm on the space of all \( n \times n \) matrices. Prove that there exists a constant \( C > 0 \) such that \( \lVert AB \rVert \le C \lVert A \rVert \lVert B \rVert \) for all matrices \( A \) and \( B \).

Answer: C


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

\[f(x, y) = \begin{cases} \frac{21}{4}x^2y & \text{for } x^2 \leq y \leq 1, \\ 0 & \text{otherwise,} \end{cases}\]

subject to the condition \(y \leq x\).

Answer: \dfrac{3}{20}


Question: Determine the relationship between $a$ and $b$ such that the series \( S = \frac{a}{b}+\frac{a(a+1)}{b(b+1)} + \frac{a(a+1)(a+2)}{b(b+1)(b+2)} + \ldots \) is convergent.

Answer: b > a + 1


Question: Let $A\subset\mathbb{R}$ be a bounded set such that for every open interval $I$, the Lebesgue outer measure of $A \cap I$ is strictly less than the Lebesgue measure of $I$. Prove or disprove: $A$ necessarily has Lebesgue measure $0$. \( \text{(True or False)} \)

Answer: B


Question: Find the angle at which the curves \(r = a\sin 2\theta\) and \(r = a\cos 2\theta\) intersect orthogonally.

Answer: \dfrac{\pi}{8}


Question: Let \(g:\mathbb{R}\to\mathbb{R}\) be a twice-differentiable function with a point of inflection at \(x=0\). Given that \(g''(0)=0\), \(g''(x) > 0\) on \((-b,0)\), and \(g''(x) < 0\) on \((0,c)\) for some \(b, c > 0\), determine if there exist \(a>0\) and \(d<0\) such that \(g(a), g(a+d), g(a+2d)\) forms an arithmetic progression.

Answer: Yes


Question:
Verify Stokes' theorem for the vector field \(\vec{F}(x,y,z) = (y, -x, xz)\) on the surface \(z = 9 - x^2 - y^2\) with \(z \geq 0\). Use the parametrization:
\[\vec{r}(\theta, r) = (r\cos\theta, r\sin\theta, 9 - r^2), \quad r\in [0, 3], \ \theta\in [0, 2\pi].\]

Calculate the curl of \(\vec{F}\) and the surface integral of the curl over the surface. Then, compute the line integral of \(\vec{F}\) over the boundary of the surface. Verify if Stokes' theorem holds by checking if these two integrals are equal.

Answer: -18\pi


Question: Find an invertible element in \( \mathbb{R} \) under the binary operation \( a * b = a + b + a^2 b^2 \) that has two distinct inverses.

Answer: -1


Question: Let $f : [0,\infty)\to\mathbb{R}$ be a non-negative continuous function. Suppose there exists a sequence $(t_{n})_{n\in\mathbb{N}}\subset[0,\infty)$ such that $t_n\to\infty$ and $f(t_n)$ is bounded. Can we conclude that $f$ is bounded from above? Answer with 'Yes' or 'No' and justify your answer.

Answer: No


Question:
Determine which of the following sets correctly describes the set \( S = \{ f : \mathbb{R} \to \mathbb{R} \mid \exists \epsilon > 0 : \forall \delta > 0, |x-y| < \delta \implies |f(x)-f(y)| < \epsilon \} \):

1. \( S = \{ f : \mathbb{R} \to \mathbb{R} \mid f \text{ is continuous} \} \)
2. \( S = \{ f : \mathbb{R} \to \mathbb{R} \mid f \text{ is uniformly continuous} \} \)
3. \( S = \{ f : \mathbb{R} \to \mathbb{R} \mid f \text{ is bounded} \} \)
4. \( S = \{ f : \mathbb{R} \to \mathbb{R} \mid f \text{ is constant} \} \)

Answer: 3


Question: Let $f:[a,b] \rightarrow \mathbb{R}$ be a bounded function. Suppose there exists a sequence $(P_{n})_{n}$ of partitions of $[a,b]$ such that $\lim_{n\rightarrow \infty} L(f,P_{n})=L= \lim_{n \rightarrow \infty} U(f,P_{n})$, where $L$ denotes the lower sum and $U$ the upper sum. Determine whether $f$ is Riemann integrable, and if so, find the value of the integral.

Answer: L


Question: Evaluate the line integral \(I=\oint_\Gamma \sin(x^3)\,dx + 6xy\,dy\) where \(\Gamma\) is the counterclockwise triangular path in \(\mathbb{R}^2\) connecting the points (0,0), (2,2), and (0,2).

Answer: 16


Question: Evaluate the finite value of the infinite series: \[ \sum_{n=2}^{\infty} \frac{\binom{n}{2}}{4^n} \]

Answer: \dfrac{4}{27}


Question: Determine whether an odd perfect number can be divisible by $5313$. Provide a true or false answer.

Answer: False


Question: Evaluate the limit: $$\lim_{n \to \infty}\frac{n^{-2n \cos\frac{1}{n}}}{n^{-3}}$$

Answer: 0


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

Answer: 1


Question: Let $x, y, z > 0$ such that $x + y + z = 3$. Find the maximum value of the expression $S = \sum_{cyc} \frac{xy}{(x+2y)^2}$. Provide your answer as a single number.

Answer: \dfrac{1}{3}


Question: Find the minimum value of \(|z|\) for complex numbers \(z\) satisfying the equation \(|(z+i)^2-3| = 2|z-i|\).

Answer: 2 - \sqrt{2}


Question: Evaluate the limit: $$\lim_{x \to \infty} \frac{1-x}{2\sqrt\pi}\exp \left(-{\left({\frac{\log (x)-a}{b}}\right)}^2\right)$$

Answer: 0


Question: Determine whether the limit \( \lim_{m \to \infty} \int_{-\infty}^{\infty} \sin(e^t) e^{-(t - m)^2} \, dt \) converges to zero.

Answer: 0


Question:
Evaluate the limit:

$$\lim_{x \to +\infty} \left(\left(x^n + \sum_{k=0}^{n-1}a_kx^k\right)^{\frac{1}{n}} - x\right)$$

where the answer is known to be $\frac{1}{n}$. Determine this limit without using L'Hopital's rule.

Answer: \dfrac{1}{n}


Question: Determine the number of continuous functions \( f: [0, 1] \to \mathbb{R} \) that satisfy the equation: \[ \int_0^1 f(x) \, \text{d}x = \frac{1}{3} + \int_0^1 f^2(x^2) \, \text{d}x. \]

Answer: 1


Question: Is it always true that for a set of points, the circular hull containing all the points must include at least one point from the given set on its boundary?

Answer: Yes


Question:
Let $f_1(x) = \sum_{n=1}^{\infty} \frac{x \sin(n^2 x)}{n^2}$ and $f_2(x) = \sum_{n=1}^{\infty} x^2 (1-x^2)^{n-1}$ for $x \in [0,1]$. Determine which of the following statements is true:

(a) $f_1$ is continuous but $f_2$ is not continuous.
(b) $f_2$ is continuous but $f_1$ is not continuous.
(c) Both $f_1$ and $f_2$ are continuous.
(d) Neither $f_1$ nor $f_2$ is continuous.

Answer: A


Question: Evaluate the integral \( \frac{1}{2\pi i}\int_{\gamma}\frac{q'(\phi)}{q(\phi)} d\phi \) where \( q(z) \) is a polynomial with a zero of multiplicity 2 at \( z_0 \), and \( \gamma \) is a simple \( C^1 \) curve that encircles \( z_0 \).

Answer: 2


Question: Find the residue of the function \( f(z) = z^{n-1} e^{\frac{pz}{2}} e^{\frac{p}{2z}} \) at the essential singularity \( z = 0 \), where \( n \in \mathbb{N} \) and \( p \in \mathbb{R} \).

Answer: I_n(p)


Question:
Calculate the Fourier coefficient \( c_1 \) for the function \( f(x) = x \cos(x) \) defined on the interval \( [-\pi, \pi] \). The Fourier coefficients are given by:
\[ c_n = \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) e^{-inx} dx \]
Given that \( c_0 = 0 \) because the function is odd, find \( c_1 \).

Answer: \dfrac{i}{4}


Question: Given that a function $f(x)$ satisfies the equation $f(x) f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right)$ for all $x \neq 0$ and $f(2) > 1$, find the value of $\lim_{x \to 1} f(x)$. Provide your answer as a single number.

Answer: 2


Question: Find the least distance from the point \( P = (-2, 10, 4) \) to the line of intersection of the planes \( M_1 : x + 4y - 5z = -3 \) and \( M_2 : 2x - y - z = 3 \).

Answer: 7\sqrt{2}


Question: Consider the parameterization of the unit circle given by \(x(t) = \frac{1-t^2}{1+t^2}\) and \(y(t) = \frac{2t}{1+t^2}\). Let \(T = \{ t \in \mathbb{Q}: 0 \leq t \leq 1 \}\). Define a sequence \((x_n)\) as \(x_0 = 0\) and \(x_n = x_{n-1} + |x(t_{n-1})-x(t_{n})| + |y(t_{n-1})-y(t_{n})|\) for some sequence \((t_n)\) in \(T\). Does this sequence converge? If so, find its limit.

Answer: 2


Question: Determine the number of trees with vertex set \( \{v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8\} \) such that at least two vertices have a degree of 3.

Answer: 80640


Question: Determine the number of automorphisms of the field $F = \mathbb{Z}_2[x]/(f(x))$, where $f(x) = x^3 + x^2 + 1$. The field $F$ has order $8$. Let $a$ be a root of $f(x)$ in $F$. How many automorphisms does this field have?

Answer: 3


Question: What is the most efficient algorithm to compute the inverse of the polynomial matrix \((xI - A)^{-1}\) when \(A\) is a triangular matrix?

Answer: (xI - A)^{-1}


Question:
Calculate the limit superior of the following expression as \( n \) approaches infinity:

\[ \limsup_{n\to\infty}\left(\frac{A^n}{n}+\frac{B^n}{n^2}\right)^{\frac{1}{n}} \]

where \( A \) and \( B \) are positive real numbers.

Answer: \max\{A, B\}


Question: Suppose that $K$ is a field of characteristic $0$ and $L/K$ is an extension such that every polynomial of degree $\leq n$ over $K$ splits in $L$. Without assuming that $L$ is algebraically closed, determine if it is true that every degree $n$ polynomial over $K[[t]]$ splits completely over $L[[t^{1/n!}]]$. Provide a justification for your answer.

Answer: Yes


Question: Find all integer solutions \((x, y)\) to the equation \(2^x + 5 = 3^y\).

Answer: (2, 2)


Question: Given the sequence defined by \(x_{n+1} = 3x_n + \sqrt{8x^2_n + 2}\), if it is known that \(x_{2017} + x_{2023} = 990\), find the value of \(x_{2020}\).

Answer: 5


Question: Determine the number of real solutions for the equation \( g(x) = 10^x - 6^x - 4^x - 2\sqrt{24^x} = 0 \).

Answer: 1


Question: Suppose $P$ is an $n \times n$ matrix such that $P^T P^3 = I$, where $I$ is the $n \times n$ identity matrix. Determine all possible values of $\det(P)$. \( \text{(A) } 1 \quad \text{(B) } -1 \quad \text{(C) } 0 \quad \text{(D) } \text{None of these} \)

Answer: D


Question: Suppose we have a filtration $(\mathcal F_k)_{k\in\mathbb N}$. If $E[X\mid\mathcal F_{k_0}]=c$ $\mathbb P$-a.s. for some constant $c$, does it follow that $E[X\mid\mathcal F]=c$ $\mathbb P$-a.s.? Provide a justification for your answer.

Answer: No


Question: Evaluate the limit: $$\lim_{(x,y)\to(0,0)} (x^2+y^2)^{x^2y^2}.$$ Determine the correct value of the limit.

Answer: 1


Question: Evaluate the contour integral \(\int_C \frac{1}{z^{2017}-1} \, dz\), where \(C\) is the contour \(|z|=2\) traced counterclockwise, using the formula \(\int_C f(z) \, dz = 2\pi i \text{Res}\left(\frac{1}{z^2}f\left(\frac{1}{z}\right); 0\right)\).

Answer: 0


Question: Determine whether there exists a continuous function \( g: \mathbb{R} \to \mathbb{R}^{\mathbb{N}} \) such that the range of \( g \) is not eventually constant in any coordinate.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to0}\frac{(1+x\cdot2^x)^{\frac{1}{x^2}}}{(1+x\cdot3^x)^{\frac{1}{x^2}}}$$ as \(x\) approaches 0.

Answer: \dfrac{2}{3}


Question: Find the limit of the function \( f(x,y) = \frac{x^2 + y^2}{1-\cos(x^2 + y^2)} \) as \((x,y) \to (0,0)\).

Answer: \infty


Question: Evaluate the integral \( \int_{\gamma} \frac{1}{z} \, dz \), where \( \gamma \) is the circle centered at the origin with radius 1.

Answer: 2\pi i


Question:
Determine the eigenvalues of the following $4 \times 4$ matrix:
$$J(\epsilon)=\begin{bmatrix} -\mu & 0 & 0 & -\frac{\beta b}{\mu}\\ 0 & -\sigma & 0 & \frac{\beta b}{\mu}\\ 0 & \sigma & -(\phi + d) & 0\\ 0 & 0 & \alpha & -(\phi + \epsilon) \end{bmatrix}$$

Answer: -\mu


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

Answer: e


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

Answer: e - 1


Question: Is the direct product \( \mathbb{Z} \times G \) Hopfian, where \( G \) is a finite group? A group is Hopfian if every surjective homomorphism from the group to itself is an isomorphism.

Answer: Yes


Question: Let \( P=\begin{pmatrix} i & 2\\ -1 & -i \end{pmatrix} \) and \( T_P\colon M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}} \) be a linear map defined by \( T_P(X)=P^{-1}XP \). Determine the minimal polynomial of \( T_P \).

Answer: x^2 - 1


Question:
Given the system of equations:

\[\begin{align*}
a &=\sqrt{4+\sqrt{5+a}},\\
b &=\sqrt{4-\sqrt{5+b}},\\
c &=\sqrt{4+\sqrt{5-c}},\\
d &=\sqrt{4-\sqrt{5-d}}.
\end{align*}\]

Compute the product $abcd$. \(\boxed{\text{Your answer here}}\)

Answer: 11


Question: A conical structure is described by the function \( z = f(x,y) = 10 - \sqrt{9x^2 +18x+9y^2 -54y + 85} \). Given that the construction process allows for a 5% error in radius and a 10% error in height, determine the maximum possible percentage error in the cone's volume using linear approximation.

Answer: 20\%


Question: Determine whether the statement is true or false: For any square matrix $A$, if $A^8x=b$, then $b$ is in the image of $A^2$.

Answer: True


Question: Determine whether a normal nilpotent linear map in a complex inner product space is always the zero map. Answer true or false.

Answer: A


Question: Evaluate the surface integral \(\int F \cdot \text{d}S\) for the vector field \(F = [2y + 3, xz, yz - x]\), where \(S\) is the surface of a unit-radius cylinder centered at the origin, with \(-1 < z < 1\).

Answer: 0


Question: Does the revised statement from the book imply that the event \(\{ B_{\tau_b + (t - \tau_b)} - B_{\tau_b} < 0 \}\) is independent of \(\mathcal{F}_{\tau_b}\) for all \(t \geq \tau_b\)? Answer 'yes' or 'no'.

Answer: yes


Question: Determine the value of \( c \in \mathbb{R} \) for which the series \( s_n = \sum_{t=1}^{n} tc^t \) converges using the limit definition of a sequence.

Answer: (-1, 1)


Question: In a circle with center $O$ and radius $7$, points $A$, $B$, and $C$ lie on the circumference. The perpendicular bisector of segment $AB$ intersects $BC$ at point $P$ and $AC$ at point $Q$. Calculate the product $OP \cdot OQ$. \( \text{Express your answer as a single number.} \)

Answer: 49


Question: Calculate the sum of the infinite series: \[ \sum_{n=0}^\infty \frac{n^3}{4^n}. \]

Answer: \dfrac{44}{27}


Question: Determine whether the function $\tan$ is bounded outside the $\epsilon$-neighborhood of its poles. Answer with True or False.

Answer: A


Question: Determine the value(s) of the constant \(c\) for which the set of differentiable functions \(u(x)\) satisfying the differential equation \(u'(x) = 2xu(x) + c\) forms a linear subspace of the space of all continuous functions \(C(\mathbb{R})\).

Answer: 0


Question: Let $X$ and $Y$ be independent and uniformly distributed random variables on $(0,1)$. Define $Z = \frac{X}{Y}$. Determine the joint probability density function of $X$ and $Z$. Provide your answer as a function of $x$ and $z$.

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


Question: Determine if there are any graphs other than the cycle on 5 vertices, $C_5$, such that changing any edge (either by adding or removing) results in an increase in either the clique number or the independence number.

Answer: C_5


Question: Let \( G \) be a finite group, and consider the commuting graph \( X(G) \) where vertices are the elements of \( G \), and two distinct vertices \( x \) and \( y \) are connected by an edge if and only if \( xy = yx \). If \( c \) denotes the chromatic number of \( X(G) \), does \( G \) necessarily contain an abelian subgroup of order \( c \)?

Answer: Yes


Question: Calculate the limit \( \lim_{x\to 0^-} \frac{e^{\frac{1}{x}}}{x^2} \) using the substitution \( y=\frac{1}{x} \).

Answer: 0


Question: Find the eigenvalues of a matrix \( B \) that satisfies the equation \( B^2 = 2B - I_n \), where \( I_n \) is the identity matrix.

Answer: 1


Question: Determine the function to which the series \( \sum_{k=-\infty}^{\infty}c_ke^{i4kx} \) converges, given that \( \sum_{k=-\infty}^{\infty}c_ke^{ikx} \) is the Fourier Series representation of a function \( f(x) \).

Answer: f(4x)


Question: In a right triangle ABC with ∠B = 90°, an interior bisector BD is drawn such that 3BI = 4ID, where I is the incenter. Determine the relationship between the circumradius (R) and the inradius (r) of triangle ABC.

Answer: R = 3r


Question: Determine the dimension of the solution space of the differential equation \(x^2y'' - 4xy' + 6y = 0\) on the interval \((-1, 1)\).

Answer: 2


Question: Compute the limit of the series \( \sum_{n=1}^\infty\frac{(-1)^n}{n} \).

Answer: -\ln 2


Question:
Determine the dimension of the vector space $V$ defined by:
\[ V = \{P \in \mathbb{R}_n [X] : P (X^2)-( X^n + 1) P (X) =0\} \] for $n \in \mathbb{N}$.

Answer: 1


Question: Calculate the remainder when $999!$ is divided by $1000$. Express your answer as $999! \mod 1000$. Use the properties of factorials to solve this problem.

Answer: 0


Question: Evaluate the integral \( \int_{-\infty}^{\infty} e^{-(x+ci)^2} \, dx \), where \( c \) is a real constant.

Answer: \sqrt{\pi}


Question: Determine whether two distinct ideals $I$ and $J$ in a ring $R$ can result in isomorphic quotient rings $R/I$ and $R/J$. Provide a justification for your answer.

Answer: Yes


Question: Determine the value of \( \alpha \) for which the limit \( \lim_{x\to0^{+}} \frac{\arctan x\log (\sin x)-x\log x}{x^{\alpha}} \) is equal to a finite non-zero number.

Answer: 3


Question: In a monoid $G=\langle A, * \rangle$ with identity element $e$, determine if an element $a \in A$ can have distinct left-inverses $L_1$ and $L_2$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \int_0^1 \frac{nx^{n-1}}{1+x} \, dx \]

Answer: \dfrac{1}{2}


Question: If $A \subseteq X \subseteq \mathbb{R}^{n}$ and $A$ is closed in $X$ with respect to the subspace topology, does it follow that the closure of $A$ in $X$, denoted $\overline{A}^{X}$, is equal to $A$?

Answer: Yes


Question: Consider the number field $K = \mathbb{Q}(\sqrt{-7})$ and let $L$ be the Hilbert Class Field of $K$. Find the minimal polynomial of a primitive element for $L$ over $K$.

Answer: x - \sqrt{-7}


Question: Calculate the limit: $$\lim_{x \rightarrow \infty}\left(\frac{2}{\pi}\arctan(x)\right)^{\frac{x^2}{1+2x}}$$

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


Question: Is it possible to create a CW decomposition of $S^n$ using only an $n$-cell and a $0$-cell?

Answer: Yes


Question: Find the minimum value of \(|f(z)|\) for the function \(f(z) = \frac{1}{z^2-2z+2}\) on the closed disk \(\overline{D(0,1)}\).

Answer: \dfrac{1}{5}


Question: Consider the sequence \( a_n = \frac{1000^n}{n!} \) for \( n \in \mathbb{N} \). Determine the value of \( n \) for which \( a_n \) is greatest. Choose the correct answer from the following options: (A) \( n=998 \) (B) \( n=999 \) (C) \( n=1000 \) (D) \( n=1001 \).

Answer: C


Question: A bounded sequence $a_n$ has exactly 2 partial limits. Determine the maximum number of partial limits that the sequence $a_{n+1} - a_n$ can have.

Answer: 3


Question: Determine if the only solution to the differential equation \( \frac{dy}{dx}=8y^2x^3 \) with the initial condition \( y(2)=0 \) is the constant solution \( y=0 \).

Answer: y = 0


Question: Find the sum of the series: \[ S = \sum_{n=2}^\infty \frac{\zeta(n)}{n!} \] where \( \zeta(n) \) is the Riemann zeta function.

Answer: e - \gamma - 1


Question: Evaluate the limit of the sequence \(\frac{c^n}{n!^{\frac{1}{k}}}\) as \(n\) approaches infinity, given that \(k > 0\) and \(c > 0\).

Answer: 0


Question: Evaluate the limit using the Maclaurin series: $$\lim_{x\to 0} \frac{\sin(x) - x + \frac{x^3}{6}}{\tan^3(x)}$$

Answer: 0


Question: Determine the value of \(\alpha\) for which the integral \(\int_{0}^{+ \infty} (\sqrt{x^2 + 2x + 2} - x - 1)^\alpha \, dx\) converges.

Answer: \alpha > 1


Question: Compute the exact value of the limit: \[ \lim_{n \to \infty} \frac{{n^2 \choose n}}{{n^2+n-1 \choose n}}. \]

Answer: \dfrac{1}{e}


Question: Determine the smallest size of a rectangular board on which a knight's tour, either open or closed, is possible.

Answer: 3 \times 4


Question: Find the limit of the sequence represented by the infinite series: $$\sum_{i=0}^\infty \frac{{(-1)}^n n! {x}^n}{10^n}$$

Answer: 1


Question: Find the limit: \[ \lim_{n\to\infty}(1+a_n)^{c_n}e^{-a_nc_n} \] where \( a_n \to 0 \) and \( c_n a_n^2 \to 0 \) as \( n \to \infty \).

Answer: 1


Question: In a reflexive Banach space $X \neq \{0\}$, does there exist an element $x \in X$ with $\|x\| = 1$ such that $f(x) = \|f\|$ for some $f \in X^*$, where $\|f\|$ denotes the operator norm?

Answer: Yes


Question: Find the common inflection point of non-equilibrium solutions for the differential equation \( \frac{dP}{dt} = 0.2P\left(1-\frac{P}{1000}\right) \), given that the equilibrium points are \(P = 0\) and \(P = 1000\).

Answer: 500


Question:
Given $a > b > c > 0$ such that $7a + 8b = 15c + 24\sqrt[3]{abc}$, determine the maximum value of the expression:

$$\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b}$$

Answer: \dfrac{32}{9}


Question: Does there exist a well-ordered proper class $P$ that has no cofinal subclasses order-isomorphic to any element of the set $Ord \cup \{Ord\}$, where $Ord$ denotes the class of all ordinals?

Answer: No


Question: Calculate \( \lim_{p \rightarrow 1 ^ {-}} f(p) \) for the function \( f(p) = (1-p) \sum_{n=0}^{\infty} y_n p^n \), where \( y_n \) is a sequence of real numbers that converges to \( y \).

Answer: y


Question: Find the infimum of the set of all real numbers of the form \( \operatorname{Re}(-iz^2+1) \) where \( |z| > 1 \).

Answer: -\infty


Question: Find the minimum value of the expression \( \left|2x-3\right|+\left|2y+1\right|+\left|2z-5\right| \) for non-negative real numbers \(x, y, z\) that satisfy the constraints \( x+y+z=2 \) and \( z^2 \le 4xy \).

Answer: 19 - 8\sqrt{3}


Question: Determine the values of \( q \in \mathbb{R} \) for which the series \( \sum^{\infty}_{n=3}{\frac{1}{n(\log n) ^ {(\log(\log n))^q}}} \) converges.

Answer: q > 0


Question: Evaluate the limit of the sequence \( a_n = \left(\frac{(n+1)!}{(n+1)^{n+1}}\right)^{1/(n+1)} \) as \( n \to \infty \).

Answer: \dfrac{1}{e}


Question: Find the minimum value of the function \((a-1)^2 + (b-1)^2 + (c-1)^2 + (d-1)^2\) subject to the constraints \(1 \le a \le b \le c \le d \le 4\) and \(abcd = 4\).

Answer: 12 - 8\sqrt{2}


Question: Find the value of $g'(0)$ for the function $g(t) = \|(t, t^2, f(3t, t-2))\|$, where $f$ is a differentiable function with $f(0, -2) = 3$ and $\nabla f(0, -2) = (-6, 2)$. Given that $g(0) = 3$, calculate $g'(0)$. Assume $g'(t) = \|(1, 2t, f'(3t, t-2))\|$. Provide the steps and final answer.

Answer: -16


Question: Evaluate the integral \( \int_c \frac{z}{z^2 + 2z + 5} \, dz \), where \( c \) is the unit circle.

Answer: 0


Question: Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n}\ d \sigma \) where \( \mathbf{F} = xy\mathbf{i}+ z^2\mathbf{j}+y\mathbf{k} \) and \( S \) is the exterior side of the cylinder \( y^2 + z^2 =9 \), bounded by \( x=0 \), \( x=1 \), \( y=0 \), and \( z=0 \) in the first octant.

Answer: \dfrac{27}{2}


Question: Consider two disjoint complete graphs on 100 vertices, $K^1_{100}$ and $K^2_{100}$. We add 5 edges between $K^1_{100}$ and $K^2_{100}$, each edge connecting a different vertex in both components. This graph is $5$-edge-connected. If 2 random edges are removed, what is the probability that the resulting graph is $3$-, $4$-, or $5$-edge-connected?

Answer: 1


Question: Let $\Gamma_1$ and $\Gamma_2$ be topologies on a set $X$ such that for every subspace $A \subseteq X$, $A$ is connected in $\Gamma_1$ if and only if it is connected in $\Gamma_2$. Are $\Gamma_1$ and $\Gamma_2$ necessarily homeomorphic? Answer yes or no.

Answer: No


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(2) = 6 \) and \( f'(2) = \frac{1}{48} \). Find \( \lim_{x \to 2} \int_{6}^{f(x)} \frac{4t^3}{x-2} \, dt \).

Answer: 18


Question: Let $G=(\mathbb{Z/2Z})^n$ be an additive group with $n>1$. Consider nonzero elements $v_i$ of $G$ for $i \in \{1, 2, \ldots, 2^n \}$ such that $\sum_{i=1}^{2^n} v_i=0$. Determine if there exists a subset $J \subset \{1, 2, \ldots, 2^n \}$ with $|J|=2^{n-1}$ such that $$\sum_{j \in J} v_j=0.$$

Answer: Yes


Question: Given the ideal $I$ in $\mathbb{C}[x,y]$ generated by $y^2-x^2$ and $y^2+x^2$, determine the dimension of the vector space $\mathbb{C}[x,y]/I$ over $\mathbb{C}$. Provide your answer as an integer.

Answer: 4


Question: Evaluate the integral \( \iiint \frac{1}{\sqrt{x^2+y^2+(z-2)^2}}\,dV \) over the region defined by \(x^2+y^2+z^2 = 1\) using cylindrical coordinates \((r, \theta, z)\). Express the integral in cylindrical coordinates and find its value.

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


Question: Let $\mathcal{B}(F)$ denote the algebra of all bounded linear operators on a complex Hilbert space $F$. Consider $M\in \mathcal{B}(F)^+$, meaning $\langle Mx, x \rangle \geq 0$ for all $x\in F$. The square root of $M$ is defined to be the unique operator $N$ such that $N^2 = M$. We represent this as $N = M^{1/2}$. If $M$ is an invertible operator, is $M^{1/2}$ also invertible? Provide a justification for your answer.

Answer: Yes


Question:
Compute the following sum for the Fourier transform of the characteristic function of the $p$-adic integers $\mathcal{O}_p$:

\[ \sum_{k \geq n} p^{-k} + \sum_{0 \leq k < n} e(p^{k-n}) p^{-k} \]

Determine the value of this sum when $n > 0$. Assume the measure is normalized such that $\mathcal{O}_p$ has measure 1.

Answer: 0


Question: Determine the values of $p \in (1, \infty)$ for which the inequality $2|x||y| \leq x^p + y^p$ holds for all $x, y \in \mathbb{R}$. Provide your answer as a range or specific values of $p$.

Answer: 2


Question: Given the joint probability density function $f_{X,Y}(x,y) = C(x+y^3)$ for $0 \leq x \leq 1$ and $0 \leq y \leq 1$, where $C$ is a constant, find the conditional probability $P(X < 0.5 \mid Y \leq 0.5)$. Determine the value of this probability.

Answer: \dfrac{9}{34}


Question: Determine whether the sequence \( t_n \) is a Cauchy sequence given that \( |t_{n+2} - t_{n+1}| < \frac{1}{n^2} \) for all \( n \in \mathbb{N} \).

Answer: Yes


Question: Determine whether the following statement is true or false: Given a positive, upper-bounded function \( F \), there exists a sufficiently small \( \epsilon > 0 \) such that the function \( F^\epsilon \) is also bounded.

Answer: True


Question: Let $h: \mathbb{C} \longrightarrow \mathbb{C}$ be a holomorphic function such that $h(0) = 0$, $h^{\prime}(0) = 1$, and $\left|h^{\prime}(z)\right| \leq 1$ for $\left|z\right| \leq 5$. Determine the function $h$.

Answer: z


Question: Given the joint probability density function of \(X\) and \(Y\) as \[ f_{(X,Y)}(x,y) = \begin{cases} 8xy, & 0 \le y \le x \le 1 \\ 0, & \text{elsewhere} \end{cases} \] find the conditional probability \( P\left( X \le \frac{1}{2} \mid Y \ge \frac{1}{4} \right) \).

Answer: \dfrac{1}{25}


Question: Let $I$ be an infinite set. Determine whether there exists a non-principal ultrafilter $F$ over $I$ that contains a finite subset of $I$. Provide a yes or no answer.

Answer: No


Question: Does there exist a group containing a pair of non-commuting infinite-order elements that are commensurable?

Answer: Yes


Question:
Calculate the determinant of the matrix $B = 3A^5 A^t A^{-1}$, where $A$ is given by:
\[ A = \begin{bmatrix} a & b & c & d+1\\ a & b & c+1 & d \\ a & b+1 & c & d \\ a+1 & b & c & d \end{bmatrix} \]
Given that $\det(A^t A^{-1}) = 1$, find $\det(B)$. Express your answer as a single number.

Answer: 81


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

Answer: \dfrac{3}{8}


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

Answer: 0


Question: Determine the minimal constant \(c\) such that the inequality \(\pi(n) \geq \frac{c\,n}{\ln n}\) holds for all integers \(n \geq N\), where \(N\) is a fixed positive integer.

Answer: 1


Question:
Determine whether there exists a sequence of continuous functions \( \{h_n\} \) that converges pointwise to the function \( h: \mathbb{R} \to \mathbb{R} \) defined by:
\[
h(x) = \begin{cases}
1/g(x) & x \in \mathbb{N} \\
0 & x \not\in \mathbb{N}
\end{cases}
\] where \( g: \mathbb{N} \to \mathbb{Q} \) is a bijection.

Answer: Yes


Question: Evaluate the limit: $$ \lim_{x\rightarrow0}\frac{1}{x^2}\int_0^{\sin x}\ln(1+t)\,dt $$

Answer: \dfrac{1}{2}


Question: Find the value of \( g^{(1000)}(0) \) for the function \( g(x) = \exp(x^{999}) \).

Answer: 0


Question: Evaluate the limit \(\lim_{x\to 0}\frac{\text{erf}(x^3)}{\text{erf}(x^2)}\) using Taylor expansion.

Answer: 0


Question: Determine whether the root of the second derivative of a third-degree polynomial function \(f(x) = ax^3 + bx^2 + cx + d\) can always be expressed as \(\frac{1}{3}\) of the sum of the roots of \(f(x)\). If not, provide a counterexample or explanation.

Answer: Yes


Question: Determine the number of solutions to the congruence \(x^2 \equiv 10 \pmod{37}\) using quadratic reciprocity.

Answer: 2


Question: Evaluate the integral \( -\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} \cos\left(be^{i\theta}\right) \, \mathrm{d}\theta,\) where \(b\) is a real number.

Answer: -1


Question: Evaluate the integral: $$\int_{0}^{\infty} \frac{1-x^2}{x^4+3x^2+1}\ dx$$

Answer: 0


Question: Determine whether the image of a Lebesgue measurable set under a Lipschitz continuous linear map is always Lebesgue measurable. Answer true or false.

Answer: B


Question: Find the value of \( F^{(66)}(0) \) for the function \( F(x) = \int_0^x \cos(t^3) \, dt \) given its series representation \( \sum_{n \ge 1} \left( \frac{(-1)^n}{(2n)!} \cdot \frac{x^{6n+1}}{6n+1} \right) \).

Answer: 0


Question: Does every Cauchy sequence in a metric space $(X,d)$ converge to a point in $X$?

Answer: No


Question: Does the space $L^1(\mathbb{X})$ have the Schur property?

Answer: No


Question: Evaluate \( \lim_{x\to 0}\frac{1}{x} \int_0^{x} \cos(t^2)\mathrm{d}t \).

Answer: 1


Question: Calculate the definite integral of the function \( f(x) = x \exp(-(x^2+x^4)) \) from \( x = -1 \) to \( x = 1 \) without using a calculator.

Answer: 0


Question: Determine whether the limit of the function \( F(x,y) = \frac{x^{3} + y^{3}}{x^{2} + y^{2}} \) exists as \((x, y)\) approaches the origin.

Answer: 0


Question: Determine whether it is possible to construct a function $y(x)$ such that $y(a) \neq 0$, $y'(a) = 0$, and $y''(a) = 0$ given that $g(x)$ has a first derivative at $a$. Provide a justification for your answer.

Answer: Yes


Question: Given a Jordan curve \(\gamma_1 \colon [a,b] \to \mathbb{C}\) and an \(\epsilon > 0\), determine if there exists a continuously differentiable Jordan curve \(\gamma_2 \colon [a,b] \to \mathbb{C}\) such that \( |\gamma_1(t) - \gamma_2(t) | < \epsilon\) for all \( t \in [a,b]\).

Answer: Yes


Question: Is it possible to traverse a 4x4x4 cube, starting at one corner and moving to adjacent 1x1x1 cubes, to reach the cube diagonally opposite the starting corner? Model this problem as a bipartite graph and determine the possibility of such a traversal.

Answer: Yes


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

Answer: e^2


Question: Evaluate the limit: \[ \lim_{x,y\to\infty}\dfrac{(xy)^{x^2}}{(x^2+y^2)^{x^2}} \] where both $x$ and $y$ approach infinity.

Answer: 0


Question: Evaluate the integral \(\int_{-\infty}^\infty (x^2 + 1) \delta(x^2 - 3x + 2) \, dx\) using the properties of the Dirac Delta function.

Answer: 7


Question: Solve the equation: \[ 2\sqrt{x+1} - \sqrt{4-x} = (x^2 - 5x + 4)^2 - 1 \]

Answer: 3


Question: Determine all polynomials \(p(x) = x^{21} + cx + d\) over \(\mathbb{Z}_{5}\) such that \(x^2 + 4\) is a factor of \(p(x)\).

Answer: x^{21} + 4x


Question: Determine the number of ways to express the polynomial $x^{2n} + 1$ as a product of $n$ quadratic polynomials with real coefficients, given $n \in \mathbb{N}$. Provide your answer as a single integer.

Answer: 1


Question: Find the first three terms of the Taylor series expansion of \(\psi(z)\) around \(z=0\), given that \((e^z-1)^2=z^2 \psi(z)\) and \(f^{(n)}(z) = 2^ne^{2z}-2e^z\).

Answer: 1 + z + \dfrac{7}{12}z^2


Question: Consider a group action \( \varphi \colon G \to \text{Homeo}(X) \) on a path-connected topological space \( X \) with basepoint \( x_0 \) and universal covering \( \pi \colon \widetilde{X} \to X \). The subgroup of lifts is defined as \( L = \{ (\widetilde{f},g) \in \text{Homeo}(\widetilde{X}) \times G : \widetilde{f} \circ \pi = \pi \circ \varphi(g)\} \), forming a group extension \( 1 \to \pi_1(X,x_0) \to L \to G \to 1 \). A right split \( \sigma \colon G \to L \) of this extension corresponds to a lift of the group action \( \varphi \) to \( \widetilde{X} \). If such a lifted group action exists and the extension is trivial (i.e., a product), does the lifted group action commute with the action of the fundamental group by deck transformations?

Answer: Yes


Question: Suppose $S$ is a totally ordered set such that for any $x, y \in S$, there are finitely many elements between them. Does it follow that $S$ is at most countable? Provide a justification for your answer.

Answer: Yes


Question:
Find a lower bound for the quadratic form \( \langle M \, y , y \rangle \), where \( y \) is a vector in \( \ell(\mathbb{Z}) \) and \( M \) is the pentadiagonal matrix given by:

\[ M= \begin{pmatrix} \ddots & \ddots & & & & \\ \ddots & 25 & -6 & 0 & 0 & \\ \ddots & -6 & 25 & -6 & 0 & \ddots \\ & 0 & -6 & 25 & -6 & \ddots \\ & 0 & 0 & -6 & 25 & \ddots \\ & & \ddots & \ddots &\ddots \end{pmatrix}. \]

Answer: 13


Question: Determine whether the following statement is true or false: If $B^A$ and $A^B$ are both countable, then $A$ and $B$ are both countable.

Answer: B


Question: Evaluate the definite integral: $$\int_0^1 \dfrac{3x^4 + 4x^3 + 3x^2}{(4x^3 + 3x^2 + 2x + 1)^2}\, dx$$

Answer: \dfrac{1}{10}


Question: Let $X_t$ be a stochastic process. Using the rules of Ito's calculus, find the value of $\langle dX_t, dX_t \rangle$ when $dX_t = Y_t \, dt$, where $Y_t$ is a progressive stochastic process.

Answer: 0


Question: Evaluate the double integral \( \iint_D xy\sqrt{1-x-y}\,dxdy \) over the domain \( D=\{(x,y):x\geq0, y\geq0, x+y\leq1\} \). Use the integration limits \( 0\leq x\leq1 \) and \( 0\leq y\leq1-x \) to solve the integral using a suitable method, such as a change of variables or the beta function.

Answer: \dfrac{16}{945}


Question: Determine whether the following statement is true or false: For any field $F$ and its algebraic closure $\bar{F}$, every element $\alpha \in \bar{F}$ is algebraic over $F$. Provide a justification for your answer.

Answer: True


Question: Find the sum of the infinite series: \[ \sum_{n=0}^\infty (n+2) \cdot (n+1)! \]

Answer: \infty


Question: Determine the largest square-free positive integer $n$ such that the ring of integers of the number field $\mathbb{Q}(\sqrt{-n})$ is a unique factorization domain.

Answer: 163


Question: Find the radius of the smallest circle centered on the $y$-axis that contains the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and intersects the ellipse only at the point $(0,b)$, where $a > b > 0$. Express the radius as a function of $a$ and $b$. \( \boxed{\text{radius}} \)

Answer: \dfrac{a^2}{b}


Question: A rock is dropped from a height \( h \). The distance fallen at time \( t \) is given by \( x(t) = \frac{1}{2}gt^2 \), where \( g \) is the acceleration due to gravity. The total flight time is \( T = \sqrt{\frac{2h}{g}} \). Express the probability density function for the distance traveled and calculate the time-averaged distance fallen by the rock.

Answer: \dfrac{h}{3}


Question: Revise the expression to accurately bound the probability $P(X > t\alpha)$, where $X$ follows a Poisson distribution with parameter $\alpha = N/M$, using Markov's inequality.

Answer: \dfrac{1}{t}


Question:
Calculate the contour integral \( \int_{|z|=1} \text{Re } z \, dz \) for a positively oriented unit circle. Identify the error in the following calculation that leads to an incorrect result of \(2\pi^2\) instead of the correct answer \(\pi i\):

\[\int_{|z|=1} \text{Re } z \, dz = \int_{|z|=1} x \, d(x+iy) = \int_{|z|=1} x \, dx\]

Answer: \pi i


Question: Let $G$ be a finite cyclic group with order $n$. For any positive integer $m$ that divides $n$, does the equation $x^m = e$ have exactly $m$ solutions in $G$, where $e$ is the identity element? Answer yes or no and justify your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{(x,y)\to(0,0)} \frac{\cos( |x| + |y|) + |x|(e^y - 1)} {|x| + |y|} \]

Answer: \infty


Question: In a finite solvable group, determine whether the smallest non-trivial group in the derived series always contains a non-trivial element whose conjugacy class generates a commutative subgroup. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Assume $X$ is a separable infinite-dimensional Hilbert space and $Y \subset X$ is a possibly non-closed subspace. Determine if $Y$ possesses a countable orthonormal basis such that: (1) the set of vectors $\{y_k\}_{k \in \mathbb{N}}$ forms a basis for $Y$; (2) this basis can be transformed into an orthonormal basis $\{z_k\}_{k \in \mathbb{N}}$ via the Gram-Schmidt process; and (3) the linear hull of $\{z_k\}_{k \in \mathbb{N}}$, denoted as $\overline{\text{lin}}\{z_k\}_{k \in \mathbb{N}}$, equals the closure of $Y$ in $X$, $\overline{Y}$.

Answer: Yes


Question: If $\lim_{x\to1} (f(x)g(x+1))$ exists and is equal to 0, and $|g(x)| \leq 1$, does it necessarily follow that $\lim_{x\to1} f(x)$ exists and is equal to 0?

Answer: No


Question: Determine the minimum value of the function \( f(a,b) = \sqrt{a^2+b^2} + 2\sqrt{a^2+b^2+2a+1} + \sqrt{a^2+b^2+6a-8b+25} \) without using calculus.

Answer: 1 + 2\sqrt{5}


Question: Determine the maximum degree of a Bezier curve that allows for an analytical solution when finding its intersection with a line.

Answer: 4


Question: Evaluate the integral \( I = \int_{0}^{1}\int_{0}^{1} \left\{\frac{x}{y}\right\}\mathrm{d}x\mathrm{d}y \).

Answer: \dfrac{3}{4} - \dfrac{\gamma}{2}


Question:
Evaluate the limit of the following expression as \( n \) approaches infinity:
\[ \lim_{n \to \infty} \frac{n^a}{2^{\sqrt{\ln n}}} \]
where \( a \in \mathbf{N} \) and \( a > 1 \). Determine whether the limit is a constant, zero, or infinity.

Answer: \infty


Question:
Given two real polynomials $f(x)$ and $g(x)$ with degrees $m \geq 2$ and $n \geq 1$ respectively, such that \(f(x^2+1)=f(x) g(x)\) for all real numbers \(x\), determine the correct conclusion about the real roots of $f(x)$:

(A) $f$ has exactly one real root $x_0$ with $f'(x_0) \neq 0$
(B) $f$ has exactly one real root $x_0$ with $f'(x_0) = 0$
(C) $f$ has $m$ distinct real roots
(D) $f$ has no real root

Given that \(\deg f(x^2+1) = 2m\) and \(\deg f(x) g(x) = m+n\), and \(2m = m+n\) implies \(m=n\), find the correct conclusion.

Answer: D


Question: Evaluate the integral \( \int^{\frac{\pi}{2}}_0 \frac{\cos^5x}{\sin^5x+\cos^5x}\,\mathrm dx \) using a suitable substitution.

Answer: \dfrac{\pi}{4}


Question: Does the Monster Group contain a subgroup isomorphic to \(\mathbb{Z_3}\)? Justify your answer.

Answer: Yes


Question: Determine which of the following expressions grows the fastest as \(n\) approaches infinity: \(n^{n+1}\), \(n!\), or \(\frac{(2n)!}{n!^2}\).

Answer: n^{n+1}