Question: In a two-player game, players A and B take turns replacing symbols \(\Box\) with positive integers in the expression \(\pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box\). Player A starts, and after all \(\Box\) are replaced, A assigns either + or - to each \(\pm\) independently. Player A wins if the resulting expression is not divisible by any number from 11 to 18, while player B wins otherwise. Determine which player has a winning strategy.

Answer: A


Question:
Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\ln \left(1 + \frac{\sin x}{2}\right)}{\sin x} \, dx \). Choose the correct answer from the following options:

A. \( \frac{\pi}{6} \)
B. \( \frac{\pi^2}{6} \)
C. \( \frac{\pi}{3} \)
D. \( \frac{\pi^2}{3} \)

Answer: B


Question: Determine the number of boolean functions of $n$ variables that can be expressed as a product of literals, excluding functions that always yield $0$. Provide your answer as a single integer.

Answer: 3^n


Question: Determine whether the measure \( \mu \) defined by \( \mu(A)=\int_A \frac{1}{x^2} d\lambda \) for a Borel measurable set \( A \) on the real line, with \( \lambda \) as the Lebesgue measure, is finite, \( \sigma \)-finite, or neither. Specifically, find the measure of the point \( \{0\} \).

Answer: 0


Question:
Let $H$ be an arbitrary Hilbert space. Suppose that for some sequence $\{a_n\} \subseteq H$ we have

$$\forall x\in H,\quad x\cdot a_n \to 0\in\mathbb K.$$

Does it follow that $a_n\to 0$ in $H$?

Answer: No


Question:
Identify the incorrect statement about holomorphic functions from the following options:

(1) If $f(z)$ is holomorphic, then $f(\bar{z})$ is not holomorphic.
(2) If $f(\mathbb{R}) \subseteq \mathbb{R}$ and $f$ is non-constant, then $f(\mathbb{R})$ is not open in $\mathbb{R}$.
(3) The map $z \rightarrow e^{f(z)}$ is holomorphic if $f(z)$ is holomorphic.
(4) If $f(\mathbb{C}) \subset \mathbb{R}$, then $f(A)$ is closed in $\mathbb{C}$ for every closed subset $A$ of $\mathbb{C}$.

Choose the incorrect statement.

Answer: 2


Question: Evaluate the limit: $$\lim_{x\to1^+} \frac{\log_{10}{(x-1)}}{\sqrt{x-1}}$$.

Answer: -\infty


Question: Determine whether it is possible to map a regular polytope's face to itself using symmetry operations, such that any vertex of that face can be mapped to any other vertex.

Answer: Yes


Question:
Evaluate the limit:

$$\lim_{x\to 0}\ \frac{\frac{\sin x}{x} - \cos x}{2x \left(\frac{e^{2x} - 1}{2x} - 1 \right)}$$

Given that as $x$ approaches $0$, $\frac{\sin x}{x} = 1$ and $\frac{e^x - 1}{x} = 1$. Determine the value of the limit.

Answer: \dfrac{1}{6}


Question: Determine if the multiplicative group of the finite field $GF(2^{2016})$ contains a subgroup of order 2016.

Answer: No


Question: Determine the number of solutions to the congruence equation \(x^2 \equiv -1211 \pmod{2020}\).

Answer: 8


Question: Calculate the limit of the sequence as \( n \) approaches infinity: \[ \lim_{n\to\infty}\sum_{k=1}^n\frac{k^r}{n^{r+1}} \]

Answer: \dfrac{1}{r+1}


Question: Determine whether there exists an irreducible element $p$ in a ring $R$ such that the ideal generated by $p$, denoted $\langle p \rangle$, is not a maximal ideal.

Answer: Yes


Question: In the first quadrant, circle $C_1$ touches the x-axis and the line $y = x \tan\theta$, where $\theta \in (0, \frac{\pi}{2})$. Circle $C_2$ touches the line $y = x \tan\theta$, the y-axis, and circle $C_1$, with the ratio of the radius of $C_1$ to the radius of $C_2$ being 2:1. Determine the value of $\tan\frac{\theta}{2} = \frac{\sqrt{a} - b}{c}$, where $a$, $b$, and $c$ are relatively prime natural numbers. Find $a + b + c$. \( \boxed{} \)

Answer: 22


Question: Is the minimal polynomial $f(x)$ of a constructible number $t$ over $\mathbb{Q}$ solvable by radicals?

Answer: Yes


Question: Determine the maximum number of triangles that can be formed by dividing a convex polygon with $M$ vertices using $N$ points inside the polygon. Assume that no three points among the $M+N$ points are collinear, and the triangles must have vertices from these points with sides intersecting only at vertices.

Answer: M + 2N - 2


Question: Determine whether every infinite-dimensional vector space has an infinite linearly independent subset that spans the space. Provide a justification for your answer.

Answer: Yes


Question: Find the greatest common divisor of the polynomials \( f(n) = n^4 - 4n^2 \) and \( g(n) = n^2 + 1 \) using the Gaussian integer method.

Answer: 1


Question: Find the positive integer $k$ such that the Chebyshev polynomial $T_k(x)$ is equal to $q(x) = 8x^4 - 8x^2 + 1$. Express your answer as a single integer.

Answer: 4


Question: Find the derivative of the expression \((X \beta)^t (X \beta)\), where \(X\) is an \(N \times P\) matrix and \(\beta\) is a \(P \times 1\) vector.

Answer: 2 X^\top X \beta


Question: Let \( f(x) = x^3 + 3x + 1 \) and \( g(x) \) be the inverse of \( f(x) \). Given that \( f(1) = 5 \) and \( g(5) = 1 \), find the value of \( g''(5) \).

Answer: -\dfrac{1}{36}


Question: Compute the limit: $$\lim_{x \to \infty}\dfrac {f^{-1}(8x)-f^{-1}(x)}{x^{1/3}},$$ where $f: \mathbb R \to \mathbb R$ is defined as $f(x)=8x^3+3x$, and $f$ is continuous, strictly increasing, and satisfies $\lim _{x\to \infty}f(x)=\infty$ and $\lim_{x \to -\infty}f(x)=-\infty$. \(f^{-1}\) denotes the inverse function of \(f\).

Answer: \dfrac{1}{2}


Question: Find the value of \( r \), where \( r > \frac{1}{2} \), such that two non-vertical normal lines from the point \((r, 0)\) to the parabola \(x = y^2\) intersect perpendicularly. One of these normal lines is the x-axis. The derivative of the parabola is \(\frac{dy}{dx} = \frac{1}{2\sqrt{x}}\), so the gradient of the normal is \(-2\sqrt{x}\). The equation of the normal can be expressed as \(y = -2\sqrt{x}(x - r)\). Determine the correct value of \( r \).

Answer: \dfrac{3}{4}


Question: Determine whether the following statement is true or false: If $f: G \to H$ is a group homomorphism and the image of $f$ contains a free group of rank $2$, then $G$ also contains a free group of rank $2$. Provide a justification for your answer.

Answer: True


Question:
Given the sequence defined by the recurrence relation \(x_{n+1}=\sqrt{\frac{1}{2}(1+x_n)}\), where \(-1<x_{0}<1\), determine the limit:

\[\lim_{n \rightarrow \infty}\cos\left(\frac{\sqrt{1-(x_{0})^2}}{x_1x_2...x_n}\right)\]

Assume that the limit of \(x_n\) as \(n\) tends to infinity is \(L=1\). Evaluate the limit.

Answer: x_0


Question: Do the sets $X_3 = \{ [a,b), \, a,b \in \mathbb{R} \}$ and $X_4 = \{ [a,b], \, a,b \in \mathbb{R} \}$ generate the Borel-$\sigma$-algebra $B(\mathbb{R})$?

Answer: Yes


Question: Calculate the flux of the vector field \( \mathbf{F}(x,y,z) = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{(x^2+y^2+z^2)^{3/2}} \) through the surface of the hemisphere defined by \( x^2 + y^2 + z^2 = 4 \), \( z \geq 0 \), oriented upwards.

Answer: 2\pi


Question: Evaluate the limit: $$\lim_{x \to 0^+} \frac{1}{x^{\beta}}\left(\left(\frac{1}{x} -1\right)^{\displaystyle\frac{1}{x}}-1\right)$$ where \(\beta \in \mathbb{R}\).

Answer: \infty


Question: Determine whether the statement \( \lim_{v \to 0} h(v, k(v)) = 0 \) is true, given that for all \( v \in (0, \infty) \cap \left(\{0\} \cup (2, 3) \right) \), \( h(v, k(v)) = 25v^2 \).

Answer: False


Question: Given that $x + 2 = 18181818\ldots$ with $n$ digits, determine the number of trailing threes in the binomial coefficient $\binom{x}{x/3}$. Provide your answer as a single number.

Answer: 0


Question: In a random walk on the 20 white faces of a truncated icosahedron (soccer ball), you move from one white face to an adjacent one at random with equal probability. Calculate the expected number of steps until you return to the starting white face.

Answer: 20


Question: Given a sequence of independent and identically distributed (i.i.d.) random variables $\{X_i\}$ with mean $0$ and variance $\sigma^2 > 0$, find the limit of the expected value of the squared fraction of the sample mean's absolute value as the number of variables approaches infinity: $$\lim_{n \to \infty} E\left(\left|\frac{\overline X_n}{\overline X_n + \frac{1}{n}}\right|^2\right)$$

Answer: 1


Question: For a one-dimensional smooth map $f(x)$, determine whether the following statement is true or false: If the geometric average of $|f'(x)|$ on a trajectory is greater than one, then the trajectory is sensitive to initial conditions.

Answer: True


Question: Given a sequence of positive integers $x_1, x_2, \dots, x_n$ and a positive integer $k \leq n$, determine if it is always possible to partition the sequence into $k$ parts such that for each part, there exists a number at one of the two ends of the part which, if removed, results in the sum of the numbers in that part being no greater than the sum of the numbers in any other part.

Answer: Yes


Question: Evaluate the definite integral: $$ \frac{24}{\pi}\int_0^\sqrt 2\frac{2-x^2}{(2+x^2)\sqrt{4+x^4}}\,\mathrm dx $$

Answer: 3


Question: Find all positive integer pairs \((a, b)\) such that \(2^a + 17 = b^4\).

Answer: (6, 3)


Question: Evaluate the limit: $$\lim_{n \to \infty} \int_1^\infty \frac{2n}{e^{2x}+n^2x}dx$$

Answer: 0


Question: Find a substitution that simplifies the integral \( \int_0^1 \int_0^{x_2} x_2^2 \cdot \sin\left(\frac{2\pi x_1}{x_2}\right) \,dx_1 \,dx_2 \) for manual computation.

Answer: 0


Question: Calculate the area under the curve of the integral \( \int_{0}^{3} x f(x^2) \, dx \), where \( f(x) = \sqrt{-x^2 + 4x} \).

Answer: \pi


Question: Express the output $Y(s)$ in terms of the input $X(s)$ for a control system with a nested feedback loop, where the plant transfer function is given by \( F(s)=\frac{R^3+R^2sL}{s^2R^2L-R^3s} \). The input $X(s)$ is subtracted from the output $Y(s)$ to form $X_1(s) = X(s) - Y(s)$, which is then multiplied by the plant function to obtain $X_2(s) = F(s) \cdot X_1(s)$. The final output is given by $Y(s) = X_2(s) + X(s)$. Unnest the feedback loop algebraically to find $Y(s)$ in terms of $X(s)$. \( Y(s) = ? \)

Answer: Y(s) = X(s)


Question: What is the formal name for an additive abelian group equipped with a biadditive multiplication that satisfies left and right distributivity over addition, without assuming associativity or the existence of a unit?

Answer: non-associative ring


Question: Let $J$ be a Jordan curve in the plane, considered as the homeomorphic image of the unit circle $T$. Suppose there exists an $\\epsilon > 0$ and another Jordan curve $K$ such that $K \subseteq V := \{z: \text{dist}(z, J) < \\epsilon\}$, the \\epsilon-neighborhood of $J$, and every point of $K$ is close to a point of $J$, and vice versa. Do there exist parametrizations $j: T \to J$ of $J$ and $k: T \to K$ of $K$ such that $|j(\\xi) - k(\\xi)| < C \\epsilon$ for every $\\xi \in T$?

Answer: Yes


Question: Determine whether the function \( f(n) = \int_0^1\cdots\int_0^1 \exp\left(-\left(\sum_{i=1}^{n}x_i\right)^{-1}\right) \, dx_1\cdots dx_n \) increases as \( n \) increases.

Answer: Yes


Question: Find the value of the expression \( \cos\left(\frac{\pi}{7}\right) + \cos\left(\frac{3\pi}{7}\right) + \cos\left(\frac{5\pi}{7}\right) \) without using trigonometric tables.

Answer: \dfrac{1}{2}


Question: Determine whether the function \( u(x,y)=\frac{(x^3+y^3)}{x^2+y^2} \) has a partial derivative with respect to \( x \) at the point \((0,0)\).

Answer: 1


Question: Determine the radius of convergence of the power series \[1+az+a(a-3b)\frac{z^2}{2!}+a(a-4b)^3\frac{z^3}{3!}+\cdots\] assuming $a$ and $b$ are fixed complex numbers with $a \neq 0$. Provide your answer as a single numerical value.

Answer: 0


Question: Given a function \( g : [0, \infty) \to [0, \infty) \) such that \( \lim_{x \to +\infty} 2g(2x) - g(x) = 0 \) and \( g\vert_{[0,M]} \) is bounded for every \( M > 0 \), does it follow that \( \lim_{x \to +\infty} g(x) = 0 \)?

Answer: 0


Question: Compute the integral \(\displaystyle \int_{\gamma} \frac{z-1}{z(z-i)(z-3i)}dz\) where \(\gamma\) is the curve defined by \(|z-i| = \frac{1}{2}\). Decompose the integrand into partial fractions to aid in the computation.

Answer: -\pi(1 + i)


Question: Let $V(\Bbb R)$ denote the vector space of all $2\times2$ matrices, and consider the matrix $M = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$. Define a linear transformation $T:V(\Bbb R)\to V(\Bbb R)$ by $T(A) = AM - MA$ for any matrix $A \in V(\Bbb R)$. If $T(I) = O$, where $I$ is the identity matrix and $O$ is the zero matrix, find the dimension of the kernel of $T$. \(\text{Express your answer as a single integer.}\)

Answer: 2


Question: Find the limit: \( \displaystyle \lim_{x \to 0} \dfrac{\sin (x \sin \frac{1}{x})} {x \sin \frac{1}{x}} \).

Answer: 1


Question: If a sequence of bounded linear operators $T_n$ converges to $T$ in the operator norm topology, determine whether it is true that $\lim\limits_{n\to\infty}\|T_n\|=\|T\|$. Provide a justification for your answer.

Answer: Yes


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

Answer: e


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

Answer: x + 6y - 2z = 3


Question: Find the limit of the series \[\lim_{n\rightarrow \infty}\left(\frac{(n+1)(n+2)...(3n)}{n^{2n}}\right)\] as \(n\) approaches infinity.

Answer: \infty


Question: Calculate the value of the contour integral \( \oint_{|z|=2\pi} \tan(z) \, dz \) using the Cauchy residue theorem, where the contour is positively oriented. The singularities of \( \tan(z) \) within the contour are at \( -3\pi/2, -\pi/2, \pi/2, \) and \( 3\pi/2 \). Determine the residues at these points and find the value of the integral.

Answer: -8\pi i


Question: Determine the real constant \( k \) for which the function \( v(x) = x_1^3 + kx_1x_2^2 \) is harmonic on \( \mathbb{R}^n \).

Answer: -3


Question: Determine whether there exists a continuous function from the interval $[0,1]$ to $\mathbb{R}$ that has an infinite length.

Answer: Yes


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

Answer: 1


Question: Given a differentiable function \( f: \mathbb{R} \to \mathbb{R} \) such that \( \lim_{{x \to \infty}} f(x) = 1 \) and \( \lim_{{x \to \infty}} f'(x) = \alpha \) exist, determine the value of \( \alpha \).

Answer: 0


Question: Calculate the flux of the vector field \( f(x,y,z) = (y,x,0) \) through the hemisphere defined by \( z = \sqrt{1-x^2-y^2} \).

Answer: 0


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

Answer: 2^{\aleph_0}


Question: Approximate \( \arctan\left(\frac{1}{2}\right) \) using the alternating Taylor series such that the error is within \( \frac{1}{33} \).

Answer: \dfrac{11}{24}


Question: For a linear operator on $\mathbb{R}^n$, where $n \geq 3$, determine if it is possible to have an invariant subspace of dimension 3 or higher.

Answer: Yes


Question: Consider a sequence of random variables $b_n$ defined as $b_1 = 1$ and $b_n = b_{n-1}$ with probability $\frac{1}{3}$ and $b_n = \frac{2}{3}b_{n-1}$ with probability $\frac{2}{3}$. Find the expected value of the sum $\sum_{n=1}^{\infty} b_n$. \( \text{Express your answer as a single number.} \)

Answer: \dfrac{9}{2}


Question: Given a set $A\subset\mathbb{R}$ with positive Lebesgue measure $\mu(A)=a\in\mathbb{R}_+$, and a value $b\in(0,a)$, determine whether there always exists a subset $B\subset A$ such that $\mu(B)=b$. Provide a justification for your answer.

Answer: Yes


Question: Determine if there exists a constant $C$ such that the inequality $m\alpha^m \le C\beta^m$ holds for all $m \in \mathbb{N}$, given that $0 < \alpha < \beta < 1$.

Answer: Yes


Question: A square with sides initially measuring $4m$ increases at a constant rate of $3m$ per second. One corner of the square moves along a line $L$ at a speed of $2m/s$ for $5$ seconds. During this time, the square makes one complete revolution about $L$, revolving at a constant rate and remaining always perpendicular to $L$. Calculate the volume of the solid generated by this motion.

Answer: 1510


Question: Determine the radius of convergence of the power series \( \sum_{n\geq 1}{n^{\log(n)}}x^n \).

Answer: 1


Question:
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x,y \in \mathbb{R} \), the following equation holds:
\[ f\bigl(xf(y)+y\bigr)+f\bigl(-f(x)\bigr)=f\bigl(yf(x)-y\bigr)+y. \]

Answer: f(x) = x + 1


Question: Identify the quotient group $\mathbb{C}^*/P$, where $\mathbb{C}^*$ is the multiplicative group of non-zero complex numbers and $P$ is the subgroup of positive real numbers.

Answer: S^1


Question: Calculate the residue of \( \frac{z}{z^4+1} \) at \( z=e^{i\pi/4} \) without using the Taylor series formula.

Answer: -\dfrac{i}{4}


Question: Let $R$ be a commutative ring, and let $P$ be a matrix in $M_n(R)$ such that $\det(P)$ is a zero divisor in $R$. Is $P$ necessarily a zero divisor in $M_n(R)$? Provide a justification for your answer.

Answer: Yes


Question: Determine the values of $k$ for which the equations $x^2 - 4x + k = 0$ and $x^2 + kx - 4 = 0$ have exactly two common roots.

Answer: -4


Question: Determine whether the functional equation $f(f(f(x))) = x$ for all $x \in \mathbb{R}$ implies that $f(x) = x$ for a continuous function $f$ on $\mathbb{R}$. Provide a justification for your answer.

Answer: f(x) = x


Question: Find the minimum value of \( \int_0^1 f(x)^2 \, dx \) for a function \( f(x) \) defined on \([0,1]\), given the constraints \( \int_0^1 f(x) \, dx = 0 \) and \( \int_0^1 x f(x) \, dx = 1 \).

Answer: 12


Question: Evaluate the integral \( \int_{\Gamma} \mathrm{Log}z \, \mathrm{d}z \) where \( \Gamma(t) = e^{2i\pi t} \) for \( 0 \leq t \leq 1 \), considering that the principal argument of \( e^{2i\pi t} \) is \( 2\pi t \).

Answer: 2\pi i


Question: Evaluate the line integral \( \int_{C}{x\,\mathrm{d}z - z\,\mathrm{d}y + y\,\mathrm{d}z} \) where the curve \( C \) is parameterized by \( \gamma(t) = (\cos t, \sin t, 2t) \) for \( t \in [0, 2\pi] \).

Answer: 0


Question: Identify the location and direction of the tangent vector $t_y$ to the surface $S$ defined by $z = \frac{1}{1+x^2+y^2}$, restricted to the plane $y=0$. The surface is given by $\mathbf{r}(x,y) = x\mathbf{i} + y\mathbf{j} + \frac{1}{1+x^2+y^2}\mathbf{k}$. Determine $t_y$ in this plane.

Answer: \mathbf{j}


Question: Consider two sequences \(x_n\) and \(y_n\) such that \(\lim_{n\to \infty} x_n = \lim_{n\to \infty} y_n = a\). If \(\lim_{n\to \infty} \sum_{i=1}^n x_i = b\), find the value of \(\lim_{n\to \infty} \prod_{i=1}^n y_i\).

Answer: 0


Question: Evaluate the infinite series: $$\sum_{n=0}^\infty\frac{3^n(n+1)}{n!}$$

Answer: 4e^3


Question: Find the volume of an n-simplex in \(\mathbb{R}^n\) defined by the set \(S(a)\) where \(0 \le x_i\) for all \(i\) and \(x_1 + \dotsb + x_n \le a\), for any \(n \ge 1\) and \(a \ge 0\).

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


Question: Determine whether the statement $S_n = \langle (i_1,i_2),(i_2,i_3),(i_3,i_4),\ldots,(i_{n-1},i_n)\rangle$ is true, where $S_n$ is the symmetric group on $n$ letters and $i_k \in \{1,2,3,\ldots,n\}$ with $i_k \neq i_l$ if $k \neq l$. Answer with "true" or "false."

Answer: true


Question: Calculate the arc length of the curve defined by the vector function \(\overrightarrow{r}(t) = \langle -t \sin t - \cos t, -t^2, t \cos t - \sin t \rangle\) from \(t = -1\) to \(t = 1\). The derivative of the curve is \(\overrightarrow{r}'(t) = \langle -t \cos t, -2t, -t \sin t \rangle\). Find the integral expression for the arc length \(L\) and evaluate it to obtain the length between the points \((-1, 0, 0)\) and \((1, -\pi^2, -\pi)\).

Answer: \sqrt{5}


Question: Suppose a tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) intersects the ellipse \(\frac{x^2}{c^2} + \frac{y^2}{d^2} = 1\) at points \(P\) and \(Q\). If the tangents drawn to the second ellipse at these points are perpendicular, find the value of \(\frac{a^2}{c^2} + \frac{b^2}{d^2}\).

Answer: 1


Question: Determine the nature of the entire function $f$ given that \( \int_{|z|=1}\frac{f(z)}{((k+1)z-1)^n}dz=0 \) for any natural number $k$ and for all positive integers $n$. What can be concluded about $f$?

Answer: f(z) = 0


Question: Determine the largest possible rank of a non-zero $4 \times 4$ complex matrix $A$ such that $A^2 = 0$. \( \boxed{} \)

Answer: 2


Question: For a difference triangle of order 8, where the total number of integers is 36, determine the closest possible total to 36 that can be achieved with either no repeated values or no missing values.

Answer: 36


Question: Is every splitting field $L$ of a polynomial $f$ over a field $K$ an algebraic extension of $K$? Provide a justification for your answer.

Answer: Yes


Question: Find the limit of the expression \( \frac{1}{n}\left[1+2\sum_{k=1}^{n-1}\sqrt{1-\frac{n-k}{n}}\right] \) as \( n \to \infty \).

Answer: \dfrac{4}{3}


Question: Find the center of mass of the lamina defined by the equation $x^2 + y^2 = 1$ with the density function $\sigma(x,y) = x+y$. Express your answer as a point $(x, y)$.

Answer: (0, 0)


Question: Determine the cardinality of the set of real-valued continuous functions \( f \) defined on \([0,1]\) such that \( f(x) \) is rational whenever \( x \) is rational.

Answer: \mathfrak{c}


Question: Determine the number of unique Tsuro tiles possible, given that each tile has 8 entry points, each of which connects to exactly one other point. Assume that every tile is unique as claimed by the game manual.

Answer: 35


Question: Calculate the degree of the map \( S: \mathbb{R}^{m+n} \rightarrow \mathbb{R}^{m+n} \) defined by \( S(x, y) = (y, x) \) using the oriented bases of the tangent spaces.

Answer: (-1)^{mn}


Question: Determine whether the following statement is true or false: In a metric space \(X\), the Borel \(\sigma\)-algebra \({\cal B}_X\) includes every countable subset of \(X\).

Answer: A


Question: In a non-equilateral triangle $\triangle ABC$, the heights $AP$ and $CQ$ are drawn, along with the interior angle bisector $BD$. Given that \( \frac{1}{AP} + \frac{1}{CQ} = \frac{2}{BD} \), find the measure of angle $\angle ABC$. Express your answer in degrees.

Answer: 60


Question: For positive integers $x$ and $y$, let $g = \gcd(x, y)$ and $\ell = \text{lcm}(2x, y)$. If the equation $xy + 3g + 7\ell = 168$ holds, find the largest possible value of $2x + y$. \( \boxed{\text{Answer}} \)

Answer: 23


Question: Evaluate the integral: $$\int\frac{\cos^2x-\sin^2x}{\cos^4x+\sin^2x\cos^2x}\mathrm d x$$

Answer: 2x - \tan x + C


Question: In a grid, each square (except corners) contains a number equal to the average of its neighboring numbers. Corners have 3 neighbors, while other squares have 4. The grid has the values -1000, 0, and 1000 on its edges. Determine the value of the number marked as 'x' in the grid.

Answer: 0


Question: In a cube $ABCDEFGH$ with side length 5, points $P$ and $Q$ are located on edges $\overline{AB}$ and $\overline{AE}$, respectively, such that $AP = 2$ and $AQ = 1$. A plane passes through points $C$, $P$, and $Q$ and intersects edge $\overline{DH}$ at point $R$. Determine the length of segment $DR$. \( \boxed{} \)

Answer: \dfrac{5}{2}


Question: Evaluate \( \lim_{x \to 0} \int_0^x \frac{\cos(t^3)}{t+x}dt \).

Answer: \ln 2


Question: In triangle ABC, perpendiculars AM and AP are drawn from vertex A to the external angle bisectors at vertices B and C, respectively. Given that the perimeter of triangle ABC is 10, find the length of segment PM.

Answer: 5


Question: Determine whether the statement "Any finite set of polynomials can span the vector space of all polynomials over a field K" is true or false. Provide a justification for your answer.

Answer: False


Question: Determine the values of $g$ and $h$ for which there exists a covering map $\Sigma_g \to \Sigma_h$ with a deck transformation group isomorphic to $\mathbb{Z}_6$.

Answer: g = 6h - 5


Question: Find a value of $k$ such that the set $S = \{A_1, A_2, A_3\}$, where each $A_i$ is a $2 \times 2$ matrix containing $k$, generates a vector space. Ensure that $k$ appears at least once in each matrix.

Answer: 0


Question: Evaluate the contour integral \( \oint_{|z|=1} |z-1|^2 \, dz \), where \( z \) is a complex number and the integration is performed along the circle of radius 1 centered at the origin.

Answer: -2\pi i


Question: Calculate \(\nabla^4 \left(\frac{1}{r^3}\right)\) in 5-dimensional space, where \(r^2 = x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2\).

Answer: 0


Question: Calculate \(15^{123456789012345} \mod 2500\).

Answer: 1875


Question: Given a complex square matrix $R$ such that $R^k = I$ for some integer $k > 1$, where $k$ is the smallest integer for which this holds, determine whether there exists a non-zero vector $x \in \mathbb{C}^n$ such that $R^j x \neq R^l x$ for all distinct integers $j, l \in \{0, 1, \ldots, k-1\}$.

Answer: Yes


Question: Three players, A, B, and C, are playing a game where the highest score wins. A's score follows a Poisson distribution with parameter 2, B's score follows a Poisson distribution with parameter 3, and C's score follows a Poisson distribution with parameter 4. The scores are independent. Given that the scores obtained were 1, 2, and 3, what is the conditional probability that player C won?

Answer: \dfrac{40}{81}


Question: Solve the equation \[8^x + 3 \cdot 2^{2-x} = 1 + 2^{3-3x} + 3 \cdot 2^{x+1}.\]

Answer: 1


Question: Let $T>0$ and $f_n : [0,T] \rightarrow \mathbb{R}$ be a sequence of continuous functions. Furthermore, let \( \sum_{k = 0}^{\infty} \sup_{t \in [0,T]} \lVert f_{k+1}(t) - f_{k}(t) \rVert < \infty. \) Does this imply that there exists a continuous function $f : [0,T] \rightarrow \mathbb{R}$ such that $(f_n)$ converges uniformly to $f$ on $[0,T]$?

Answer: Yes


Question: Determine whether the sequence \( \{\cos(\pi\sqrt{n^2+n})\}_{n=1}^\infty \) is convergent.

Answer: 0


Question: Determine whether the following statement is true or false: Given functions $f, g$, and $g'$ in $L^1(\mathbb{R})$, is it true that $(f \ast g)' = f \ast g'$?

Answer: True


Question: Evaluate the triple integral \( \iiint_V xyz \, dxdydz \) in spherical coordinates, where \( V \) is the upper octant of the unit sphere.

Answer: \dfrac{1}{48}


Question: Evaluate the integral \( \oint_{|z|=3} \frac{1}{(z+1)(z-1)} \, dz \) along the circle of radius 3 centered at the origin, oriented clockwise, using Cauchy's Integral Formula.

Answer: 0


Question: Evaluate the infinite series \(\sum_{n=0}^\infty\frac{n^2}{n!}\) and express it in the form \(\sum_{n=1}^\infty\frac{n}{(n-1)!}\). Find the value of this series.

Answer: 2e


Question: Determine the values of \(\alpha\) for which the double integral \(\iint_D\frac{1}{(x+y)^{\alpha}}\ dxdy\) converges, where \(D=\{(x, y): 0\leq y \leq 1-x, \quad 0 \leq \ x \leq 1\}\). Express the integral as an iterated integral: \(\int_{0}^{1}\left(\int_{0}^{1-x} \frac{1}{(x+y)^{\alpha}} \ dy\right)dx\).

Answer: \alpha < 2


Question: Calculate the limit: $$\lim_{x \to \infty}((x^3-x^2+\frac{x}{2})e^{\frac{1}{x}}-\sqrt{x^6+1})$$

Answer: \dfrac{1}{6}


Question: Evaluate the improper integral \(\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} \, dx\) using complex analysis techniques.

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


Question: Determine the number of ordered pairs of nonempty sets \( \mathcal{A} \) and \( \mathcal{B} \) such that \( \mathcal{A} \cup \mathcal{B} = \{1,2,\ldots,10\}\), \( \mathcal{A} \cap \mathcal{B} = \emptyset\), the number of elements of \( \mathcal{A} \) is an element of \( \mathcal{B} \), and the number of elements of \( \mathcal{B} \) is an element of \( \mathcal{A} \).

Answer: 186


Question: Determine whether there exists a non-abelian finite simple group $S$ such that the second cohomology group $H^2(S, \mathbb{Z}/p\mathbb{Z})$ is non-trivial, where $p$ is an odd prime.

Answer: Yes


Question: Evaluate the integral \(\int_{-\infty}^{\infty}\frac{\sin(cx) - \sin(dx)}{x^2} \mathrm{d}x\) where \(c\) and \(d\) are positive, non-zero constants, using a suitable indented contour.

Answer: 0


Question: Is a finite-dimensional inner product space \(V\) with the induced norm \(||x|| = (x|x)^{1/2}\) automatically a Hilbert space, given that it is isomorphic to \(\mathbb{R}^n\)?

Answer: Yes


Question: Given a set \( S = \{1, 2, 3\} \), calculate the total sum of the expression \((\text{sum of all numbers in the piece}) \times (\text{length of the piece})\) over all possible contiguous partitions of the set. For example, for the set \( A = \{1, 3, 6\} \), the total sum is 73. Find the total sum for the set \( S = \{1, 2, 3\} \).

Answer: 44


Question: Determine the number of non-isomorphic random graphs that can be formed with a set of cardinality \( \kappa \), where \( \kappa \) is a regular cardinal.

Answer: 2^\kappa


Question: Evaluate the integral: $$ \int_0^2 \sqrt{x+\sqrt{x+\sqrt{x+\dotsb}}}\,dx $$

Answer: \dfrac{19}{6}


Question: Evaluate the limit: $$\lim_{x\to1}\left(\tan\frac{\pi x}{4}\right)^{\tan\frac{\pi x}{2}}$$

Answer: \dfrac{1}{e}


Question:
Solve for $x$ in the equation:

$$\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+\ldots+x^{2008}\right)=2010x^{2009}$$

Answer: 1


Question: Let $R$ be the radius of convergence of the power series $\sum_{n=0}^{\infty}a_nz^n$. Determine the radius of convergence of the series $\sum_{n=0}^{\infty}a_{kn}z^n$ for a fixed positive integer $k$.

Answer: R^k


Question: Given the sequence $(a_n)_{n=1}^{\infty}$ in $\mathbb{C}$ defined by $a_1 = i$ and $a_{n+1} = \frac{3}{2 + a_n}$, and knowing that $\lim(a_n)$ exists, determine the limit of the sequence if $\lim(a_n) \in \{-3, 1\}$.

Answer: 1


Question: Let \(a, b, c, d\) be non-negative real numbers such that \(a^5 + b^5 \le 1\) and \(c^5 + d^5 \le 1\). Determine the maximum possible value of \(a^2c^3 + b^2d^3\).

Answer: 1


Question: Find the limit of the sequence \(a_n = \sqrt[n]{1^n + 2^n + \dots + n^n}\) as \(n\) approaches infinity using the squeeze theorem.

Answer: \infty


Question: Find the Maclaurin series expansion of the function \( g(x) = \frac{\sin(2x^3) - x^3}{x^5} \). Using this series, determine the value of \( g^{(12)}(0) \).

Answer: 0


Question: Determine the total charge on the spherical surface defined by \(x^2+y^2+z^2=R^2\) with a charge density given by \(\rho(x,y,z)=|z|\) C/m\(^2\).

Answer: 2\pi R^3


Question: Given a commutative Noetherian ring $R$, consider the categories Mod$(R)$ (all $R$-modules) and mod$(R)$ (finitely generated $R$-modules). Both categories are abelian and have enough projective objects. Let $F: \text{Mod}(R) \to \text{Mod}(R)$ be an $R$-linear, right exact functor such that $F(M) \in \text{mod}(R)$ for every $M \in \text{mod}(R)$. Define the restricted functor $F|_\text{mod}: \text{mod}(R) \to \text{mod}(R)$. If $F$ has left derived functors $L_i F: \text{Mod}(R) \to \text{Mod}(R)$ and $L_i (F|_\text{mod}): \text{mod}(R) \to \text{mod}(R)$, are they isomorphic for all $i \ge 0$? Provide a justification for your answer.

Answer: Yes


Question: Given the statistic \( T_c = \sum_{j=1}^n \frac{{(X_j - \bar X)}^2}{c} \), where \( c \) is a constant, which serves as an estimator of variance \( \sigma^2 \) for a random sample \( X_1, \ldots, X_n \) from a normally distributed population with an unknown mean \( \mu \) and variance \( \sigma^2 \). If \( T_c \) follows a chi-squared distribution with \( n-1 \) degrees of freedom, find the expected value \( \mathbb{E}[T_c] \).

Answer: n - 1


Question:
Find a value $p > 1$ such that the series $\sum_n f_n$ converges in $L^p(\mathbb{R})$, where the functions are defined as:
$$ f_n(x) = \frac{1}{1 + n^2\sqrt{x}}\chi_{[\exp(2n), 2\exp(2n+2)]} $$

Answer: 2


Question: Let $\mathcal{H}$ be a separable Hilbert space and $\mathfrak{B}(\mathcal{H})$ be the Banach algebra of bounded operators on $\mathcal{H}$. If $e$ is a unit vector in $\mathcal{H}$, determine whether it is possible to find unitary operators $U_n$ in $\mathfrak{B}(\mathcal{H})$ such that $U_n e$ form a maximal orthonormal set in $\mathcal{H}$. Provide a justification for your answer.

Answer: Yes


Question: Can a graph with 20 vertices and 25 edges be embedded in a surface with genus 1?

Answer: Yes


Question: Determine whether every completely metrizable normed vector space is a Banach space. Provide a justification for your answer.

Answer: Yes


Question: Find the solution set of the equation \( \sin x \cdot \sin 2x \cdot \ldots \cdot \sin 100x = 1 \).

Answer: \emptyset


Question: Evaluate the integral \( \int_{|z-2| = 3} e^{1/z} \, dz \).

Answer: 2\pi i


Question: Does the series \( \sum_{n\ge 1} \frac{\sin nx}{n} \) converge to a function \( f \) at points in \((-\pi,\pi)\) other than \(0\) according to the Dirichlet conditions?

Answer: Yes


Question: Given a sequence $a_n$ defined by $a_n = (a_{n-1} + a_{n-2} + a_{n-3} + a_{n-4}) \mod 10$, determine if the sequence will ever contain the subsequence $7, 3, 6, 7$. Provide a mathematical justification for your answer.

Answer: Yes


Question: Evaluate the second derivative \( f''\left(\frac{8}{\pi x}\right) \) at \( x = 2 \) for the function \( f(x) \) defined by the integral \( \int \sec^{-1}[-\sin^2x]dx = f(x) + c \), where \([y]\) denotes the greatest integer function.

Answer: 0


Question: Given a sequence of independent random variables \(\{X_i, i \geq 1\}\), where \(\mathbb{P}(X_i = i^2 - 1) = i^{-2}\) and \(\mathbb{P}(X_i = -1) = 1 - i^{-2}\), determine the constant \(c\) to which \(\frac{1}{n} \sum_{i=1}^n X_i\) converges almost surely.

Answer: -1


Question: Determine the asymptotic growth rate of the recurrence relation \(T(n) = 2T\left(\frac{n}{3}\right) + T\left(\frac{n}{4}\right) + n^2\) using the Akra-Bazzi theorem.

Answer: \Theta(n^2)


Question: Calculate the sum of the squares of the distances from a point $P$ to the vertices of a regular 11-gon inscribed in a circle with radius 2, given that the distance from $P$ to the center of the circle is 3.

Answer: 143


Question:
Find the derivative of the function \( f(x) \) defined as follows:

\[
f(x) = \begin{cases}
\ln(\sin(x)) - \ln(x), & \text{if } 0 < x < \pi \\
0, & \text{if } x = 0 \\
\ln(-\sin(x)) - \ln(-x), & \text{if } -\pi < x < 0
\end{cases}
\]

Determine if the derivative exists at \( x = 0 \).

Answer: 0


Question: Evaluate the limit \( \lim_{n\to\infty} \int_1^2 \frac{e^{-nx}+n}{x+n}dx \) using the dominated convergence theorem.

Answer: 1


Question:
Given a symmetric adjacency matrix $A \in \{0, 1\}^{n \times n}$, where the degree is defined as $D_{ii} = \sum_j A_{ij} = \sum_j A_{ji}$, determine the derivative of the degree matrix with respect to the adjacency matrix. Represent this derivative as a 4-dimensional tensor. For an example with $n = 3$, which of the following derivative matrices is correct?

1. $\frac{\partial D_{00}}{\partial A} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$
2. $\frac{\partial D_{00}}{\partial A} = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$
3. $\frac{\partial D_{00}}{\partial A} = \begin{pmatrix} 1 & 1/2 & 1/2 \\ 1/2 & 0 & 0 \\ 1/2 & 0 & 0 \end{pmatrix}$
4. $\frac{\partial D_{00}}{\partial A} = \begin{pmatrix} 0 & 1/2 & 1/2 \\ 1/2 & 0 & 0 \\ 1/2 & 0 & 0 \end{pmatrix}$

The derivative should be symmetric, and the diagonal elements should be 1 or 0.

Answer: 3


Question: Determine the dimension of the range space of the linear operator $T: C[0,1] \rightarrow C[0,1]$, defined by $\displaystyle T(f(x))= \int_{0}^{1} \sin(x+y)f(y)dy$.

Answer: 2


Question: Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \[ xf(x) - yf(y) = (x-y)f(x+y) \; \text{for all} \; x, y \in \mathbb{R}. \]

Answer: f(x) = ax + b


Question: Determine the number of integer solutions \((x, y)\) to the equation \((x-2)(x-10) = 3^y\).

Answer: 2


Question: Let $A$ be a commutative ring with unit and $M$ a module over $A$. Suppose the rank of $M_{P}$ over $A_{P}$ is constant for every $P \in \mathrm{Spec}(A)$. Does this property hold for $M_{Q}$ over $A_{Q}$ for any prime ideal $Q$ of $A$?

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty}^\infty \frac{x^2}{x^4+4} \, dx \) using complex analysis techniques.

Answer: \dfrac{\pi}{2}


Question: Determine the number of natural numbers \( n \) for which \( n! + 10 \) is a perfect square.

Answer: 1


Question: In the ring $\mathbb{Z}[\sqrt{23}]$, the ideal $\langle 2 \rangle$ factors as $\langle 2, 1 + \sqrt{23} \rangle^2$. Since $\mathbb{Z}[\sqrt{23}]$ is a principal ideal domain, the ideal $\langle 2, 1 + \sqrt{23} \rangle$ is generated by a single element. Given that $(5 - \sqrt{23})(5 + \sqrt{23}) = 2$, determine whether $5 - \sqrt{23}$ is contained in the ideal $\langle 5 + \sqrt{23} \rangle$.

Answer: Yes


Question: Find a tight asymptotic upper bound for the recurrence relation \( p(n) = \sqrt{n}p(\sqrt{n}) + \sqrt{n} \).

Answer: O(n)


Question: Find the radius of convergence for the series \(\sum \limits _{n=2} ^\infty \frac{(\ln x)^n} n\).

Answer: 1


Question: Given the matrix \( A = \begin{bmatrix} 1 &0 &0 \\ i& \frac{-1+ i\sqrt 3}{2} &0\\0&1+2i &\frac{-1- i\sqrt 3}{2} \end{bmatrix} \), find the trace of \( A^{102} \).

Answer: 3


Question: Calculate the sum of the infinite series \( M = \cos(2x) + \cos(4x) + \cos(6x) + \cos(8x) + \ldots \). Express your answer in terms of complex exponentials.

Answer: -\dfrac{1}{2}


Question: For a compact Riemann surface $X$ of genus $g$, determine the degree of the anti-canonical line bundle $K_X^{-1}$.

Answer: 2 - 2g


Question: Determine the partial limit of the function \( f(x, y) = e^{\frac{2x}{\sqrt{x^2 + y}}} \) at the point \((0, 0)\) along the set \( A = \{(x, x) : x > 0\} \).

Answer: 1


Question: Find the value of $y$ such that the equation $2\arccos\left(\frac{4}{5}\right) - \arcsin\left(\frac{4}{5}\right) = \arctan(y)$ holds true.

Answer: \dfrac{44}{117}


Question: Consider an open connected set $S \subseteq \mathbb{C}$. Determine whether it is possible to connect any two points $a, b \in S$ using a finite sequence of line segments that are either horizontal, vertical, or diagonal (with slope $\pm 1$).

Answer: Yes


Question: Given the functions \(f(t)=(\int_0^{\sqrt{t}}e^{-x^2} dx)^2\) and \(g(t)=\int_0^1\frac{e^{-t(1+x^2)}}{1+x^2} dx\) for \(t>0\), find \(f'(t)+g'(t)\).

Answer: 0


Question: Consider two continuous functions $f$ and $g$ defined on the interval $[0,1]$, with values in $[0,1]$, such that $g(0) < f(0)$ and $g(1) > f(1)$. Determine if there exists a smallest $t \in (0,1)$ for which $g(t) \geq f(t)$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the improper integral \( \int_{-2}^2 \frac{\sqrt{2-x}-3\sqrt{2+x}}{\sqrt{4-x^2}}\,dx \).

Answer: -8


Question: Find the determinant of the matrix representing the linear transformation $x \mapsto \beta x$ in the vector space $\mathbb{Q}(\sqrt{d})$ with basis $\{1,\sqrt{d}\}$, where $\beta$ is an element of $\mathbb{Q}(\sqrt{d})$. Express your answer as a single number.

Answer: a^2 - b^2 d


Question: Determine for which values of $a \in \mathbb{R}$ the series \( \sum_{n=1}^{\infty}\left(e^{\frac{1}{n^a}} - 1 + \log{n} - \log{(n+1)}\right) \) converges.

Answer: 1


Question: Given a non-recursive set $A$, consider the sets $B = \{x \in A \mid x \text{ is odd}\}$ and $C = \{x \in A \mid x \text{ is even}\}$. Is it necessarily true that at least one of $B$ and $C$ is non-recursive? Answer yes or no.

Answer: Yes


Question: Given the equations $2005 + x = y^2$ and $2005 + y = x^2$, find the value of $xy$. Provide your answer as a single number.

Answer: -2004


Question: If $g(x)$ and its derivative $g'(x)$ are both integrable, and $g(x)$ converges as $x \to \infty$, does this imply that $g(x) \to 0$ as $x \to \infty$? Answer yes or no and justify your answer.

Answer: Yes


Question: Is the function \( g(z) + g(z^3) + \cdots + g(z^{2n+1}) \) harmonic in the domain \( |z| < 1 \), given that \( g(z) \) is harmonic in \( |z| < 1 \) and \( g(0) = 0 \)?

Answer: Yes


Question: Evaluate the limit: $$ \lim_{n \to \infty}\sum_{k=3}^{n} \frac{1}{\sqrt[k]{n^k+2n+3}-1} $$

Answer: 1


Question: Find the minimum number of squares needed to completely fill a rectangle with a 3:4 ratio, without using the greedy algorithm.

Answer: 4


Question: Determine whether $y + x = 3$ is a solution to the differential equation \( \frac{dy}{dx} = \frac{y+x-2}{y+x-4} \).

Answer: Yes


Question:
Determine the correct Jordan normal form of the matrix \(\begin{bmatrix}1&1&0\\-1&3&0\\-1&1&2\end{bmatrix}\) given that its characteristic polynomial is \((X-2)^3\) and its minimal polynomial is \((X-2)^2\). Choose from the following options:

1) \(\begin{bmatrix}2&1&0\\0&2&0\\0&0&2\end{bmatrix}\)
2) \(\begin{bmatrix}2&0&0\\0&2&1\\0&0&2\end{bmatrix}\)

Answer: 1


Question:
Given positive numbers $x, y, z$ such that $x + y + z = 1$, determine the minimum value of the expression:
\[ P = \left(\frac{x+1}{x}\right)^3 \cdot \frac{z^2}{z^2+1} + \left(\frac{y+1}{y}\right)^3 \cdot \frac{x^2}{x^2+1} + \left(\frac{z+1}{z}\right)^3 \cdot \frac{y^2}{y^2+1} \]

Answer: \dfrac{96}{5}


Question: Determine the number of integer pairs \((p, q)\) such that \(3^p - 2^q = 19\) and both \(p\) and \(q\) are positive integers.

Answer: 1


Question: Evaluate the limit: $$\lim\limits_{x \to +\infty}\dfrac{\ln x}{\displaystyle \int_0^x \dfrac{|\sin t|}{t}{\rm d}t}.$$

Answer: \dfrac{\pi}{2}


Question: Determine whether the Baire Category Theorem applies to the set of continuous functions \( C: \mathbb{R} \to \mathbb{R} \).

Answer: Yes


Question:
Determine the behavior of the partial Bell polynomial \(B_{n,0}(x_1, x_2, \ldots, x_n)\) given the conditions:
\[ j_1 + j_2 + \cdots = 0 \]
\[ j_1 + 2j_2 + 3j_3 + \cdots = n \]
where the sum is taken over all sequences \(j_1, j_2, \ldots, j_{n+1}\) of non-negative integers. How can the second condition be satisfied when \(k=0\)?

Answer: 0


Question: Evaluate the limit: $$\lim_{x \to 1} (x - 1)^n \sin \left(\frac{1}{x - 1}\right)$$ without using L'Hopital's rule.

Answer: 0


Question: Let $(X, d)$ be a compact metric space and $f:X \rightarrow X$ be a function such that $d(f(x), f(y)) < d(x, y)$ for all distinct $x, y \in X$. Determine whether $f$ always has a unique fixed point.

Answer: Yes


Question: Determine the number of elements in the ring \( \mathbb{Z}_2[X]/(x^5 + x + 1) \), given that the polynomial \( x^5 + x + 1 \) can be factored as \((x^2 + x + 1)(x^3 + x^2 + 1)\) in \( \mathbb{Z}_2[X] \). Note that both factors are irreducible over \( \mathbb{Z}_2 \).

Answer: 32


Question: Evaluate the limit: $$\lim _ { n \rightarrow \infty } \sum _ { k = 0} ^ { n } \arctan \left( \frac { k + 1} { n ^ { 2} } \right)$$

Answer: \dfrac{1}{2}


Question: Calculate the limit: \[ \lim_{n \rightarrow \infty} \left( \frac{n \pi}{4} - \left( \frac{n^2}{n^2+1^2} + \frac{n^2}{n^2+2^2} + \cdots + \frac{n^2}{n^2+n^2} \right) \right). \]

Answer: \dfrac{1}{4}


Question: Let $\beta_{1} = \{v_1, v_2, \ldots, v_n\}$ and $\beta_{2} = \{u_1, u_2, \ldots, u_n\}$ be two bases for a vector space $V$. If the coordinate vectors of every vector $x \in V$ are the same with respect to both bases, does it imply that $v_i = u_i$ for all $i = 1, 2, \ldots, n$? Answer yes or no.

Answer: Yes


Question: Let $A$ be an $n \times n$ matrix over $\mathbb{C}$ such that every non-zero vector of $\mathbb{C}^n$ is an eigenvector of $A$. Determine the eigenvalues of $A$.

Answer: \lambda


Question: Evaluate the integral \( \int _{0}^{2\pi} e^{it} e^{( e^{-it})} \, dt \) using the method of Taylor series expansion.

Answer: 2\pi


Question: Evaluate the limit \( \lim_{n \to \infty} I_n \) where \( I_n = \int_{\mathbb{R}} \frac{1}{x^2+n^2} \cos(\sqrt{x^2+n^2})~d\lambda. \)

Answer: 0


Question: Determine the order of the pole at \(z=\infty\) in the Laurent expansion of \(\frac{z^2}{(z^2-1)^{\frac{1}{2}}}\) in terms of \(z\).

Answer: 1


Question: Determine whether the statement is true or false: A ring $R$ is local if its Jacobson radical is a maximal ideal.

Answer: A


Question: Let \(A\) be a \(3 \times 3\) matrix with complex entries whose eigenvalues are \(1, \pm 2i\). Suppose that for some \(a, b, c \in \mathbb{C}\), the equation \(aA^{-1} = A^{2} + bA + cI\) holds, where \(I\) is the \(3 \times 3\) identity matrix. Determine the values of \((a, b, c)\) from the following options: \((A)\ (-1, -4, 4), (B)\ (4, -1, 4), (C)\ (-1, 4, -2), (D)\ (-1, -2, 4)\).

Answer: B


Question: Determine the value of $m$ that maximizes the sum \( \sum_{i = 0}^{m} {10 \choose i}{20 \choose m - i} \), where \({p\choose q} = 0\) if \(p < q\).

Answer: 15


Question: An integral curve of the differential equation \((y-x)\frac{dy}{dx} = 1\) passes through the points \((0,0)\) and \((a,1)\). Determine the value of \(a\).

Answer: \dfrac{1}{e}


Question: Determine the radius of convergence for the series \(\sum\limits_{n=1}^\infty (-1)^{n+1} \left(\left|\frac{1-x}{1+x}\right|-1\right)^n \frac{1}{n}\), which represents \(\ln\left|\frac{1-x}{1+x}\right|\). Given that the ratio test yields the inequality \((x<-3) \cup (-\frac{1}{3}<x)\), find the radius of convergence.

Answer: \dfrac{1}{3}


Question: Determine the modulo classes for which the product of two twin primes plus 2 is always congruent to 1.

Answer: 4


Question: For a function $g(z)$ with a removable singularity at $z = z_1$, determine if the integral of $g(z)$ around any closed contour enclosing $z_1$ is always zero.

Answer: Yes


Question: Evaluate the expression: $$[(\nabla \times \nabla)\cdot \nabla](x^2 + y^2 + z^2)$$

Answer: 0


Question: Find a primitive root of 17 using its quadratic residues.

Answer: 3


Question: Determine the values of $x > 0$ for which the series \( \sum_{n=1}^{\infty} x^{1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+...+\frac{1}{\sqrt n}} \) converges.

Answer: (0, 1)


Question: Evaluate the limit \( \lim_{x\to1}(3 \sqrt[3]{x}-2\sqrt{x})^{\frac{1}{\ln x}} \).

Answer: 1


Question: Consider a circular table with \( n \) chairs, where \( n \geq 3 \). Each person seated has a unique height, and they sit randomly. Calculate the expected number of people who are shorter than both of their neighbors.

Answer: \dfrac{n}{3}


Question: Determine the highest power of two that divides the expression \(32! + 33! + 34! + \dots + 90!\).

Answer: 32


Question: In a game with two players, each player selects a number between 0 and b (inclusive). The payoff for each player is determined by the function \( Z = u(x_1 + x_2) + v(2b - x_1 - x_2) \), where \( u \) and \( v \) are strictly concave functions. Derive the slope of the best response function for player 2.

Answer: -1


Question: Let $f:\mathbb{R}\to \mathbb{C}$ be a continuous function. If $\int_{-\infty}^{\infty}|f(x)|\,dx$ exists, does $\int_{-\infty}^{\infty}f(x)\,dx$ necessarily exist?

Answer: Yes


Question: Evaluate the limit of the expression \( \lim_{n\to\infty}\frac{[x] + [2^2x] + \dots + [n^2x]}{n^3} \) as \( n \) approaches infinity, where \([y]\) denotes the greatest integer less than or equal to \(y\). Consider both cases when \(x\) is an integer and when \(x\) is not an integer.

Answer: \dfrac{x}{3}


Question: Let $X$ and $Y$ be two random variables such that for all $a, b \in \mathbb{R}$, \( E[(X-a)(Y-b)] = E(X-a) E(Y-b) \). Does this imply that $X$ and $Y$ are independent random variables?

Answer: No


Question: Determine if the equation \( 3x^3 + 5y^5 + z^3 + 4z = 13 \) defines a function \( z(x, y) \) at the point \( A(1, 1, 1) \).

Answer: Yes


Question: Find the general solution to the partial differential equation $u_x^2 + u_y^2 = 0$.

Answer: u(x, y) = C


Question: Determine the points at which the partial derivative $f_x$ is continuous for the function \( f(x,y) = \frac{x^2 - y^2}{(x^2 + y^2)^{1/3}} \) when \((x,y) \neq (0,0)\), and \( f(x,y) = 0 \) when \((x,y) = (0,0)\).

Answer: \mathbb{R}^2


Question: In a Hilbert space $X$, consider the elements $x_1, x_2, x_3 \in X$. Suppose that for all $y \in X$, the inequality \(\langle x_1, y \rangle_X \leq \langle x_2, y \rangle_X + \langle x_3, y \rangle_X\) holds. Does this imply the norm inequality \(\|x_1\|_X \leq \|x_2\|_X + \|x_3\|_X\)? Answer with 'yes' or 'no'.

Answer: yes


Question: Is it possible to find 2041 distinct natural numbers such that the sum of their squares is a perfect square?

Answer: Yes


Question: Determine a tight upper bound for the expression \(-p \log p\) in terms of the probability \(p\), where \(0 < p < 1\), especially when \(p\) is very close to 0, such as \(p < 1/e\).

Answer: \dfrac{1}{e}


Question: In triangle ABC, where \( \angle A = 120^\circ \), the lines AD, BE, and CF are the angle bisectors of \( \angle A \), \( \angle B \), and \( \angle C \), respectively. Determine the measure of \( \angle FDE \).

Answer: 90^\circ


Question: In the localization $U^{-1}R$ of a ring $R$ with respect to a multiplicatively closed subset $U$, can every element $\frac{r}{s}$ be expressed with $s \in U$?

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{1}{x} \int_{x}^{2x} e^{-t^{2}}\,dt$$.

Answer: 1


Question: Can any of the 52! possible orderings of a 52-card deck be achieved through a series of perfect in and out shuffles?

Answer: No


Question: Let \( w \) be a primitive 8th root of unity. Determine the irreducible polynomial of \( w + w^3 \).

Answer: x^2 + 2


Question: Evaluate the limit \(\lim_{a\rightarrow \infty} \int_{-a}^a \cos(x+b)\,dx\) and determine the values of \(b\) for which it exists.

Answer: 0


Question: Consider a sequence of random variables $Y_n$ where $E(|Y_n|) \rightarrow 2$ and $E(Y_n^2) \rightarrow 5$. Determine whether $Y_n$ can converge in square mean.

Answer: Yes


Question: Is the definition of a map $f : M \to N$ being of class $C^r$ at a point $p \in M$ independent of the choice of charts $(U, x)$ from $A_M$ and $(V, y)$ from $A_N$?

Answer: Yes


Question: Assume that $F$ is a field, and $f \in F[x]$ is a polynomial. If $f$ is surjective for every sufficiently large $n$ when regarded as a map from $M_n(F)$ to $M_n(F)$, must $f$ be linear? Answer yes or no.

Answer: Yes


Question: Determine the number of roots of the polynomial \( P(z) = 2z^4 + z^3 - 5z^2 + z + 2 \) that lie in the region where \( \operatorname{Re}(z) < 0 \).

Answer: 2


Question: Evaluate the limit of the integral as $n$ approaches infinity: $$I = \lim_{n \to \infty} \int_0^1 \frac{1 + nx^2}{(1 + x^2)^n} \log(2 + \cos\left(\frac{x}{n}\right))\,dx.$$

Answer: 0


Question: Let $G$ be a finite group, and let $\chi$ be a faithful irreducible character of $G$ of degree $d$. Suppose there exists a $d \times d$ unitary matrix $V$ of order $n$ such that $G$ contains a conjugacy class $[g]$ with elements of order $n$ and $\chi([g]) = \text{Trace}(V)$. Determine whether there always exists a faithful irreducible representation $\pi: G \to \text{U}(d)$ with character $\chi$ for which $V$ is in the image of $\pi$. Provide a justification for your answer.

Answer: No


Question: Determine the values of $i$ for which the isomorphism $H_i(K^{(p)}) \simeq H_i(K)$ holds, where $K$ is a topological space.

Answer: i < p


Question: Find the smallest positive integer \( n \) such that the number \( a_n = 10^{2n} + 10^n + 1 \) has repeated prime factors.

Answer: 14


Question: Determine the appropriate limits for the parameter $t$ in the parametrization $(a\cos t, a\sin t, b-b\cos t)$ to describe the curve of intersection between the cylinder $x^2+y^2=a^2$ and the plane $\frac{x}{a}+\frac{z}{b}=1$.

Answer: [0, 2\pi]


Question:
Determine whether the following claim is true or false: For any odd integer $n \geq 3$ and integer $a$ coprime to $n$, $n$ is prime if and only if $P_n^{(a)}(x) \equiv x^n \pmod{n}$ holds, where $P_n^{(a)}(x)$ is defined as:

$$P_n^{(a)}(x)=\left(\frac{1}{2}\right)\cdot\left(\left(x-\sqrt{x^2+a}\right)^n+\left(x+\sqrt{x^2+a}\right)^n\right)$$

Answer: True


Question: Consider the function \( f(x) = ax + \frac{1}{x+1} \), where \( a \) is a positive constant. Let \( M \) be the maximum value of \( f'(x) \) for \( x \in [0, 1] \). Determine the minimum value of \( M \) for any \( a > 0 \).

Answer: -\dfrac{1}{4}


Question: Find the coefficient of \(x^{17}\) in the expansion of \((3x^7 + 2x^5 - 1)^{20}\).

Answer: -41040


Question: Determine whether the sequence \((x_n)\), defined by \(x_1=\sqrt{2}\), \(x_2=\sqrt{2}^{\sqrt{2}}\), \(x_3=(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}}\), and \(x_{n+1} = (\sqrt{2})^{x_n}\), converges.

Answer: 2


Question: Let $G$ be a group and $H$ be a normal subgroup of $G$ generated by an element $a$. If $a$ has order $n$, determine the possible orders of $bab^{-1}$ for $b \in G$.

Answer: n


Question: Let $f(x)$ be a monic cubic polynomial with $f(0) = -2$ and $f(1) = -5$. If the sum of all solutions to $f(x+1) = 0$ and to $f(\frac{1}{x}) = 0$ are the same, find $f(2)$. \( \boxed{} \)

Answer: -6


Question: Consider the ring \( S=\begin{Bmatrix}\begin{pmatrix}a&b\\0&a\end{pmatrix}: a,b\in \mathbb{Z}\end{Bmatrix} \). Determine the number of endomorphisms of \( S \).

Answer: \infty


Question: Evaluate the norm of the linear operator $T: C[a,b] \to C[a,b]$ defined by $(Tf)(x) = \int_a^x f(t) \, dt$, where $C[a,b]$ is the set of continuous functions on $[a,b]$ with the supremum norm.

Answer: b - a


Question: Determine the precise decay rate of the sequence \(a_n = \min_{1 \le N \le n} |\langle N \alpha \rangle|\) for an irrational number \(\alpha\).

Answer: \dfrac{1}{n}


Question: Determine whether the following statement is true or false: Any section of a vector bundle over a scheme induces a closed embedding if the projection of the vector bundle is a separated morphism.

Answer: True


Question:
Given the equation \(\frac {5030}{5555}=\frac{a+b}{10}+\frac{b}{10^3}+\frac{a}{10^4}+\frac{a+b}{10^5}+\frac{b}{10^7}+\frac{a}{10^8}+\frac{a+b}{10^9}+...\), where the right-hand side terms have been classified and partial sums \(S_1\), \(S_2\), and \(S_3\) have been calculated as follows:

\[S_1=\frac{10^3}{10^4-1}\]
\[S_2=\frac 1{1000}S_1\]
\[S_3=\frac 1{100}S_1\]

With the relationship \((a+b)S_1+aS_2+bS_3=\frac {5030}{5555}\), find the product \(ab\) assuming \(a\) and \(b\) are positive integers.

Answer: 20


Question: In the diagram, a circle \(\omega\) with center \(P\) is tangent to line segment \(AB\) at point \(T\), and \(\angle APB = 90^\circ\). Point \(K\) is on circle \(\omega\), and a semicircle with diameter \(BK\) intersects \(PB\) at point \(Q\). Given that the radius \(R\) of the semicircle satisfies \(4R^2 - AT \cdot TB = 10\), and \(PQ = \sqrt{2}\), find the length of \(BQ\).

Answer: 2\sqrt{3}


Question: Calculate the expected value of the random variable $X$ given the joint density function: $$f(x,y) = \frac{1}{x}e^{-\left(\frac{y}{x}+x\right)} \text{ for } x,y > 0.$$

Answer: 1


Question: Determine whether an uncountable ring can have a countable proper subring. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{x^{(\sin x)^{x}}-(\sin x)^{x^{\sin x}}}{x^3} \] without using Taylor's formula.

Answer: \dfrac{1}{6}


Question: Find an epimorphism from the free group $F_2$ to the symmetric group $S_3$. Determine the kernel of this homomorphism and the rank of the kernel.

Answer: 7


Question: Can the sum of two subsets of real numbers, each without a least element, have a least element?

Answer: No


Question: Determine whether there exists a non-constant function \( f: \mathbb{R} \to \mathbb{R} \) that has arbitrarily small periods. Specifically, is there a function such that for any given \( \epsilon > 0 \), there exists a \( 0 < a < \epsilon \) for which \( f(x) = f(x + a) \) for some \( x \in \mathbb{R} \)?

Answer: Yes


Question:
Determine the sup-norm, \(\|f\|_{\sup}\), for the function \(f(x)\) defined as:

\[f(x)=\begin{cases}
0, & x \in \mathbb{Q} \\
-x^2, & x \not\in \mathbb{Q}
\end{cases}\]

Answer: \infty


Question:
Determine the values of $a$ for which the following limit evaluates to $-\infty$:
\[ \lim_{n\to\infty} \sqrt{n^2-n+1}-an-b \]

Answer: a > 1


Question: Find the dimension of the set \( S \) of all \( n \times n \) matrices over a field of prime elements that commute with every matrix in \( M_{n \times n} \).

Answer: 1


Question: Let $A$ be a square matrix. If $1$ is an eigenvalue of $A^n$, is it true that $A$ has an eigenvalue which is a root of unity?

Answer: Yes


Question: Calculate the residue of the function \(\frac{1}{\cos(z)+1}\) at the pole \(z=\pi\).

Answer: 0


Question: Determine all real numbers \(a\) for which there exists a function \(f: \mathbb{R} \to \mathbb{R}\) such that \(f(x + 2) = -f(x)\) and \(f(x + 3) = f(x) + a\) for every real number \(x\).

Answer: 0


Question: Determine the values of $n$ such that $$(1+2i)^{4n}=(1-2i)^{4n}.$$

Answer: 0


Question: If a closed symmetric operator $T$ has at least one complex number in its resolvent set such that both the operator and its complex conjugate map onto the Hilbert space, does it follow that $T$ is self-adjoint? Answer yes or no.

Answer: Yes


Question: Given that the number of Master's degrees awarded by faculty F of university U follows a Poisson distribution, and the funding for faculty F in a given year is proportional to the sum of the number of degrees awarded in the three preceding years, find the variance of the funding for year \((2000 + i)\), where \(i \ge 3\).

Answer: 3\lambda


Question: Evaluate the limit: $$\lim_{n \to \infty} \frac{\frac{\pi^2}{6} - \sum\limits_{k=1}^n \frac{1}{k^2} - \frac{1}{n}}{\frac{1}{n^2}}$$

Answer: -\dfrac{1}{2}


Question: Determine whether the following statement is true or false: For any linear operator $T$ on a finite-dimensional complex vector space $V$, if the dimension of $V$ is $n$, then $T^n$ must have $n$ distinct eigenvalues.

Answer: False


Question: Calculate the genus of the projective line \(\mathbb{P}^1\) using the Čech complex.

Answer: 0


Question: Let $X$ be a set. Suppose there exists a bijection $f:X \rightarrow \{ \emptyset \} \cup X \cup X^2$. Determine whether there exists a bijection $g: X \cup X^3 \rightarrow X^2 \cup X^4$. Provide a justification for your answer.

Answer: Yes


Question: Suppose a Riemannian surface $M$ is isometrically embedded in $\mathbb{R}^n$ such that any metric circle on $M$ is also a circle in $\mathbb{R}^n$. Does this imply that $M$ must have constant nonnegative Gaussian curvature? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x\to\frac{\pi}{2}}\left(\frac{1}{\tan x}\right)^{\frac{2\tan x}{\tan^2 x-1}}$$ without using L'Hopital's rule.

Answer: 1


Question: Does the von Neumann algebra $\mathcal{M} = M_2(vN(\mathbb{F}_2))$, associated with the free group on two generators, admit a separating vector in its standard representation?

Answer: Yes


Question: Calculate the total number of geometric progressions that can be formed from the set \(\{2^1, 2^2, \ldots, 2^{2000}\}\) such that the three elements have different powers of 2 as their difference factors.

Answer: 999000


Question: Find the limit of \( \frac{n}{\sqrt[n]{n!}} \) as \( n \to \infty \).

Answer: e


Question: Determine whether the set $N = \{(x,y,z) \in \mathbb{R}^3 \mid x, y, z \text{ are distinct}\}$ is homeomorphic to a finite union of products of one-dimensional manifolds.

Answer: Yes


Question: Evaluate the integral \( \int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx \) using Euler's formula or other techniques to simplify the process.

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


Question: Does a polynomial with $d$ roots over the arithmetic modulo a prime $p$ necessarily have a degree of at most $d$?

Answer: No


Question: In a finite semigroup, if for every elements $a$ and $b$, both equations $ax = b$ and $ya = b$ have at least one solution, does this imply that the semigroup is a group? Provide a justification for your answer.

Answer: Yes


Question: In the ring \( \mathbb{Z}_n \), is it true that every ideal can be expressed as a principal ideal \( \langle m \rangle \) for some \( m \in \mathbb{Z}_n \)?

Answer: Yes


Question: Evaluate the double integral \( \iint_D \frac{1}{\sqrt{u^2+v^2}} \, du \, dv \) where \( D \) is the region in the first quadrant bounded by the lines \( u=0 \), \( v=0 \), and the circle \( u^2+v^2=4 \).

Answer: \pi


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

Answer: \infty


Question: Given a Poisson distribution with parameter $\lambda$, where the probability density function (PDF) of $\lambda$ is $f(\lambda) = e^{-\lambda}$, and the Poisson distribution's PDF is $p(x, \lambda) = e^{-\lambda} \frac{\lambda^x}{x!}$, find the expectation of the random variable $X$ given that $\lambda$ is also a random variable.

Answer: 1


Question: In ZF theory, determine whether there exists a well-ordered set \( A \) such that the cardinals of all Dedekind-finite sets below the cardinal of \( A \) form a linear order.

Answer: Yes


Question: Compute the surface integral \( \iint_S F \cdot dS \) where \( F = \langle y, z, -x \rangle \) and \( S \) is the portion of the paraboloid \( z = x^2 + y^2 \) that lies below the plane \( z = 4 \).

Answer: 0


Question: Find the derivative of the function \( f(x) = \arctan\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right) \) with respect to \( \arccos(x^2) \).

Answer: -\dfrac{1}{2}


Question: Is the axiom of replacement sufficient to ensure that the domain and range of a relation, defined as a collection of ordered pairs, are sets?

Answer: Yes


Question: Calculate the fundamental group of the space \( \mathbb{R}^5 \setminus \mathbb{R}^3 \).

Answer: \mathbb{Z}


Question: Consider a compact, connected Riemannian manifold $(M^n,g)$ with a concave boundary. Determine whether it is possible for the infimum of chord lengths in $M$, denoted as $\ell(M,g)$, to be zero. Justify your answer.

Answer: Yes


Question: Suppose $Q \in M_{3 \times 3}(\mathbb{R})$ is a matrix of rank $2$. Consider the linear transformation $T: M_{3 \times 3}(\mathbb{R}) \to M_{3 \times 3}(\mathbb{R})$ defined by $T(P) = PQ$. Determine the rank of the transformation $T$. \( \boxed{2} \)

Answer: 6


Question: For a linear transformation $T$ on an $n$-dimensional vector space $V$, if there exists a vector $\mathbf{v}$ such that the set $\{\mathbf{v}, T(\mathbf{v}), T^2(\mathbf{v}), \ldots, T^{n-1}(\mathbf{v})\}$ is linearly independent, does it imply that each eigenvalue of $T$ has a unique eigenvector (up to scalar multiplication)? Answer yes or no and justify your answer.

Answer: Yes


Question: Let $X$ be an irreducible topological space of dimension 1, where every nonempty open subset is dense and points are closed. Suppose $F$ is a sheaf of abelian groups on $X$, and $s \in F(X)$ is a global section such that $s|_{X-\{p\}} = 0$ for some point $p \in X$. Does it follow that $s = 0$? Specifically, consider the case where $X$ is a smooth curve and $F$ is the sheaf of holomorphic 1-forms, $\Omega^1_X$. Answer with 'Yes' or 'No' and justify your answer.

Answer: Yes


Question: Evaluate the limit: \[ I = \lim_{x \to \infty} \left(\sin{\frac{2}{x}} + \cos{\frac{1}{x}}\right)^x \]

Answer: e^2


Question: Evaluate the limit \[\lim_{n \to \infty} \frac{1^{b-1} + 2^{b-1} + \cdots + n^{b-1}}{n^{b}(1^{b} + 2^{b} + \cdots + n^{b})}.\]

Answer: 0


Question:
Consider the circle given by the equation \(x^2 + y^2 + 2gx + 2fy + c = 0\). This circle intersects three other circles, whose equations are \(x^2 + y^2 = 5\), \(x^2 + y^2 - 8x - 6y + 10 = 0\), and \(x^2 + y^2 - 4x + 2y - 2 = 0\), at the endpoints of their diameters. Determine the correct relationship among \(c\), \(f\), and \(g\):

A) \(c = – 5\)
B) \(fg = \frac{147}{25}\)
C) \(g + 2f = c + 2\)
D) \(4f = 3g\)

Choose the correct option.

Answer: D


Question: Determine the positive values of \( a \) for which the equation \( x^a = a^x \) has exactly three distinct real solutions.

Answer: 2


Question:
Consider the functions \(u(z)\) and \(v(z)\) defined as follows for complex numbers \(z\):

\[u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z} \quad \text{for} \quad -\pi \leqslant \arg \left(z+\tfrac{1}{2}\right) < \pi\]
\[v(z)=\frac{\log z}{z} \quad \text{for} \quad -\pi \leqslant \arg z < \pi\]

The function \(f(z) = u(z) - v(z)\) is holomorphic in the annulus \(1<|z|<R\) for any \(R>1\). Explain how the arguments of \(u(z)\) and \(v(z)\) effectively "cancel out" when crossing the interval \((-\infty, -1)\), which lies within the intersection of the branch cuts \((-\infty, 0]\) and \((-\infty, -\frac{1}{2}]\). Additionally, find the first few terms of the Laurent expansion of \(f(z)\) about \(0\) in the same annulus, and determine the integral of \(f\) around a circle of radius \(1<r<R\) centered at \(0\).

Answer: 0


Question: Evaluate the series \( \displaystyle \sum^{n}_{k=0}(-4)^k\binom{n+k}{2k} \) using combinatorial methods or generating functions.

Answer: (-1)^n (2n + 1)


Question: Evaluate the limit: $$\lim \limits_{n\to \infty }\sin^2 \left(\pi \sqrt{(n!)^2-(n!)}\right).$$

Answer: 1


Question: Determine the set of values for $p$ such that the variance of a random variable $X$ with probability density function $f(x) = \frac{k(p)}{x^p}$ for $x > 1$ is finite, but the fourth moment does not exist.

Answer: (3, 5]


Question: Evaluate the contour integral $\int_{\gamma}\frac{dz}{z}$, where $\gamma$ is a closed curve in a simply connected region of the complex plane that does not include the origin.

Answer: 0


Question: Evaluate the integral: \[ \int_{0}^{2\pi} \ln|10\cos x+6\sin x|\;\mathrm{d}x \]

Answer: \pi \ln 34


Question: Find the antiderivative of the function \( y = x^x(1+\ln(x)) \).

Answer: x^x + C


Question: Determine the lower bound for the expression \( \left(\left\lfloor \frac{x}{3} \right\rfloor - \left| x - 3\left\lfloor \frac{x}{3} \right\rfloor \right| \right)^2 \) for \( x \geq 0 \), where \( \left\lfloor x \right\rfloor \) denotes the floor function of \( x \).

Answer: 0


Question: Determine the number of solutions to the equation \(3\log_{3}(3^x+2x)=3^x+2\).

Answer: 0


Question: Determine whether all the roots of the polynomial equation \( z^5 + 5z^2 + 2z - 3 = 0 \) lie within the circle centered at the origin with radius 3.

Answer: Yes


Question: Is it possible to construct two functions that have the same limit at every point but differ at an uncountable number of points?

Answer: Yes


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

Answer: 0


Question: Consider a sequence of non-empty, monotonic decreasing, open, and bounded subsets $A_1, A_2, \ldots$ of $\mathbb{R}^n$ such that $A_1 \supset A_2 \supset \cdots$ and their intersection $\bigcap_{i=1}^{\infty} A_i$ is non-empty and compactly embedded in an open subset $B$ of $\mathbb{R}^n$ (i.e., $\bigcap_{i=1}^{\infty} A_i \subset \subset B$). Determine whether there exists an integer $m$ such that $A_m \subset B$. Provide a proof or counterexample.

Answer: Yes


Question: Find the limiting value of the Beta function \( B(a,b) = \int_0^1 x^{a -1}(1-x)^{b-1}\,d x \) as \(a \to \infty\), \(b \to \infty\), and \(a,b \to \infty\).

Answer: 0


Question: Evaluate the double integral \( \iint_{D} \arcsin(x^2+y^2)\, dx\,dy \) where the region \( D \) is defined by the polar equation \( \rho=\sqrt{\sin \theta} \) for \( 0\le\theta\le\pi \).

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


Question: Consider a sequence of tosses of a balanced die. Let $X_k$ be the result of the $k$-th toss and $Y_n=\max_{k\le n} X_k$. Determine the almost sure convergence of the sequence $(Y_n)_{n\ge1}$.

Answer: 6


Question: Let $G = \langle X_1, X_2 \rangle$. Determine if $G = X_1X_2$ when either $X_1$ or $X_2$ is a normal subgroup of $G$. Provide a proof or counterexample.

Answer: G = X_1 X_2


Question: Three persons $A$, $B$, and $C$ flip three fair coins $n$, $n+1$, and $n+2$ times, respectively. Let $X$, $Y$, and $Z$ be the number of heads for $A$, $B$, and $C$, respectively. Find the probability $P(X<Y<Z)$. Express your answer as a single probability value.

Answer: \dfrac{1}{8}


Question: Determine the minimum value of the expression $x^q + (1 - x)^q$ for any $q \ge 1$ and $x \in [0, 1]$. Provide your answer as a single value.

Answer: 2^{1 - q}


Question: If a function $f$ has $n$ distinct roots, does its $(n-1)$th derivative necessarily have at least one root? Answer yes or no.

Answer: Yes


Question: Given a natural number $k$, consider a function $f: \mathbb{R}^d \rightarrow [0, \infty)$ that is lower semicontinuous. Does there exist an increasing sequence of $k$-times continuously differentiable functions $(g_n)_n \subset C^k(\mathbb{R}^d ; [0, \infty))$ that converges pointwise to $f$, satisfying $g_n(x) \leq g_{n+1}(x)$ for all $x \in \mathbb{R}^d$ and $n \in \mathbb{N}$, as well as $\lim_{n \rightarrow \infty} g_n(x) = f(x)$ for all $x \in \mathbb{R}^d$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_{C}\frac{1}{z^3-1}dz \) where \( C \) is the circle centered at \( z=1 \) with radius \( 2 \), using Cauchy's Integral Formula.

Answer: 0


Question: Evaluate the limit: $$\lim _{x\to \infty }\frac{\left(\int _{10}^x\frac{e^{\sqrt{t}}}{t}dt\right)^2}{e^x}$$

Answer: 0


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

Answer: 1


Question: Determine whether there exists a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that is continuous only at the irrational numbers.

Answer: Yes


Question: Consider the cubic equation \(x^3 - 3x + 1 = 0\), which has three real roots \(x_1, x_2, x_3\) such that \(x_1 < x_2 < x_3\). Determine the value of \(\{x_1\} + \{x_2\} + \{x_3\}\), where \(\{x\}\) denotes the fractional part of \(x\).

Answer: 1


Question: Find the maximum possible value of the determinant of a $3 \times 3$ matrix whose elements are the numbers $1, 2, \dots, 9$. Provide a formal proof to establish this as the optimal value.

Answer: 412


Question: Evaluate the integral \( \int_C w = \int_0^{2\pi}\sin^2(t)(3\cos^2(t)-2\cos(t)+1)dt \) using Green's theorem, where \( C \) is the ellipse \( x^2+4y^2=4 \) with parametrization \( \alpha(t)=(2\cos(t),\sin(t)) \).

Answer: \dfrac{7\pi}{4}


Question: Does there exist a "zebra set" in \(\mathbb{R}^2\), defined as a set with alternating sections of constant positive measure and zero measure?

Answer: Yes


Question: Let $X_1,X_2,\dots,X_n$ be i.i.d. random variables with a uniform distribution on $(0,1)$. Define $X_{(n)}=\max\{X_1,X_2,\dots,X_n\}$. Find the variance of the asymptotic distribution of $(n+1)(1-X_{(n)})$. Provide your answer as a numerical value.

Answer: 1


Question: Determine if the sequence \( \{Y_n\} \) defined by \( Y_n = \frac{1}{n^3} \sum_{k=1}^n \frac{X_k^2}{k} \), where \( X_k \) are independent uniform random variables on the interval \((0,k)\), converges almost surely. If it does, find its limit.

Answer: 0


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( f(f(x) + 3y) = 12x + f(f(y) - x) \).

Answer: f(x) = 3x + c


Question: Are all subgroups of the additive group of rational numbers cyclic?

Answer: No


Question: Calculate the area enclosed between the curve \( y = \left[\sqrt{2 + 2\cos(2x)}\right] \) and the \( x \)-axis, for \( x \) in the interval \([-3\pi, 6\pi]\), where \([\cdot]\) denotes the greatest integer function.

Answer: 6\pi


Question: Given the curve $y = rac{4	ext{π}}{3} 	imes 	ext{sin}(x)$, let $s \in \mathbb{Q}$ with $s > 1$. The arc length $L$ from $0$ to $2\pi s$ is expressed as an elliptic integral of the second kind using the formula: $L = \int_0^{2\pi s} \sqrt{1 + \cos^2 x}\, dx$. If the period is changed to $\frac{4\pi}{3}$, determine the new amplitude of $\sin x$ so that the arc length from $0$ to $2\pi s$ remains $L$. Assume the current amplitude is $|A| = 1$. What should the new amplitude be?

Answer: \dfrac{2}{3}


Question: Evaluate the limit \( \lim\limits_{k\to\infty}\mathbb{P}\left(|X-\mathbb{E}[X]|>\epsilon\cdot\mathbb{E}[X]\right) \) for an Erlang-distributed random variable \( X\sim\textrm{Erlang}(k,\lambda) \), where \( \epsilon>0 \) is small, using the Central Limit Theorem.

Answer: 0


Question: Can a uniformly continuous function on the open interval $(0,1)$ be extended to the closed interval $[0,1]$ such that it is right-continuous at $x=0$ and left-continuous at $x=1$? Justify your answer.

Answer: Yes


Question: Find the limit of the expression \(((x^4-x^2)^\frac{1}{4}-\sqrt{x^3+x})\) as \(x\) approaches infinity without using L'Hôpital's rule or conjugation.

Answer: -\infty


Question: Determine whether every integer $n > 2$ has a "reciprocating" pseudoprime $a$ coprime to $n$ such that both conditions $a^{n - 1} \equiv 1 \pmod{n}$ and $n^{a - 1} \equiv 1 \pmod{a}$ hold, including even pseudoprimes. Answer "Yes" or "No".

Answer: Yes


Question: Find a function $h$ such that $h$ is orthogonal to both $x$ and $x^2$ with respect to the inner product $\langle f,g \rangle = \int_{-1}^{1} g(x)f(x) \, dx$ defined on $\mathcal{C}[-1,1]$. Normalizing $h$ is not required.

Answer: 5x^3 - 3x


Question: Evaluate the improper integral \( I = \int_0^\infty \frac{1}{x^2+b^2} \, dx \) where \( b > 0 \).

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


Question: Determine whether non-commuting matrices can share a simultaneous eigenvector. Provide a justification for your answer.

Answer: Yes


Question: True or False: Let $A$ be a square matrix with non-negative entries. If there exists a positive integer $m$ such that $A^m = A$, then the entries of $A$ must be either 0 or 1.

Answer: B


Question:
Let $a, b, c, d$ be real numbers with $a < c < d < b$. Consider the ring $C[a, b]$, which consists of continuous functions on the interval $[a, b]$ with pointwise addition and multiplication. If $S = \{f \in C[a, b]: f(x) = 0 \text{ for all } x \in [c, d]\}$, determine which of the following statements about $S$ is true:

(A) $S$ is NOT an ideal of $C[a, b]$.
(B) $S$ is an ideal of $C[a, b]$ but NOT a prime ideal of $C[a, b]$.
(C) $S$ is a prime ideal of $C[a, b]$ but NOT a maximal ideal of $C[a, b]$.
(D) $S$ is a maximal ideal of $C[a, b]$.

Choose the correct option.

Answer: B


Question: Determine the precise value of the uniform density function $f(y_1, y_2)$ for the continuous bivariate random variable $(Y_1, Y_2)$ with support $S = \{(y_1, y_2) : y_2 \leq 1-y_1^2, y_1 \leq 0, y_2 \leq 0\}$.

Answer: \dfrac{3}{2}


Question: In triangle $ABC$, where $AB = AC$ and $I$ is the in-center, it is given that $BC = AB + AI$. Determine the measure of angle $\angle BAC$. \( \text{Express your answer in degrees.} \)

Answer: 90


Question: Find the limit as \( n \to \infty \) of the probability \( P((x-n)^2 + (y-n)^2 \leq n^2) \), where \( x \) and \( y \) are randomly chosen from the set \( S = \{0, 1, \ldots, 2n\} \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit \[\lim_{n\to \infty} \left(\frac{(n+1)(n+2) \ldots(n+m)}{n^m}\right)^{\frac{1}{n}}\] where \(m\) is a fixed positive integer.

Answer: 1


Question: Determine whether the matrices \(A=\begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}\) and \(B=\begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix}\) are conjugate in the general linear group \(GL_2(\mathbb{R})\). Specifically, does there exist a matrix \(X=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in GL_2(\mathbb{R})\), with \(ad-bc \neq 0\), such that \(XAX^{-1}=B\)?

Answer: Yes


Question: Determine the possible values of the integral \( \int_{\gamma} \frac{dz}{\sqrt{1-z^2}} \) for a closed curve \( \gamma \) in \( \mathbb{C} \) that intersects the path \( \sigma \) joining \(-1\) and \(+1\).

Answer: 2\pi n


Question: Let $K_1$ and $K_2$ be two field extensions of a field $L$ with $[K_2:L]=3$. If $K_1 \cap K_2 = L$ and $K_2 = L(\beta)$, determine $[K_1(\beta):K_1]$. \( \text{Express your answer as a single integer.} \)

Answer: 3


Question: Evaluate the limit \(\displaystyle\lim_{x\to 0}^{}(1+\sin{x})^{\cot{x}}\).

Answer: e


Question:
Let \(\{x_{n}\}\) be a sequence of non-negative real numbers such that \(x_{n + 1}^2 =6x_{n} +7\) for all \(n \geq 2\). Determine which one of the following statements is true:

\(\quad(A).\space\) If \(x_{2} > x_{1} > 7\) then \(\displaystyle\sum_{n=1}^{\infty} x_{n}\) is convergent

\(\quad(B).\space\) If \(x_{2} > x_{1} > 7\) then \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n}}{x_n} \) converges

\(\quad(C).\space\) If \(7 > x_{2} > x_{1} > 6\) then \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{x_n}\) is convergent

\(\quad(D).\space\) None of these is convergent

Answer: D


Question: Evaluate the limit: \[ \lim_{x\rightarrow 0}\frac{x^{6000}-(\sin x)^{6000}}{x^{6002}} \] without using series expansion or L'Hôpital's rule.

Answer: 1000


Question: Let $f: M \to \mathbb{R}$ be a positive integrable function over a compact domain $M$ with measure $\mu(M)$. The arithmetic mean of $f$ is defined as $A(f) = \frac{1}{\mu(M)}\int_M f$, and the geometric mean is given by $G(f) = \exp\left(\frac{1}{\mu(M)}\int_M \log f\right)$. Prove or disprove that $A(f) \geq G(f)$ for every positive integrable function $f$ on $M$.

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


Question: In the commutative ring \( \left( \mathscr{C} \left[ 0, 1 \right], +, \cdot \right) \), where \( \mathscr{C} \left[ 0, 1 \right] \) denotes the set of all continuous functions from the interval \( \left[ 0, 1 \right] \) to \( \mathbb{R} \) with point-wise addition and multiplication, consider the ideal \( I_{x_0} = \left\lbrace f \in \mathscr{C} \left[ 0, 1 \right] \mid f \left( x_0 \right) = 0 \right\rbrace \) for some \( x_0 \in \left[ 0, 1 \right] \). Determine the annihilator of \( I_{x_0} \), denoted as \( \text{Ann}(I_{x_0}) \).

Answer: \{0\}


Question: Calculate the number of permutations of the letters ABCDEFGHI such that the substrings ABC, DEF, and GHI do not appear consecutively.

Answer: 348114


Question: For a given fixed integer $N$, find the largest integer $n$ such that the symmetric group $\Sigma_n$ can be embedded as a subgroup of the orthogonal group $O(N)$. What is the maximum value of $n$?

Answer: N + 1


Question: Evaluate the integral \( \int_0^{2 \pi} (\cos\theta)^{2015} \, d\theta \).

Answer: 0


Question: Can a non-surjective function have more than one distinct left inverse?

Answer: Yes


Question: Determine whether the equation \( 2x^3 - 1 = z^3 \) has any positive integer solutions.

Answer: (1, 1)


Question: Given the differential equation \( X''+u^2X=0 \), find the condition for nontrivial solutions given the boundary conditions. Show that this condition can be expressed as \( \tan u = -u \).

Answer: \tan u = -u


Question: In a circular sequence of 2019 numbers, the sum of any five consecutive terms is 70. Find all possible values for these 2019 numbers.

Answer: 14


Question: Find the norm of the integral operator $A$ in $L_2(0,1)$, where $Ax(t) = \int_0^t x(s) \, ds$.

Answer: \dfrac{2}{\pi}


Question: Evaluate the limit: \[ \lim_{x,y\to 0,1} \frac{x^2y^3 - 2xy}{\tan(x^2+y^2)} \] if it exists.

Answer: 0


Question: Determine whether there exists a sequence in a Banach space that converges to the norm of a bounded linear operator.

Answer: Yes


Question: Find the dimension over \(\mathbb{R}\) of the quotient ring \(\mathbb{R}[x, y] / (x^2, y^2)\).

Answer: 4


Question: Determine the number of real solutions to the equation \( \sin x \cdot \sin 2x \cdot \sin 3x = 1 \) for all \( x \in \mathbb{R} \).

Answer: 0


Question: Determine all natural numbers \( n \) such that \( \phi(n^4+1) = 8n \), where \( \phi(n) \) is Euler's totient function.

Answer: 2


Question: Evaluate the double integral \( \iint_S \sqrt{4-x^2 - y^2} \, dA \), where \( S \) is the region in the first quadrant bounded by the circle \( x^2 + y^2 = 4 \), the line \( y = 0 \), and the line \( y = x \).

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


Question: Determine whether the function \( f(x,y) = \begin{cases} (x-1)^2(y-1)^2\sin\left(\frac{1}{(x-1)^2+(y-1)^2}\right), & (x,y) \neq (1,1) \\ 0, & (x,y) = (1,1) \end{cases} \) is differentiable at \((x,y) = (1,1)\).

Answer: Yes


Question: If a function $h$ is both $a$-strongly convex and $b$-strongly convex, determine the strong convexity constant of $h$.

Answer: \max\{a, b\}


Question: Find a diagonal quantum gate with eigenvalues \(\pm 1\) that can be decomposed into a finite number of Z and controlled-Z gates.

Answer: CZ


Question: Determine the cardinality of the center of the group of all $2 \times 2$ orthogonal matrices over $\mathbb{R}$, denoted as $\Bbb O_2(\mathbb{R})$. Provide your answer as a single number.

Answer: 2


Question: Find the value of $x$ that satisfies the equation $\sin(x) = x + 0.5$. Express your answer to three decimal places.

Answer: -1.497


Question: Find the number of distinct elements in the group defined by the presentation \( \langle x,y \mid x^3=y^2=(xy)^3=e \rangle \) by determining a normal form for its elements.

Answer: 12


Question: Evaluate the sum \( \sum_{k=80}^{167} \frac{1}{\sin k^\circ \sin (k+1)^\circ} \).

Answer: 280


Question: Find the radical of the ideal $J = (ab, ac-bc)$. Express your answer in terms of generators.

Answer: (ab, ac, bc)


Question:
Evaluate the limit of the following expression, assuming that $f$ is differentiable at $a$ and $p, q \neq 0$, but $f'$ may not be continuous:

$$\lim_{h\to0}\frac{f(a+ph)-f(a)}{ph}-\lim_{h\to0}\frac{f(a-qh)-f(a)}{-qh}$$

Answer: 0


Question: In an acute triangle ABC inscribed in a circle with radius R, the orthocenter H of triangle ABC lies on the circle, and AH = R. Determine the measure of angle BAC.

Answer: 60^\circ


Question: Given the parametric equations of a curve at any point \(\theta\): \[x = a\cos(\theta) + a\theta\sin(\theta)\] \[y = a\sin(\theta) - a\theta\cos(\theta)\] find the distance from the origin to the normal of this curve at any \(\theta\).

Answer: a


Question: Determine the number of positive real roots of the equation \(x^6 - x - 1 = 0\).

Answer: 1


Question: Consider a matrix with a row of all zeros. Is it possible for this matrix to have linearly independent eigenvectors that span the entire vector space? Provide a justification for your answer.

Answer: Yes


Question: Determine the number of combinations where 16 trials are evenly distributed among 4 individuals, each conducting 4 trials, such that there are a total of 5 successes across all trials and no individual has more than 2 successes in their trials.

Answer: 3264


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

Answer: \dfrac{2}{3}


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

Answer: 0


Question: Determine whether for every natural number \(k \geq 3\) and a prime number \(p \geq 7\), there exist natural numbers \(w_1, w_2, \ldots, w_k\) (all different from \(1\)) such that \(p\) divides \(\sum_{i=1}^k w_i^2\) but does not divide any of the numbers \(w_1, w_2, \ldots, w_k\).

Answer: Yes


Question: Suppose that whether or not it rains today depends on weather conditions through the previous two days. If it has rained for the past two days, then it will rain today with probability 0.7. If it did not rain for any of the past two days, then it will rain today with probability 0.4. In any other case, the weather today will, with probability 0.5, be the same as the weather yesterday. Describe the weather condition over time with a Markov chain. Given that it rained today and yesterday, calculate the expected number of days until it rains three consecutive days for the first time in the future.

Answer: \dfrac{59}{14}


Question: Given that $V$ is a subspace of $\mathbb{R}^m$ spanned by $\{v_1, v_2, v_3\}$ with $m \geq 3$, and $W$ is a subspace of $\mathbb{R}^n$ spanned by $\{w_1, w_2, w_3\}$ with $n \geq 3$, where $V$ has dimension 2 and $W$ has dimension 3, determine the rank of the $m \times n$ matrix $A$ defined by $A = v_1w_1^T + v_2w_2^T + v_3w_3^T$, given that one of the $v$ vectors can be expressed as a linear combination of the other two, and the $w$ vectors are linearly independent.

Answer: 2


Question: Evaluate the integral \( \int_0^n \frac{(1-\frac{x}{n})^n}{ne^{-x}}dx \) as \( n \to \infty \).

Answer: 0


Question:
Let $f(x)$ be the $2\pi$-periodic function defined as follows:
$$ f(x)= \begin{cases} 1+x & \text{if } x \in [0, \pi) \\ -x-2 & \text{if } x \in [-\pi, 0) \\ \end{cases} $$

Determine the correct statement regarding the Fourier series of $f$:
(A) It converges to $-\frac{1}{2}$ at $x = 0$.
(B) It does not converge pointwise at all points where $f$ is discontinuous.
(C) It does not converge at $x = \frac{\pi}{2}$.
(D) It converges to $1$ at $x = 0$. \(\text{Choose the correct option.}\)

Answer: A


Question: Find the value of $a$ such that the $y$-coordinate of the point on the curve $f(x) = e^x$ that is closest to the line $y = \frac{x}{a} - 1$ is $\frac{1}{e}$.

Answer: e


Question: Determine whether the series \( \sum_{k=1}^{\infty} \frac{1}{(k+\sqrt{k})^p} \) converges or diverges, and specify the values of \( p \) for which it converges.

Answer: p > 1


Question: Determine the number of distinct ways an $n$-cycle can be expressed as the product of a $2$-cycle and an $(n-1)$-cycle in cycle notation.

Answer: n


Question: Given two non-zero matrices $A$ and $B$ such that $A \neq B$, $\text{tr}(A) = 1$, $\text{tr}(B) = 1$, and $\text{tr}(A-B) = 0$, determine a lower bound for the Schatten 1-norm $\|A-B\|_1$.

Answer: 0


Question: Find the maximum value of the function \( f(x,y) = xy \) subject to the constraints \( y + x^2 \leq 6 \) and \( x, y \geq 0 \) using the Lagrange multiplier method.

Answer: 4\sqrt{2}


Question: Evaluate the double integral \( \iint_R (x^2 - y^2) \, dA \), where \( R \) is the region defined by the inequality \( x^2 - y^2 \geq a^2 \).

Answer: \infty


Question: Is a group of order 185 cyclic? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \iiint_{[0,1]^n} \max \{u_1,u_2,...,u_n\} \,du_1 \dots du_n \).

Answer: \dfrac{n}{n+1}


Question: Find the volume of the solid generated by revolving the curve \( y = 2\cos(x) \) from \( x = 0 \) to \( x = \frac{\pi}{2} \) around the \( x \)-axis.

Answer: \pi^2


Question: Given a discrete random variable $X$ with probability generating function $G_X(\theta) = \frac{2}{3-\theta}$, find the probability mass function (PMF) of $Y = X^2$. Then, calculate the expected value $E(Y)$.

Answer: 1


Question: Evaluate the limit: $$\lim_{x \to 0^{+}}\frac{\sin(x)^x − x^ {\sin(x)}}{\tan(x) ^x − x^{\tan (x)}}$$ without using Taylor expansions as much as possible.

Answer: -\dfrac{1}{2}


Question: Determine whether the sequence of functions \( f_n(x) = \sqrt{n}x^2(1 - x^3)^n \) converges pointwise for \( 0 < x < 1 \). If it does, find the limit function.

Answer: 0


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty}\frac{1}{2n}\ln \binom{2n}{n}.$$

Answer: \ln 2


Question: Evaluate the limit: $$\lim_{x \rightarrow 0}\frac{(1+2x)^\frac{1}{x}-(1+x)^\frac{2}{x}}{x}$$

Answer: -e^2


Question: Calculate the limit of \( \sqrt{n}(A_{n+1} - A_n) \) as \( n \to \infty \), where \( A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n) \) and \( |a_n| \le 1 \) for all \( n \).

Answer: 0


Question: Calculate the product of the non-zero eigenvalues of the matrix \(\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}\).

Answer: 6


Question: Does a projective transformation that maps an ellipse to itself necessarily map the center of the ellipse to itself? Answer yes or no.

Answer: No


Question: Evaluate the limit: \[ \lim_{n\to\infty} n^{\frac{4}{3}} \left(\sqrt[3]{n^2 + 1} + \sqrt[3]{3 - n^2}\right) \] as \( n \) approaches infinity.

Answer: \dfrac{4}{3}


Question: Let \( f_n \in C([0,2014]) \). Show that if \( f_n \rightrightarrows f \) and for all \( n \), \( \int_{[0,2014]} f f_n \, dl_1 = 0 \), then \( f \equiv 0 \).

Answer: f \equiv 0


Question: Determine whether the paths $p \ast f \ast \bar{p}$ and $h \ast f \ast \bar{h}$ are path homotopic, where $p(s) = (1, s)$, $f(s) = (e^{2\pi is}, 1)$, and $h(s) = (e^{2\pi is}, s)$ for $s \in [0, 1]$. Here, $g \ast k$ denotes the path $\sigma(s) = \begin{cases} g(2s), & s \in [0, 1/2] \\ k(2s-1), & s \in [1/2, 1] \end{cases}$, and $\bar{g}(s)$ denotes the path $g(1-s)$. Answer "yes" if they are path homotopic and "no" otherwise.

Answer: yes


Question: Find the radius of convergence of the series \( \sum_{n=0}^{\infty}{n^{1/n}x^{n}} \).

Answer: 1


Question: Evaluate the integral \( \int_{-2}^{2}\frac{1+x^2}{1+2^x}dx \) analytically.

Answer: \dfrac{14}{3}


Question: Calculate the area bounded by the curve $y = g(x)$, the x-axis, and the lines $x = -2$ and $x = 3$, where $g(x)$ is defined as follows: \[g(x)=\left\{\begin{array}{ll}\max \{f(t):-2 \leq t \leq x\}, & \text { where }-2 \leq x<0 \\ \min \{f(t): 0 \leq t \leq x\}, & \text { where } 0 \leq x \leq 3\end{array}\right.\] with $f(x) = x^2 - |x|$. Express your answer as a fraction $\frac{a}{b}$.

Answer: \dfrac{113}{24}


Question: Calculate the conditional expectation $E(X | X<Y)$ for a discrete random variable $X$ with $P(X=0)=P(X=1)=P(X=2)=\frac{1}{3}$ and $Y$ uniformly distributed on $(0,2)$. What is the correct value of $E(X | X<Y)$?

Answer: \dfrac{1}{3}


Question: Consider a function with a zero of order \(m\) in the numerator and a zero of order \(n\) in the denominator at a point \(z_0\). Determine the order of the pole or singularity at \(z_0\) when \(m > n\).

Answer: m - n


Question: Determine whether the existence of a function $f: \mathbb{R} \to \mathbb{R}$ such that $Y = f(X)$ implies the existence of a Borel-measurable function $f: \mathbb{R} \to \mathbb{R}$ such that $Y = f(X)$ almost surely. In other words, if $X$ determines $Y$ in the set theory sense, does $X$ also determine $Y$ in the probability theory sense?

Answer: Yes


Question: Find the minimum values of the functions \(f(x) = 2\log_{10}(x) - \log_x(0.01)\) and \(g(x) = e^x + e^{-x}\). Determine the value of \(\frac{k_1}{k_2}\), where \(k_1\) and \(k_2\) are these minimum values, respectively. Use the Arithmetic Mean-Geometric Mean (AM-GM) inequality to assist in finding the minimum values.

Answer: 2


Question: Prove that one less than a perfect square is a prime number only when the prime is 3.

Answer: 3


Question: Evaluate the limit: \( \lim_{z\rightarrow 0^{-}} \frac{e^{-a/z}-(1-\frac{a}{z})}{z^2} \) without using L'Hopital's rule.

Answer: \infty


Question: Find the number of normal subgroups in the direct product of $k$ isomorphic non-abelian simple groups, considering all possible projections and intersections.

Answer: 2^k


Question: Determine the values of $n$ for which the following statement is true: If $H$ is a normal subgroup of $O(n)$ and contains an element $h$ that acts as the negative identity on a one-dimensional subspace and as the identity on its orthogonal complement, then $H$ must be equal to $O(n)$. What are these values of $n$?

Answer: n \geq 1


Question: Evaluate the integral: $$\int_0^{\pi}\frac{\sqrt{2 - 2\cos(x)}}{\sqrt{2 + 2\cos(x)}}\,dx$$

Answer: \infty


Question: Determine the maximum value of the function \( g(a) = \sqrt[3]{a + \frac{a+1}{3}\sqrt{\frac{8a-1}{3}}} + \sqrt[3]{a - \frac{a+1}{3}\sqrt{\frac{8a-1}{3}}} \) for \( a \geq \frac{1}{8} \).

Answer: 1


Question: Find the intersection point of any two circles that pass through the origin and the points of intersection of lines with slope 1 and the parabola \( y^2 = ax \), where \( a \) is a constant.

Answer: (a, -a)


Question: Let $X$ be an uncountable set with the cocountable topology. Determine whether $(X, T)$ is a $T_1$ space.

Answer: T_1


Question: Determine the limit of the sequence \(\frac{n!}{n^n}\) as \(n\) approaches infinity.

Answer: 0


Question: Determine whether the ratio of any two short chord lengths of a regular n-gon can be expressed in terms of trigonometric ratios.

Answer: Yes


Question: Calculate the limit: \[ \lim_{x \to +\infty } \left(e^{x^{2}\sin \frac{1}{x}}-e^{x}\right) \]

Answer: -\infty


Question: Suppose \( E \) is an extension field of \( \mathbb{Z}_7 \) and \( d \in E - \mathbb{Z}_7 \). If \( d^5 = 2 \), determine the degree of the minimal polynomial \( \deg(d/\mathbb{Z}_7) \).

Answer: 4


Question:
Find the maximum value of $2c + 2a - b$ subject to the following conditions:
1. $a^2 + b^2 + c^2 = 2018$
2. $2a + 2b - c = 15$
3. $2b + 2c - a = 36$

Answer: 129


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

Answer: 0


Question: Evaluate the limit \(\displaystyle\lim_{{x \to 0}} \frac{\arcsin(x) - \arctan(x)}{x^3}\) without using L'Hôpital's rule.

Answer: \dfrac{1}{2}


Question: Calculate the remainder when \(3^{999}\) is divided by 1000.

Answer: 667


Question: Determine the degree of the field extension $[\mathbb{Q}(i,\sqrt{2},\sqrt{3}):\mathbb{Q}]$. Explain any discrepancy between the degree calculated using the tower law and the degree suggested by the minimal polynomials of $i$, $\sqrt{2}$, and $\sqrt{3}$.

Answer: 8


Question: Evaluate the limit of the sequence \(a_n = \left( \left(1 + \frac{1}{n}\right) \left(1 + \frac{2}{n}\right) \cdots \left(1 + \frac{n}{n}\right) \right)^{\frac{1}{n}}\) as \(n\) approaches infinity.

Answer: \dfrac{4}{e}


Question: Let $\Omega$ be a bounded region and $\{f_n\}$ a sequence of functions holomorphic in $\Omega$. If $f_n$ converges uniformly on the boundary of $\Omega$ to a continuous function, does it imply that $f_n$ converges uniformly on $\overline{\Omega}$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the equation \(a^\pi + b^\pi = c^\pi\) can hold true for any natural numbers \(a, b, c\).

Answer: No


Question:
Find the limit as \( n \to \infty \) of the sequence:
\[ \frac{\cos(1)}{n^2 + 1} + \frac{\cos(4)}{n^2 + 4} + \dots + \frac{\cos(n^2)}{n^2 + n^2} \]

Answer: 0


Question: Determine whether the following statement is true or false: If a matrix representing a bilinear form is singular, then there exists a non-zero vector \(x_0\) such that the form evaluated at \((x_0, y)\) is zero for all vectors \(y\).

Answer: True


Question: Find the maximum value of \(Q = \sin(x)\sin(y)\sin(z)\) given that \(x + y + z = \frac{\pi}{2}\) and \(x, y, z > 0\).

Answer: \dfrac{1}{8}


Question: Let $G$ be a Lie group, and $H \subseteq G$ a closed normal subgroup. Given the quotient map $\pi: G \to G/H$, determine if for every smooth path $\gamma: [0,1] \to G/H$, there exists a smooth path $\tilde{\gamma}: [0,1] \to G$ such that $\pi \circ \tilde{\gamma} = \gamma$. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Determine the possible values of $F(0)$ if $F(x)$ is a function that satisfies the equation \( F(x)F(y) - F(xy) = x + y \) for all real numbers $x$ and $y$. \( \boxed{} \)

Answer: 1


Question: Find the coefficient of \(x^{21}\) in the expansion of \((x^3 + x^4 + x^5 + \ldots + x^{10})^4\).

Answer: 204


Question: Calculate the expected value \( E(X(X-1)(X-2)) \) for a Poisson random variable \( X \) with parameter \( \lambda = \frac{4}{3} \).

Answer: \dfrac{64}{27}


Question: For a finite field $\mathbb{F}$ of size $q$, determine if there exists a non-trivial bound for the Weil sum $W_{\mathbb{F},d}(a)$ when $d$ is smaller than $\sqrt{q}$. Provide a justification for your answer.

Answer: Yes


Question:
Arrange the following functions in decreasing order of growth rate:

(a) $n^{n^n}$
(b) $(n!)^n$
(c) $n^{2^n}$
(d) $2^{2^n}$
(e) $n!$

Answer: a > c > d > b > e


Question: Determine the cohomology groups $H^q(G, M)$ for a free group $G$ with $n$ free generators and a trivial $G$-module $M$, where $q \geq 3$. Provide your answer in terms of $n$ and $M$.

Answer: 0


Question: Find the singularities and their corresponding residues for the function \( \cos \frac{1}{z-1} \).

Answer: 0


Question: Evaluate the limit of the sequence given by the general term \( \frac{1}{n+1-k}\left(\frac{n+1}{2}\right)^n \) as \( n \to \infty \), where \( k \) is a fixed natural number.

Answer: \infty


Question: Evaluate the integral of the function \( \frac{z^3}{2z-i} \) around the unit circle.

Answer: \dfrac{\pi}{8}


Question: In a Poisson process $\{X_t\}$, given that $P(X_s=i,X_t-X_s=k)=P(X_s=i)P(X_t-X_s=k)$ for fixed time points $s$ and $t$, determine if it follows that $P(X_u-X_s=h,X_t-X_u=k)=P(X_u-X_s=h)P(X_t-X_u=k)$ for any $s<u<t$. Provide a justification for your answer.

Answer: Yes


Question: Find the equation of the tangent line to the curve \((x^2 + y^2 - 3)^3 = (xy)^3 - x^2 - y^2\) at the point \((1, 1)\).

Answer: y = -x + 2


Question: Determine whether the space of self-adjoint trace class operators over a separable Hilbert space is separable with respect to the trace norm.

Answer: Yes


Question: Find the order of the quotient group \( \mathbb{Z}^3 / \langle (2,1,5), (1,2,10), (2,1,7) \rangle \). Use a method other than the presentation matrix approach.

Answer: 6


Question: Evaluate the double summation: $$ \sum_{m=0}^{n} \sum_{k=0}^{m}(m-k) \binom{n}{m} \binom{m}{k} $$

Answer: n \cdot 3^{n-1}


Question: Determine whether a group $G$ can have two proper subgroups $H$ and $K$ of the same order such that $H \cap K$ is nontrivial.

Answer: Yes


Question: Determine the number of distinct real roots of the determinant \[ \begin{vmatrix} \tan x & \cot x & \cot x \\ \cot x & \tan x & \cot x \\ \cot x & \cot x & \tan x \\ \end{vmatrix} \] in the interval \(-\frac{\pi}{4} \le x \le \frac{\pi}{4}\).

Answer: 2


Question: Let $n \geq 2$ be an integer and let $a_1, \ldots, a_n \in \mathbb{Z}$ with $\gcd(a_1, \ldots, a_n) = 1$. Determine whether the equation \[ \begin{vmatrix} a_1 & \cdots & a_n \\ x_{11} & \cdots & x_{1n} \\ x_{21} & \cdots & x_{2n} \\ \vdots & \ddots & \vdots \\ x_{n-1,1} & \cdots & x_{n-1,n} \end{vmatrix} = 1 \] always has an integer solution for the $x_{kl}$'s.

Answer: Yes


Question: Determine whether the function \( f(x,y) = -\frac{2xy}{(\frac{3}{2}\sqrt{|y|}+1+x^2)} \) satisfies the local Lipschitz condition \( |f(x,y_2)-f(x,y_1)| \leq M|y_2-y_1| \) for \( x,y \in \mathbb{R} \).

Answer: Yes


Question: Given the functions $f: \mathbb{R}^2 \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}^2$ where $g(0) = (1,2)$, $g'(0) = (-1,4)$, and $(f \circ g)'(0) = -3$, and that $\frac{df}{dx}(1,2) = \frac{df}{dy}(1,2)$, determine the value of $\frac{df}{dx}(1,2)$. \( \boxed{} \)

Answer: -1


Question: Given that $|q(z)| \leq 3$ for all $z$ on the unit circle, where $q(z) = d_nz^n + d_{n-1}z^{n-1} + \ldots + d_1z + d_0$ is a polynomial, determine if the inequality $|d_k| \leq 3$ holds for all $k = 0, 1, \ldots, n$. Provide a justification for your answer.

Answer: Yes


Question: Determine the period of the continuous time signal \(y(t)=\sum_{n=-\infty}^{+\infty}{e^{-|8t+n|}}\).

Answer: \dfrac{1}{8}


Question: Evaluate the line integral \( \int_\Gamma F(x) \, dx \) where \( F(x) = (x,y) \), along the path \( \Gamma \) from \((0,1)\) to \((-1,0)\) on the unit circle.

Answer: 0