Question: Consider the function \( f(x) = \frac{x}{\sqrt[6]{1 + x^6}} \). Determine the value of the repeated composition \( f \circ f \circ \cdots \circ f \) as \( n \to \infty \).

Answer: 0


Question: Determine the number of onto group homomorphisms from \((\mathbb{Q}, +)\) onto \((\mathbb{Q} \setminus \{0\}, \cdot)\).

Answer: 0


Question: In the context of the National Eligibility and Screening Test (NEST) 2018, consider the equation \((x^2+px+1)(x^2-2x+q)=0\), where \(p\) and \(q\) are integers, and \(i=\sqrt{-1}\). Given that \(2+\sqrt{3}\) and \(1-i\) are roots of this equation, find the value of \(p+q\).

Answer: -2


Question: Evaluate the double integral \( \int \int_R xy \, dA \), where \( R \) is the square with vertices \((0,0), (1,1), (2,0), (1,-1)\), using the change of variables formula and an appropriate transformation.

Answer: 0


Question: Determine the condition that must be satisfied for the matrix equation \((I-T)^2 = I - T\) to hold.

Answer: T^2 = T


Question: Is it always possible to extract a subsequence from any sequence in a compact metric space that converges to a unique point in the space?

Answer: Yes


Question: Calculate the expectation of a continuous random variable $X$ given that $P(X \ge a) = 1 - \frac{1}{4}a^2$ for $0 \le a \le 2$. Provide your answer as a single numerical value.

Answer: \dfrac{4}{3}


Question: Let \( X_1, X_2, \) and \( X_3 \) be three independent exponential random variables with rate \( \lambda \). Find the probability that \( \min(X_1, X_2, X_3) \leq X_1 + X_2 + X_3 - \min(X_1, X_2, X_3) \).

Answer: 1


Question: Evaluate the improper integral \( \int_0^{\infty} \frac{\log(x)}{1+x^2}dx \) using contour integration.

Answer: 0


Question: Evaluate the line integral \( \oint_C P \, dx + Q \, dy \), where \( P(x, y) = xe^{-y^2} \) and \( Q(x, y) = -x^2ye^{-y^2} + \frac{1}{x^2+y^2+1} \). The curve \( C \) is the boundary of the square with vertices at \((-a, -a)\), \((-a, a)\), \((a, a)\), and \((a, -a)\), oriented counterclockwise.

Answer: 0


Question: Find all complex numbers $z$ such that $|z| + \overline{z} = 9 - 3i$. \( \overline{z} \) denotes the complex conjugate of \( z \).

Answer: 4 + 3i


Question: Determine the interval $I \subset \Bbb{R}$, which may be unbounded, where the function $f(x) = \frac{\ln(t)}{1+t^{x}}$ is differentiable.

Answer: (-\infty, \infty)


Question: Calculate the limit of the series as \( n \) approaches infinity: \[ \lim_{n\to \infty} \frac{\frac{2}{1}+\frac{3^2}{2}+\frac{4^3}{3^2}+\dots+\frac{(n+1)^n}{n^{n-1}}}{n^2}. \]

Answer: \dfrac{e}{2}


Question: Determine whether the process \( \{X_t = W_t + W_{t/2}, t \geq 0\} \), where \( \{W_t, t \geq 0\} \) is a Brownian motion, is a Gaussian Process.

Answer: Yes


Question: Determine whether the countable union of sets, each with cardinality equal to that of the real numbers \(|\mathbb{R}|\), also has cardinality \(|\mathbb{R}|\).

Answer: |\mathbb{R}|


Question: Evaluate the integral \( \int_1^\infty (\ln x)^n \, dx \) for positive integers \( n \).

Answer: \infty


Question: Derive a reduction formula for \(I_{n} = \int_{0}^{1} x^3(\ln x)^n \, dx\) and use it to evaluate \(I_4\).

Answer: \dfrac{3}{128}


Question: Evaluate the power series \( \sum_{n = 1}^{\infty}\dfrac{n}{9^{n}} \) without using the formula for infinite geometric series.

Answer: \dfrac{9}{64}


Question: Find the radius of convergence for the Taylor series of the function \( \frac{1}{1+z^3} \).

Answer: 1


Question: Is it possible for two rings $R$ and $T$ to be Morita equivalent if $T$ is simple but $R$ is not?

Answer: No


Question: Evaluate the integral \( \int_{0}^{\infty}\frac{\ln(x)}{1+x^{2}}dx \) using Feynman's trick.

Answer: 0


Question: Evaluate the contour integral \( \oint_{C} \frac{2\,dz}{(2i+1)z^2 + 6iz + 2i - 1} \) where \( C \) is the unit circle.

Answer: \pi


Question: Given an increasing sequence of functions $f_n:E\subseteq\mathbb{R}^n\rightarrow \mathbb{R}$, where the sequence of integrals $\left\{\int_E f_n\right\}$ is bounded, determine if there exists a non-negative, increasing sequence of functions $g_n:E\rightarrow \mathbb{R}$ such that $|f_n|\leq g_n$ for all $n$ and $\left\{\int_E g_n\right\}$ remains bounded.

Answer: Yes


Question: If $X \times X$ is a Hausdorff space, does it follow that $X$ is also a Hausdorff space?

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} S(n) = \lim_{n \to \infty} \int_0^1 x^n n f(x) \, dx \), where \( f \) is a continuous function on \([0,1]\).

Answer: f(1)


Question: Find the maximum possible dimension of a subspace in $V_n = \{(a_1,a_2,\ldots,a_n) \mid a_i \in \mathbb{GF}(2)\}$, given $k$ linearly independent vectors that are not part of this subspace, where $k < n$.

Answer: n - 1


Question: Find all real numbers \(x\) that satisfy the equation \(|x^2 - \lfloor x \rfloor| = 1\), where \(\lfloor x \rfloor\) is the greatest integer less than or equal to \(x\).

Answer: \sqrt{2}


Question: Let $a$ be a zero of the polynomial $x^3+x^2+1$ in some extension of the finite field $\mathbb{Z}_2$. Find the multiplicative inverse of $a+1$ in this extension.

Answer: a^2


Question: Integrate the rational function \( \frac{z}{(z+2)(z+1)(z-1)} \) over a disk of radius 3 centered at the origin using Cauchy's Formula.

Answer: 0


Question:
Determine the value of \( n \) such that the following limit holds:
\[
\lim\limits_{x \to 1}\frac{\pi/4-\tan^{-1}x}{e^{\sin(\ln x)}-x^n} =\frac 18.\]

Answer: 5


Question: Evaluate the line integral \( \int_C y^2 x^2 \, dy + 2xy^3 \, dx \) using Green's Theorem, where \( C \) is the negatively oriented triangle with vertices at \((0,0), (3,1), (3,0)\).

Answer: \dfrac{12}{5}


Question: Evaluate the integral: $$\int_{-5}^5\int_0^{\sqrt{25-x^2}}\int_0^6\frac1{1+x^2+y^2}\,dz\,dy\,dx$$

Answer: 3\pi \ln 26


Question: Find the harmonic conjugate of the function \( w(x, y) = 2x - y \).

Answer: x + 2y


Question: Evaluate the limit: $$\lim_{x\rightarrow 0^-} \, \frac{\sqrt[3]{x^2} \ln(|\ln (-x)|)}{\sqrt[3]{(x+4)\ln (-x)}}$$

Answer: 0


Question: Determine an approximate upper bound for the maximum number of balls in a bin when dropping \( n \log n \) balls into \( n \) bins randomly, ignoring distribution dependencies and integer constraints.

Answer: O(\log n)


Question: Evaluate the limit: $$\lim\limits_{n\to\infty}\prod_{n=1}^{n}\left(1+\left(\frac{1}{2}\right)^{2^{n}}\right)$$

Answer: \dfrac{4}{3}


Question: Determine the order of the pole for the complex function \( \frac{z}{\sin\pi z^{2}} \) at the following points: \( z=0 \), \( z=\pm\sqrt{n} \), and \( z=\pm i\sqrt{n} \), where \( n \in \mathbb{Z}^{+} \).

Answer: 1


Question: Determine whether there exists a regular curve \( \gamma : \mathbb{R} \to \mathbb{R}^{3} \) and a rigid motion \( f \) such that \( f \) is not the identity motion and \( f \circ \gamma \) is a reparametrization of \( \gamma \).

Answer: Yes


Question: Given the equation \(\int_{0}^{\frac{\pi}{2}}(\sin x + a\cos x)^3dx - \frac{4a}{\pi - 2}\int_{0}^{\frac{\pi}{2}}x\cos x dx = 2\), where the second integral evaluates to \(\frac{\pi - 2}{2}\), find the sum of the roots \(a_1 + a_2 + a_3\) that satisfy the equation.

Answer: -\dfrac{3}{2}


Question: Find a compact form for the series: $$\sum_{i=j}^{n}\binom{n}{i}\binom{i}{j}\frac{(-1)^{i+j}}{i}, \quad n\ge 1, \quad j\le n-1$$ where $j$ is a constant.

Answer: \dfrac{1}{j}


Question: Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $g(x+y)=g(x)g(y)$ for all $x, y \in \mathbb{R}$. Given that $g'(0) = 2$, find the equation of the tangent line to the graph of $g$ at the point $(0, g(0))$. \( \text{Express your answer in the form } y = mx + c. \)

Answer: y = 2x + 1


Question: Determine whether it is possible to construct a subfield of \( \mathbb{C} \) that is isomorphic to a proper subfield of \( \mathbb{R} \).

Answer: Yes


Question: Is every real number a limit point of irrational numbers?

Answer: Yes


Question: Solve for rational numbers the equation \(x\lfloor x\rfloor\{x\}=58\), where \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x\) and \(\{x\}\) is the fractional part of \(x\).

Answer: \dfrac{29}{3}


Question:
Evaluate the limit:

$$\lim_{n\rightarrow \infty}\sqrt[n]{f\left(a+\frac{b-a}{n}\right)g\left(b\right)+f\left(a+\frac{2(b-a)}{n}\right)g\left(b-\frac{b-a}{n}\right)+...+f\left(b\right)g\left(a+\frac{b-a}{n}\right)}$$

Answer: 1


Question: Determine whether the function \( \mu(x,y) = e^{\int h(xy) d(xy) } \) can be expressed as a function of the product \( xy \). Specifically, can we write \( \mu(x,y) = \mu(xy) \)?

Answer: Yes


Question: Find all real numbers $x$ such that $x - \lfloor x \rfloor = \frac{2}{\frac{1}{x} + \frac{1}{\lfloor x \rfloor}}$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Determine if your solution set includes integers.

Answer: \emptyset


Question: Determine the number of different Euclidean divisions possible for a given pair of elements in the Gaussian integers, $\mathbb{Z}[i]$. Provide your answer as a single integer.

Answer: 4


Question:
Consider the system of equations given by:

$$ \prod_{i=1}^n p_i = \prod_{i=1}^n (1-p_i)= \frac{1}{2^n} $$

where $p_i \in (0,1)$ for $i = 1, 2, \ldots, n$. Determine if the only solution to this system is $p_1 = p_2 = \ldots = p_n = \frac{1}{2}$ for all $n \in \mathbb{N}$. Is this statement true?

Answer: True


Question: Determine whether a function with an infinite number of jump discontinuities can still be increasing on an interval. Provide a justification for your answer.

Answer: Yes


Question: Evaluate \( \lim_{n\to\infty}\left(\left(\frac{10}{3}\right)^n+\left(1+\frac{2}{n}\right)^{n^2}\right)^{\frac{1}{n}} \).

Answer: e^2


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

Answer: \dfrac{3}{2}


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

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


Question: Determine whether the following statement is true or false: A series \( s_n = \sum a_i \) converges if and only if there exist indices \( n_k \) such that \( s_{n_k} \) converges. Specifically, is the implication correct that if there exist indices \( n_k \) such that \( s_{n_k} \) converges, then the series \( s_n \) converges?

Answer: False


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

Answer: \dfrac{\pi}{4}


Question: Find the limit of \( n!^{1/n^2} \) as \( n \to \infty \).

Answer: 1


Question: Evaluate the line integral \( \oint_C P \, dx + Q \, dy \) using Green's theorem, where \( P(x,y) = \frac{y}{x^2+y^2} \) and \( Q(x,y) = \frac{-x}{x^2+y^2} \), and \( C \) is a piecewise smooth Jordan curve contained in \( S = \{(x,y) | x^2 + y^2 > 0\} \) such that the origin \((0,0)\) lies interior to \( C \).

Answer: -2\pi


Question: Evaluate the improper integral \( \int_0^{\pi/2} \frac{\sin x+\cos x-1}{\sin^2x}dx \).

Answer: \infty


Question: Verify the Divergence Theorem for the vector field \( \mathbf{F}(x,y,z) = x^2\hat{\imath} + z\hat{\jmath} \) over the region \( D \), where \( D \) is the upper half-ball defined by \( x^2 + y^2 + z^2 \leq 1, z \geq 0 \).

Answer: 0


Question:
Given a partially separable 3-qubit state, where the first qubit is separable and the second and third qubits are entangled as follows:

$$\phi = (a_0|0\rangle + a_1|1\rangle) \otimes (b_{00}|00\rangle + b_{01}|01\rangle + b_{10}|10\rangle + b_{11}|11\rangle)$$

Expressed in its unseparated form:

$$\phi = \sum_{i,j,k} a_i b_{jk} |ijk\rangle$$

Suppose a 4x4 unitary transformation $U = (u_{nm})$ is applied to the first two qubits. Determine the overall 8x8 matrix that captures the effect of this transformation on all three qubits, considering that the third qubit remains entangled with the transformed pair.

Answer: U \otimes I


Question: Given that the radius of convergence of the power series \(\sum_{n=0}^{\infty} a_n x^n\) is 6, determine the radius of convergence of the series \(\sum_{n=0}^{\infty} a_n x^{\frac{n}{2}}\).

Answer: 36


Question: Given the equation \(\int_{a_s(\mu_1)}^{a_s(\mu_2)} \frac{d a_s}{\beta(a_s)} = \ln \left(\frac{\mu_1}{\mu_2}\right)\), where \(\beta(a_s) = \beta_1 a_s^2 + \beta_2 a_s^3 + \beta_3 a_s^4 + \beta_4 a_s^5\) with known coefficients \((\beta_1, \beta_2, \beta_3, \beta_4)\), and given \(a_s(\mu_1)\), \(\mu_1\), and \(\mu_2\), calculate \(a_s(\mu_2)\) numerically.

Answer: a_s(\mu_2)


Question: Find the volume of the solid $S$ described by the inequalities $\frac{x+y}{2} < z < \sqrt{x+y}$ in the first quadrant of the $(x, y)$-plane.

Answer: \dfrac{32}{15}


Question: Evaluate the limit: $$\lim_{x\to 0^+}\sum\limits_{r=1}^\infty x^{\frac{1}{r}}.$$

Answer: \infty


Question: Evaluate the expression \( \sum_{m=0}^n (-1)^m m^n {n \choose m} \) for a given integer \( n \).

Answer: (-1)^n n!


Question: Express the sum of the alternating series \( \sum_{k=1}^{2n} \frac{(-1)^{k-1}}{\binom{2n}{k}} \) in terms of \( n \), given that for \( n=1 \) and \( n=2 \), the sum equals \(-\frac{n}{n+1}\).

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


Question: Let $g(x)$ be a function defined on $\mathbb{R}$ that is Lebesgue integrable on every compact interval. Given that \(\int_{-\infty}^\infty|g(x)|dx<\infty\) holds as an improper Riemann integral, can we conclude that the Lebesgue integral and improper Riemann integral of $g(x)$ over $\mathbb{R}$ are equal?

Answer: Yes


Question: Calculate the limit of \((\cos x)^y\) as \((x,y)\) approaches \((0,0)\).

Answer: 1


Question: Determine the radius of convergence for the power series \( \sum_{n>0} \frac{z^{n!}}{n} \).

Answer: 1


Question: Is it possible to construct a set $A \subset \mathbb{R}^2$ that is $L^2$-measurable such that its projection onto the first coordinate is not $L^1$-measurable?

Answer: Yes


Question: Suppose the function $g(x)$ has complex roots of the form $r + is$, where $r$ and $s$ are real numbers. If the integration path $\Omega$ on the real line does not intersect with these roots, determine whether the integral $\int_{\Omega} f(x)\delta(g(x)) \mathrm{d}x$ is zero.

Answer: 0


Question: Calculate the limit: \[ \lim_{n \to \infty} \frac{\sqrt[5]{n^5+2n} - \sqrt[4]{n^4+2n}}{\sqrt[3]{n^3+1} - \sqrt[3]{n^3+n}} \cdot \frac{1}{n^2} \]

Answer: 0


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

Answer: 1


Question: Determine whether there exists an irrational number $b$ such that $b^{b^b}$ is a rational number.

Answer: Yes


Question: Given the equations \(y = (f(u) + 3x)^2\) and \(u = x^3 - 2x\), where \(f(4) = 6\), find \(f'(4)\) if \(\frac{dy}{dx} = 18\) when \(x = 2\).

Answer: -\dfrac{9}{40}


Question: Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty} a_n x^n \), where \( a_n \) is the \( n \)-th prime number.

Answer: 1


Question: Evaluate the integral \( \iint_\omega |xy|\,dx\,dy \), where \( \omega \) is the region defined by \( x^2+y^2=a^2 \).

Answer: \dfrac{a^4}{2}


Question: Compute the limit, if it exists: $$\lim_{r\rightarrow +\infty} \int_{B(0,r)} \frac{y}{1+\sqrt{(x^2+y^2)^5}} \, dx \, dy,$$ where $B(0,r)$ is the open ball of radius $r$ centered at the origin. Use polar coordinates to determine the limit.

Answer: 0


Question: Determine the order of convergence of the fixed point iteration \(x_{n+1} = 2 \sin x_n + \tan x_n - 3x_n\) at the fixed point \(\alpha = 0\).

Answer: 5


Question: In the Zariski topology on \(\mathbb{R}\), determine whether every open set containing a point \(x\) also contains all but finitely many points of the sequence \(x_n = n\).

Answer: Yes


Question: Find the coefficient of $x^{25}$ in the expansion of $(1+x^3 + x^8)^{10}$.

Answer: 2520


Question: Is the space of continuous functions on an arbitrary non-compact subset $X$ of the real numbers, denoted by $C(X)$, complete under the supremum norm?

Answer: No


Question: In a normed vector space, determine whether compact sets, which are both closed and bounded, are necessarily finite-dimensional.

Answer: No


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

Answer: 589824


Question: Let $(b_n)$ be a non-increasing sequence of non-negative numbers. If the series $\sum_{n=1}^\infty b_n$ is divergent, does it necessarily follow that $\lim\limits_{n\to\infty}{n b_n} \neq 0$? Provide a proof or counterexample.

Answer: No


Question: Evaluate the definite integral \(\int_{0}^{\infty} x^{m-1}e^{-\frac{x}{a}}dx\), where \(m\) and \(a\) are positive constants.

Answer: a^m \Gamma(m)


Question: Determine whether the following statement is true or false: If \(x \in y\), then \(\mathcal{P}(x) \in \mathcal{P}(y)\), where \(\mathcal{P}(A)\) denotes the power set of \(A\).

Answer: B


Question: Assume there are $n+1$ nodes $V = \{v_0,v_1,\dots,v_n\}$, with each node $v_i$ having a value $f_i$. The weight between $v_0$ and each node $v_i \in \{v_1,\dots,v_n\}$ is $w_{0i}$, and the values and weights are sampled from a normal distribution with mean 0 and variance 1. Define the difference between $v_0$ and $v_i$ as $\delta_i := w_{0i} \cdot |f_0-f_i|$. If $n$ is divisible by 5, what is the expected value of the sum of the weights of the nodes in the set $V'$ of $\frac{4}{5}n$ nodes with the smallest $\delta_i$?

Answer: 0


Question: Evaluate the limit: $$ \lim_{(x, y)\to(0, 0)} \frac{e^{xy}\sin(xy)}{xy} $$

Answer: 1


Question: Determine the limit and cluster points of the sequence \( b_n = \frac{1}{2 \sqrt{n+1}} - \frac{(-1)^n}{n+1} + o \left( \frac{1}{\sqrt{n+1}}\right) \).

Answer: 0


Question: Evaluate the limit using Riemann sums: $$\lim_{n\to\infty} \left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n}\right)$$

Answer: \ln 3


Question: Determine if the function \( g(x,y) = \begin{cases} \frac{x^{8/3} \, y^{2/3}}{x^2 + y^4} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y) = (0,0) \end{cases} \) is Gateaux differentiable at the origin. If it is, find the Gateaux derivatives in all directions.

Answer: 0


Question:
Given an orthogonal $3 \times 3$ matrix $R$ represented as:

$$R = \begin{bmatrix}
r_{11}&r_{12}&r_{13} \\
r_{21}&r_{22}&r_{23} \\
r_{31}&r_{32}&r_{33} \\
\end{bmatrix}$$

where the parameters $r_{11}, r_{21}, r_{31}, r_{12}, r_{22}, r_{32}$ are known, determine how many unique solutions exist for the remaining parameters $r_{13}, r_{23}, r_{33}$. If there is a unique solution, describe the method to find it using the orthogonality condition, considering that $R$ represents a rotation.

Answer: 1


Question: Let $R$ be a commutative ring and $I$ and $J$ be two ideals of $R$. Define the homomorphism $\phi:R \to R/I \times R/J$ by $\phi(x) = (x+I, x+J)$. Determine the conditions under which the induced morphism $R/\ker\phi \to R/I \times R/J$ is an isomorphism.

Answer: I + J = R


Question: Find all functions \( g: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+} \) that satisfy the equation \( g\left ( g(x)^{3}y \right )=x^{2}g(xy) \) for all \( x,y \in \mathbb{Q}^{+} \).

Answer: g(x) = x


Question: Determine the values of \( \alpha > 0 \) for which the sequence of functions \( \bigl (f_n(x) \bigr)_{n=0}^{\infty} \), defined by \( f_n(x) = n^\alpha x^n (1-x) \) for \( x \in [0,1] \), converges uniformly on the interval \( [0, \frac{1}{2}] \).

Answer: \alpha > 0


Question: Are all the measures in \( \mathcal{M}(\mathbb{R}^d) \), the space of real-valued regular Borel measures on \( \mathbb{R}^d \), finite?

Answer: Yes


Question: Determine the correct operating point \((x_{10}, x_{20}, u_{0})\) for the dynamic system described by the ODE \(10\ddot{y}(t) + \dot{y}(t) = u^{2}(t)\) when linearizing around \(u=1\).

Answer: (0, 1, 1)


Question: Is the expression \(A=\{x \mid x \in A\}\) a valid set definition in ZFC set theory?

Answer: No


Question:
Given a complex matrix \(\mathbf{X} \in \mathbb{C}^{m \times 2}\) with full column rank, determine the matrix \(\mathbf{M}\) that completes the following valid equality:

\[ \mathbf{I}_{m} - \mathbf{X} \left( \mathbf{X}^{*} \mathbf{X} \right)^{-1} \mathbf{X}^{*} = \mathbf{M} \]

Choose the correct option:

(a) \(\mathbf{I}_{m}\)

(b) \(\mathbf{0}\)

(c) \(\mathbf{M}\)

(d) None of the options will work.

Answer: D


Question: Calculate the limit without using L'Hopital's rule or Taylor polynomials: \[\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan(x)}{x - \frac{\pi}{4}}\]

Answer: -2


Question: Given a Hamiltonian circuit, if the maximum weighted edge is removed, does this always result in a minimum weight Hamiltonian path? Answer yes or no.

Answer: No


Question: Determine the maximum number of non-empty subsets that can be chosen from a set with 20 elements such that no two subsets have more than two elements in common.

Answer: 1350


Question: Determine whether the following statement is true or false: If a sequence of continuous functions converges uniformly on a compact set, then the sum of the absolute values of the functions also converges uniformly on that set.

Answer: B


Question: If $p$ and $q$ are primes with $q > p$, and $n^2(1 + kq) = p^2$ for some positive integer $k$, determine the possible values of $1 + kq$.

Answer: 4


Question: Evaluate the limit: \[ \lim_{n \to \infty} n\left(\frac{1}{(n+3)^2} + \frac{1}{(n+4)^2} + \ldots + \frac{1}{(2n+2)^2}\right) \]

Answer: \dfrac{1}{2}


Question: Evaluate the double integral \( \iint_A (x^2 + 4y^2) \, dx \, dy \) where the region \( A \) is enclosed by the parabolas \( y = x^2 \) and \( x = y^2 \).

Answer: \dfrac{3}{7}


Question: Calculate the integral \( \int_\gamma \frac{e^{3z}}{(z+2)^2}dz \) using Cauchy's Integral Formula, where \( \gamma \) is a circle of radius 1 centered at \(-1\).

Answer: 6\pi i e^{-6}


Question: Determine the set of real values of $x$ for which the series \( \sum_{n=2}^\infty\frac{1}{n\sqrt{\ln (n+x^2)}} \) converges.

Answer: \emptyset


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{n!-n^3}{n^5-e^n+3} \]

Answer: -\infty


Question: Calculate the limit: $$\lim \limits_{n \to \infty}\left(1-e^{\left(\frac{t+ \log^2(n)}{t-\log(n)}\right)}\right)^n$$

Answer: e^{-e^{-t}}


Question: In a polyhedron with congruent and regular (equilateral) faces, does this condition imply that the dihedral angles are also congruent? Answer yes or no.

Answer: No


Question: Evaluate the integral \( \int \int_R (x^2 + y^2) \, dA \), where \( R = \{ (x, y) \in \mathbb{R}^2 : |x| \leq |y| \leq 2 \} \).

Answer: \dfrac{64}{3}


Question: Determine whether every power of 16 greater than \(16^4\) contains at least one of the digits 1, 2, 4, or 8 when expressed in base 10.

Answer: Yes


Question: Suppose $X$ and $Y$ are continuous random variables on the same probability space. If $\mathbb{P}(X=x)=\mathbb{P}(Y=x)=0$ for all $x\in\mathbb{R}$, does it follow that $\mathbb{P}(X=Y)=0$? Answer 'yes' or 'no' and justify your answer.

Answer: no


Question: Compute the integral \( \mathcal{J}_2 = \int_{0}^{2\pi} \cos(\cos t)\,dt \).

Answer: 2\pi J_0(1)


Question: Evaluate the limit: $$\lim_{x\to \infty} x^2 \int_{0}^{x} e^{t^3 - x^3}dt$$

Answer: \dfrac{1}{3}


Question: Let $(A_n)$ be a non-decreasing sequence of natural numbers and $p$ be a probability such that $p \in [0,1]$. Prove that if $p < \frac{1}{\liminf_{n}(A_n)^{\frac{1}{n}}}$, then $A_n p^n$ approaches 0 as $n$ approaches infinity.

Answer: 0


Question: Calculate the value of the expression \( \frac{1}{n^2} \sum_{i = 0}^{n} \ln\binom{n}{i} \).

Answer: \dfrac{1}{2}


Question: Determine whether it is true that for every natural number $n$ greater than 2, there exists at least one $n \times n$ magic square.

Answer: True


Question: Three points are randomly selected from independent uniform distributions between 0 and 1. What is the expected area of the triangle formed by these points as vertices in a 2D plane?

Answer: \dfrac{11}{144}


Question: Find the residue of the function \( f(z) = \frac{z}{1-\cos(2z)} \) at each pole, where the poles are of order 2 and located at \( z = \pi k \) for integer \( k \neq 0 \).

Answer: \dfrac{1}{2}


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

Answer: 1


Question: Evaluate the integral \(\int_{0}^{+\infty }\frac{x \log(x)}{(x^{2}+1)^{2}}dx\) and determine whether it converges or diverges.

Answer: 0


Question: Determine whether the statement is true or false: For an irreducible polynomial $f \in \mathbb{C}[x,y]$, the domain $\mathbb{C}[x,y]/(f)$ is normal if and only if it is a Unique Factorization Domain (UFD).

Answer: B


Question: Determine whether there exists a graph that is bipartite, has an Euler circuit, but does not have a Hamiltonian circuit.

Answer: Yes


Question: Determine the number of ways to partition a set with $N$ elements into non-empty, pairwise disjoint subsets using the recurrence relation for Bell numbers.

Answer: B_N


Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a strictly increasing function such that $f(f(x)) = f(x)$ for all $x \in \mathbb{R}$. Find all possible functions $f$. \(\circ\) denotes the composition of two functions.

Answer: f(x) = x


Question: Calculate the sectional curvature of the Poincaré half-plane model using the basis vectors $\partial_1=\partial/ \partial x$ and $\partial_2=y\partial/ \partial y$. Provide your answer as a single numerical value.

Answer: -1


Question: Evaluate the limit: $$\lim_{{x\to0}}\frac{1-\frac{1}{2}x^2-\cos\left(\frac{x}{1-x^2}\right)}{x^4}$$

Answer: \dfrac{23}{24}


Question: Consider a chain of subrings $C \subset B \subset A$, where $A, B, C$ are commutative Noetherian rings with unity. If $A \cong C$ and $A, B, C$ are Artinian, determine whether it is true that $A \cong B$. Provide a justification for your answer.

Answer: A \cong B


Question: Let $p: \tilde{X} \to X$ be a covering space where both $\tilde{X}$ and $X$ are path-connected. If $p(x_1) = p(x_2) = x$, is it always true that $p_\ast(\pi_1(\tilde{X}, x_1)) = p_\ast(\pi_1(\tilde{X}, x_2))$?

Answer: No


Question: Evaluate the limit of the integral \( \int_0^{\pi/2}\dfrac {dx} {\sqrt{\sin^2(x)+\epsilon \cos^2(x)}} \) as \( \epsilon \to 0^{+} \).

Answer: \infty


Question: Evaluate the limit \( \lim_{x \to 0^+} \left( \frac{\ln(1+x)}{3(1+x)^{1/3}-1}\right)^{\frac{x}{\sin^2x} } \) using L'Hopital's rule.

Answer: 0


Question: Evaluate the integral \( \iint_{S}\sin\left(\frac{y - x}{x + y}\right)\,\mathrm{d}A \) where \( S \) is the trapezoidal region of the plane bounded by the lines \( x + y = 1 \), \( x + y = 2 \) and the coordinate axes.

Answer: 0


Question: Evaluate the contour integral \( \oint_{|z|=2} \frac{f'(z)}{f(z)} \, dz \), where \( f(z)=\frac{2 z^{4}-7 z^{3}-2 z-i}{(z-i)^{2}(3 z^{2}-z+9)^{2}} \). The function \( f(z) \) has no zeros or poles on the contour \( |z|=2 \), but may have poles inside the contour. Use the argument principle to determine the value of the integral.

Answer: -6\pi i


Question: Let $A$ be a matrix with linearly independent columns over $\mathbb{Q}$. If $A$ can be transformed into a matrix in reduced row-echelon form with a pivot in each column using only rational elementary row-operations, do the columns of $A$ remain linearly independent over $\mathbb{C}$?

Answer: Yes


Question: Suppose $f$ and $g$ are entire functions and $g(z) \neq 0$ for all $z \in \mathbb{C}$. If $|f(z)| \leq |g(z)|$, which of the following must be true about $f$? (a) $f(z) \neq 0$ for all $z \in \mathbb{C}$ (b) $f$ is a constant function (c) $f(0) = 0$ (d) For some $c \in \mathbb{C}$, $f(z) = c g(z)$

Answer: D


Question: Estimate the sum \(\sum_{n=1}^\infty\frac{(-1)^n}{n^3}\) correct to three decimal places.

Answer: -0.902


Question: Determine the geometric object that represents the solution set of the equation \( \overline{z} = \frac{2}{z} \), where \( z \) is a complex number.

Answer: |z| = \sqrt{2}


Question: Determine the number of continuous and bounded functions on the interval $[0,1]$ that satisfy the functional equation $f(x) = f\left(\frac{x}{2}\right)\frac{1}{\sqrt{2}}$. Provide your answer as a single integer.

Answer: 1


Question: Determine the orbits of the symmetric group $S_{n}$, where $n \geq 4$, acting on the set of ordered pairs of distinct 2-subsets of the set $\{1,2,\ldots,n\}$. An ordered pair of 2-subsets is a pair $(\{a, b\}, \{c, d\})$ where $a, b, c, d$ are distinct elements from the set $\{1, 2, \ldots, n\}$.

Answer: 1


Question: Determine whether the sequence of random variables $X_1, X_2, X_3, \ldots$, where each $X_n$ is a Bernoulli random variable with parameter $\frac{1}{2}$ and $X_n = X_{n-1}$ for $n > 1$, converges almost surely to a random variable $X$ that is also a Bernoulli random variable with parameter $\frac{1}{2}$. Provide a justification for your answer.

Answer: Yes


Question: Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be a polynomial function satisfying the equation \[ f(f(x) - 2y) = 2x - 3y + f(f(y) - x), \] where \( x, y \in \mathbb{R} \). Determine the value of \( f(21) - f(14) \).

Answer: 7


Question: Let $X_1, X_2, \ldots$ be a sequence of independent and identically distributed random variables. Define $M$ as the smallest index $n$ such that $X_n > X_{n-1}$. Find the expected value $\mathbb{E}[M]$. \( \text{Express your answer as a single number.} \)

Answer: e


Question:
Determine if either of the following functions can be a valid probability density function (PDF) or can be made into one by multiplying with a non-negative constant $c$:

$$f_1(z)=\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{\max (1,z^2)}{2}\right)$$

$$f_2(z)=\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{\min (1,z^2)}{2}\right).$$

Answer: f_1


Question: Does there exist an element $y$ in the quaternion group $Q_8$ such that $Q_8$ can be generated by $y$ and its center $Z(Q_8)$, where $Z(Q_8)$ is isomorphic to the cyclic group $C_2$?

Answer: No


Question:
Suppose we have a nonnegative sequence $b_m \in \ell^1$. Define a new sequence $d_n$ as:

$$d_n = \alpha^n \sum\limits_{m=n}^{\infty} b_m \binom{m}{n}\beta^{m-n}.$$

where $0 < \alpha < 1$ and $0 \leq \beta < 1$ satisfy $\alpha + \beta \leq 1$. Find a constant $L$ such that $\|d_n\|_{\ell^1} \leq L \|b_n\|_{\ell^1}$. Determine the value of $L$.

Answer: 1


Question: Evaluate the contour integral \( \oint_{|x+i|=\rho} \frac{1}{(x^2+4) \sqrt{ x^2+4}} \, dx \) as \( \rho \rightarrow 0 \) using the residue theorem.

Answer: 0


Question: A stone block has a square base with corners at coordinates \((0,0,0)\), \((0,3,0)\), \((3,3,0)\), and \((3,0,0)\). The height of its top surface is given by the equation \(z = 2 + xy\). If the density of the stone is \(5000\, \text{kg/m}^3\), calculate the x-coordinate of the block's center of mass.

Answer: \dfrac{30}{17}


Question: Let $X$ and $Y$ be two independent random variables with probability density functions given by $$f_X(t)=f_Y(t)=\begin{cases} e^{-t}, & \text{if } t \geq 0; \\ 0, & \text{otherwise.} \end{cases}$$ Find the probability that $X \leq 2Y$. \(\boxed{\text{Answer}}\)

Answer: \dfrac{2}{3}


Question: Determine the number of roots of the equation \(z^4 - 8z + 10 = 0\) that are located within the annulus defined by \(1 < |z| < 3\).

Answer: 4


Question: Find the limit of the function \( g(x,y) = \frac{x^4y^2}{x^6+y^4+3x^2y^2} \) as \((x,y) \rightarrow (0,0)\).

Answer: 0


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

Answer: 0


Question: Is it possible to construct an uncountable collection $E \subset 2^{\mathbb{N}}$ such that for any two distinct elements in $E$, one is a subset of the other?

Answer: Yes


Question: In a compact Hausdorff uniform space $(X, \mathcal{U})$, does there exist a nested sequence $\{U_n\}_{n\in\mathbb{N}}$ in the uniformity $\mathcal{U}$ such that $U_{n+1}\subseteq U_n$ and the intersection of this sequence is the diagonal $\Delta_X$?

Answer: No


Question: Let \(X\) and \(Y\) be independent random variables uniformly distributed on the interval (0,1). Calculate the probability that the integer closest to the random variable \(N = \frac{X}{Y}\) is 0.

Answer: \dfrac{1}{4}


Question: Calculate $2000! \pmod{2003}$. Use concepts from finite fields to find the solution, given that $2003$ is a prime number.

Answer: 1001


Question: Given a non-trivial $k$-term arithmetic progression ($k$-AP) in the set $[n] = \{1,2,\dots, n\}$, where $a_1, a_2, \dots, a_k$ are numbers in $[n]$ with a common difference $d > 0$ and $k \ge 3$, determine if $n$ is a valid upper bound on the number of $k$-APs in $[n]$ that contain a specific number $a$. A current estimate is $kn/(k-1)$, derived from the $k$ possible positions for $a$ within the progression and an upper limit of $n/(k-1)$ for the common difference.

Answer: n


Question: Given topological spaces $U$ and $V$, and their respective topological subspaces $A \subset U$ and $B \subset V$, let $f:A \to B$ be a homeomorphism. Determine if it is true that if $a \in \delta A$, then $f(a) \in \delta B$. Here, $\delta X$ denotes the boundary of $X$. Provide a justification for your answer.

Answer: No


Question: Consider the theorem: If $f$ is a polynomial of degree $\leq n$, all divided differences $f[x_0,x_1,\cdots, x_i]$ are zero for $i\geq n+1$. Is the converse true? If $f$ is $n$-times continuously differentiable on $[a,b]$ with all divided differences $f[x_0,x_1,\cdots, x_i]=0$ for $i\geq n+1$, can we conclude that $f$ is a polynomial of degree $\leq n$? Answer yes or no.

Answer: Yes


Question: Given an integrated-circuit production line with a long-term yield of 80% (defective rate of 20%) and a standard deviation of 0.04 in the proportion defective from day to day, assume a Beta prior distribution for the proportion defective ($\theta$). On a specific day, a sample of 100 units is inspected, and 38 defective units are found. Calculate the mean of the posterior distribution of $\theta$ for this day.

Answer: 0.2905


Question: Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a non-constant, thrice differentiable function such that \(f\left(\frac{1}{n} + 1\right) = 1\) for all integers \(n\). Determine the value of \(f''(1)\).

Answer: 0


Question: Suppose we have a sequence of identically distributed random variables, $X_1, X_2, \ldots, X_N$, which are not necessarily independent. Given that $a_1 + a_2 + \ldots + a_N = 1$, let $S = a_1 X_1 + a_2 X_2 + \ldots + a_N X_N$. Does $S$ have the same distribution as any individual $X_i$? Provide a justification for your answer.

Answer: No


Question: Find the maximum value of the series \( G = 1 + \frac{1}{2}\cos \theta + \left(\frac{1}{2}\right)^2\cos(2\theta) + \left(\frac{1}{2}\right)^3\cos(3\theta) + \cdots \).

Answer: 2


Question: Evaluate the line integral \( \int_C x^2 y^2 \, dx + 4xy^3 \, dy \) using Green's Theorem, where \( C \) is the positively oriented triangle with vertices at \( (0,0), (1,3), \) and \( (0,3) \).

Answer: \dfrac{318}{5}


Question: Determine whether a non-simple group must have a normal Sylow subgroup. Answer with 'Yes' or 'No'.

Answer: No


Question: Evaluate the integral \( \int^{4}_{0}\int_{0}^{\sqrt{4x-x^2}}\sqrt{x^2+y^2}\, \text{dy dx} \).

Answer: \dfrac{128}{9}


Question: Given the binary operation $*$ defined on the integers $\mathbb{Z}$ by $x*y=xy-x-y+c$ for all $x, y \in \mathbb{Z}$, where $c$ is a constant, find the value of $c$ such that $*$ is associative.

Answer: 2


Question: Determine whether every open set in a topology can be expressed as a union of elements from any base of that topology. Provide a justification for your answer.

Answer: Yes


Question: Determine the minimum entropy of a probability vector using Lagrange multipliers, given the convention that $0\log{0} = 0$. Provide your answer as a single value.

Answer: 0


Question: Determine whether it is possible to construct a function that equals 1 at all rational points and 0 at all irrational points, without using a Dirac delta function.

Answer: Yes


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

Answer: \dfrac{1}{2}


Question: Determine the coefficient of \(x^{30}\) in the expansion of \(\frac{1}{1 - x^2 - x^5 - x^{15}}\) using a recurrence relation.

Answer: 241


Question: Consider the sequence \((x_n)_{n \ge 0}\) defined by \(x_0 > \sqrt{\alpha}\), where \(\alpha > 1\), and the recursive relation \(x_{n+1} = \frac{x_n + \alpha}{x_n + 1}\) for all \(n \in \mathbb{N^0}\). Determine whether this sequence converges, and if it does, find the limit.

Answer: \sqrt{\alpha}


Question: In a regular nonagon (9-sided polygon) inscribed in a circle, let AB and BC be two adjacent sides, and let M be the midpoint of AB. Let N be the midpoint of the radius OT, where OT is perpendicular to BC. Find the measure, in degrees, of the angle OMN.

Answer: 30


Question: Solve for $x$ in the equation: \[ \sqrt[3]{1+\sqrt{x}}+\sqrt[3]{1-\sqrt{x}}=\sqrt[3]{5} \] using the hint that if we set $a=\sqrt[3]{1+\sqrt{x}}$ and $b=\sqrt[3]{1-\sqrt{x}}$, we have $a+b=\sqrt[3]{5}$ and $a^3+b^3=2$. Find the value of $x$.

Answer: \dfrac{4}{5}


Question: Calculate the volume of the solid that lies above the region $\Omega=\{(x,y)\in \mathbb{R}^2:0<y<x<2y<2\}$ and below the plane $z = y$. Provide your answer as a single numerical value.

Answer: \dfrac{1}{3}


Question: Find the Radon-Nikodym derivative \( \frac{d\mathbb{P}_1}{d\mathbb{P}_0} \) where \( \mathbb{P}_0 \) is the Lebesgue measure on the interval \([0,1]\) and \( \mathbb{P}_1 \) is the probability measure given by \( \mathbb{P}_1([a,b])=\int_a^b (4w-1) \, d\mathbb{P}_0(w) \).

Answer: 4w - 1


Question:
Let $g:\mathbb{R} \to \mathbb{C}$ be a function such that $\lim_{x \to 0} |g(x)| = +\infty$. Determine which of the following statements is false:

(a) For every $\varepsilon \in (0,1)$, there exists $|x| \leq \frac{\varepsilon}{2}$ such that $|g(x)| \geq 2$.

(b) There exists $\varepsilon \in (0,1)$ such that $\sinh(\Re(g(x)) + |\Im(g(x))|^2) > \varepsilon$ if $x \in (\varepsilon, 2\varepsilon)$.

Answer: B


Question: Let $S$ be a smooth, compact hypersurface without boundary in the Euclidean space $\mathbb{R}^N$. Assume that the interior region of $S$, which is the bounded connected component of $\mathbb{R}^N \setminus S$, is contractible. Does this imply that $S$ is diffeomorphic to a sphere?

Answer: Yes


Question: Determine whether there exists a power series of the form \( \sum_{n=1}^{\infty}a_nx^n \) for which the ratio test, given by \( \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| \), fails to converge.

Answer: Yes


Question: In a simple group of order 168, there are 8 Sylow 7-subgroups. Determine the order of the normalizer of a Sylow 7-subgroup.

Answer: 21


Question: Let \( X \) be a non-empty subset of \( \mathbb{R}^n \). Is it true that \( X \) is necessarily the union of its convex components, where the convex components are the maximally convex subsets of \( X \) ordered by inclusion?

Answer: Yes


Question: Compute the line integral of the vector field \( F(x,y) = (x^2y, y^2x) \) along the path \( r(t) = (\cos t, \sin t) \) for \( t \in [0, 2\pi] \).

Answer: 0


Question: Given that $A$, $B$, and $C$ are angles of a triangle, and the equations $5\sin(A) + 12\cos(B) = 15$ and $12\sin(B) + 5\cos(A) = 2$ hold, find the measure of angle $C$.

Answer: 30^\circ


Question: Compute \( E(Y|X+Y=m) \) using the method of conditional expectation, where \( X \) and \( Y \) are independent geometrically distributed random variables with the same parameter \( p \).

Answer: \dfrac{m}{2}


Question: Determine whether the infimum \( \inf_{f \in X} J(f) \) exists for the functional \( J(f) = \int_0^1 \left(f'(x)^2 - 4\pi^2 f(x)^2\right) dx \), where \( X = \{ f \in C^1[0,1] : f(0) = 0 = f(1) \} \). If it exists, find its value.

Answer: -\infty


Question: Evaluate the limit: $$ \lim _{x\to 0} \frac{x^2\sin(\frac{1}{x})}{\sin x} $$

Answer: 0


Question: Compute the size \(\alpha\) of a Wald test for the null hypothesis \(H_0: \lambda = \lambda_w\) against the alternative hypothesis \(H_1: \lambda \neq \lambda_w\), where \(\lambda_w > 0\) is a given value, for a random sample \(X_1, \ldots, X_n\) from a Poisson distribution with parameter \(\lambda\).

Answer: \alpha


Question:
Solve the following system of congruences for the integer $n$:
\[ \begin{align*}
2n &\equiv 3 \pmod{5} \\
3n &\equiv 4 \pmod{7} \\
4n &\equiv 5 \pmod{9} \\
5n &\equiv 6 \pmod{11}
\end{align*} \]
Find the smallest non-negative integer $n$ that satisfies all these congruences.

Answer: 3464


Question: Suppose we have \(n\) independent variables from \(U[0,1+\theta]\) and let \(\lambda=1+\theta\). The estimator of \(\theta\) is given by \(\bar{\theta}=2 \bar{X} - 1\). Find the bias of \(\bar{\theta}\) given that the distribution of \(\bar{X}\) for \(n\) variables from \(U[0,1]\) is the Bates distribution.

Answer: 0


Question: Evaluate the limit: $$\lim_{x\rightarrow -1^+}\frac{{e^{\frac{x}{x+1}}}}{x+1}$$

Answer: 0


Question: Determine whether the following statement is true or false: In every weighted graph with only positive edges and at least 11 vertices, there exist at most $2n^2$ shortest paths between every two vertices, where $n$ is the number of vertices.

Answer: False


Question: Let $k \in \mathbb{N}$ and let $f : \mathbb{R}^d \rightarrow [0,\infty)$ be a lower semicontinuous function. Determine whether there exists an increasing sequence of $k$-times continuously differentiable functions $(g_n)_n \subset C^k(\mathbb{R}^d ; [0,\infty))$ that converges pointwise to $f$. Specifically, does there exist such a sequence where $g_n(x) \leq g_{n+1}(x)$ for all $x \in \mathbb{R}^d$ and $n \in \mathbb{N}$, and $\lim_{n \rightarrow \infty} g_n(x) = f(x)$ for all $x \in \mathbb{R}^d$?

Answer: Yes


Question: Determine whether every normal space is regular if the T1 axiom (every singleton set is closed) is not assumed. Provide a true or false answer.

Answer: False


Question: Calculate the expected number of tosses required to obtain either $n$ consecutive heads or $n$ consecutive tails in a sequence of fair coin tosses.

Answer: 2^n - 1


Question: For square matrices A and B, if the determinant of A is nonzero and the determinant of B is the square of the determinant of A, can we always conclude that A and B have the same rank? Answer 'yes' or 'no' and justify your answer.

Answer: no


Question: Evaluate the limit: $$\lim_{x\to\infty}{\left({{(x!)^2}\over{(2x)!}}\right)}.$$

Answer: 0


Question:
Suppose a continuous function $y = f(x)$ satisfies the equation: $$f(x) = \int_{0}^{x} f(t) \, dt.$$ Determine which of the following statements does not hold true for $f(x)$:

1. $f$ is periodic.
2. $f$ is differentiable for all $x \in \mathbb{R}$.
3. The range of $f$ is a singleton.
4. $f$ is neither odd nor even.

Answer: D


Question: Let $X$ be a metric space and $A$ be an open subset of $X$. If $B$ is a compact subset of $A$, does there exist an open set $C$ such that $B \subseteq C \subseteq A$? Answer yes or no.

Answer: Yes


Question: A coffee shop roasts a random amount of coffee beans each morning, denoted by the random variable \( R \) (in hundreds of kilograms), with a probability density function given by \( f(r) = -6(r-1)^2+1.5 \) for \( 0.5 < r < 1.5 \). The amount of coffee sold on any given day, \( S \) (in hundreds of kilograms), is uniformly distributed such that \( S < R \). Calculate the expected value \( E[S] \).

Answer: 0.5


Question: Calculate the limit \(\lim_{{x\to 1^+}} \frac{\sin(x^3-1)\cos\left(\frac{1}{1-x}\right)}{\sqrt{x-1}}\) using the substitution \(x-1=y\) and applying relevant theorems to determine the limit.

Answer: 0


Question: Is the ideal $I = \langle 5, 2 + \sqrt{145} \rangle$ in $\mathbb{Z}[\sqrt{145}]$ a principal ideal?

Answer: Yes


Question: Find the maximum value of the expression \[ f(\theta) = \frac{\tan\left(\theta + \frac{2\pi}{3}\right) - \tan\left(\theta + \frac{\pi}{6}\right) + \cos\left(\theta + \frac{\pi}{6}\right)}{\sqrt{3}} \] when \( \theta \in \left[-\frac{5\pi}{12}, -\frac{\pi}{3}\right] \). If this maximum value can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are coprime integers, determine the value of \( a - b \).

Answer: 5


Question: In an equilateral triangle $\triangle{ABC}$, points $E$, $F$, and $G$ are located inside the triangle such that they form four smaller triangles of equal area. Determine the ratio $\frac{AF}{FG}$.

Answer: 2


Question: Determine whether there exists a sequence of consecutive numbers such that each number in the sequence can form a perfect rectangle.

Answer: Yes


Question: Evaluate the integral \( \int_0^\pi \sin{(x \cos{t})}\cos{t}\; dt \) using series representation and integration by parts.

Answer: \pi J_1(x)


Question: Evaluate the series \(\sum_{n=2}^\infty \frac{2n+5}{n^3+4n^2+3n}.\)

Answer: \dfrac{67}{72}


Question: Determine whether the following statement is true or false: If a random variable $X$ has a finite expected value $E[X] < \infty$, then $E[|X|] < \infty$. In other words, does having a finite mean imply that $X$ is in $L^1$?

Answer: True


Question: Let $H$ be a normal subgroup of $G$ with index $[G: H] = m$. Prove or disprove that for any $g \in G$, $g^m \in H$. \( \text{(True or False)} \)

Answer: A


Question: For a given base $b$, determine the probability that a normal number contains a sequence of digits that repeats infinitely often.

Answer: 1


Question: Using Burnside's Lemma, calculate the number of distinct ways to color the 5 regions of a regular pentagon using 4 different colors, considering two colorings equivalent if they can be transformed into each other by the pentagon's symmetries.

Answer: 136


Question: Evaluate the integral \( I = \int_{0}^{\infty} \cos(x) e^{-x} \, dx \).

Answer: \dfrac{1}{2}


Question: Is it always possible to find a closed affine sub-variety in $\mathbf{A^{n+1}}$ such that its coordinate ring is isomorphic to the coordinate ring of the non-zero locus of a single polynomial in $K[x_1, ..., x_n]$?

Answer: Yes


Question: Is the map \( T: C \to C \) defined by \( A \mapsto AX \cap C \) (where \( X \) is a fixed point not on \( C \) and we take the point which isn't \( A \), except when \( AX \) is a tangent) a projective transformation?

Answer: Yes


Question: Evaluate the limit: $$\lim_{\alpha\to 0}\int_0^{\infty}\sin(\alpha x) \,\mathrm{d}x$$

Answer: 0


Question: Evaluate the integral \( 4\pi R \int_{-r}^{r}\sqrt{r^2-y^2}dy \) to find the volume of a torus.

Answer: 2\pi^2 R r^2


Question: Calculate \((2023)^{2024} \mod{2025}\).

Answer: 1741


Question: Find the equation of a function $f(x)$ that satisfies the following conditions: $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$, $f'(0) = -1$, and $f(0)=1$. Provide your answer in the form of an equation.

Answer: f(x) = -x + 1


Question: Find the sum of all positive integers $a \leq 100$ that satisfy the equation \( \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2} \) for some integers $b$ and $c$, where $a$, $b$, and $c$ are coprime.

Answer: 100


Question: Determine whether an algebraically closed field $K$ can be a subset of a division ring $D$, with $K$ contained in the center of $D$ ($K \subseteq Z(D)$), given that $D$ is finite-dimensional over $K$. Provide a justification for your answer.

Answer: No


Question: Given a compact, closed non-convex body $K \subset \mathbb{R}^n$ that contains no holes, determine if there exists a finite set of compact convex closed bodies $\{C_i\}_{i=1}^N$ such that $K = \bigcup_{i=1}^N C_i$ and $C_i \subset K$ for each $i$. Note that $N$ may vary depending on $K$. Provide a justification for your answer.

Answer: Yes


Question: Determine if there exists a formal power series $S(x)$ such that $[S(x)]^2 = 2x + 2$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: The cardinality of the Cartesian product of a countable set with itself an uncountable number of times is always greater than the cardinality of the Cartesian product of an uncountable set with itself a countable number of times.

Answer: B


Question: Let \( h(x) = \int_{-x}^x f(t) \, dt \) where \( f \) is an even function. Given that \( h(x-3) = f(x) \), find the value of \( \int_{-x}^{x-3} h(t) \, dt \).

Answer: 0


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

Answer: 0


Question: Determine whether the closure of every bounded convex set with non-empty interior in \(\mathbb{R}^n\) (where \(n > 1\)) is homeomorphic to a closed ball \(B[a,r] = \{x \in \mathbb{R}^n : d(x,a) \le r\}\), where \(a \in \mathbb{R}^n\) and \(r > 0\).

Answer: Yes


Question: Let $N$ be a $3\times3$ matrix such that $N^3 = O_3$ and $N^2 \neq O_3$. Determine the number of linearly independent eigenvectors of $N$. \( \boxed{} \)

Answer: 1


Question: Determine the periodicity of the entire function $g(z)$, given that $g(iy)$ and $g(2+iy)$ are real for all $y \in \mathbb{R}$. Provide your answer as the period of the function.

Answer: 4


Question: Determine whether the pointwise minimum of two Riemann integrable functions is always Riemann integrable. Provide a justification for your answer.

Answer: Yes


Question: Find all natural numbers \( n \) such that \( S_1 > S_2 \), where \( S_1 = \sum_{k=1}^{4n^2} \frac{1}{k^{2/3}} \) and \( S_2 = \sum_{k=1}^{n} \frac{1}{k^{1/2}} \).

Answer: \mathbb{N}


Question: Let $\Omega = [0,1] \times [0,1]$ with $P$ as the Lebesgue Measure. Define $X(x,y)=x$ and $Y(x,y)=y$. Given $f(x,y)=x^2y$ and $\mathcal{G}=\sigma(Y)$, find $E[f(X,Y)|\mathcal{G}]$. Provide your answer as a function of $y$.

Answer: \dfrac{y}{3}


Question: Let $X$ be an integral scheme and let $\mathcal{I}$ and $\mathcal{J}$ be coherent ideal sheaves on $X$. Suppose there exists a point $x \in X$ such that the stalks $\mathcal{I}_x$ and $\mathcal{J}_x$ are equal as ideal sheaves in $\mathcal{O}_{X,x}$. Does it follow that there is an open set $U$ containing $x$ with $\mathcal{I}\vert_U = \mathcal{J}\vert_U$?

Answer: Yes


Question: Given two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the parabola $y^2 = 4ax$, a circle is drawn with $AB$ as its diameter and touches the parabola at a third point $C$. Find the value of $|y_1 - y_2|$. \( \boxed{} \)

Answer: 8a


Question: Calculate the covariance between $X$ and $Y^3$ for a bivariate normal distribution $(X,Y)$ with mean vector $\begin{pmatrix} 0\\ 0 \end{pmatrix}$ and covariance matrix $\begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix}$.

Answer: 3\rho


Question:
In a study of product lifespan, consider the hypothesis test:

\[H_0: \mu = 2\]
\[H_1: \mu < 2\]

where \(\mu\) is the expected lifetime of the product, assumed to follow an exponential distribution with parameter \(\lambda\). The probability density function is:

\[f_X(x) = \begin{cases} \lambda e^{-\lambda x} & \text{for } x \geq 0 \\ 0 & \text{otherwise} \end{cases}\]

Given that \(E[X] = \mu = \frac{1}{\lambda}\), a sample of 10 products yields a sample mean \(\bar{X}\). Define \(c > 0\) such that if \(W \geq c\), the null hypothesis is accepted; otherwise, it is rejected. Determine the critical value \(c\) for a significance level of 95%.

Answer: 1.085


Question: Consider a group $G$ with subgroups $H$ and $K$, where $K$ is normal in $H$ and maximal in $H$. Does it follow that $K$ is also normal in $G$? Provide a justification for your answer.

Answer: No


Question: Find the sum of the infinite series: \[ 1 + \frac{1}{1!} \cdot \frac{1}{4} + \frac{1 \cdot 3}{2!} \cdot \left(\frac{1}{4}\right)^2 + \frac{1 \cdot 3 \cdot 5}{3!} \cdot \left(\frac{1}{4}\right)^3 + \cdots \] where the dot represents multiplication and the exclamation mark denotes factorial.

Answer: \sqrt{2}


Question: Determine if the Banach algebra $(\ell^2, \|.\|_2)$ with the coordinatewise product is reflexive. If it is reflexive, find its dual space.

Answer: \ell^2


Question:
In a vector space with a weighted Hermitian inner product defined as \((f,g)_w=\int_0^1 f(x) \overline{g(x)} w(x) dx\), where the weight function is \(w(x)=x^2\), determine a function \(g(x)\) such that the Cauchy-Schwarz-Bunyakovsky inequality results in the following inequality:

$$ \left| \int ^1 _0 x^2 f(x) dx \right|^2 \le \frac{1}{3} \int^1_0x^2|f(x)|^2dx $$

Answer: 1


Question: Find the floor of the limit \( L = \displaystyle\lim_{m\to \infty}\sum_{p=1}^m \frac{p}{2p+m+m^2} \).

Answer: 0


Question: Calculate the total curvature of the curve given by the parameterization \( \exp(t) \cdot (\cos(t), \sin(t)) \), where \( t \in [0, 2\pi] \).

Answer: 2\pi


Question: Find the value of \( x \) such that the infinite power tower \( x^{x^{x^{.^{.^.}}}} = 2 \).

Answer: \sqrt{2}


Question: Consider a sequence where the first value $a_1$ is the sum of two numbers drawn from a normal distribution with mean $\mu$ and variance $\sigma^2$. The second value $a_2$ is the product of $a_1$ and the sum of two other numbers drawn from the same distribution. The third value $a_3$ is the product of $a_2$ and the sum of two more numbers drawn from the distribution, and so on. Specifically, $a_1 = X_1 + X_2$ and $a_2 = a_1 \cdot (X_3 + X_4) = (X_1 + X_2)(X_3 + X_4)$. What is the expected value of $a_n$?

Answer: (2\mu)^n


Question: Given the matrix \( M = \begin{bmatrix} -1 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 \\\ 0 & -1 & 0 & 0 & 0 & -1 & 0 & 1 & 0 \\\ 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \), its Smith normal form is \( S = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \). Determine the cokernel of the group homomorphism \( \varphi: \mathbb{Z}^9 \to \mathbb{Z}^3 \) induced by this matrix.

Answer: 0


Question: Find the function $f(x)$ given that it is continuous, differentiable, and satisfies the equation $f(x + y) f(x - y) = f^2(x)$ for all $x$ and $y$. Additionally, it is given that $f(0)$ is non-zero and $f(1) = 1$. What is $f(x)$?

Answer: 1


Question: Determine the number of fixed points for the function \( F(x) = e^{2x-3} \) by solving the equation \( F(x) = x \) without using graphical methods.

Answer: 2


Question: Let $p(x)$ be the characteristic polynomial of an $n \times n$ matrix $A$ with real entries. If $p(x)$ has no repeated roots, can a basis of $\mathbb{R}^n$ consisting of eigenvectors of $A$ always be found?

Answer: No


Question: Calculate the sum of the series \( \sum_{r=0}^{n-1}\cos^2(\theta+ \alpha r ) \), where \( \alpha \) is the exterior angle of a regular polygon with \( n \) sides, for any real number \( \theta \).

Answer: \dfrac{n}{2}


Question: Suppose $f:[0,\infty)\rightarrow \mathbb{R}$ is a twice differentiable function and $f''$ is bounded. Prove that if $\lim_{x\rightarrow \infty}f(x)$ exists and is finite, then $\lim_{x\rightarrow \infty}f'(x)=0$.

Answer: 0


Question: Consider the sequence $a_n = (-1)^n$. Does this sequence satisfy the conditions that the series $\sum |a_n|$ converges to a rational number, while $\sum a_n$ diverges?

Answer: No


Question: Determine the connected component $C_{(0,0)}$ of the point $(0,0)$ in the set $X = \{(0,0),(0,1)\} \cup \bigcup_{n \in \mathbb{N}} \left(\left\{\frac{1}{n}\right\} \times [0,1]\right)$. Explain why $C_{(0,0)} = \{(0,0)\}$, even though $(0,0)$ is not an open set in $X$. Provide a detailed justification for your answer.

Answer: \{(0,0)\}


Question: Compute the limit: $$\lim_{n \rightarrow \infty} \left(\left(\frac{9}{4} \right)^n+\left(1+\frac{1}{n} \right)^{n^2} \right)^{\frac{1}{n}}$$

Answer: e


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

Answer: \dfrac{e}{2}


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

Answer: \dfrac{4}{e}


Question:
Let $G$ be a Lie group, $\Gamma$ a lattice in $G$, and $\mu$ the $G$-invariant Haar probability measure on $X:=G/\Gamma$. Let $f$ be a continuous function on $X$. If for a one-parameter subgroup $(g_t)$ of $G$, we have

$$\lim_{T\to \infty}\frac{1}{T} \int_0^T \mathbf{1}_{F_{g_t}}(x) \, dt >0$$

for $\mu$-almost every $x\in X$, where $F_{g_t}:=\{x\in X:\liminf_{t\to \infty}f(g_t.x) < \delta\}$, determine if it follows that

$$\mu\{x\in X:\limsup_{t\to \infty} f(g_t.x) < \delta\}>0.$$

Answer: No


Question: Determine the constant $B$ that minimizes the error between the prime counting function $\pi(x)$ and the approximation $\frac{x}{\ln x - B}$ for large $x$, given that the error is bounded by $O\left(\frac{x}{\ln^3 x}\right)$. What is the value of $B$?

Answer: 1


Question: Determine the minimum value of $n$ for which the probability $P(0.39 \leq \overline{Y} \leq 0.41)$ is at least $0.95$, where $\overline{Y}$ is the sample mean of $n$ independent and identically distributed Bernoulli random variables with success probability $p = 0.4$. Assume the mean and variance of a Bernoulli variable are $np$ and $np(1-p)$, respectively.

Answer: 9220


Question: Evaluate the double integral \( \int\int_{D} \frac{xy}{y^{2}-x^{2}}\,dx\,dy \) over the region \( D \) bounded by the curves \( x^{2}-y^{2} = 1 \), \( x^{2}-y^{2} = 4 \), \( \frac{x^{2}}{16}+\frac{y^{2}}{4} = 1 \), and \( \frac{x^{2}}{4}+y^{2} = 1 \).

Answer: 0


Question: Find the maximum value of the function \( f(x, y, z) = xy + 2yz \) subject to the constraint \( x^2 + y^2 + z^2 = 36 \).

Answer: 18\sqrt{5}


Question: Find the maximum likelihood estimator (MLE) of \( \theta \) for an i.i.d. sample \( X_1, \ldots, X_n \) from a uniform distribution on the interval \((0, \theta)\), where \( \theta > 0 \) is unknown.

Answer: X_{(n)}


Question:
Determine if an error has been made in calculating the inward flux of the vector field \( F=\langle y,-x,z \rangle \) over the surface \( S \) of the solid bounded by the paraboloid \( z=\sqrt{x^2+y^2} \) and the plane \( z=3 \). The integral simplifies to:

\[-\int_0^{2\pi} \int_0^3 r^2 \, dr \, d\theta\]

The calculated flux is \(-18\pi\), but the provided answer is \(-9\pi\). Has an error been made?

Answer: Yes


Question: Determine whether the statement "For any open subset $A$ of a topological space, $A' \subseteq A$" is true or false. Provide a proof or a counterexample.

Answer: False


Question: Find the limit of the expression \( \left(\cos\left(\frac{\pi}{x}\right)\right)^{x^2} \) as \( x \to 0 \) using a method other than L'Hopital's Rule.

Answer: 1


Question: Find the value of $c_0$ in the power series representation of the function $e^{-a(gx-b)^{2}}$.

Answer: e^{-a b^2}


Question: Find the minimum area of the image of a univalent function $g: \mathbb{D} \rightarrow \mathbb{C}$ with $g(0) = 0$ and $|g'(0)| = k$, where $k$ is a positive constant.

Answer: \pi k^2


Question: Determine whether a matrix can have multiple square roots, even if it is not positive definite.

Answer: Yes


Question: Find the limit of the function \( f(x) = \frac{2x^3+x^2+x\sin(x)}{(\exp(x)-1)^2} \) as \( x \) approaches 0 without using L'Hôpital's rule.

Answer: 2


Question: Determine the limit of \( \left(\frac{1}{n^2}+\frac{2}{n^2-1}+\frac{3}{n^2-2}+ \cdots + \frac{n}{n^2-n+1}\right) \) as \( n \to \infty \).

Answer: \dfrac{1}{2}


Question: Calculate the Lie derivative of the metric tensor \(g_{ab} = \begin{pmatrix} 1 & 0 \\ 0 & x^2 \end{pmatrix}\) along the vector field \(X = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\).

Answer: \begin{pmatrix} 0 & 0 \\ 0 & 2x \end{pmatrix}


Question: Consider a Markov chain with a one-periodic, irreducible, and diagonalizable transition matrix $P$. If $P$ converges exponentially, is the rate of convergence (in terms of total variation norm) still determined by the second-largest eigenvalue?

Answer: Yes


Question: Given a matrix $A$ such that $A^2 = A - I$, where $I$ is the identity matrix, find the value of $A^{15}$.

Answer: -I


Question:
Determine which of the following groups is isomorphic to $S_3 \oplus (\mathbb{Z}/2\mathbb{Z})$:
1. $\mathbb{Z}/12\mathbb{Z}$
2. $\mathbb{Z}/6\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$
3. $A_4$, the alternating group of order $12$
4. $D_6$, the dihedral group of order $12$

Answer: D


Question: Evaluate the limit \( \lim_{x\rightarrow 0}\frac{f(x)}{x^{8}} \), where \( f(x)=\int_{0}^{x^{4}}e^{\sqrt t }\cdot\sin t \ dt \).

Answer: \dfrac{1}{2}


Question: Given four random variables $W, X, Y, Z$ where $W$ and $X$ have the same distribution, and $Y$ and $Z$ have the same distribution, and $E(W+Z) = 0$. If $E(X+Y)$ exists, is it necessarily true that $E(X+Y) = 0$?

Answer: 0


Question: Given three complex numbers with unit modulus, $z_1$, $z_2$, and $z_3$, where \( |z_1 - z_2|^2 + |z_1 - z_3|^2 = 4 \), determine the possible values of \( |z_2 + z_3| \).

Answer: [0, 2]


Question: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed (i.i.d.) random variables with probability density function $f(x) = e^{-x}$ for $x > 0$. Given that $X_n = \max(X_1, X_2, \ldots, X_n)$, find $\lim_{n \to \infty} P(X_n < 2 + \log n)$. Assume that the distribution of $X_n$ approaches a Normal distribution as $n$ grows large.

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


Question: Solve the equation \(-2^x + 3^{x/2} + 1 = 0\) for integer values of \(x\).

Answer: 2


Question: Determine whether the following statement is true or false: Let $f$ be a differentiable function on $\mathbb{R}$. If $f \circ f \circ f$ is increasing and $f \circ f$ is decreasing, then $f$ is decreasing.

Answer: True


Question: Calculate the expected value $E(X^3)$ for a random variable $X$ that follows a normal distribution with mean $\mu$ and variance $\mu^2$, i.e., $X \sim N(\mu, \mu^2)$. Use this to find the covariance $\operatorname{Cov}(X, X^2)$. Express your answer in terms of $\mu$. \( \operatorname{Cov}(X, X^2) = E(X^3) - \mu(\mu^2 + \mu^2) \).

Answer: 2\mu^3


Question:
Determine which of the following sets has a cardinality greater than the cardinality of the set of real numbers, $\mathbb{R}$.

(A) The set of all functions from $\mathbb{N}$ to $\mathbb{N}$
(B) The set of all subsets of $\mathbb{R}$
(C) The set of all polynomials with real coefficients
(D) The set of all finite subsets of $\mathbb{R}$

Answer: B


Question: If a function \(f:\mathbb{T}\to\mathbb{C}\), given by \(f(t)=\sum_{k}a_ke^{ikt}\), has Fourier coefficients \(a_k\) that belong to \(\ell^1\), meaning \(\sum_{k}|a_k|<\infty\), is \(f\) necessarily a continuous function?

Answer: Yes


Question:
Determine the radius of convergence for the power series solution of the Legendre differential equation given by:
\[ y(x) = c_{1}y_{1}(x) + c_{2}y_{2}(x) \]
where
\[ y_{1} = x + \sum_{n = 1}^{\infty} \left [ \frac{(-1)^{n}}{(2n+1)!} \prod_{k=1}^{2n} \left ( l+(-1)^{k}k \right ) \cdot x^{2n+1} \right ]\]
\[ y_{2} = 1 + \sum_{n = 1}^{\infty} \left[ \frac{(-1)^{n}}{(2n)!} \prod_{k=1}^{2n} \left ( l+(-1)^{k}(k-1) \right ) \cdot x^{2n} \right] \]

Answer: 1


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

Answer: \sin 1


Question: Find the last two digits of the number $3^{100} \cdot 11^{50} + 7^{518}$. What is the remainder when this expression is divided by 100?

Answer: 50


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

Answer: s > 1


Question:
Find the exact value of the limit as \( n \) approaches infinity:

\[ \lim_{n\to\infty}\prod_{k=1}^{\lceil\alpha\sqrt{n}\rceil-1}\frac{2n-2k}{2n-k} \]

for any \( \alpha>0 \).

Answer: e^{-\alpha^2/4}


Question: In a unitary ring \( R \), if elements \( a \) and \( b \) satisfy \( ab = 1 \), is \( a \) necessarily invertible?

Answer: No


Question: Evaluate the limit: \[\lim_{x \to 0}\left(\frac{1+ \tan x}{1+ \sin x}\right)^{\frac{1}{\sin^3x}}.\]

Answer: \sqrt{e}


Question: Determine the conditions on the parameter $p$ such that the curve $y = x^p\cos\left(\frac{\pi}{x}\right)$, for $0 < x \leqslant 1$, has a finite length.

Answer: p > 1


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

Answer: \dfrac{4}{27}


Question: Evaluate the limit \(\lim_{x\to 0}\left(1+\sin(x)-x\right)^{\frac{1}{x^3}}\).

Answer: e^{-\frac{1}{6}}


Question: A triangular prism has infinite height, with three of its edges parallel to the z-axis passing through the points (0, 0, 0), (3, 0, 0), and (2, 1, 0). Calculate the volume of the region bounded by the prism's side surfaces and the planes z = 2x + 3y + 6 and z = 2x + 7y + 8.

Answer: 5


Question: Determine the conditions on the symmetric matrix $A \in \mathbb{R}^{n \times n}$ such that the set $Z = \{x \in \mathbb{R}^n \mid x'Ax = 0\}$ has Lebesgue measure zero.

Answer: A \neq 0


Question: Find the limit of \((1+1/n^2)^{n^3}\) as \(n\) approaches infinity.

Answer: \infty


Question: Find the dimension of the space of cubic polynomials in \( \mathbb{P}^4 \) that pass through three general subspaces \( L, M, N \) of codimension 2 and vanish at six general points on these subspaces.

Answer: 5


Question: Determine the number of distinct terms in the expansion of \((x+\frac{1}{x} + x^2 + \frac{1}{x^2})^{15}\), where distinct terms have different powers of \(x\).

Answer: 61


Question: Determine whether the sequence \( b_n = \sum_{i=1}^n \frac{1}{\sqrt{n^2 + 2i}} \) converges. If it converges, find its limit.

Answer: 1


Question: Find all non-constant functions \( g: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation \( g(a+b) = g(a) + g(b) \) and the condition \( \lim\limits_{x\rightarrow \infty} g(x) = 0 \).

Answer: 0


Question: Given that the eigenvalues of matrix $A$ are $1, 2, 3$, find the determinant of the matrix $A^2 + A^T$.

Answer: 144


Question: Let \( f : [0, 1] \to \mathbb{R} \) be a differentiable function such that \( f'(x) = \sqrt{1 + f^2(x)} \) for all \( x \in [0, 1] \). Given that \( f(0) = 1 \) and \( f(1) = -1 \), find the value of the integral \( \int_{0}^{1} f'(x) \, dx \).

Answer: -2


Question: Does the closure of the set \( T = \left \{ (x, \cos\left(\frac{1}{x}\right)) ; 0 < x \leq 1 \right \} \) include the line segment \( L = \{0\} \times [-1,1] \)?

Answer: Yes


Question: Calculate the sum \(\sum_{k=0}^{n}\binom{n}{k}\left(\binom{m+k+1}{m-n}+(-1)^{k+1}\binom{m+2(n-k)+1}{m}\right)\) for integers \(m\) and \(n\) where \(m \ge n \ge 0\).

Answer: 0


Question: Compute the first Chern class \( c_1(L_D) \) of the line bundle \( L_D \) associated with the effective divisor \( D = [F] \) on \( \mathbb{C}\mathbb{P}^3 \), where \( F = z_0z_1^2 + z_2z_3(z_2 - z_3) \).

Answer: 3H


Question: Determine if the sequence $(N_n)_{n \ge 2}$, where $N_n=(X_{n-2}, X_{n-1}, X_n)$ and $(X_n)$ is an independent sequence with $P(X_n=1)=P(X_n=0)=\frac{1}{2}$, is a Markov chain.

Answer: Yes


Question: Determine whether a function can have the intermediate value theorem property without having any jump discontinuities or discontinuities of the second kind.

Answer: Yes


Question: For any integer \(n > 2\), determine if there exists a pair of primes \(p\) and \(q\) such that \(p \cdot q \equiv 1 \pmod{n}\) and \(p \cdot q > n^2\).

Answer: Yes


Question: Consider an analytic function $f(z)$ defined for $\text{Im}(z) \leq 0$. Its conjugate $g(z) = \overline{f(\bar{z})}$ is analytic for $\text{Im}(z) \geq 0$. Define $h(z)$ as $f(z)$ for $\text{Im}(z) \leq 0$ and $g(z)$ for $\text{Im}(z) > 0$. Is the contour integral of $h(z)$ around any closed loop enclosing the origin equal to zero?

Answer: No


Question: Determine if the set of all squares of permutations in $S_5$ is equal to the set of all even permutations in $S_5$. Provide a justification for your answer.

Answer: Yes


Question: Is it possible to construct a sequence of countably infinite subsets of the natural numbers such that the intersection of any two of these subsets is empty and the union of all of these subsets is not the entire set of natural numbers?

Answer: Yes


Question: Evaluate the limit: $$\lim \limits_{(x,y) \to (0,0)}\frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4}$$

Answer: 0


Question: Determine the limit function of the series \( \sum\limits_{k=0}^{\infty}\frac{x^k}{k!}k^2 \), where \( x \in (0,1) \).

Answer: x(x + 1)e^x


Question: Calculate the area enclosed by the curve given by the polar equation \( r = \Theta(3\theta - \pi) \Theta(\sin(3\theta)) \), where \( \Theta(x) \) is the Heaviside step function.

Answer: \dfrac{\pi}{3}


Question: If the sequence $\{\ln(a_n)\}$ of positive terms converges to $a$, does it imply that the sequence $\{a_n\}$ converges to $e^a$? Justify your answer with a formal $\epsilon-\delta$ proof.

Answer: e^a


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

Answer: 0


Question: Let $F$ be an algebraically closed field. Determine the dimension of the algebraic set $X = V(y - x^2, y^2 - y + z^2)$.

Answer: 1


Question:
Given the system of equations for real numbers $x, y, z$:

1) $x^2 + xy + yz + zx = 30$
2) $y^2 + xy + yz + zx = 15$
3) $z^2 + xy + yz + zx = 18$

Find the value of $x^2 + y^2 + z^2$. \( \boxed{} \)

Answer: 21


Question: Determine all surjective functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), the following equation holds: \[ f(x + f(x) + 2f(y)) = f(2x) + f(2y) \]

Answer: f(x) = x


Question: Find the critical points of the function defined by the following system of equations: \(2(x-1)=-2\lambda x\), \(2(y-3)=-2\lambda y\), \(2(z-1)=2\lambda z\), and \(z^2 - x^2 - y^2 =0\). Determine the minimum distance between these critical points.

Answer: \sqrt{11}


Question: Find the limit of the sequence \[ \left[ \prod_{k=1}^{n} \left(1+\frac{k}{n}\right) \right]^{1/n} \] as \( n \to \infty \).

Answer: \dfrac{4}{e}


Question:
Let $X$ be a standard Borel space, and denote by $\mathcal P(X)$ the space of Borel probability measures on $X$, equipped with the topology of weak convergence. Define the total variation metric $d: \mathcal P(X) \times \mathcal P(X) \to [0,1]$ as:

$$ d(p,q) := \sup_{A \in \mathcal B(X)} |p(A) - q(A)| $$

where $\mathcal B(X)$ is the Borel $\sigma$-algebra of $X$. Determine whether $d$ is a measurable function.

Answer: Yes


Question: Given three sequences of positive real numbers, $(a_n)_{n\in \mathbb{N}}$, $(b_n)_{n\in \mathbb{N}}$, and $(c_n)_{n\in \mathbb{N}}$, where $\displaystyle \limsup_{n\to \infty} \frac{a_n}{b_n} < \infty$ and $\displaystyle \lim_{n\to \infty} \frac{b_n}{c_n} = 0$, determine if it is true that $\displaystyle \limsup_{n\to \infty} \frac{a_n}{c_n} = 0$. Provide a justification for your answer.

Answer: 0


Question: Can a Turing machine with an unbounded tape length generate the decimal representation of any computable number?

Answer: Yes


Question: Is every finite, connected set with a binary relation of the form \( \text{Aut}(X)\setminus X \) isomorphic to the orbit set \( (Y_*) \) of some finite, connected set \( Y \) with a binary relation? Here, a related set \( (X, R_X) \) consists of a set \( X \) and a binary relation \( R_X \subseteq X^2 \), denoted by \( X \) for simplicity and \( R \) for \( R_X \). A map \( f:X\to Y \) is a morphism if \( xRx' \) implies \( f(x)Rf(x') \) for all \( x,x'\in X \). Isomorphism and automorphism are defined conventionally. A set \( X \) is connected if the equivalence relation generated by \( R \) has a single class.

Answer: Yes


Question: Given that \((\tan\alpha)^2 (\tan\beta)^2 + (\tan\beta)^2 (\tan\gamma)^2 +(\tan\alpha)^2 (\tan\gamma)^2 + 2(\tan\alpha)^2(\tan\beta)^2 (\tan\gamma)^2 = 1\), where \(\alpha, \beta, \gamma\) are angles within the domain of the tangent function, find the value of \(\cos2\alpha +\cos2\beta+\cos2\gamma\).

Answer: 1


Question: Evaluate the integral \( \int_0^{2\pi}\int_0^R\frac{rx}{\sqrt{R^2-x^2-y^2}}\,dx\,dy \) by converting to polar coordinates.

Answer: 0


Question: Given a set of $n$ independent and identically distributed (I.I.D.) random variables $X_1, \ldots, X_n$, where each $X_i \sim N(\theta, \theta^2)$, calculate the expected value $E[X_i^2]$. This value is needed to find the Fisher Information $i(\theta)$. Provide your answer as a single expression.

Answer: 2\theta^2


Question: Evaluate the limit \( \lim_{{x \to 0}} \frac{1}{x^2} \cdot \int_{0}^{x} \frac{t+t^2}{1+\sin(t)}dt \) by finding the correct derivative of the expression using L'Hôpital's rule.

Answer: \dfrac{1}{2}


Question: Determine the maximum possible value of the ratio \(\frac{\phi(n)}{n}\), where \(\phi(n)\) is Euler's totient function, representing the number of positive integers less than \(n\) that are relatively prime to \(n\).

Answer: 1


Question: Using the Weierstrass product representation of the Gamma function, derive an alternative expression for the derivative of the Gamma function at \( z = 1 \).

Answer: -\gamma


Question: Determine the dimension of the $R$-module $M$, where $R = \mathbb{C}[T]$ and $M$ is generated by elements $x$ and $y$ subject to the relations $Tx = y$ and $T^2x = y$, as a vector space over $\mathbb{C}$.

Answer: 2


Question: Determine whether the following statement is true or false: For any finite group $G$, if the exponent of $G$ is even, then $G$ must be abelian.

Answer: False


Question: Find the linearization of the differential equation \( y''(t) + y(t) e^{0.02y} = 0 \) near \( y = 0 \).

Answer: y''(t) + y(t) = 0


Question: Evaluate the limit: $$ \lim_{x\to0} \left( \frac{e^x-x^x}{x^2} \right) $$

Answer: \infty


Question: Evaluate the integral \( \int \limits_{0}^{2\pi} g(\cos x) \sin x \, dx \) where \( g \) is a continuous function on \([-1,1]\).

Answer: 0


Question: Find the minimum value of the expression \( y = \sqrt{x^2 + 4x + 13} + \sqrt{x^2 - 8x + 41} \).

Answer: 10


Question: Let $A$ be a square matrix in $\mathbb{R}^{n \times n}$ such that $A^2 = I_n$. Determine whether $A$ has a basis of eigenvectors.

Answer: Yes


Question: Find the total length of the curve defined by the parametric equations \( \gamma(t) = (t - \cos(t), 1 + \sin(t)) \) for \( t \in (-\pi, \pi) \). Use a suitable substitution to evaluate the integral.

Answer: 8


Question: A cylinder has a fixed diameter of 50 mm and a length of 500 mm. Spherical balls have a diameter of 49.5 mm with a standard deviation of 0.1 mm. Estimate the probability that 12 of these balls will fit into the cylinder, considering both the width and length dimensions.

Answer: 0


Question: Does the equation $\mathbb{Z} * \mathbb{Z} / N = \mathbb{Z} * \mathbb{Z}$ imply that the normal subgroup $N$ of $\mathbb{Z} * \mathbb{Z}$ is trivial?

Answer: Yes


Question: Find the number of elements in the conjugacy class of the permutation $\partial = (1\ 10)(2\ 6\ 3\ 7\ 6\ 8\ 12)(4)(9\ 11)$ in the symmetric group $S_{12}$.

Answer: 4989600


Question: For random variables X and Y with a joint normal distribution, if the sample mean is independent of the sample standard deviation, does it imply that X and Y are independent?

Answer: No


Question: Find the limit of the sequence \( \lim\limits_{n\to\infty}\frac{(-2)^{n}n!}{n^{n}} \).

Answer: 0


Question: Determine if the initial value problem \(\begin{cases} y’=|y|\\ y(0)=0 \end{cases} \) has a unique solution. If it does, find the solution.

Answer: 0


Question: Given that $X$ and $Y$ are independent identically distributed (i.i.d.) random variables, and $P(X+Y=0)=1$, determine the characteristic function of $X$. Provide a hint to approach this problem.

Answer: 1


Question: Determine the radius of convergence for the power series $$\sum \left(\frac{n^3}{2^n}\right)^{n^2} x^{2n^2+1}$$.

Answer: \infty


Question: Let $X$ and $Y$ be two independent normal random vectors. If an affine transformation is applied to the joint vector $(X^T, Y^T)^T$, does the resulting vector follow a multivariate normal distribution?

Answer: Yes


Question: Let $M \in \mathbb{C}^{n \times n}$ be a complex-symmetric $n \times n$ matrix, meaning $M$ is equal to its own transpose. If the real part of $M$ is positive-definite, is $M$ necessarily diagonalizable?

Answer: Yes


Question: Evaluate the double integral \( \int \int_R 1 \, dA \), where the region \( R \) is defined by the boundaries \( 0 \le x \le x^2 + y^2 \le 1 \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the triple integral \( \iiint_E z \, dV \) where \( E \) is the region bounded by \( x = 4y^2 + 4z^2 \) and \( x = 4 \).

Answer: 0


Question: Evaluate the contour integral \( I = \oint_C \frac{\sin(\pi z)}{(z-1)^5} \, dz \), where \( C \) is the circle \( |z|=2 \).

Answer: 0


Question: Calculate the limit: $$\lim_{n \to \infty} \frac{\sqrt[n]{n(n+1)\cdots(2n)}}{n}$$

Answer: \dfrac{4}{e}


Question: Find the set of all real numbers $x$ such that $\sin x > \ln x$. Express your answer in interval notation.

Answer: (0, 2.219)


Question: Calculate the value of the expression: \[ \arctan \frac{21 \pi}{\pi^2-54} + \arctan \frac{\pi}{18} + \arctan \frac{\pi}{3} \] without using a calculator or Maclaurin series.

Answer: 0


Question: Calculate the $L^3$ norm of the martingale $(X_n)$, given that the martingale $(M_n)$ is bounded by a constant $M$.

Answer: M


Question: Evaluate the integral \( \int_{\gamma} \frac{z^2 + 2z}{z^3 - 1} \, dz \) where \( \gamma = C(1,2) \) is a circle centered at 1 with radius 2 and positive orientation, using partial fraction decomposition.

Answer: 2\pi i


Question:
Given the matrix \( A = \begin{pmatrix} 0&0&0&-4 \\ 1&0&0&0 \\ 0&1&0&5 \\ 0&0&1&0 \end{pmatrix} \), determine the Jordan canonical form of \( A \). Choose the correct option:

(a) \( \begin{pmatrix} -1&0&0&0 \\ 0&1&0&0 \\ 0&0&2&0 \\ 0&0&0&-2 \end{pmatrix} \)

(b) \( \begin{pmatrix} -1&1&0&0 \\ 0&1&0&0 \\ 0&0&2&0 \\ 0&0&0&-2 \end{pmatrix} \)

(c) \( \begin{pmatrix} 1&1&0&0 \\ 0&1&0&0 \\ 0&0&2&0 \\ 0&0&0&-2 \end{pmatrix} \)

(d) \( \begin{pmatrix} -1&1&0&0 \\ 0&-1&0&0 \\ 0&0&2&0 \\ 0&0&0&-2 \end{pmatrix} \)

Answer: a


Question: Find the point on the surface defined by the function \( f(x,y) = 4x^2+7y^2+5xy+14 \) where the vector \( 6\hat{i} + 69\hat{j} + 3\hat{k} \) is normal to the tangent plane.

Answer: (1, -2, 36)


Question: Given that $f(x,y)=u(x,y)+iv(x,y)$ is an entire function of $z=x+iy$ such that $f(0)=-1$, $\frac{\partial u}{\partial x}=(e^y+e^{-y})\cos x $ and $\frac{\partial u}{\partial y}=(e^y-e^{-y})\sin x $, find $f(\pi/3)$. \(z = x + iy\)

Answer: \sqrt{3} - 1


Question: Given two finite-dimensional real smooth manifolds $M$ and $N$, and a local diffeomorphism $f:M\to N$ at a point $p\in M$, is the induced map $f_{\star p}:T_pM\to T_{f(p)}N$ an isomorphism of tangent spaces?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\infty} \cos(pr)p\sqrt{p^2 + m^2} \, dp \) given that \( \int_{-\infty}^{\infty} e^{-ipr}p\sqrt{p^2 + m^2} \, dp = 0 \), where \( r \) and \( m \) are constants.

Answer: 0


Question: Find the closed form of the recurrence relation given by \( h(0) = 1, h(1) = 4, h(n) = \frac{h(n-1)^2}{h(n-2)} \).

Answer: 4^n


Question: Does the simple continued fraction \( r_n = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{\ddots_{a_n+0}}}} \) with an infinite sequence of positive integers \(<a_0;a_1,a_2,\dots>\) always converge to a real number?

Answer: Yes


Question: Find the value of \( \alpha \) such that the following equation holds: \[ \sum _ { n = 1 } ^ { \infty } \tan ^ { -1 } \left( \frac { 2 } { n ^ { 2 } + n + 4 } \right)= \tan ^ { -1 } \left( \frac { \alpha } { 5 } \right) \]

Answer: 10


Question: Determine the values of \(\alpha > 0\) for which the fixed point iteration \(x_{k+1} = e^{-\alpha x_k}\) converges, given that the initial value \(x_0\) is in the interval \([0,1]\).

Answer: (0, e)


Question: Evaluate the integral \(\int(x^2-2xy)\,dx + (y^2+2xy)\,dy\) over the region bounded by the ellipse \(x^2+4y^2=4\) using Green's theorem.

Answer: 0


Question: Determine the number of distinct cyclic subgroups of order 12 in the group \( \mathbb{Z}_6 \times \mathbb{Z}_{10}^\times \).

Answer: 2


Question: Determine the total number of non-real complex solutions to the equation \(z^{2018} = 2018^{2018} + i\), where \(i = \sqrt{-1}\).

Answer: 2018


Question: Evaluate the limit: $$\lim_{(x,y)\to (0,0)}\frac{\log\left(x^2+2^{y^2}\right)} {\sqrt{x^2+4y^2}}$$ Determine whether the limit exists or not.

Answer: 0


Question:
Let $H$ be a finite group of odd order. Determine which of the following statements is true:
1. The number of elements of order 2 in $H$ is odd.
2. The number of elements of order 2 in $H$ is even.
3. $H$ has no elements of order 2.
4. None of the above.

Answer: C


Question: Solve the recurrence relation \(T(n)=1+2\sum_{i=1}^{n-1}T(i)\) for \(n > 1\) with the initial condition \(T(1)=1\).

Answer: 3^{n-1}


Question: Evaluate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) for the vector field \( \mathbf{F} = x\,\mathbf{i} + 3y\,\mathbf{j} + 3x^2\,\mathbf{k} \), where \( C \) is the circle defined by \( x^2 + y^2 = 16 \) in the plane \( z = 3 \), oriented clockwise.

Answer: 0


Question:
Let $f(x)$ and $g(x)$ be differentiable functions on $\mathbb{R}^+$ satisfying the equations:

$$xf'(x) - g(x) = 0$$
$$xg'(x) - f(x) = 0$$

Given that $f(2) + g(2) = 8$, find the value of $f'(1) \cdot g'(1)$. \( \boxed{} \)

Answer: 4


Question:
Find the minimum value of the integral

$$ \mathcal {I} [y] = \int_{- \infty}^{\infty} \dfrac{1}{2} y'^2 + 1 - \cos y \; \mathrm d x $$

subject to the boundary conditions $y(-\infty) = 0$ and $y(\infty) = 2 \pi$. \( \boxed{\text{Minimum value}} \)

Answer: 8


Question: Calculate the area of a geodesic triangle on a sphere with radius 2, given that the interior angles of the triangle are 1.5 radians, 2 radians, and 2.5 radians.

Answer: 24 - 4\pi


Question: Find the supremum of the set of real numbers of the form \( \text{Im}(iz^2 + 2) \) where \(|z| \leq 2\).

Answer: 4


Question: Determine the number of real solutions to the equation \(2\cos^2x+5x^2=\sin x\).

Answer: 0


Question: In a normed linear space $X$, specifically a Banach space, determine whether there exists an element $x$ such that $\|x\| \ge N$ for any given natural number $N$. Provide a justification for your answer.

Answer: Yes


Question: Find the largest integer \( n \) such that for all integers \( b \) that are relatively prime to \( n \), it holds that \( b^{24} \equiv 1 \pmod{n} \).

Answer: 131040


Question: Given that $a$, $b$, and $c$ are positive real numbers satisfying $a^2 + b^2 + c^2 = 27$, find the minimum value of $a^3 + b^3 + c^3$. Express your answer in simplest form.

Answer: 81


Question: Evaluate \( \lim_{x \to 0} \frac{P_n}{x} \) where \( P_n = a^{P_{n-1}} - 1 \) for all \( n = 2, 3, 4, \ldots \) and \( P_1 = a^x - 1 \), with \( a \in \mathbb{R}^+ \).

Answer: (\ln a)^n


Question:
Given that \(A + B + C = \pi\), find the maximum value of the expression:
\[
\tan\left(\frac{A}{2}\right)\cdot \tan\left(\frac{B}{2}\right) + \tan\left(\frac{B}{2}\right)\cdot \tan\left(\frac{C}{2}\right) + \tan\left(\frac{C}{2}\right)\cdot \tan\left(\frac{A}{2}\right)
\]

Answer: 1


Question: Find the minimum value of the greatest integer function \([e_1 + e_2]\), where \(e_1\) and \(e_2\) are the eccentricities of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) and a hyperbola \(\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\), respectively. The vertices of the hyperbola are the foci of the ellipse, and the foci of the ellipse are the vertices of the hyperbola. Assume \(A\), \(B\), \(a\), and \(b\) are positive with \(b = B\).

Answer: 2


Question: Evaluate the infinite series \( \sum_{n=1}^{\infty} \frac{2n}{(n+1)!} \). Determine the sum of this series.

Answer: 2


Question: Are two independent random variables $X$ and $W$, where $W = Y$ almost surely and $Y$ is independent of $X$, also independent?

Answer: Yes


Question: Solve the equation: \[ \frac{1}{\log_{2x-1}{(x)}} + \frac {1}{\log_{x+6}{(x)}}=1+\frac{1}{\log_{x+10}{(x)}} \]

Answer: 2


Question: Determine whether the process $(B_{t}^{2})$ is a Markov process.

Answer: Yes


Question: Determine whether a logarithm can have a negative base in the field of complex numbers, \( \mathbb{C} \).

Answer: Yes


Question: Evaluate the improper integral \( \int_0^\infty \frac{\sqrt{x}}{x^3+1}\ \mathrm{d}x \) using residue theory.

Answer: \dfrac{\pi}{3}


Question: Evaluate the integral \( \int_{|z-2|=1} \frac{z^3}{(z+1)(z-4)} \, dz \) by deforming the contour.

Answer: 0


Question: Determine if the evaluation of the sum \(\sum_{n=0}^k \frac{(\frac{1}{2})_n}{n!} \frac{(\frac{1}{2})_{k-n}}{(k-n)!}\) is correct.

Answer: 1


Question: Given that the greatest common divisor (gcd) of two numbers a and b is 1, determine the number of ways the number \(3^3\cdot 5^7 \cdot 11^{13}\) can be expressed as a product of two factors, \(m\) and \(n\), such that \(\text{lcm}(m,n)=3^3\cdot 5^7\cdot 11^{13}\) and \(\gcd(m,n)=1\).

Answer: 4


Question: Determine all possible values of \(a > 0\) for which the series \(\sum_{n=1}^{\infty}\frac{5^n}{(4^n+a^n)}\) is absolutely convergent.

Answer: (5, \infty)


Question:
Compute the following limit without using l'Hopital's Rule:

$$\lim\limits_{x\to 0} \frac{e^x-1-x-\frac{1}{2}x^2}{x-\sin{x}}$$

Answer: 1


Question: Determine whether the monic polynomial \(x^2-1\) has a root in \(\mathbb{Q}_2\). Justify your answer using Hensel's Lemma.

Answer: Yes


Question: Determine the maximum possible rank of a nilpotent $n \times n$ matrix $A$ given that $A^3 \neq 0$. Provide your answer as an integer.

Answer: n-1


Question: Given that $X$ and $Y$ are independent Poisson random variables with parameters $\lambda_X = 1$ and $\lambda_Y = 2$, respectively, calculate the conditional probability $\mathbb{P}(X = 1 \mid Z = 2)$, where $Z = \frac{X + Y}{2}$. \(\boxed{\text{Your answer here}}\)

Answer: \dfrac{32}{81}


Question: Suppose that \(|A|+|C|=|B|+|C|\), and that both sides are strictly larger than \(|C|\). Does this imply \(|A|=|B|\) in the absence of the axiom of choice?

Answer: No


Question: Evaluate the limit: \[ \lim\limits_{n\to \infty}\lim\limits_{x\searrow 0}f^{(n)}(x) \] where the function \( f:[0,\infty)\to \mathbb{R} \) is given by \( f(x)=\exp(\sqrt{x})+\exp(-\sqrt{x}) \). Assume that \( f \) and its derivatives satisfy the differential equation \( 2f'(x)+4xf''(x)=f(x) \).

Answer: 0


Question: Determine the number of real solutions to the equation \[2\sqrt{1-\tfrac{2}{x}}+\sqrt{x-\tfrac{4}{x}}=x.\]

Answer: 0


Question: Calculate the limit of the function \( \frac{(1.000002)^n}{n^{1.000002}/\log n} \) as \( n \to \infty \).

Answer: \infty


Question: Determine whether the equality \( \limsup\sqrt[n]{|a_{n+1}|}=\limsup\sqrt[n]{|a_n|} \) holds for any arbitrary sequence \( \{a_n\}_{n=1}^\infty \) of complex numbers.

Answer: Yes


Question: Determine the radius of convergence of the power series for $e^{-x^{2}}$. Provide your answer as a single value.

Answer: \infty


Question: If a function $f: X \to X$ is continuous and bijective, does the image of a set $I \subset X$ under $f$ equal the pre-image of $I$ under $f^{-1}$? Answer yes or no.

Answer: Yes


Question: Determine the function \( f: \mathbb{N} \to \mathbb{N} \) that satisfies the functional equation \( f(m^2 + f(n)) = f(m^2) + n \) for all natural numbers \( m \) and \( n \).

Answer: f(n) = n


Question: Determine whether a continuous function $f \in C_c([a,b])$ with compact support in $[a,b]$ must satisfy $f(a) = f(b) = 0$. Provide a justification for your answer.

Answer: No


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

Answer: (-2, 2)


Question: Consider the function \( f: \Bbb{N}^\Bbb{N} \rightarrow \mathcal{P}(\Bbb{N}) \). Is it possible for different arguments of \( f \) to be mapped to the same value? If so, provide an example.

Answer: Yes


Question: Find the minimal polynomial of the real number \( u = \sqrt[3]{2+\sqrt{5}} + \sqrt[3]{2-\sqrt{5}} \) over the field \( \mathbb{Q} \). Show that \( u = 1 \) without directly computing the cube roots.

Answer: x - 1


Question: Calculate the limit of the expression \( \lim_{n\rightarrow \infty }\sum_{k=n}^{5n}\binom{k-1}{n-1}\left( \frac{1}{5} \right )^n\left( \frac{4}{5} \right )^{k-n} \) as \( n \) approaches infinity.

Answer: \dfrac{1}{2}


Question: Given a set, such as \( \{1, 7, 4, 90, 23\} \), and a value \( A \), determine the number of subsets of the set whose sum is a multiple of \( A \).

Answer: 8


Question: Determine whether the differentiation operator, which maps an entire function $f$ to its derivative $f'$, is continuous with respect to the metric $d(f, g)=\sum_{n=1}^{\infty} \min \left\{\frac{1}{2^{n}}, \max _{|z| \leq n}|f(z)-g(z)|\right\}$ on the space of entire functions. Justify your answer.

Answer: Yes


Question: Find the asymptotes for the curve given by the equation \(x_1=\frac{C}{x_{1}^{2}-3x_1x_2+3x_{2}^{2}}\), where \(C\in \mathbb{R}\), in the \((x_1,x_2)\)-plane.

Answer: x_1 = 0


Question: Suppose the function \( f(z) \) has a pole of order \( m \) at \( z = 0 \), and satisfies the inequalities \( \frac{1}{|z|^{3/2}} \le |f(z)| \le \frac{1}{|z|^{5/2}} \) for all \( z \) with \( 0 < |z| \le 1 \). Determine the value of \( m \).

Answer: 2


Question: Given non-negative real numbers $a, b, c, d$ such that $a^5 + b^5 \leq 1$ and $c^5 + d^5 \leq 1$, find the maximum possible value of the expression $a^2c^3 + b^2d^3$. \( \text{Express your answer in the form } \boxed{\text{value}}. \)

Answer: 1


Question: Determine the number of binomial coefficients \( \binom{200}{k} \) that are divisible by 3, where \( k \) ranges from 0 to 200.

Answer: 165


Question: Evaluate the limit \( \lim\limits_{x\to 0}\frac{(1+x)^\frac{1}{x}-e}{\ln(1+x)} \).

Answer: -\dfrac{e}{2}


Question: Determine whether a sequence $a_n$ converges to 0 if, for every real number $x > 1$, the subsequence $a_{\lfloor{x^n}\rfloor}$ converges to 0.

Answer: No


Question:
Consider the function $f: \mathbb{R^2} \rightarrow \mathbb{R}$ defined by
$$f(x, y) = \begin{cases} \dfrac{xy}{x^2 + y}, & x^{2} \neq -y\\ 0, & x^2 = y \end{cases} $$
Determine which of the following properties holds at the point $(0, 0)$:

A) The directional derivative does not exist.
B) The function is continuous.
C) The function is differentiable.
D) Each directional derivative exists, but the function is not continuous.

Answer: D


Question: Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function that is differentiable on $\mathbb{R} \setminus \{0\}$. If $\lim_{x \rightarrow 0}f'(x)$ exists and is finite, can it be concluded that $f$ is differentiable at $0$? Provide a justification for your answer.

Answer: Yes


Question: In \(\mathbb{R}^n\), determine the limiting value of the angle between the longest space diagonal of an \(n\)-cube and a vector perpendicular to one of its faces as \(n\) approaches infinity.

Answer: \dfrac{\pi}{2}


Question: Calculate the directional derivative of the function \( f(x, y, z) = x^2 + y^2 - z^2 \) at the point \((3, 4, 5)\) along the curve formed by the intersection of the surfaces \( 2x^2 + 2y^2 - z^2 = 25 \) and \( x^2 + y^2 = z^2 \).

Answer: 0


Question: A polynomial $f(x) = x^{50} + 3x^{49} + 3x + 12$ leaves a remainder of $3$ when divided by $x - a$. When its quotient is further divided by $x - b$, it leaves a remainder of $5$. Additionally, when $f(x)$ is divided by $x^2 - (b + a)x + ab$, the remainder is $x + 6$. Determine the value of $a$. \( \boxed{} \)

Answer: -3


Question: Determine whether the inequality \( f(\{a_i\}) = (\sum_{i=1}^n a_i^{1.5})^2 - \sum_{i=1}^n a_i \; \sum_{i=1}^n a_i a_{i+1} > 0 \) holds for positive real numbers \(\{a_i\}\) where not all \(a_i\) are equal, and \(n \le 8\). Assume \(a_{n+1} = a_1\).

Answer: Yes


Question: In the inner product space $\mathcal{P}[0,1]$ of all real polynomials on $[0,1]$, with the inner product defined as $\langle f,g\rangle =\int_{0}^{1}f(t)g(t)dt$, consider the subspace $V=\operatorname{span}\{t^2\}$. Find the polynomial $h(t) \in V$ such that the distance from $2t-1$ to $h(t)$ is minimized among all polynomials in $V$. Express $h(t)$ in the form $\alpha t^2$ and determine the value of $\alpha$. \( \boxed{\alpha} \)

Answer: \dfrac{5}{6}


Question: Evaluate \(\lim_{n\to\infty}{\sum_{0 \leq k \leq n}} \binom{n}{k}\frac{1}{n^{k}(k+3)}.\)

Answer: e - 2


Question:
Simplify the expression in the finite field \(\mathbb{F}_{2^4}\) using the irreducible polynomial \(f(x) = x^4 + x + 1\):

\[\left( \frac{\alpha^{10} + \alpha^{15}}{\alpha^8+\alpha^5} \right)^2 + \frac{\alpha^{10} + \alpha^{15}}{\alpha^8+\alpha^5} + \alpha^8 + \alpha^5 + \alpha^6\]

Answer: \alpha^3 + 1


Question: Let $f(x)=6x^3-11x^2-2$, where $x\geqslant 1.5$. Calculate $\frac{\text{d}f^{-1}}{\text{d}x}$ at $x=1517$, given that $1517=f(7)$. \(\text{Provide your answer in the form of a single value.}\)

Answer: \dfrac{1}{728}


Question: Calculate the sum of the squares of the distances between all pairs of vertices of a regular $n$-gon inscribed in the unit circle using complex numbers.

Answer: n^2


Question: Find the volume bounded by the surfaces \(x^2+z=1\) and \(y^2+z=1\) in the first octant.

Answer: \dfrac{1}{2}


Question: Let \( S \) be the open unit disk in \( \mathbb{C} \) and \( f: S \to \mathbb{C} \) be a real-valued analytic function with \( f(0) = 1 \). Determine the nature of the set \( \{z \in S : f(z) \neq 1\} \). Is it (a) empty, (b) non-empty finite, (c) countably infinite, or (d) uncountable?

Answer: a


Question: Find the sum of the infinite alternating series: $$\sum_{n=1}^\infty \dfrac{(-1)^{n +1} (n+1)}{n!}$$.

Answer: 1


Question: What is the formula for the surface area of a 4-dimensional sphere?

Answer: 2\pi^2 r^3


Question: In triangle ADC, G is the centroid, and AE is perpendicular to FC. Given that BD = DC, AC = 12 units, and the midpoint of AC is H, with D, G, and H being collinear, find the length of AB.

Answer: 36


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

Answer: e


Question: Determine whether there exists a constant $C > 0$, independent of $f$, such that for all $f \in L^p(\Omega)$, the inequality $\| f - \tilde{f} \|_p \leq C \| f \|_p$ holds, where $\tilde{f} = \frac{1}{|\Omega|}\int_{\Omega} f$ and $\Omega \subset \mathbb{R}^N$ is open and bounded.

Answer: Yes


Question:
What is the 100th derivative of the function \( y=\ln(2x-x^2) \) evaluated at \( x=1 \)?

Choose the correct answer from the options below:

(a) \( 2 \times 99! \)

(b) \( -2 \times 99! \)

(c) \( 2 \times 101! \)

(d) \( -2 \times 101! \)

Answer: B


Question: Consider the function \( f(x) = x\sqrt{x\sqrt{x\sqrt{x...\infty}}} \). If \( f(x) \) is defined as the limit of the expression as the number of roots approaches infinity, determine if \( f'(5) \) exists. If it does, find its value.

Answer: 10


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

Answer: 0


Question: Solve for $x$ in the equation: $$(1+x)^{299} - 124x(1+x)^{299} = 1$$

Answer: 0


Question: Determine whether the sequence \( \sum^n_{k=0}\frac{x^k}{k^2} \) is a Cauchy sequence in \( n \) for \( x \in [0,1] \).

Answer: Yes


Question: Determine the asymptotic growth rate of the sequence defined by the recurrence relation: \[ a(n+1) = a(n) + \frac{1}{1+2^{a(n)}} \]

Answer: \log_2 n


Question: Let $T$ be a linear transformation from a finite-dimensional real vector space $V$ to itself such that $T^2 = I$. Does this imply that $T$ has a real eigenvalue? Provide a justification for your answer.

Answer: Yes


Question: Find the distributional derivative of the indicator function $\Bbb1(Y \ge a)$. Provide your answer in terms of distributions.

Answer: \delta(Y - a)


Question: Let $a < b$ be positive real numbers. Determine if there exist $x, y \in (a, b)$ such that \( \left(\frac{a+b}{2}\right)^{x+y} = a^x b^y \).

Answer: Yes


Question: Compute the 32nd mixed partial derivative of the function \( e^{\frac{x^2}{2}}y \) with respect to \( x \) 16 times and \( y \) 16 times.

Answer: 0


Question: Determine the number of values of $x$ for which the function $f(x) = \cos x + \cos(\sqrt{2}x)$ achieves its maximum value.

Answer: 1


Question: Given two mappings $f$ and $g$ of a set $S$ into itself, where $f^2 = g^2 = I$ and $f \circ g = g \circ f$, determine if it is always true that $(f \circ g)^4 = I$. Provide a proof or counterexample.

Answer: (f \circ g)^4 = I


Question: Let $A$ be an $n \times n$ hyperbolic matrix (i.e., no eigenvalues of absolute value $1$). Consider the action of $A$ on $(\mathbb{R} \backslash \mathbb{Z})^n$. Does this action always have periodic orbits with minimal period being an even number $k \geq 2$? Answer yes or no.

Answer: No


Question:
Evaluate the limit as \(x\) approaches 0 from the left:
\[\lim_{x\to0^-}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2}\]

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


Question: Determine the largest positive integer $m$ such that $m$ divides $a^2b^2c^2d^2$ for all positive integers $a, b, c, d$ satisfying $a^2 + b^2 + c^2 = d^2$. \(N\) is the set of all positive integers.

Answer: 144


Question: Determine the number of maximal ideals in the ring of integers modulo $p^4$, where $p$ is a prime number.

Answer: 1


Question: Let $R$ be a ring and $I$ be an ideal of $R[x]$. Determine if $I$, which is not a principal ideal, can contain monic polynomials.

Answer: Yes


Question: Evaluate the line integral of the vector field \( \boldsymbol{F}(x,y,z) = (3x^2y^{10}z^{10}, 10x^3y^9z^{10}, 10x^3y^{10}z^9) \) along the straight line path from \((0,0,0)\) to \((1,1,1)\).

Answer: 1


Question: Given four random variables, $W$, $X$, $Y$, and $Z$, where $W$ and $X$ share the same distribution, and $Y$ and $Z$ share the same distribution, and $E(W+Z) = 0$. If $E(X)$ and $E(Y)$ are undefined, but $E(X+Y)$ exists, can we conclude that $E(X+Y) = 0$?

Answer: 0


Question: Given the functional equation \(f(x)f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right)\) and that \(f(3) = 28\), find the value of \(f(4)\).

Answer: 65


Question: Evaluate the integral \(\int_{-e}^e \left(x+x^3+x^5+\cdots+x^{999}\right) dx\).

Answer: 0


Question: Find the minimum value of the function \(f(x,y,z) = x^2 + y^2 + z^2\) subject to the constraint \(4x^2 + 2y^2 + z^2 = 4\) using the Lagrange multiplier method.

Answer: 1


Question: Let $H$ be a Hilbert space, and consider a set of vectors $\{h_s: s \in S\}$ in $H$ such that $\sum_{s \in S}\|h_s\| < \infty$. Here, the summation refers to the generalized convergence of nets of finite partial sums ordered by inclusion. Does this imply that $\sum_{s \in S} h_s$ converges in $H$ with respect to the norm topology?

Answer: Yes


Question: Consider the sequence defined recursively by \(b_{n+1}=b_n+\frac{1}{b_n}\) with the initial condition \(b_0=1\). Determine a function \(f(n)\) such that \(0<\lim_{n \to \infty}\frac{b_n}{f(n)}<\infty\).

Answer: \sqrt{2n}


Question: Suppose random variables $Z_i$ are distributed according to $p(z)$ and $X_i$ are distributed according to $p(x|z)$. Does the sequence $\frac{\sum_{i=1}^n X_i}{n}$ converge to the expected value $E(X)$?

Answer: Yes


Question: Determine the real number $y$ such that the inequality \( \frac{a+c}{b+d} \geq y \cdot \left ( \frac{a}{b} + \frac{c}{d} \right) \) holds for all real numbers $a, b, c, d$ where $a, c \geq 0$ and $b, d > 0$. Provide your answer as a single real number.

Answer: 0