Question: Evaluate the triple integral \( \int_{0}^{5}\int_{-z}^{z}\int_{0}^{\sqrt{z^2-y^2}}3xz~dxdydz \) and verify if the result is 2500.

Answer: 1250


Question: Determine whether the modular arithmetic equation \((33^{40})^{111}33^4\equiv 33^4 \pmod{100}\) implies that \(m=\frac{(33^{40})^{111}}{100}\) is an integer, i.e., \(m \in \mathbb{Z}\).

Answer: No


Question: Determine the Galois group of the polynomial $T^3-2$ over the finite field $\mathbb{F}_5$. Provide your answer as a group description or notation.

Answer: \mathbb{Z}/2\mathbb{Z}


Question: Find the supremum of the integral \( \int_{-1}^1 f \, dg \), where \( g(x) = x^2 \) on \([-1, 1]\) and \( \mathcal{F} \) is the set of all continuous functions \( f \) on \([-1, 1]\) such that \(|f(x)| \leq 5\) for all \( x \in [-1, 1] \).

Answer: 10


Question: Suppose $T$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$. If for any $x \in \mathcal{H}$ and all $z \in \mathbb{C}$, the inequality $\| (z-T)x \| \ge | \text{Im}(z) | \| x \|$ holds, can you conclude that $(z-T)$ is a bounded operator?

Answer: Yes


Question: Calculate the integral \( \oint_{|z|=3} \frac{1}{z^2-2z+2} \, dz \) using Cauchy's integration formula.

Answer: 0


Question: Given a figure with an incircle of radius $R$, find the length of the segment $IN$.

Answer: R


Question: Identify the smallest group that is not isomorphic to a cyclic group, a direct product of cyclic groups, or a semi-direct product of cyclic groups.

Answer: Q_8


Question: Let $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ be a measure space. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a bounded function with \[f_n(x)=f(x^n)\mathbb{1}_{[-1,1]}(x), \qquad x\in \mathbb{R}, n\in \mathbb{N}.\] Calculate \[\lim_{n\rightarrow \infty} \int f_n \, d\lambda.\]

Answer: 2f(0)


Question: Calculate the volume of the solid enclosed by the surface \((x^2 + y^2 + z^2)^2 = x\).

Answer: \dfrac{\pi}{3}


Question: Given a sequence of complex numbers $a_n$ such that $\lim_{n \to \infty} \sqrt[n]{|a_n|} = 0$, determine whether the shifted sequence $\lim_{n \to \infty} \sqrt[n]{|a_{n+1}|}$ also tends to $0$. In other words, if a power series $a_0 + a_1z + a_2z^2 + \cdots$ defines an entire function, does the shifted power series $a_1 + a_2z + a_3z^2 + \cdots$ also define an entire function?

Answer: Yes


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

Answer: \dfrac{2}{3}


Question: Determine whether every orthogonal operator on a subspace of \(\mathbb{R}^n\) is surjective. Provide a justification for your answer.

Answer: Yes


Question: In a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),P)$ with a right-continuous filtration, let $M$ be a supermartingale and $N$ be a cadlag modification of $M$ that is also a supermartingale. Determine if the mapping $t \mapsto E[M_t]$ is right-continuous.

Answer: Yes


Question:
Determine the rank of the matrix \(A\) where each entry \(a_{ij} = i + j\) for an \(n \times n\) matrix:

$$A = \begin{bmatrix}
2 & 3 & \cdots & n+1\\
3 & 4 & \cdots & n+2\\
\vdots & \vdots & \ddots & \vdots\\
n+1 & n+2 & \cdots & 2n\\
\end{bmatrix}$$

Find the maximum number of linearly independent columns in this matrix.

Answer: 2


Question: Determine whether a linear subspace of a Banach space that contains the unit ball must be equal to the entire space. Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit inferior of the sequence \( \left(-1-\frac{2}{n}\right)^n \) as \( n \to \infty \). Verify that the result is \( -e^2 \).

Answer: -e^2


Question:
Consider a three-state continuous-time Markov chain with states $1$, $2$, and $3$. The holding rates are $\lambda_1 = 3$, $\lambda_2 = 4$, and $\lambda_3 = 5$. The transition matrix for the underlying discrete Markov chain is:

$$\mathbf P = \begin{pmatrix} 0 & 1/2 & 1/2 \\ 1/3 & 0 & 2/3 \\ 2/5 & 3/5 & 0 \end{pmatrix}$$

Calculate the probability that the chain is in state $2$ as $t \rightarrow \infty$. \(\boxed{\text{Answer}}\)

Answer: \dfrac{6}{17}


Question: Given the function \( f(a) \), where \( f(a) \) is the unique real solution to the equation \( x(1+\ln x) = a \) for any real number \( a \geq 1 \), find the value of the limit \[ \lim_{a \to \infty} \frac{f(a)\ln a}{a}. \]

Answer: 1


Question: In the vector space $V$ of real-valued, infinitely differentiable functions on $\mathbb{R}$ that are periodic with period 1, with the inner product defined as $\langle f, g \rangle = \int_0^1 f(x)g(x) \, dx$, find the adjoint map $D^*: V \rightarrow V$ of the differentiation map $D: V \rightarrow V$, where $Df = f'$, using the property $\langle Df, g \rangle = \langle f, D^*g \rangle$. Provide your answer as an expression for $D^*$. \( \text{Note: } f(x) = f(x+1) \text{ for all } x \in \mathbb{R}. \)

Answer: -D


Question: Determine the value of the limit \(\alpha\), given by \[\alpha = \lim_{n\to\infty}\sum_{m = n^2}^{2n^2}\frac{1}{\sqrt{5n^4 + n^3 + m}}.\] Express \(10\sqrt5\,\alpha\) in simplified form.

Answer: 10


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{(x!)^2}{(2x)!} \]

Answer: 0


Question: Calculate the value of \( R = \frac{1}{\lim\limits_{n \to \infty} \sup \frac{1}{\sqrt[n]{n!}}} \). Explain why \( \sqrt[n]{n!} \to +\infty \) as \( n \to \infty \).

Answer: \infty


Question: Let $X$ be a matrix with rank $p$, and let $H$ be its corresponding projection matrix. If $O$ is an orthogonal matrix such that $HOX = OX$, does it follow that $O^TH = H$? Provide a justification for your answer.

Answer: No


Question: Find the general formula for the sequence $a_n$ defined by $a_0 = 3$ and $a_n = 2 + a_0 a_1 a_2 \ldots a_{n-1}$ for $n \geq 1$. Express your answer in terms of $n$.

Answer: 2^{2^n} + 1


Question:
Compute the limit \( \displaystyle \lim_{\omega\rightarrow\infty}{u(t,0)} \) for the given Fourier series representation of a solution to the heat equation:

\[u(t,x) = \sin \omega t + \sum_{n = 1}^{\infty}{\frac{4(-1)^{n}}{(2n - 1)\pi}\omega\left[\frac{\omega\sin \omega t + \left( n - \frac{1}{2} \right)^{2}\pi^{2}\left( \cos \omega t - e^{- \left( n - \frac{1}{2} \right)^{2}\pi^{2}t} \right)}{\omega^{2} + \left( n - \frac{1}{2} \right)^{4}\pi^{4}}\right]\cos\left[\left( n - \frac{1}{2} \right)\pi x \right]}\]

Answer: 0


Question: Evaluate the integral \( \oint_C \frac{1}{z-i} \, dz \) where \( C \) is the circle \( \left\vert z \right\vert = 2 \).

Answer: 2\pi i


Question: Determine the number of ways to arrange the five vowels—A, E, I, O, U—along with 20 X's in a row such that no X is at an extreme position and there are at least 4 X's between any two vowels.

Answer: 4200


Question: Compute the limit: \[\lim_{n\to\infty}\max_{x\in[0,\pi]}\sum_{k=1}^n\frac{\sin(kx)}{k}\] Choose the correct option from the following: \(\text{A) } \int_0^\pi\frac{\cos^2(x)}{x^2}~\mathrm dx\) \(\text{B) } \int_0^\pi\frac{\sin^2(x)}{x^2}~\mathrm dx\) \(\text{C) } \int_0^\pi\frac{\sin(x)}{x}~\mathrm dx\) \(\text{D) } \int_0^\pi\frac{\cos(x)}{x}~\mathrm dx\)

Answer: C


Question: Evaluate the limit \(\lim_{x \to 0} \frac{1-\cos(x)}{x^2}\) using the \(\varepsilon\)-\(\delta\) definition of a limit.

Answer: \dfrac{1}{2}


Question: Let $x, y, z$ be positive real numbers such that $x + y + z = xyz$. Find the minimum value of the expression \[ \sqrt{\frac{1}{3}x^4+1}+\sqrt{\frac{1}{3}y^4+1}+\sqrt{\frac{1}{3}z^4+1}. \]

Answer: 6


Question: Evaluate the limit: $$\lim \limits_{(x,y) \rightarrow (0,0) }\sqrt{x^2+y^2} \sin \left(\frac{1}{x^2+y^2}\right)$$ as $(x,y)$ approaches $(0,0)$.

Answer: 0


Question: Evaluate the improper integral: $$\int_{0}^{\infty}\text{sech}^2(x+\tan(x))dx$$

Answer: 1


Question: Determine whether it is true that for any measure space $(X, \mathcal{M}, \mu)$, the measure of any open ball $B_\varepsilon(x)\subset X$ is nonzero.

Answer: False


Question: Calculate the limit: \[ L = \lim \limits_{n \to \infty} -(n-n^{n/(1+n)}) \]

Answer: -\infty


Question: Evaluate the limit: $$\lim_{n\to\infty} \int_{0}^{\pi/4}\tan^n x \ dx$$

Answer: 0


Question: Determine the number of hyperreal numbers that have a standard part equal to 0.

Answer: 2^{\aleph_0}


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

Answer: \dfrac{1}{e}


Question: Determine all polynomials \( P(x) \) with real coefficients such that \( xP(x-3) = (x-2)P(x) \) for all real numbers \( x \).

Answer: 0


Question: Determine whether every affine $k$-variety, given by an affine scheme that is reduced and of finite type over a field $k$, corresponds to the Spec of a finitely generated $k$-algebra. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the boundary of a solid cube in 3-space is a 2-manifold.

Answer: Yes


Question: Given two $n \times n$ matrices $A$ and $B$ such that $(A^2)(B^2) = (B^2)(A^2)$, determine whether this necessarily implies that $AB = BA$. Provide a justification for your answer.

Answer: No


Question: Given a set of $n \geq 3$ points in the Cartesian plane in general position (no three points are collinear), determine if it is always possible to connect all the points with $n - 1$ line segments, drawn continuously without lifting the pencil, such that every subsequence of three consecutively connected points is clockwise oriented, and no two line segments intersect. Provide a justification for your answer.

Answer: Yes


Question: Find all integer roots of the equation \(5^{2(x+1)} + 621 \times 10^x = 100 \times 4^x\), given that \(x = 0\) is not a solution. Determine the negative integer root.

Answer: -2


Question: Let $g:\mathbb{R} \rightarrow \mathbb{R}$ be a non-constant, twice differentiable function. If $g(1+\frac{1}{n})+g(1-\frac{1}{n})=2$ for all integers $n$, find $g''(1)$. \( \boxed{} \)

Answer: 0


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

Answer: e^{\frac{3}{2}}


Question: Determine whether the following statement is true or false: In a metric space, if a subset has an accumulation point, then it must have a subsequence that converges to that point.

Answer: A


Question:
Evaluate the limit using Lebesgue's theorem:

$$ \lim_{n\to\infty}\int_{0}^{n}\left(1+\frac{x}{n}\right)^{n+1}e^{-2x}\,dx $$

Hint: For $x \in [0,\infty)$,

$$
\lim_{n \to \infty}\left(1+\frac{x}{n}\right)^n=e^x
$$

and for $x \in [0,\infty)$,

$$
1_{[0,n]}(x)\left(1+\frac{x}{n}\right)^ne^{-2x}\le e^{-x}, \quad n\ge1.
$$

Answer: 1


Question: Determine if there exists a constant $d$ such that the inequality \( \sum_{i=0}^{k}\sqrt{3^i} \geq d \sqrt{3^k} \) holds for all integers \( k \geq 0 \).

Answer: 1


Question: Determine the number of distinct ways to color a cube using four different colors, with each color appearing on at least one face of the cube. Consider two colorings to be the same if one can be obtained from the other by rotating the cube.

Answer: 68


Question: Determine whether there exists a bounded continuous function $f: \mathbb{R} \to \mathbb{R}$ such that the $p$-variation of $f$, defined as \[ \overline{V}_p(f)_T := \left(\sup_{\pi} \sum_{j=1}^n |f(t_j) - f(t_{j-1})|^p \right)^{1/p}, \] is infinite for any $p \geq 1$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_C \frac{z^{n}(z^{2n} -1)^2}{i(z-1)^2} \, dz \) using Cauchy's integration formula, where \( C \) is the unit circle.

Answer: 0


Question: Determine the minimum possible degree of a polynomial function $f(z)$ on the complex plane, given that $f(z) = \delta$ whenever $z^3 = \gamma$ and $z^3 = \gamma$ has exactly 3 distinct solutions.

Answer: 3


Question: Determine the positive value of $a$ such that the family of ellipses defined by the equation $x^2 + ay^2 = r^2$ is orthogonal to the parabola $y = 5x^2$, where $r > 0$. Provide your answer as a single positive number.

Answer: 2


Question: Evaluate the residue of the function \( \omega(z) = \frac{\sqrt{(z-2)(z-6)}}{z+1} \) at the point \( z = -1 \), where the branch of \( \sqrt{(z - 2)(z - 6)} \) is defined to have negative imaginary values on the southern shore of the cut [2, 6].

Answer: \sqrt{21}


Question: Provide an example of an ideal \( A \) in the ring of polynomials \( \Bbb{C}[X] \) such that \( A \) is a proper subset of the ideal \( I(Z(A)) \), where \( Z(A) \) is the set of common zeroes of all polynomials in \( A \), and \( I(Z(A)) \) is the ideal of polynomials vanishing on \( Z(A) \).

Answer: (X^2)


Question: Determine the set of real numbers \(x\) for which the power series \[x + \sum_{k \geq 2} (-1)^{k-1} \frac{2k x^{2k-1}}{(2k-1)!}\] converges.

Answer: \mathbb{R}


Question: Is the functor of taking $G$-invariants exact on the category of $R[G]$-modules, where $G$ is a finite group with order $n$ and $R$ is a ring in which $n$ is invertible?

Answer: Yes


Question: If $f: E \to \mathfrak{M}$ is continuous almost everywhere, is the preimage of an open set under $f$ Lebesgue measurable? Answer yes or no.

Answer: Yes


Question: Suppose $F$ is a field, and $\alpha,\beta$ are algebraic, separable elements over $F$. Determine whether it is always possible to find an element $c \in F$ such that $F(\alpha,\beta) = F(\alpha + c\beta)$. Provide a justification for your answer.

Answer: Yes


Question: In a non-commutative ring with an identity element, can a non-unit element generate a left ideal that is equal to the entire ring?

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to\infty} \frac{x^{100n}}{n!} \) for \( x > 0 \).

Answer: 0


Question: In a totally ordered abelian group, does the equation $nx = 0$ imply that $x = 0$ for any positive integer $n$? Answer yes or no.

Answer: Yes


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

Answer: -e^2


Question: Consider distinct natural numbers $a, b, c \neq 0$. Express them in the form $3^a + 3^b + 3^c$. When these numbers are ordered in increasing order, what is the sum $a + b + c$ for the 101st number in this sequence?

Answer: 19


Question:
Calculate the variance of the random variable $Y$ given that $X > 3$ and $Y > 3$, where the joint probability density function for $X$ and $Y$ is given by:

$$f_{XY}(x,y)=\begin{cases} 2e^{-(x+2y)}, & x>0,y>0\\ 0, & \text{otherwise} \end{cases}$$

Answer: \dfrac{1}{4}


Question: Determine the values of \( n \) for which \( 2^n \) divides the product \( 19! \cdot 18! \cdot \ldots \cdot 1! \) without any remainder.

Answer: 150


Question: Let $F$ be a field and $g$ a polynomial of degree $n$. Determine the length of the $F$-module $F[X]/\langle g \rangle$. Express your answer as a single integer.

Answer: n


Question: Find the residue of the function \( f(z)=\frac{\cos(\pi z)}{z^2 \sin(\pi z)} \) at \( z=1 \), given that it is a pole of order 2.

Answer: \dfrac{1}{\pi}


Question: Find the maximum value of the function \( f(x) = \frac{x^2 + x + 1}{x^2 - x + 1} \) using the AM-GM inequality.

Answer: 3


Question: Determine whether there exists a basis for $\mathbb{C}^n$ (or $\mathbb{R}^n$) consisting solely of eigenvectors that are common to two commuting $n \times n$ matrices $A$ and $B$, given that $A$, $B$, and $AB$ each have $n$ distinct eigenvectors forming a basis for $\mathbb{C}^n$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( \int_0^{2\pi} \cos(\cos x)\,dx \).

Answer: 2\pi J_0(1)


Question: Determine the set of real values of $x$ such that \( x^{(\log_{10}x)^2-3\log_{10}x+1} > 1000 \).

Answer: (1000, \infty)


Question: Calculate the integral \( \int_{-1}^{1} P_m(x) P_n(x) \, dx \) where \( P_m(x) \) and \( P_n(x) \) are Legendre polynomials and \( m \neq n \).

Answer: 0


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

Answer: \dfrac{1}{e}


Question: Let $f(x)$ be a real-valued function that is twice differentiable on the interval $[1,5]$, with $f(1) = f(5) = 0$, and satisfies the differential equation $f(x) = f'(x) + f''(x)$ for all $x \in [1,5]$. Find the value of $f(2) + f(4) - f'(3)$. \( \boxed{} \)

Answer: 0


Question:
Determine the coefficient $k_1$ in the Taylor series expansion of the integral \( F(c) = \int_{0}^{1} \sqrt{x^{2p} + c^2} \, dx \) about \( c = 0 \). The expansion is given by:

\[ F(c) = \frac{1}{p+1} + k_1c + k_2c^2 + \cdots \]

Answer: 0


Question: Determine the value of $A$ such that the function $f(x) = \lfloor{x}\rfloor \cos(Ax)$ is continuous for all $x \in \mathbb{R}$.

Answer: \dfrac{\pi}{2}


Question: Determine the number of natural number triples \((a, b, p)\) with \(a > b > 0\) such that \[ \left(\sum_{k=1}^n k^a\right)^p = \sum_{k=1}^n k^b \] holds for all natural numbers \(n\). Identify these triples.

Answer: 0


Question: Suppose \(\{a_n\}\) is a sequence of non-negative real numbers such that \(a_n = O(n^M)\) for a positive number \(M\), and its Dirichlet series \(L(s)=\sum \frac{a_n}{n^s}\) has an analytic continuation to a meromorphic function on \(\mathbb{C}\) with only a simple pole at a positive real number \(a\) with residue \(A\). Using a standard Tauberian theorem, determine the maximum value of \(\delta\) for which the asymptotic formula \(\sum_{n\leq x} a_n = Ax^a + O(x^{a-\delta})\) holds.

Answer: 1


Question: Find the slope of the curve formed by the intersection of the surface \( z = x^3 + xy^2 \) and the plane \( 2x - 2y = 1 \) at the point \( (1, \frac{1}{2}) \). Use the parametrization \( x = 1 + t \) and \( y = \frac{1}{2} + t \), and calculate the derivative with respect to \( t \) at \( t = 0 \).

Answer: \dfrac{17}{4}


Question: Determine whether \(2^{30} \equiv 1\) or \(-1\) modulo 61 using Euler's Criterion.

Answer: -1


Question: Let $U$ be a Riemann surface and $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. Let $Q\in U$ be another point and $V \subset U$ be an open subset such that its image under $z$ is an open disc around $z(Q)$ contained in $B(0,1)$. Consider the morphism $z': V\longrightarrow B(0,1)$ defined by $z'(x) = z(x) - z(Q)$. Determine the relationship between the differentials $dz$ and $dz'$.

Answer: dz' = dz


Question: Suppose $g$ is a nonnegative radial function with compact support on $\mathbb{R}^{3}$, and $f$ is a radial solution of $(-\Delta + g)f = 0$ with $f(0) = 0$. Given that $\frac{df(0)}{dr} = 0$, determine whether it follows that $\lim_{r\to 0}r^{2}\frac{df}{dr} = 0$. Provide a justification for your answer.

Answer: 0


Question: Determine whether the following statement is true or false: If a sequence of continuous functions \( f_n \) converges pointwise to a function \( f \), then \( \|f_n\|_{\infty} \to \|f\|_{\infty} \).

Answer: B


Question: Evaluate the limit \( \lim\limits_{(x,y) \to (0,0)} \dfrac{2 - \cos(x) - \cos(y)}{x^2+y^2} \).

Answer: \dfrac{1}{2}


Question: Determine the minimum number of linear measurements required to uniquely identify a matrix $A$ with null trace. A linear measurement is defined as knowing the quantities $A\vec{x}_i \cdot \vec{h}_j$ for given vectors $\vec{x}_i$ and $\vec{h}_j$.

Answer: n^2 - 1


Question: Suppose the equation \((x^2+px+1)(x^2-2x+q)=0\) has roots \(2+ \sqrt{3}\) and \(1-i\), where \(p\) and \(q\) are integers and \(i=\sqrt{-1}\). Find the value of \(p+q\).

Answer: -2


Question: If $f$ is analytic in a simply connected domain $D$, and for any closed loop in $D$, the integral of $f(z)dz$ around the loop is 0, is this sufficient to conclude that $f$ is analytic in the closed disk centered at 0 with radius 2? Answer "Yes" or "No".

Answer: No


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

Answer: 1


Question: Let $f(x) = g(x) \cdot |x-1| \cdot |x-2| \cdot |x-3|^2 \cdot |x-4|^3$, where $g(x) = x^3 + ax^2 + bx + c$. If $f(x)$ is differentiable for all $x$ and $f'(3) + f'''(4) = 0$, find the third factor of $g(x)$ given that $x-1$ and $x-2$ are factors and $f'(3) = 0$.

Answer: x - 4


Question: Determine whether the Volterra operator $T: L^1([0,1]) \rightarrow C([0,1])$, defined by $T(f)(x) = \int_0^x f(t) \, dt$, is weak-strong continuous. Specifically, if a sequence $(f_n)_n$ in $L^1([0,1])$ weakly converges to $f$, does it follow that $\lim_{n\to\infty}\left\lVert T(f) - T(f_n) \right\rVert_\infty = 0$?

Answer: Yes


Question: Let \((M,d)\) be a complete metric space and let \(T:M\to M\) be a mapping such that \(T^n\) is a strict contraction for some \(n > 1\), but \(T^{n-1}\) is not a strict contraction. Determine the number of fixed points that \(T\) can have.

Answer: 1


Question: Consider a function $f: S^0 \to S^0$. Determine whether the attaching space $D^1 \cup_f D^1$ is always homeomorphic to $S^1$, regardless of whether $f$ is the identity or swaps the points in $S^0 = \{1, -1\}$. Provide a justification for your answer.

Answer: S^1


Question:
Let \(B = (B_t)_{t \ge 0}\) be a standard Brownian motion starting at zero. Consider the non-negative stochastic process \(X_t\) that solves the stochastic differential equation:

$$dX_t = \frac{1}{X_t} dt + dB_t$$

Calculate \(E[\sigma]\), where \(\sigma = \inf \{ t \ge 0 : X_t = 1 \}\).

Answer: \dfrac{1}{3}


Question: Let $f$ and $g$ be continuous functions defined on $[0, 1]$. If the maximum value of $f(x)$ is equal to the maximum value of $g(x)$, does there necessarily exist a point $c$ in $[0, 1]$ such that $(f(c))^2 - (g(c))^2 = 2f(c) - 2g(c)$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question:
Determine if the random variable $T$, which follows a Bernoulli distribution with parameter $p \in (0,1)$ and is defined as:

$$T= \left\{ \begin{aligned} &1 &\text{ with probability of }1-p,\\ &1-\frac{1}{p} &\text{ with probability of }p. \end{aligned} \right.$$

is a sub-Gaussian random variable. If it is, find the variance proxy $\sigma^2$ in terms of $p$. Provide your answer for $\sigma^2$.

Answer: \dfrac{1}{4p^2}


Question: Evaluate the integral \( \int_{0}^{2\pi}\cos(m\theta)\,d\theta \) using path integration, where \( m \) is a nonzero integer.

Answer: 0


Question: What is the smallest possible size of an $n \times n$ matrix that is nilpotent with a nilpotence degree of $k$?

Answer: k


Question: Determine the number of distinct isomorphism classes of Abelian groups of order $p_1^4p_2^4\ldots p_n^4$, where $p_1, p_2, \ldots, p_n$ are distinct primes.

Answer: 5^n


Question: Is the Möbius function the unique inverse of the constant function 1 under Dirichlet convolution?

Answer: Yes


Question: Determine whether the following statement is true or false: A continuous bijection of compactly generated spaces is a homeomorphism.

Answer: B


Question: Find the function $f: \mathbb{R} \to \mathbb{R}$ that satisfies the equation \( f(x^2+x+3) + 2f(x^2-3x+5) = 6x^2 - 10x + 17 \) for all \( x \in \mathbb{R} \).

Answer: 2x - 3


Question: Find the limit: $$ \lim_{x\to 0}\ (\sqrt {2x+1}\ -\ \sqrt[3]{1-3x})^{x} $$ as \( x \) approaches 0.

Answer: 1


Question: Given two field extensions $E_1$ and $E_2$ of a field $F$, determine whether there always exists a field $E$ that is an extension of both $E_1$ and $E_2$. Provide a justification for your answer.

Answer: Yes


Question: Let $A\subseteq \mathbb{R}_{\ge 0}$ be a non-empty set. If there exists a sequence $\{x_n\}\subseteq A$ such that $x_n\to L$ for some $L \in [0,\infty]$, does it imply that $L$ is the least upper bound of $A$? Answer yes or no and justify your answer.

Answer: No


Question: Express \(\cos(x^2)\) in terms of \(\cos(x)\) and \(\sin(x)\), if possible.

Answer: \cos(x^2)


Question: Evaluate the limit: $$\lim_{x \to 0} \left(a^{x}+b^{x}-c^{x}\right)^{\frac{1}{x}}$$ where $a, b, c > 0$ are real numbers.

Answer: \dfrac{ab}{c}


Question: Suppose you have an $n \times n$ matrix $A$ with full rank. Let $B$ be another $n \times n$ matrix. Determine if there always exists a constant $c$ sufficiently small such that $A + cB$ also has full rank.

Answer: Yes


Question: Determine whether the following statement is true or false: If the minimal polynomials of the restriction of a linear operator to two invariant subspaces are coprime, then the two subspaces are independent.

Answer: True


Question: In Fraenkel-Mostowski set theory, determine if it is possible to construct a model where the set of atoms can be linearly ordered but not well-ordered.

Answer: Yes


Question: Determine whether the following statement is true or false: Any two matrices $A$ and $B$ with the same characteristic polynomial are similar.

Answer: False


Question: Find a polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $\mathbb{Q}(\sqrt{1+\sqrt{5}})$ is ring isomorphic to $\mathbb{Q}[x]/\langle p(x)\rangle$. Provide the polynomial $p(x)$.

Answer: x^4 - 2x^2 - 4


Question: Is the set of all Turing Machine specifications recursively enumerable?

Answer: Yes


Question: Determine whether a doubly stochastic matrix $A \in M_n$ can have exactly $n + 1$ positive entries. Answer with 'Yes' or 'No'.

Answer: No


Question: Find all positive integer solutions \((x, y)\) to the Diophantine equation \[24x^4 + 1 = y^2.\]

Answer: (1, 5)


Question: Determine the number of distinct ways to make a total of 24 using an unlimited supply of coins with values $1^2, 2^2, 3^2, \ldots, 17^2$. Provide your answer as a single integer.

Answer: 16


Question: In any ring \( R \) with unity, if there exists an element \( c \) such that \( ca = 1 \) and \( ab = 1 \), must \( c \) be equal to \( b \)? Determine whether this statement is true or false.

Answer: True


Question: Determine the minimum value of the function \( G(x) = \frac{(x^2+\frac{1}{x^2})^6-(x^{12}+\frac{1}{x^{12}})-2}{(x^2+\frac{1}{x^2})^3+(x^6+\frac{1}{x^6})} \) for \( x > 0 \), without using calculus.

Answer: 6


Question: Find the value of $p \in \mathbb{R}$ such that the function \( g(x) = f(\sin x) - x + \frac{1}{3}x^2 + \frac{1}{3}x^3 \), where \( f(x) = x + px^2 + O(x^5) \), has a local maximum, a local minimum, and an inflection point at \( x = 0 \).

Answer: -\dfrac{1}{3}


Question: Evaluate the sum \( \tan\left(\frac{\pi}{16}\right)+\tan\left(\frac{5\pi}{16}\right)+\tan\left(\frac{9\pi}{16}\right)+\tan\left(\frac{13\pi}{16}\right) \) using tangent addition formulas.

Answer: -4


Question: Determine all natural numbers \( n \) such that \( w = \frac{2^{n+1}(1+i)}{(1-i)^{3n+5}} \) is a purely real number.

Answer: n \equiv 2 \pmod{4}


Question: Given unit vectors \(\vec{u}\) and \(\vec{v}\), and a vector \(\vec{w}\) such that \(\vec{u} \times \vec{v} + \vec{u} = \vec{w}\) and \(\vec{w} \times \vec{u} = \vec{v}\), find the value of the scalar product \(\vec{u} \cdot (\vec{v} \times \vec{w})\).

Answer: 1


Question: Determine the value of $m > \frac{1}{2}$ that minimizes the area of the triangle formed by the lines $y = 10 - 2x$, $y = mx$, and $y = -\frac{x}{m}$. Provide your answer as a single value of $m$.

Answer: 3


Question:
Find the value of $x$ that satisfies the equation $x^{x^3} = 3$.

Express your answer as a decimal rounded to three decimal places.

Answer: 1.442


Question:
Determine the pointwise limit of the Fourier series for the function \( f(x) = x \sin(\pi x) \) on the interval \([-1, 1]\):

\[ \frac{1}{\pi} - \frac{\cos(\pi x)}{2\pi} + \sum_{k=2}^{\infty} \frac{2\cos(k\pi x)}{\pi(k^2-1)}(-1)^{k+1} \]

as the number of terms approaches infinity.

Answer: x \sin(\pi x)


Question: Determine the value of \(\alpha\) for which the series \(\displaystyle\sum_{n=1}^{\infty} \left(n^{1/n} - 1\right)^{\alpha}\) converges. Express \(n^{1/n}\) as \(e^{\frac{\ln n}{n}}\) and use the Taylor Series expansion to obtain \(n^{1/n} = 1 + \frac{\ln n}{n} + \frac{(\ln n)^2}{2n^2} + \ldots\).

Answer: \alpha > 1


Question: Given two infinite sets $A$ and $B$, if there exist injective functions from $A$ to $B$ and from $B$ to $A$, can we conclude that $A$ and $B$ have the same cardinality? Justify your answer.

Answer: Yes


Question: Evaluate the limit as \( n \rightarrow \infty \) of the series: \[ \sum_{p=1}^{n} \sin\left(\frac{p}{n^2}\right) \]

Answer: \dfrac{1}{2}


Question:
Determine the asymptotic behavior of the iterated sum function \( f(x) \) as \( x \to \infty \), where:
\[
f(x)=\sum_{i_{u-1}=1}^{x}\sum_{i_{u-2}=1}^{i_{u-1}} \cdots \sum_{i_2=1}^{i_3} \sum_{i_1=1}^{i_2}i_1
\]

Answer: \dfrac{x^u}{u!}


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the inequality \( |f(x) - f(y)|^2 \leq |x - y|^3 \) for all \( x, y \in \mathbb{R} \).

Answer: f(x) = c


Question: Find the minimum value of the positive integer \( a \) such that there exists a positive integer \( b \) with \( a < b \), and the arithmetic mean \( \frac{a+b}{2} \) and the geometric mean \( \sqrt{ab} \) are positive integers with the same two digits in reverse order.

Answer: 32


Question: For an invertible matrix $A$, if $A^n$ is diagonal for some positive integer $n$, does it imply that $A$ is diagonalizable?

Answer: Yes


Question: Let \(0 < a < b < 1\). Let \(g(x)\) be a non-decreasing function on \((a, b)\) such that \(g(a) = 0\). Given that \(\int_a^b xg(x) \, dx > b^2\), determine if there exists a constant \(C > 0\) such that \(\int_a^b (x-a) \cdot g(x) \, dx > Cb^2\).

Answer: No


Question: A function $f$ is defined by \(f(z)=(4+i)z^2+a z+b\) for all complex numbers $z$, where $a$ and $b$ are complex numbers. Given that $f(1)$ and $f(i)$ are both purely real, find the minimum value of $|a|+|b|$. \(i=\sqrt{-1}\)

Answer: \sqrt{2}


Question:
Given the system of equations:
\[\begin{cases} x + y + z = 12 \\ x^2 + y^2 + z^2 = 12 \\ x^3 + y^3 + z^3 = 12 \end{cases}\]
Find the value of \(x^4 + y^4 + z^4\).

Answer: 1992


Question: Determine if the Taylor series method can be used to evaluate the limit \(\lim_{x\to 1^{-}}\frac{\pi-\arccos(x)}{\sqrt{1-x^2}}\).

Answer: \infty


Question: Evaluate \( \lim_{t \to \infty} u(1,t) \) for the function \( u(x,t) \) defined by the partial differential equation \( \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \) with initial condition \( u(x,0)=\frac{e^{2x}-1}{e^{2x}+1} \) and boundary conditions \( u(0,t)=u(\pi,t)=0 \).

Answer: 0


Question: Evaluate the limit \(\lim_{n\to \infty}{\frac {n^2}{(n_p)!}}\), where \((n_p)!\) represents the product of the first \(n\) prime numbers.

Answer: 0


Question: Find the limit of the sequence \(a_n = \frac {2^{2n} \cdot1\cdot3\cdot5\cdot...\cdot(2n+1)} {(2n!)\cdot2\cdot4\cdot6\cdot...\cdot(2n)} \) as \(n\) approaches infinity.

Answer: 0


Question: Evaluate the limit: $$\lim_{b\to -\infty}b^4 \int_{b}^{a}\left(\frac{a^2-b^2}{a-b}(x-b)-x^2\right){\rm d}x$$

Answer: -\infty


Question: Solve the equation $x = i^{\frac{1}{i}}$.

Answer: e^{\frac{\pi}{2}}


Question: Determine the limit of the sequence \( a_n = \sqrt[n!]{\frac{1}{2^{(n!)}}-\frac{1}{3^{(n!)}}} \) as \( n \to \infty \).

Answer: \dfrac{1}{2}


Question: For the matrix $A(6)$, verify whether the sum of the squares of the magnitudes of its eigenvalues satisfies the equation $|\lambda_1|^2 + |\lambda_2|^2 + |\lambda_3|^2 = 8$. Is this statement true?

Answer: True


Question: Evaluate the integral \( \lim_{n\rightarrow \infty} \int_{\mathbb{R}^2}e^{-(x^2+y^2)^n}dxdy \) using polar coordinates.

Answer: \pi


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \frac{\sin((2n+1)x)}{2n+1} \, dx \) for any non-negative integer \( n \).

Answer: 0


Question: Solve the equation for $x$ without using inverse functions or a calculator: $$2 - 3 \cos 2x - 5 \sin 2x + \cos 4x - \sin 4x = 0$$

Answer: x = \pi n


Question: Given a smooth manifold $M$, an open subset $O \subseteq M$, and a closed subset $B \subseteq M$ with $\text{closure}(O) \subseteq \text{interior}(B)$, determine whether there exists a smooth function $f: M \rightarrow \mathbb{R}$ such that $0 \le f \le 1$, $f|_{M-B} \equiv 0$, and $f|_O \equiv 1$. Provide a justification for your answer.

Answer: Yes


Question: Let $\{W_t\}$ be a standard Brownian motion and define the process $\{X_t\}$ by $X_t= {W_t}^{2}-t ,\ \ t\geq0.$ Calculate $P(X_5>0\mid W_3=1)$. Provide your answer as a probability value.

Answer: 0.202


Question: Determine the derivative of the function \( f(x) = x^2 \) at the endpoint \( x = 0 \) for the domain \( [0,1] \).

Answer: 0


Question: Evaluate the limit of the infinite product: \[ \lim _{x\to \infty }\left(\prod_{n=2}^{x}\left(1-\frac{1}{n^2}\right)\right) \]

Answer: \dfrac{1}{2}


Question:
Simplify the following summation:

$$ \sum_{n=3}^{\infty} \frac{1}{2^n} \sum_{i=3}^{n} \left\lceil\frac{i-2}{2}\right\rceil $$

Answer: \dfrac{2}{3}


Question: Evaluate the integral \( \displaystyle\int_{\gamma}\frac{1}{z^2 + 9}\:dz \) where \( \gamma := \lvert z - 2i \rvert = 3 \) using partial fractions and Cauchy's theorem.

Answer: \dfrac{\pi}{3}


Question: Are all norms equivalent on a finite-dimensional vector space?

Answer: Yes


Question: Determine the number of elements of order 8 in the direct product of the dihedral group $D_{16}$ and the cyclic group $Z_4$. Provide your answer as a single integer.

Answer: 16


Question: Evaluate the limit: $$ \lim_{n \to \infty}(n^2+1)\cdot\left(\ln(n^2-4)-2\ln(n)\right) $$.

Answer: -4


Question: Evaluate the following summation modulo 7: \( \sum_{k=1}^{300} k^k \pmod{7} \).

Answer: 5


Question: Evaluate the limit: $$\lim_{x\to 0} \frac{1}{x^3} \int_0^x\frac{t\ln(1+t)}{t^4+4}dt.$$

Answer: \dfrac{1}{12}


Question: Suppose $A$ is a real $n \times n$ matrix of rank $r$. Let $V$ be the vector space of all real $n \times n$ matrices $X$ such that $AX=O$. Determine the dimension of the vector space $V$.

Answer: n(n - r)


Question: Compute the complex integral \( \int \frac{e^z}{z^2 + 1} \, dz \) along a circle with center \(-i\) and radius \(2\), using the residue theorem.

Answer: -\pi e^{-i}


Question: Find the last two digits of the expression \(3^{3^{3^{\cdots}}}\) where there are a total of two thousand and fifteen 3's in the power tower.

Answer: 87


Question: If $f(x) \to L$ as $x \to \infty$ and $f$ is of class $C^1$ (i.e., $f'$ is continuous), does it imply that $f'(x) \to 0$ as $x \to \infty$? Provide a justification for your answer.

Answer: No


Question: Consider the sequence $(q_n)_{n\in\mathbb{N}}$ of all rational numbers in the interval $[0, 1]$. Determine whether this sequence has any convergent subsequences, and if so, identify the range of possible limit points for these subsequences.

Answer: [0, 1]


Question: Let $g:[0,+\infty) \to \mathbb{R}$ be a measurable function such that $g(x) \geq 0$ for almost every $x \in [0,+\infty)$ and $\int_0^{+\infty} g(x) \, dx < +\infty$. Determine whether it is true that $\liminf_{x \to +\infty} g(x) = 0$. Provide a justification for your answer.

Answer: 0


Question: Determine the minimum period of the periodic function $f(x)$ that satisfies the equation $$\forall x :f(x+346)=\frac{1+f(x)}{1-f(x)}.$$

Answer: 1384


Question: Determine whether the following statement is true or false: If \(b_n\) is a positive sequence and \(\lim_{n\to\infty}\frac{b_{n+1}}{b_n}=k\), then \(\lim_{n\to\infty} b_n^{\frac{1}{n}}=k\).

Answer: True


Question: Determine the number of real roots of the polynomial \(x^4 - 2x^3 + 5x^2 - 4x + 4\) in the interval \((-1, 1)\) using Sturm's theorem.

Answer: 0


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

Answer: -1


Question: Given the equations \(xy(z^2-1)+z(x^2-y^2)=10\) and \((x+y)z+(x-y)=5\), find the value of \((x^2+y^2)(z^2+1)\).

Answer: 5


Question: Given a prime number $P$, a group is called a P-group if its order is $P^r$ for some $r \geq 0$. If $G$ is a group with order $P^r \cdot m$, where $m$ is an integer and $P$ does not divide $m$, can it be concluded that a Sylow P-subgroup of $G$ is a P-group?

Answer: Yes


Question: Determine the point of contact between the plane \(2x - 2y + z + 12 = 0\) and the sphere \(x^2 + y^2 + z^2 - 2x - 4y + 2z = 3\).

Answer: (-1, 4, -2)


Question: Determine the smallest number of sets required to generate the power set of a finite set $A$ where $|A| = n$.

Answer: \lceil \log_2 n \rceil


Question: Is the set \(T^{\mathbb{N}}x\), where \(T : \mathbb{S}^1 \to \mathbb{S}^1\) is multiplication by \(e^{i\alpha}\) with \(\alpha \notin \pi\mathbb{Q}\) and \(x \in \mathbb{S}^1\), dense in \(\mathbb{S}^1\)?

Answer: Yes


Question: Evaluate the limit using Taylor series: $$\lim_{h\to0}\frac{f(a-5h^2)-f(a+3h)}{h}$$

Answer: -3f'(a)


Question: Calculate \(\lim_{n \to \infty} (n!e - \lfloor n!e \rfloor)\).

Answer: 0


Question: Let $\Delta u = 0$ in the unit ball $B_1(0)$ in $\mathbb{R}^3$, with the boundary condition $u(1, \varphi, \theta) = \sin^2 \varphi$. Find the value of $u$ at the origin $(0,0,0)$ using the mean value property for harmonic functions.

Answer: \dfrac{2}{3}


Question: Let $M$ be a compact connected orientable $3$-manifold without boundary. If $H_1(M; \mathbb{Z}) = 0$, does it follow that $M$ has the same homology groups as a $3$-sphere? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following statement is true or false: If $G$ and $H$ are groups and $\phi : H \rightarrow G$ and $\psi : G \rightarrow H$ are homomorphisms with $\psi \circ \phi = Id_{H}$, then $G \cong H \oplus \ker \psi$. Provide a proof or counterexample.

Answer: False


Question: Does the derivative of a differentiable function $f:(a,b) \rightarrow \mathbb{R}$, where $(a,b)$ is an open interval, correspond to the limit of a sequence of continuous functions?

Answer: Yes


Question: Calculate the nominal annual discount rate, compounded semi-annually, that investors are applying to the revised cash flows of a bond issued by Alpha Inc. The bond, bought at its face value of $1,000 on January 1, 2017, initially had an 8% semi-annual coupon but was renegotiated to a 6% semi-annual coupon and will redeem at $800 at maturity. The yield to maturity on January 2, 2017, is 24.43%.

Answer: 24.43\%


Question: In a directed layered graph with $m$ nodes and $n+1$ layers, determine the tightest upper bound on the number of node removals (excluding cascaded removals) that can be performed without disconnecting the source and target nodes, based on the graph parameters.

Answer: m - n - 1


Question: Evaluate the integral \( \iint_R y\ dx\ dy \) where \( R \) is the region bounded by the \( x \)-axis and the parabolas \( y^2 = 4-4x \) and \( y^2=4+4x \).

Answer: 2


Question: Find the value of the sum \( \sum_{k=1}^{n-1}\ln|1-a^k| \) where \( a \) is a primitive \( n \)th root of unity.

Answer: \ln n


Question: Let \( x_0 \in (0,1) \) and define the sequence \( x_{n+1} = x_n(1-x_n) \) for \( n \geq 0 \). Prove that \( \lim_{n \to \infty} nx_n = 1 \) without using Stolz's theorem.

Answer: 1


Question: Determine whether there exists a subset \( \mathfrak{F} \) of the set of all functions from \( \omega \) to \( \omega \) (denoted \( \omega^\omega \)), with cardinality \( \aleph_1 \), such that no subset \( \mathfrak{G} \) of \( \mathfrak{F} \) with cardinality strictly less than \( \aleph_1 \) can generate \( \mathfrak{F} \). This means every function in \( \mathfrak{F} \) can be expressed as a finite composition of functions in \( \mathfrak{G} \).

Answer: Yes


Question: Find the critical points of the function \( F(x, y, z) = xy + yz + xz + \lambda(x + y + z - 3) \) using the method of Lagrange multipliers.

Answer: (1, 1, 1)


Question:
Suppose $\Omega$ is an open subset of the complex plane and $K$ is a compact subset of $\Omega$. Given a holomorphic and bounded function $f: \Omega \setminus K \to \mathbb{C}$, consider a point $z \in \Omega \setminus K$. Let $\Gamma$ be a contour in $\Omega \setminus (K \cup \{z\})$ with winding number 1 around $z$ and winding number 0 around $K$ and $\mathbb{C} \setminus \Omega$. Define the function $F(z)$ as
$$F(z)=\int_\Gamma \frac{f(\zeta)}{\zeta-z} d \zeta.$$ Determine whether $F(z)$ is holomorphic in $\Omega \setminus K$ or in $\Omega$. \(\text{Choose one: } \Omega \setminus K \text{ or } \Omega\)

Answer: \Omega


Question: If $K$ is an algebraically closed field, determine whether there exists a point $w$ in $K^n$ that is Zariski-closed.

Answer: Yes


Question: Let \( D^2 \) denote the closed unit disk in \( \mathbb{R}^2 \). Let \( \omega = dx \wedge dy \) denote the standard area form on \( \mathbb{R}^2 \) (and on \( D^2 \) by restriction). Let \( \phi \) be a diffeomorphism of \( D^2 \) which is equal to the identity in a neighborhood of \( \partial D^2 \), and which preserves area; i.e. \( \phi^*\omega = \omega \). We denote the group of such diffeomorphisms by \( \text{Diff}_\omega(D^2, \partial D^2) \). Given that there is a 1-form \( \alpha \) with \( d\alpha = \omega \), determine if \( \phi^* \alpha - \alpha \) is exact. Specifically, is \( \phi^*\alpha - \alpha \) equal to \( df \) for some smooth function \( f \)?

Answer: Yes


Question: Given three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ such that $|\vec{a}| = 3$, $|\vec{b}| = 4$, and $|\vec{c}| = 5$, and each vector is perpendicular to the sum of the other two, find the magnitude of the vector $|\vec{a} + \vec{b} + \vec{c}|$. \(\boxed{\text{Your answer here}}\)

Answer: 5\sqrt{2}


Question: Evaluate the limit: \[ \lim_{m\rightarrow \infty}m^3\int^{2m}_{m}\frac{xdx}{x^5+1} \]

Answer: \dfrac{7}{24}


Question: Given \(Q(x)=\sum^{n}_{i=1} P_{i}(x)\), where each \(P_{i}(x)\) is a cubic polynomial of the form \(P_{i}(x)=x^{3}+a_{i}x^{2}+b_{i}x+c_{i}\), and assuming all roots of \(Q(x)\) are real, determine the maximum value of the real constant \(T\) such that \((a_{1}^2+\ldots+a_{n}^2)\geq T(b_{1}+\ldots+b_{n})\).

Answer: 3


Question: Determine the distribution to which the sequence of random variables \( \frac{\zeta_n}{n} \) converges, where \( \zeta_n \) follows a Poisson distribution with parameter \( \lambda = 2n \).

Answer: 2


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

Answer: \dfrac{3}{4}


Question: A fire department receives calls at a Poisson rate of three per day, with one third of them being false alarms. Under the existing system, firefighters earn $140 per day. A new plan suggests firefighters receive a random reward with an expected value of $60 for each fire they extinguish. Calculate the expected total reward up to time \( t \) using Wald's identity, \( E(R_t)=E(N_t)E(Y_i) \), where \( E(N_t) \) is the expected number of fires and \( E(Y_i) \) is the expected reward per fire.

Answer: 120t


Question: Calculate the limit: $$\lim_{x\to\infty}x\left(\int_0^x te^{-2t}\,dt-\dfrac{1}{4}\right)$$

Answer: 0


Question: Given the permutations \(\sigma = (1, 2, 3, 4, 5)(6, 10)(7, 11)(8, 12)(9, 13)\) and \(\tau = (2, 5)(3, 4)(6, 7, 8, 9, 10, 11, 12, 13)\) with orders 10 and 8, respectively, and subgroups \(H = \langle \sigma \rangle\) and \(K = \langle \tau \rangle\) within the group \(G\) generated by \(\sigma\) and \(\tau\), determine the size of \(G\) if \(\langle H \cap K \rangle\) has order 2 and consists of the identity and the cycle \((6, 10)(7, 11)(8, 12)(9, 13)\).

Answer: 40


Question: Evaluate the integral \( \int_\gamma \frac{1}{t}\cdot\frac{t+\zeta}{t-\zeta}\frac{dt}{t} \) where \( \gamma \) is a unit circle in the complex plane, and \( \zeta \) is an arbitrary point inside the unit circle. Determine whether the result of the integral is zero or \( \frac{4\pi i}{\zeta} \).

Answer: 0


Question: Find the number of elements of order 8 in the multiplicative group of the finite field $GF(p^n)$, where $p \equiv 3 \pmod{4}$ and $n$ is even.

Answer: 4


Question: Determine whether the set \( \left\{ 1, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}, \frac{1}{16}, \frac{3}{16}, \ldots \right\} \) forms a dense subset of the interval [0,1].

Answer: Yes


Question: Suppose $B$ is a 3 by 3 matrix with eigenvalues 3, 5, and 7. Calculate the determinant of the matrix $B^4 - 3B^3 + 2B^2 - 6I$, where $I$ is the identity matrix.

Answer: 5164992


Question: Suppose $R$-module homomorphisms $T_1, T_2: R^m \rightarrow R^n$ are represented by matrices $A$ and $B$, respectively. If $B$ is obtained from $A$ through elementary column and row operations, does it follow that $\text{Im}(A) = \text{Im}(B)$, where $\text{Im}(A)$ is the subspace spanned by the columns of $A$? Answer yes or no.

Answer: No


Question: Determine the curvature of the cycloid defined by the parametric equations \( x = t - \sin(t) \) and \( y = 1 - \cos(t) \) at the point where \( t = 0 \). Note that the standard curvature formula \( \kappa = \frac{||\det (r'(t), r''(t))||}{||r'(t)||^3} \) cannot be applied directly at \( t=0 \) because \( r'(t) \) is zero. Calculate the curvature around this point.

Answer: \infty


Question: Determine the number of partitions of a positive integer \( n \) without using summation or product symbols.

Answer: p(n)


Question: Determine the order of the element $32$ in the multiplicative group $(\mathbb{Z}/2015\mathbb{Z})^\times$. Provide your answer as a single integer.

Answer: 12


Question: Determine whether any prime number raised to a power greater than or equal to 2 can be expressed as the sum of two prime numbers.

Answer: No


Question: Classify the singularity of the function \( f(z) = \frac{\sin z}{\cos(z^3) - 1} \) at \( z = 0 \) and compute its residue using series expansion.

Answer: -\dfrac{1}{60}


Question: Find the limiting value of the Beta function \( B(a, b) \) as either \( a \) or \( b \) approaches zero, while the other parameter remains fixed.

Answer: \infty


Question: Given that $xyz = (1-x)(1-y)(1-z)$ with $x, y, z \in (0,1)$, find the minimum value of the expression $x(1-z) + y(1-x) + z(1-y)$. Use the AM-GM inequality to determine this minimum value.

Answer: \dfrac{3}{4}


Question: Determine the number of imaginary roots of the equation \( \frac{A_1^2}{x-a_1} + \frac{A_2^2}{x-a_2} + \frac{A_3^2}{x-a_3} + \cdots + \frac{A_n^2}{x-a_n} = k \), where \(a_1, a_2, \ldots, a_n, A_1, A_2, \ldots, A_n,\) and \(k\) are real numbers.

Answer: 0


Question:
Let $f \geq 0$ be a measurable function on a measure space $(X, \mathcal{F}, \mu)$. Suppose that
$$ \mu\left(\{x: f(x) > t\}\right) = \frac{1}{t^2 + 1} $$ for all $t \geq 0$. Compute the integral $$ \int_X f\,d\mu $$.

Answer: \dfrac{\pi}{2}


Question: Determine the dimension of the vector space $\Bbb Q[x,y]/J$ over $\Bbb Q$, where $J=\langle(x-1)^3,(x-1)(y+2),(y+2)^2\rangle$. Provide your answer as a single integer.

Answer: 4


Question: Find the volume of the elliptical cone bounded by the surface \( z=\sqrt{9x^2+y^2} \) and the plane \( z=2 \).

Answer: \dfrac{8\pi}{9}


Question: Determine whether there exists a system of irreducible polynomial equations over $\mathbb{Q}$ that has more than 5 but less than 50 $\mathbb{Q}$-solutions.

Answer: Yes


Question: Evaluate the limit \( \lim\limits_{(x,y)\to(1,-1)} (x-y+3)e^{-1/(x^2+y^2-2x+2y+2)} \) using the triangle inequality and a suitable substitution.

Answer: 0


Question: Find the chromatic number of the graph G, where the vertices of G are the integers from 1 to 64, and two vertices are adjacent if and only if one of the corresponding numbers divides the other.

Answer: 7


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

Answer: e^{\frac{3}{2}}


Question: Determine whether the following statement is true: In topology, for a finite set \( X \), a basis for a topology on \( X \) can be constructed as the collection of sets \( \{ \{y \mid y \text{ is in every open set containing } x\} \mid x \in X \} \), after possibly removing duplicates. This suggests that the minimum number of basis elements is bounded above by the cardinality of \( X \).

Answer: True


Question: Find the value of \(x\) that satisfies the equation: $$\sum_{n=1}^{\infty }\sum_{k=1}^{\infty }\frac{1}{n}\left(\frac{x}{n}\right)^k=0$$

Answer: 0


Question: Can a conditionally convergent series be rearranged to converge to a different value?

Answer: Yes


Question: Evaluate the limit: $$\lim_{n\to\infty} \frac{1}{\sqrt{n}}\int_{1/{\sqrt{n}}}^{1}\frac{\ln(1+x)}{x^3}\,dx.$$

Answer: 1


Question: Evaluate the complex line integral \(\int_{C_2(0)}\frac{1}{z^2+z+1}\ dz\), where \(C_2(0)\) is the open disk of radius 2 centered at the origin in the complex plane. Simplify or evaluate the integral using partial fractions and Cauchy's integral formula.

Answer: 0


Question: Given a finite set $S \subset \mathbb{Z}$, we can easily obtain its median $\text{median}(S)$. Consider a set of sets $T = \{ S_1, S_2, \cdots, S_n \}$, where $S_i \cap S_j = \emptyset$ for all $S_i, S_j \in T$. Define two medians: $M_1 = \text{median}(\cup_{S \in T} S)$ and $M_2 = \text{median}(\{ \text{median}(S) \mid S \in T \})$. Determine if $M_1 = M_2$. Provide a justification for your answer.

Answer: M_1 \neq M_2


Question: Find the coefficient of $x^{26}$ in the series expansion of $(1+x+x^{10})^{20}$. Express your answer as a single integer.

Answer: 3546540


Question:
Given real and positive numbers \(a\), \(b\), and \(c\), consider the quadratic equation in \(x\):
\[
(a+b+c)x^2 - 2(ab+bc+ca)x + ab^2 + bc^2 + ca^2 = 0
\]
If this equation has at least one real solution, determine the value of \(\frac{a+5b}{c}\).

Answer: 6


Question: Find a non-constant function that is harmonic in the punctured unit ball $B(0,1)\setminus\{0\}$ and satisfies $\lim_{x\to 0}u\!\left(x\right)/\lvert x \rvert^{2-N}=0$. Provide the function.

Answer: u(x) = x_1


Question: Evaluate the integral \( \int_{\gamma} \tanh(z) \, dz \) where the contour \( \gamma(t) = e^{it} \) for \( 0 \leq t \leq 2\pi \). Use the residue theorem to determine the poles and compute the integral.

Answer: 0


Question: Determine the supremum of the function \( f(x,y) = x^2y^2(x^2+y^2) \) on the set \( S = \{(x,y) \in \mathbb{R}^2 : x > 0, y > 0, x+y = 2n\} \).

Answer: 2n^6


Question: Evaluate the integrals \( I = \int_0^\infty x^{1/2}e^{-x^3}\,dx \) and \( J = \int_0^1 x^{-2/3}(1-x)^{-1/3}\,dx \), and find the ratio \( \frac{J}{I} \).

Answer: 2\sqrt{3\pi}


Question: Evaluate the limit: $$\lim_{{x\to 0}} \left(\frac{{e^{x^2}+2\cos(x)-3}}{{x\sin(x^3)}}\right).$$

Answer: \dfrac{7}{12}


Question: Determine whether the sequence \( b_n = \arctan \left( \ln \left( \frac{n^2+1}{n^2+3} \right) \right) \), where \( n \in \mathbb{N} \), converges. If it converges, find its limit.

Answer: 0


Question: Let $R$ be a commutative ring and $M$ be a module over $R$. If $M$ is simple as an $R/I$-module for some ideal $I$ of $R$, does it imply that $M$ is simple as an $R$-module? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit as \( x \) approaches 0 for the function: \[ \lim_{x \to 0} \frac{|x|}{\sqrt{x^{4}+4 x^{2}+7}} \sin \left(\frac{1}{3 \sqrt{x}}\right) \]

Answer: 0


Question: Evaluate the integral \( \int_1^4 \int_{2}^{\sqrt{x}} (xy + 2y) \, dy \, dx \).

Answer: -9


Question: Determine the parity of the sum of the degrees of the vertices in a graph where each vertex represents a square on a chessboard, and each edge represents a neighborhood relation between squares, given that every vertex except the first has an even number of previously visited neighbors in a tour.

Answer: even


Question: Determine the number of sets \(\mathcal{A}\) such that \(1 \leq |\mathcal{A}| \leq 999\) and \(2^{|\mathcal{A}|} = z|\mathcal{A}|\) for some positive integer \(z\).

Answer: 10


Question:
Let $a, b, c$ be non-negative real numbers satisfying $a^2 + b^2 + c^2 = 3$. Find the minimum value of the expression:
\[ P = ab + bc + 3ca + \frac{3}{a + b + c}. \]

Answer: \sqrt{3}


Question: Find the value of $k$ if the angle between the planes given by the equation $6x^2+4y^2-10z^2+3yz+4zx-11xy=0$ is $\cos^{-1}(k)$. Use the appropriate concept to determine the angle between these planes.

Answer: 0


Question: Determine whether there exists a sequence defined by $A_0=0$, $A_1=1$, and $A_{n+2}=pA_{n+1}-qA_{n}$, where $p$ and $q$ are fixed integers, that contains infinitely many different square numbers.

Answer: Yes


Question:
Determine the number of continuous functions \( f: \left[0, \frac{3}{2}\right] \rightarrow (0, \infty) \) that satisfy the equation:
\[ 4\int_{0}^{\frac{3}{2}}f(x)dx + 125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}} = 108 \]

Answer: 1


Question: Evaluate the integral \( \int_{|z+1|=2} \frac{z^2}{4-z^2}dz \) using Cauchy's Integral Formula.

Answer: 2\pi i


Question: Determine the order of growth of the entire function defined by the infinite product \( \prod_{n=0}^{\infty} \left(1 - \frac{z}{(n+1)!}\right) \).

Answer: 0


Question: Suppose we have a matrix $A$ with distinct eigenvalues $\lambda_1, \dots, \lambda_r$. Let $S_i$ be linearly independent sets of eigenvectors corresponding to $\lambda_i$ for $i = 1, \dots, r$. If we combine all the eigenvectors in these sets into one set, is this combined set also linearly independent? Provide a justification for your answer.

Answer: Yes


Question: Determine the set of values for $p$ such that the points of extremum of the function \( f(x) = x^3 - 3px^2 + 3(p^2 - 1)x + 1 \) lie within the interval \((-2,4)\).

Answer: (-1, 3)


Question: Evaluate the limit of the sequence \( a_n = \sin(n) \cdot \sin\left(\frac{1}{n}\right) \) as \( n \to \infty \).

Answer: 0


Question:
Which of the following assertions is true for absolutely continuous functions $g \in {\rm W}^{1,1}(0,1) \cap {\rm C}^1(0,1) \cap {\rm C}[0,1]$? Choose the correct option from the list below:

1. If $F \subseteq (0,1)$ is a set of full measure in $(0,1)$, then $g(F)$ is a set of full measure in ${\rm Im}(g)$.
2. For every sequence of measurable subsets $(E_n)$ in $(0,1)$ such that $\lim_{n\rightarrow+\infty}\lambda(E_n)=0$ it holds that $\lim_{n\rightarrow+\infty}\lambda(g(E_n))=0$.
3. For every $\varepsilon>0$ there exists $\delta>0$ such that: for every countable family of non-overlapping intervals $([a_k,b_k])_{k=1}^{+\infty}$ such that $\sum_{k=1}^{+\infty}|b_k-a_k|\leq\delta$ we have $\sum_{k=1}^{+\infty}|g(b_k)-g(a_k)|\leq\varepsilon$.
4. For every $\varepsilon>0$ there exists $\delta>0$ such that: for every countable family of (possibly overlapping) intervals $([a_k,b_k])_{k=1}^{+\infty}$ such that $\sum_{k=1}^{+\infty}|b_k-a_k|\leq\delta$ we have $\sum_{k=1}^{+\infty}|g(b_k)-g(a_k)|\leq\varepsilon$.
5. For every $\varepsilon>0$ there exists $\delta>0$ such that: for every measurable set $E\subset (0,1)$ such that $\lambda(E)\leq\delta$ we have $\lambda(g(E))\leq\varepsilon$.
6. If $(K_m)$ is an increasing sequence of compact sets in $(0,1)$ such that $\lim_{m\rightarrow+\infty}\lambda((0,1)\backslash K_m)=0$, then $\lim_{m\rightarrow+\infty}\lambda\{\xi\in\mathbf{R}: (g_{\vert K_m})^{\leftarrow}(\xi)\neq\emptyset\}=\lambda({\rm Im}(g))$, where, as before, ${\rm Im} (g):=g([0,1])=[m,M]$.
7. Measure preserving mapping: for every $0<\theta<1$ there exists $0<\varphi_{\theta}<1$ such that we have: for every measurable $E\subseteq (0,1)$ such that $\lambda(E)=\theta$, it holds that $\lambda(g(E))=\varphi_{\theta}\lambda({\rm Im}(g))$.

Select the correct assertion.

Answer: 3


Question: Determine whether the following statement is true or false: In a poset $(P, \leq)$, if every sequence $p_1 \leq p_2 \leq \cdots$ in $P$ stabilizes (meaning that for some $n$, $k > n$ implies $p_k = p_n$), then every non-empty subset of $P$ contains a maximal element.

Answer: True


Question: Evaluate the limit \( L = \lim_{x\rightarrow 0}\left(\frac{1}{\ln(1+x)}-\frac{1}{\ln(x+\sqrt{x^2+1})}\right) \) using L'Hôpital's rule. Then, find the value of \( L +\frac{153}{L} \).

Answer: \dfrac{613}{2}


Question:
For each positive real number $\lambda$, let $A_\lambda$ be the set of all natural numbers $n$ such that $|\sin\sqrt{n+1}-\sin\sqrt n|<\lambda$. Let $A_\lambda^c$ be the complement of $A_\lambda$ in the set of all natural numbers. Determine which of the following is true:

(A) $A_{1/2},A_{1/3},A_{2/5}$ are all finite sets
(B) $A_{1/3}$ is a finite set but $A_{1/2},A_{2/5}$ are infinite sets
(C) $A_{1/2}^c,A_{1/3}^c,A_{2/5}^c$ are all finite sets
(D) $A_{1/3},A_{2/5}$ are finite sets and $A_{1/2}$ is an infinite set

Answer: C


Question: Determine whether every closed, orientable 3-manifold can be embedded in Euclidean 4-space.

Answer: Yes


Question: Calculate the expected number of steps, $E[\tau]$, for a symmetric random walk starting at 0 to reach either level 3 or level -1.

Answer: 3


Question: Determine the number of ways to completely cover a $3 \times 10$ rectangle using $3 \times 2$, $3 \times 3$, and $3 \times 4$ rectangles.

Answer: 17


Question: Find the limit of the sequence \( S_n = \int_0^1 \frac{nx^{n-1}}{1+x} \, dx \) as \( n \to \infty \).

Answer: \dfrac{1}{2}


Question: Determine whether it is possible to construct a $C^*$-algebra that contains $L^1(G)$ as a subalgebra, where $G$ is a locally compact abelian group, even though $L^1(G)$ itself is not a $C^*$-algebra.

Answer: Yes


Question: If a sequence $(a_n)$ satisfies $a_n > 0$ for all $n$ and $\frac{a_{n+1}}{a_n} \to 1$, does it follow that $(a_n)^{1/n} \to 1$? Answer 'yes' or 'no' and justify your answer.

Answer: yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} n \log\left(1+ \left(\frac{f(x)}{n}\right)^p\right) \] where \(0<p<1\).

Answer: \infty


Question: Let \( f(x) \) be a four-times differentiable function such that \( f(2x^2-1)=2xf(x) \). Find the value of \( f''''(0) \).

Answer: 0


Question: Suppose the point $z$ lies on a circle centered at the origin in the complex plane. The vertices of a triangle are $z, \omega z,$ and $z + \omega z$, where $\omega$ is a complex cube root of unity. If the area of this triangle is $4 \sqrt{3}$ square units, find the radius of the circle.

Answer: 4


Question: Determine the values of $a$ and $b$ such that the polynomial identity $x^6+1=(x^2+1)(x^2+ax+1)(x^2+bx+1)$ holds. Then, find the product $ab$.

Answer: -3


Question:
Consider the exact sequence of abelian groups:

$$0 \to \mathbb{Z}^k \to \mathbb{Z}^m \to \mathbb{Z}^n \to 0.$$ After tensoring this sequence with $\mathbb{Q}$, determine the relationship between $k$, $m$, and $n$.

Answer: m = k + n


Question: Evaluate the limit: $$\lim_{(x,y,z)\to (1/2,1/2,1/2)}{x\cos(\pi y)\sin\left(\frac{1}{2z-1}\right)}$$

Answer: 0


Question: Determine whether the projection of an open set in \(\mathbb{R}^N\) onto a subspace \(\mathbb{R}^n\) is always an open set.

Answer: Yes


Question: In a random graph with $n$ nodes, where each node is connected to $k$ randomly chosen nodes, determine the probability that the graph is connected if $k = \frac{\log n + c}{n}$, where $c$ is a constant that tends to infinity.

Answer: 1


Question: Determine whether the sequence \( \left(\alpha^{1/(n \cdot \ln n)}-1\right)^{1/n} \) converges or diverges for \( 2<\alpha<3 \).

Answer: 1


Question: Determine the structure of the quotient ring $R[[x]]/(g)$, where $g = x^2-x$ and $R$ is any coefficient ring.

Answer: R \times R


Question: Let $X$ and $Y$ be topological spaces, and let $f, g \in C(X, Y)$ be continuous functions. If $Y$ is a normal space, is the set $\{x \in X : f(x) = g(x)\}$ necessarily closed?

Answer: No


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

Answer: 1


Question: Determine the values of $p$ for which the integral \( \int_1^\infty \frac{1}{x^p(1+\ln x)} \, dx \) converges.

Answer: p > 1


Question: Evaluate the limit \( \lim_{r\to0}\int_{\sigma_{r}}\frac{e^{iz}}{z}\, dz \), where \( \sigma_{r}:[0,\pi]\to\mathbb{C} \) is defined by \( \sigma_{r}(t)=re^{it} \).

Answer: i\pi


Question: Evaluate the integral \( \int_E \cos(x^2 + y^2) \, dA \), where \( E = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 4\} \), by converting to polar coordinates.

Answer: \pi \sin(4)


Question: Suppose \(\{X_i\}_{i=1}^n\) is a sequence of independently distributed random variables that take values in \([0,1]\). Let \(\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i\) denote the average of the sequence. Find an upper bound for \(\text{Var}(\bar{X}_n)\).

Answer: \dfrac{1}{4n}


Question: Evaluate the double integral: \[ \int_{y=0}^{y=1} \int_{x=y}^{x=1} \sin(\pi x^2) \,dx \,dy. \]

Answer: \dfrac{1}{\pi}


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

Answer: -\dfrac{1}{2}


Question: Calculate the expected value of the area of the rectangle EFDG formed inside a unit square ABCD, where point E is randomly and uniformly chosen within the interior of the square.

Answer: \dfrac{1}{4}


Question: Let $T$ be the shift modulo 1 of the Cantor set by an irrational number $\alpha \in (0,1)$. Consider the measure $\mu$ on the interval $[0,1]$ generated by the Cantor staircase. Determine whether $\mu(T(C) \cap C) = 0$.

Answer: 0


Question: Consider a directed graph $G(V,A)$ with arc capacities $c_i$ and a source/sink. Let $f$ be the max flow in $G$. If we modify the capacities such that $c_i^{'} = c_i$ for all arcs where $f_i = c_i$ and $c_i^{'} = c_i + 1$ for all arcs where $f_i < c_i$, is $f$ still a max flow in $G$ with the new capacities $c^{'}$?

Answer: Yes


Question:
Given the partial derivatives of a function:
\[\frac{\partial M}{\partial y} = (4x^2+2xy+6y)a(x+y)^{a-1} + (x+y)^a(2x+6)\]
\[\frac{\partial N}{\partial x} = (2x^2+9y+3x)a(x+y)^{a-1} + (x+y)^a(4x+3)\]
The condition for the function to be conservative is that \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\). Applying this, we get:
\[(4x^2+2xy+6y)a(x+y)^{a-1} + (x+y)^a(2x+6) = (2x^2+9y+3x)a(x+y)^{a-1} + (x+y)^a(4x+3)\]
By simplifying the equation, find the value of \(a\).

Answer: 1


Question: Three circles have radii 1, 2, and 3, and are externally tangent to each other. Determine the radius of a smaller circle that is tangent to all three of these circles.

Answer: \dfrac{6}{23}


Question: Determine the dominant term in the asymptotic behavior of the sequence \( u_n = \sum_{k=0}^{n} \frac{1}{k^2+(n-k)^2} \) as \( n \rightarrow +\infty \).

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


Question: Evaluate the integral \( \frac{1}{2 \pi i} \oint_{C} \frac{z^{2} d z}{z^{2}+4} \), where \( C \) is the square with vertices at \( \pm2, \pm2+4i \). Use the Cauchy integral theorem and formula to find the value of the integral.

Answer: i


Question:
Given that \( x^5 = 1 \) and \( x \neq 1 \), evaluate the expression:

\[ \frac{x}{1+x^2}+\frac{x^2}{1+x^4}+\frac{x^3}{1+x}+\frac{x^4}{1+x^3} \]

Use the identity \( 1+x+x^2+x^3+x^4 = 0 \) to simplify the expression.

Answer: 2


Question: Consider two matrices $M$ and $N$ of the same dimension. The symmetric sum is defined as $L = MN + NM$. Determine if there exists a transformation $T$ such that $T[M] = M'$ and $T[N] = N'$ satisfy $M'N' + N'M' = MN + NM$. Is the symmetric sum invariant under the transformation $T$?

Answer: Yes


Question: Determine whether the space $B^{s;1,\theta}(\Omega)$ is compactly embedded in $L^{r}(\Omega)$ under the conditions $0<s<1$, $1\leq \theta \leq \frac{N}{N-s}$, $r<\frac{N}{N-s}$, and $\Omega$ is an open, bounded subset of $\mathbb{R}^N$ with a smooth boundary.

Answer: Yes


Question: Given a function \( f(z) \) with a zero of order \( n \) at \( z=a \), where \( f(a) = f'(a) = \cdots = f^{(n)}(a) = 0 \) but \( f^{(n+1)}(a) \neq 0 \), and another function \( g(z) \) such that \( g(a) \neq 0 \), determine the order of the zero of the function \( \frac{f}{g} \) at \( z=a \).

Answer: n


Question: Determine the order and structure of the factor group $G/N$, where $G$ is a group of order 24 and $N = \{ e, a^{12} \}$, with $e$ as the identity element and $a$ as an element of $G$. What is the structure of $G/N$?

Answer: \mathbb{Z}_{12}


Question: Find all holomorphic functions \( f = u + iv \) on the unit disk \( \mathbb{D}^2 \) such that the imaginary part \( v \) is non-positive and \( v(0) = 0 \).

Answer: f(z) = c


Question: Is it possible for a self-complementary graph to have 10 vertices? Provide a justification for your answer.

Answer: No


Question: Evaluate the contour integral \( \int_{|z|=3}\frac{2z^2-z-2}{z-\omega}dz \) given that \(|\omega| > 3\).

Answer: 0


Question: Find the residue of the function \( \frac{z}{\sin z} \) at \( z = \pi \).

Answer: -\pi


Question: Suppose that $X_i$ are i.i.d. random variables with finite absolute moment $E|X_1| < \infty$. Determine whether the statement $\max(X_1, \ldots, X_n) / n \to 0$ almost surely is true.

Answer: True


Question: Given a $12$-cycle permutation $\sigma = (1, 2, \dots, 12)$, determine the conditions under which $\sigma^i$ is also a $12$-cycle. Consider the relationship between the order of a permutation and its cycle structure.

Answer: \gcd(i, 12) = 1


Question: Calculate the limit: $$ \lim_{n\rightarrow\infty}\ln\left(\prod_{k=1}^{n}\frac{2k+1}{2k}\right) $$

Answer: \infty


Question: Determine whether a commutative ring with unity, whose spectrum contains only the zero ideal, must be a field. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: $$\lim_{x \to \infty}{x^aa^x} $$ where $0<a<1$.

Answer: 0


Question: Determine whether there always exists a field extension $E$ that contains two given field extensions $E_1$ and $E_2$ of a base field $F$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether every prime number greater than 2 has a prime primitive root in the range from 2 to p-1.

Answer: Yes


Question: Find the reflection of the point $(-1, -1, -1, 3)$ with respect to the hyperplane defined by the normal vector $N$, given that the reflection of the point $(-2, 2, 2, -3)$ is $(-4, 0, -2, 1)$. How can this reflection be calculated?

Answer: (1, 1, 3, -1)


Question: What is the probability that in a sequence of coin flips, there will be a point where the number of heads is at least twice the number of tails?

Answer: \dfrac{5 - \sqrt{5}}{4}


Question: Determine whether there exists a topology on the interval $[0,1]$ that makes it pseudocompact but not compact.

Answer: Yes


Question: Determine the range of the largest root of the polynomial \(x^{a+2} - x^{a+1} - 1\) as \(a\) approaches infinity.

Answer: 1


Question: Evaluate the improper integral \( \int_{0}^{\infty}\frac{\sin^2 t}{t^2} dt \) using Laplace transforms.

Answer: \dfrac{\pi}{2}


Question: Determine the values of $x$ for which the expression $(A^TA)^x$ is well-defined, where $A$ is a real matrix.

Answer: x > 0


Question: Determine whether the sequence $\left( x_n \right)_{n \in \mathbb{N}}$, where $x_n = \sqrt[n]{n!}$, converges or diverges using the concepts and tools introduced in Chapter 3 of Rudin's Principles of Mathematical Analysis.

Answer: \infty


Question: Evaluate the infinite series \( S = 1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \cdots \) for \(|x| < 1\).

Answer: 1 - \ln(1 - x)


Question: Suppose that \( A \) is a square, invertible matrix such that \( A^4 = A \). Find all real eigenvalues of \( A \).

Answer: 1


Question: Let $P(x)$ be a polynomial with non-negative integer coefficients such that $P(0)=33$, $P(1)=40$, and $P(9)=60,000$. Find the value of $P(2)$. \( \boxed{} \)

Answer: 87


Question: Find the limit: $$\lim_{n \to +\infty} n^2x(1-x)^n, \text{ where } x \in [0,1].$$

Answer: 0


Question: Given the vector space $V = M_{2\times 3}(\mathbb{R})$ and the bilinear form $f: V \times V \to \mathbb{R}$ defined by $f(X, Y) = \text{Tr}(X^T A Y)$, where $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, find the matrix representation of $f$ with respect to the standard basis of $M_{2\times 3}(\mathbb{R})$. What is the order of this matrix?

Answer: 6


Question: Determine the number of continuous functions on the interval $[0,1]$ that satisfy the equation $f(x) = \frac{1}{\sqrt{2}}f\left(\frac{x}{2}\right)$ and are bounded on this interval.

Answer: 1


Question: Find the fundamental group of a topological space formed by connecting six 3-spheres in a cyclic order with line segments, where each sphere also touches a common 3-manifold.

Answer: \mathbb{Z}


Question: A broken $12$-hour analog clock has an hour hand that moves forward by one hour every sixty minutes with a probability of $0.99$, and remains stuck with a probability of $0.01$. The movements are independent across hours. If it is currently noon (12pm), what is the probability that the hour hand points towards 3 o'clock as the number of hours, $N$, approaches infinity?

Answer: \dfrac{1}{12}


Question: Let $F$ be a nonnegative integrable function on $\mathbb{R}$. Suppose there exists a constant $C$ such that \( \int_{\mathbb{R}} Ff \leq C\int_{\mathbb{R}} f \) for all nonnegative continuous functions $f$ with compact support. Is it true that $F(x) \leq C$ almost everywhere?

Answer: Yes


Question: Determine the maximum number of solutions possible for a system of two non-linear equations in the plane, where each equation can be a quadratic, circle, hyperbola, ellipse, or a rotated version of these curves.

Answer: 4


Question: Let $a, b, c$ be the roots of the polynomial $f(x) = x^3 + 3x^2 - 7x + 1$. Calculate the product $f'(a)f'(b)f'(c)$. Provide your answer as a single number.

Answer: -1300


Question: Determine the number of distinct $k$-dimensional subspaces in an $n$-dimensional vector space $V$ over a finite field with $q$ elements.

Answer: \dbinom{n}{k}_q


Question: Given two random variables $X$ and $Y$ with a joint distribution, where $E(X) = E(Y)$, $E(X^2) = E(Y^2)$, and $Cov(X, Y) = 2$, determine whether the statement $Var(X) < 2$ is true or false.

Answer: B


Question: Find all natural numbers \(n\) such that if \(1 \le a_1 < a_2 < \ldots < a_k = n\) are all the divisors of \(n\), then \(a_2^3 + a_3^2 - 15 = n\).

Answer: 2009


Question: Determine the kernel of the ring homomorphism \( g: k[x_1,x_2] \to k[u,v] \) defined by \( x_1 \mapsto u^2+v^2 \) and \( x_2 \mapsto uv \).

Answer: (0)


Question: Evaluate the infinite series: $$\sum_{n=1}^{\infty} \left[\sin\left(\frac{4}{n}\right)-\sin\left(\frac{4}{n+1}\right)\right].$$

Answer: \sin 4


Question: Let $f(x)$ be a polynomial with integer coefficients. If $f(r)$ is divisible by a prime number $p$ for some integer $r$, determine if there exists a polynomial $g(x)$ such that $f(x) - (x-r)g(x)$ is divisible by $p$.

Answer: Yes


Question: Determine whether the integers $yz$ and $x(y+z)$ are coprime, given that $x$, $y$, and $z$ are pairwise coprime integers, meaning $\gcd(x,y) = \gcd(x,z) = \gcd(y,z) = 1$. Provide a justification for your answer.

Answer: Yes


Question: Let \((X_n)_n\) be i.i.d. random variables and define \(W_n = \sum_{k=1}^n X_k\). Suppose there exist \(u_n > 0\) and \(v_n \in \mathbb{R}\) such that \(\frac{1}{u_n}W_n - v_n\) converges in distribution to a non-degenerate random variable \(W\). If \(\frac{u_{2n}}{u_n} \to c\), where \(c\) is a positive constant, determine the behavior of the sequence \(\frac{u_{n+1}}{u_n}\) as \(n\) approaches infinity.

Answer: 1


Question: Determine whether the following statement is true or false: Let \( \mathbb{D} = \{ z \in \mathbb{C} \mid |z| < 1 \} \) and \( f: \mathbb{D} \to \mathbb{C} \) be an analytic function. For any closed curve \( \gamma \) in the unit disc \( \mathbb{D} \), the integral \( \int_{\gamma} \frac{f(z)}{(z-a)^2} \, dz = 0 \) for all \( a \in \mathbb{C} \) with \( |a| \ge 1/2 \).

Answer: B


Question: Find the limit of the sequence \( q_n = \frac{n^2}{(2n+3)^2} \cdot \frac{1}{\frac{1}{n}\sum_{d=1}^n\frac{1}{(\frac{d}{n}+3)^2}} \) as \( n \to \infty \).

Answer: 3


Question: Are there locally compact subsets of \(\mathbb{R}^n\) that are not compact? Justify your answer.

Answer: Yes


Question: For two jointly normal random variables $X$ and $Y$ with means $E[X] = \mu_X$ and $E[Y] = \mu_Y$, variances $Var(X) = \sigma_X^2$ and $Var(Y) = \sigma_Y^2$, and correlation $\rho(X,Y) = \rho$, find the partial derivative $\frac{\partial M_{XY}(s,t)}{\partial s}$ of the moment generating function $M_{XY}(s,t) = E[e^{sX+tY}]$ at the point $(s,t) = (0,0)$. \( \boxed{} \)

Answer: \mu_X


Question:
Given two matrices $A$ and $B$ of the form:

$$
A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\
a_{12} & a_{22} & a_{23} \\
a_{13} & a_{23} & a_{33}
\end{pmatrix},
\quad
B = \begin{pmatrix} b_{33} & b_{23} & b_{13} \\
b_{23} & b_{22} & b_{12} \\
b_{13} & b_{12} & b_{11}
\end{pmatrix}.
$$

If $A$ is positive definite, does it necessarily follow that $B$ is also positive definite? Provide a justification for your answer.

Answer: Yes


Question: Determine the measure of the set \( A = \left \{x \in [0,1] \cap (\mathbb{R} - \mathbb{Q}) \mid \text{in the decimal expansion of } x, \text{ the first 1 (if it exists) is followed by 2} \right \} \).

Answer: \dfrac{1}{10}


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function with \( f(-2) = 7 \) and \( f(0) = 13 \). If \( h(x) = \int_{-2}^0 f(x-t)\,dt \), find \( h'(1) \).

Answer: 6


Question:
In the ring of $2 \times 2$ matrices over the finite field $\Bbb{Z}_p$, consider the set of matrices of the form
$$
\begin{bmatrix} a & b \\ -b & a \\ \end{bmatrix}
$$
where $a, b \in \Bbb{Z}_p$. Determine the order of the centralizer of this set.

Answer: p^2


Question: Determine whether the following statement is true: If $f:\mathbb{R} \to \mathbb{R}$ is a strictly increasing function with $f(0) = 0$, and $a > 0$, $b > 0$, then $\int_0^a f(x) \, dx + \int_0^b f^{-1}(x) \, dx \geq ab$. Assume no continuity for $f$. Is this statement true?

Answer: True


Question: Evaluate the limit: $$\lim_{n \to \infty}\left( \frac{3^{n+1} \cdot (n+1)^{2017}}{2^n} \right)$$ using a method other than L'Hôpital's rule or the Stolz Cesaro theorem.

Answer: \infty


Question: Evaluate the integral \( \int_{C} \frac{e^z}{z^3} \, dz \), where the contour \( C(t) = 2e^{2\pi it} + 1 + i \) for \( 0 \leq t \leq 1 \).

Answer: \pi i


Question: Determine whether there exists a differentiable convex function, other than constant functions, that has more than one global minimum point.

Answer: Yes


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

Answer: 2


Question:
Consider the system of differential equations given by:

$$x' = ax + ay$$
$$y' = ax - (a + 2b)y$$

Determine which condition ensures that each solution $\begin{pmatrix}x(t)\\y(t)\end{pmatrix}$ satisfies $\lim_{t\to \infty}x(t)=\lim_{t\to \infty}y(t)=0$.

(a) $a^2 + b^2 > 0$
(b) $a, b > 0$
(c) $-b < a < 0$
(d) $a > 0 > b$

Answer: C


Question: Suppose you have a function $f$ on $\mathbb{R}$ such that \(\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty\) and \(\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)\) as \(u \to 0\). Is it true that $f$ is in $L^p(\mathbb{R})$ for any $p \geq 1$?

Answer: No


Question: Determine whether the series \(\sum_{n=1}^\infty \frac{(x-2)^n}{n^2 \cdot 3^n}\) converges.

Answer: [-1, 5]


Question: Find the equation of the plane \(\pi\) such that the line \(l_1 : (x,y,z) = t(1,2,-2)\) reflects to become the line \(l_2: (x,y,z) = t(2,-1,2)\) upon reflection from the plane. The intersection point of the two lines is \((0,0,0)\). Express the equation of the plane in the form \(Ax + By + Cz = 0\).

Answer: x - 3y + 4z = 0


Question:
Find an integer pair \((a, n)\) such that:

\[\begin{cases} 5^a+1 \equiv 0 \pmod {3\cdot 2^n-1} \\
3\cdot 2^{n-1}-1 \equiv 0 \pmod a\end{cases}\]

where \(n \equiv 3 \pmod 4\).

Answer: (11, 3)


Question: Let $x_1, x_2, \ldots, x_n$ be the roots of the equation $1 + x + x^2 + \ldots + x^n = 0$. Calculate the value of the expression \[ P(1) = \frac{1}{x_1 - 1} + \frac{1}{x_2 - 1} + \ldots + \frac{1}{x_n - 1}. \]

Answer: -\dfrac{n}{2}


Question: Let $F$ be a finite set with two operations, addition "+" and multiplication "\cdot", satisfying the field axioms except for the requirement that for every $a \in F\setminus\{0\}$, there exists an element $b\in F$ such that $b\cdot a =1$. Instead, it is required that for all $a,b \in F\setminus \{0\}$, we have $ab\in F \setminus \{0\}$. Prove that $F$ is a field.

Answer: F


Question: Find the Fourier series of the function \( f(x) = \cos(x) \) on the interval \([-\pi, \pi]\).

Answer: \cos(x)


Question: Calculate the limit of the expression as $n$ approaches infinity: $$ \lim_{n \to \infty} \left( 1 + \frac{2}{n} \right)^{n^2} e^{-2n}. $$

Answer: e^{-2}


Question: Determine the remainder when \(7^{369}\) is divided by 350.

Answer: 7


Question: Determine whether the image of an $F_\sigma$ set under a homeomorphism in general topological spaces is also an $F_\sigma$ set.

Answer: Yes


Question: Calculate the value of the integral \( \int_0^1 \ln^{2020} (x) \, dx \).

Answer: 2020!


Question: A giant kangaroo, with its tail attached to a stake via a rubber band, leaps one mile away. A persistent flea starts on the stake and each time the kangaroo leaps, it jumps one inch further along the stretched rubber band. Assuming a flat and unbounded earth, will the flea eventually catch up to the kangaroo if this pattern persists indefinitely?

Answer: Yes


Question: What is the probability that the sum of 10 exponentially distributed random variables, each with a mean of 20 minutes, exceeds 200 minutes?

Answer: 0.458


Question: Suppose $a, b, c > 0$ and $a^2 + b^2 + c^2 = 2abc + 1$. Determine the maximum value of $(a - 2bc)(b - 2ca)(c - 2ab)$.

Answer: \dfrac{1}{8}


Question: Let $A$ be a $4\times 4$ real matrix with eigenvalues $1, -1, 2, -2$. If \( B = 2A + A^{-1} - I \), find the determinant of $B$. \( \boxed{?} \)

Answer: 154


Question: Find the limit of the function \( f(x) = \frac{\cosh(x)}{\sinh(x)} - \frac{1}{x} \) as \( x \) approaches 0 by expressing \( f(x) \) as a quotient of two power series.

Answer: 0


Question: Given \(A = \frac{(p^2 + q^2 + r^2)^2}{p^2q^2 + q^2r^2 + r^2p^2}\) and \(B = \frac{q^2 - pr}{p^2 + q^2 + r^2}\), where \(p + q + r = 0\), find the value of \(A^2 - 4B\).

Answer: 14


Question: Find the limit of the sequence \( \left\{ \frac{1}{n}\left(\frac{a_1}{1} + \frac{a_2}{2} + \cdots + \frac{a_n}{n}\right) \right\} \) as \( n \to \infty \), given that \( a_n \to a \) as \( n \to \infty \) for some \( a \in \mathbb{R} \).

Answer: 0


Question: Find the limit of \( \frac{\sin_{n+1}x}{\sin_nx} \) as \( x \to 0 \), where \( \sin_n \) is defined as in the original question.

Answer: 1


Question: A candle has a lifespan $X$ that is uniformly distributed over the interval $(0, 60)$ minutes. You must leave after 30 minutes or when the candle burns out, whichever comes first. You receive a cash reward $M$ equal to half the time the candle was lit. Find the cumulative distribution function (CDF) of $M$ and calculate the expected value $E[M]$. Provide your answers in terms of $M$.

Answer: \dfrac{45}{4}


Question: Calculate the residue of the function \( f(z) = \frac{z^5}{(z-i)^3} \) at the point \( z=i \).

Answer: -10i


Question: Calculate the line integral \( \int _C (-x ^ 2y) \, dx + (y ^ 2 x) \, dy \) over the path \( C \), which is the triangle formed by the lines \( y = x \), \( x = 1 \), and \( 0 \leq y \leq 1 \). Use Green's Theorem to evaluate the integral.

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{n\to\infty} n\sin(2\pi en!) \] where \( e = \sum_{k=0}^\infty \frac{1}{k!} \) and \( n!\sum_{k=0}^n \frac{1}{k!} = m \) for integer \( m \).

Answer: 2\pi


Question: Consider two joint normal random variables, \(X\) and \(Y\), each with a mean of 0, variance of 1, and a covariance of \(\frac{1}{2}\). Calculate the probability that their sum, \(X + Y\), is less than or equal to \(\sqrt{3}\). Express your answer in terms of the standard normal cumulative distribution function, \(\Phi\).

Answer: \Phi(1)


Question: Determine whether the following statement is true or false: For every nonempty set $A$ of real numbers having an upper bound, and for every $d \in \mathbb{R}$, the statement $$[(\sup A = d, d \notin A) \implies (\exists_{N \in \mathbb{N}}{\forall_{n \ge N}{\exists_{a \in A}{\;d - \tfrac{1}{n} < a \le d - \tfrac{1}{n -1}}}})]$$ holds. Provide an example to support your answer.

Answer: False


Question: Determine whether the Krull dimension of any commutative semilocal Hilbert ring is equal to zero.

Answer: 0


Question: Evaluate the integral \( \int_{0}^{\infty} e^{-x} \sum_{k=0}^{\infty} \frac{(-1)^k 2017^{2k} x^{2k}}{(2k)!^2} \, dx \).

Answer: \cos(2017)


Question: Determine whether the following statement is true or false: For a polynomial $f(x) \in \mathbb{Q}[X]$, there exists a subfield of $\mathbb{C}$ that is a splitting field for that polynomial.

Answer: True


Question: Find the strong convexity parameter for a function that is a convex combination of $\beta$-strongly convex functions, given that the coefficients are non-negative and sum to 1.

Answer: \beta


Question: Determine the number of connected components of the set \( F = \{R^{-1}AR: R \in \text{ker}(\psi) \cap GL_n(\mathbb{C})\} \), where \( \psi: M_n(\mathbb{C}) \to \mathbb{C}^n \) is defined by \( S \mapsto (AS-SA)e_1 \), \( A \in M_n(\mathbb{R}) \) is a fixed real matrix, and \( e_1 = (1, 0, \dots, 0) \).

Answer: 1


Question: Find the sum of the infinite series: $$ \sum_{k=0}^{\infty}\frac{(-1)^k}{2k+1} $$.

Answer: \dfrac{\pi}{4}


Question: Determine whether the function \((1+x)^{\frac{1}{x}}\) can be extended to a real analytic function for \(|x|>1\) by defining a suitable value at \(x=0\). If possible, find the value at \(x=0\) that allows this extension.

Answer: e


Question: Evaluate the integral \( \int_0^\infty \frac{1}{x^p(1+\ln(x)^2)^p} \, dx \) and determine the values of \( p \) for which it converges.

Answer: 1


Question: Evaluate the integral \( \int_{c}\frac{3z^2+2}{(z-1)(z^2+9)} \, dz \) where \( c \) is the circle defined by \( |z-2| = 2 \).

Answer: \pi i


Question: Calculate the flux of the vector field \( F(x,y,z) = \langle 9x, 9y, 4 \rangle \) across the part of the paraboloid \( z = x^2 + y^2 \) that lies between the planes \( z = 0 \) and \( z = 4 \). Assume the outward orientation for the closed surface, and take into account that the normal vector \( n \) points downwards.

Answer: 128\pi


Question: Evaluate the triple integral \( \iiint\limits_{D} \left(-4z + y^2 + x^2\right) \, \mathrm{d}V \) over the region \( D = \{(x,y,z) \in \mathbb{R}^3 \mid x \ge 0, 0 \le y \le x, x^2 + y^2 \le 16, 0 \le z \le 5\} \).

Answer: -20\pi


Question: Determine whether the open unit ball in \(\mathbb{R}^n\), defined as \(B(0,1) = \{ x \mid \left|x\right| < 1 \}\), can be covered by a countable, disjoint collection of half-open cubes with their maximum edge length strictly less than 1, where each cube's edges can be either open or closed.

Answer: Yes


Question: Evaluate the integral \( J = \int_{0}^{2}\int_{0}^{1}\left\{\dfrac{x+y}{2}\right\} dy dx \), where \( \{\cdot\} \) denotes the fractional part of a number.

Answer: 1


Question:
Determine whether the following conditions are sufficient for a metric $d$ on a vector space $(X,d)$ over either the field of real numbers $\mathbb{R}$ or the complex numbers $\mathbb{C}$ to be induced by a norm:

1. $d(u+x, u+y) = d(x, y)$ for all $u, x, y \in X$.
2. $d(\alpha x, \alpha y) = |\alpha| \cdot d(x, y)$ for all $x, y \in X$ and for all scalars $\alpha$.

Provide a justification for your answer.

Answer: Yes


Question: Given that \(X \sim Be(p_1)\) and \(Y \sim Be(p_2)\) with \(\text{Cov}(X,Y) = c\), find the formula for the joint probability \(\Pr(X=1 \wedge Y=1)\).

Answer: p_1 p_2 + c


Question: Determine whether the natural functor from the category of sets (Set) to the category of relations (Rel) is conservative. Specifically, does it preserve isomorphisms, meaning that if a morphism $f$ in Set is mapped by the functor to an isomorphism in Rel, then $f$ is also an isomorphism in Set?

Answer: Yes


Question: Compute the derivative of the expression \( n_\theta^T K^2 n_\theta \) with respect to the matrix \( N_\theta \), where \( N_\theta \) is constructed from the outer product \( n_\theta n_\theta^T \) and \( K^2 \) is a positive semi-definite matrix.

Answer: K^2


Question: Determine the value of $r$ such that the series $K(x)=\sum_{n=1}^{\infty}\frac{1}{n^r}\cos(nx)$ converges uniformly for $x \in \mathbb{R}$.

Answer: r > 1


Question: Evaluate the contour integral \(\int_C \frac {3z^5 - 10z^3} {z^6-5z^4+10}\ dz\) where \(C\) is the circle \(|z| = 2\) in the complex plane.

Answer: 4\pi i


Question: Determine whether the cardinality of the Cartesian product of two equinumerous infinite sets is the same as the cardinality of one of the sets.

Answer: Yes


Question:
Suppose that $x_n$ is a sequence of complex numbers converging to $x \in \mathbb{C}$. Verify whether $|x_n\overline{x} - \overline{x}_n x| \to 0$ as $n \to \infty$ by considering the expression:

\[ |x_n\overline{x} - \overline{x}_n x| = |\overline{x}(x_n-x) + \overline{x}_n(x_n-x) + |x|^2 - |x_n|^2| \]

Does this expression approach zero as $x_n \to x$?

Answer: 0


Question: Consider the expansion \((1+x+x^2)^n = \sum_{r=0}^{2n} a_r x^r\). Find the sum of the coefficients \(a_0 + a_3 + a_6 + a_9 + \ldots\) where the powers of \(x\) are multiples of 3.

Answer: 3^{n-1}


Question: Evaluate the integral \( I = \int_0^{\pi} \lim_{n \to \infty} S_n \, dx \), where the sequence \( S_n \) is defined recursively by \( S_1 = \sin{x} \), \( S_2 = \sin(\sin{x}) \), \( S_3 = \sin(\sin(\sin{x})) \), and so on. Prove that \( S_n \to 0 \) as \( n \to \infty \).

Answer: 0


Question: Find the smallest integer $n$ such that the Maclaurin polynomial $P_n(x)$ of $f(x) = e^{-x}$ satisfies $|P_n(x) - e^{-x}| < 10^{-5}$ for all $x \in [0,1]$. The Maclaurin series of $f(x)$ is $\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}x^n$. Ensure that the error term $\left|\frac{x^{n+1}}{(n+1)!}\right| < 10^{-5}$ for $x \in [0,1]$. Determine the smallest such $n$. \(\xi\) is a value in $(0, x)$ and $|f^{(n+1)}(\xi)| \leq 1$. \(n\) is the smallest integer satisfying this condition.

Answer: 8


Question: Find the minimum sample size $n$ such that the probability of the absolute error between the true probability $p$ of a coin coming up heads and the estimator $\hat{p} = X/n$ is less than 0.01 is at least 0.95. That is, determine the smallest $n$ for which $$\mathbb{P}[|p-\hat{p}| < 0.01] \ge 0.95.$$

Answer: 9604


Question:
Compute the following limit:
\[ \lim_{x\rightarrow0} \lim_{y\rightarrow x} \frac{y^2 f(x) - x^2 f(y)}{(1-\cos x)\sin (x-y)} \] where $I$ is an open interval containing $0$, and $f: I \rightarrow \mathbb{R}$ is differentiable with $f'$ continuous.

Answer: -2f'(0)


Question: Is the mean of the real parts of all the roots (including complex roots) of a polynomial function \(P(x)\) equal to the mean of the real parts of all the roots of its derivative \(P'(x)\)?

Answer: Yes


Question: In a row of 49 caves, each cave contains a non-negative integer number of bats. It is known that for any continuous subset of 7 caves, the sum of bats in these caves is exactly 77. Moreover, every cave has at least 2 bats. What is the maximum number of bats that can be in the second cave?

Answer: 65


Question: Evaluate \( \lim_{x \to 1^-} \ln x \cdot \ln(1-x) \) using the change of variable method.

Answer: 0


Question: Determine whether both $p^{\frac{r}{q}}$ and $q^{\frac{r}{p}}$ can be rational numbers if $p$, $q$, and $r$ are distinct prime numbers.

Answer: No


Question: Evaluate the limit \( \lim_{n\to\infty} n f\left(\frac{1}{n}\right) \) for a differentiable function \( f: \mathbb{R} \to \mathbb{R} \) with \( f(0) = 0 \).

Answer: f'(0)


Question: Determine whether there exists an entire function $g(z)$ such that $\lim_{x\to+\infty} \frac{g(x)}{\exp(x^2)} = 1$ along the real axis.

Answer: Yes


Question: Evaluate the series \(\sum_{n=1}^\infty (-1)^{n+1}\frac{2n+1}{n(n+1)}.\)

Answer: 1


Question: Determine if there exists a number in the set $A = \{1,5,9,13,17,21,25,29,33,\ldots\}$ that can be expressed as a product of two numbers from the set, where one of the numbers is a square. If such a number exists, provide an example.

Answer: 45


Question: Calculate the limit: \[ \lim_{x\to \pi}\frac{\cos2x+2(\pi-x)\sin x-1}{(\cos2x+\cos x)^2} \]

Answer: \dfrac{4}{27}


Question: What is the probability of hitting a specific point on a dartboard with an uncountable number of points when a dart is thrown randomly at the board?

Answer: 0


Question: Determine the first non-zero digit from the right in the base-10 representation of the product \( \prod\limits_{i=1}^{n/2} (i)(n-i+1) \) for even numbers \( n \geq 6 \).

Answer: 2


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

Answer: 0


Question:
Determine whether the following identity holds for every $0 \leq i \leq l \leq r$:
\[ \sum_{j=0}^i\binom{r-l+i-j}{i-j}\binom{l-i+j}{j}=\binom{r+1}{i} \]

Answer: \binom{r+1}{i}


Question: Determine the maximum value of the function \( D(\underline{u},\underline{v}) = \|\underline{u} - \underline{v}\|_p \) for vectors \( \underline{u}, \underline{v} \) in the first orthant with unit length, where \( T = \{\underline{v} \in \mathbb{R}^n \mid v_i \geq 0, \|\underline{v}\|_p = 1\} \).

Answer: 2^{1/p}


Question: Determine whether the multiplicative group $U(103)$ contains an element of order 51. Justify your answer using number theory principles.

Answer: Yes


Question: Let $k$ and $p$ be natural numbers, and let $A = \{1, 2, \ldots, p\}$. Determine the number of ways to select $k$ (not necessarily distinct) numbers from $A$ such that their sum is congruent to $0$ modulo $p$. Provide your answer as a single integer.

Answer: p^{k-1}


Question: Given that \(a \ge b \ge c \ge 0\) and \(ab+bc+ca=3\), find the minimum value of the expression \(T = a+b+(a-b)(c-b)\).

Answer: 2


Question: Consider a commutative, finite-dimensional algebra $A$ and a finitely generated $A$-module $M$ with its projective cover $P(M)$. If the annihilator of $M$, denoted by $\operatorname{Ann}(M)$, is non-zero, is it always true that the annihilator of $P(M)$, $\operatorname{Ann}(P(M))$, is also non-zero? Assume that $P(M)$ is the projective cover of $M$. Provide a justification for your answer.

Answer: No


Question: Determine the values of \( a \) such that the function \( f(x) = (x^2 + 2ax + a^2 - 1)^{\frac{1}{4}} \) has a domain and range that together cover all real numbers.

Answer: (-\infty, -1]


Question: Determine whether there exist two matrices $A, B \in M_3(\mathbb{R})$ with rank 2 such that the determinant of $(A + \lambda B)$ is a non-zero polynomial in $\lambda$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether the space $c_{0}-\oplus_{n=1}^{\infty} B(\ell^{p}_{n})$ can be embedded into $c_{0}-\oplus_{n=1}^{\infty} K(\ell^{p})$, where $p \in (1, \infty)$. Provide a justification for your answer.

Answer: Yes


Question: Calculate the expected value of \((a^{\top}x)^{4}\) where \(x\) is a fixed unit vector and \(a \sim \mathcal{N}(0, I)\).

Answer: 3


Question: Alice and Bob are playing a coin-flipping game. Alice wins when the sequence "h, h, h" appears, and Bob wins when the sequence "t, h, t" appears. What is the expected number of coin flips until one of them wins?

Answer: \dfrac{35}{6}


Question: Evaluate the limit of the improper integral: $$\lim_{n\rightarrow \infty }\int_{n}^{n+1}\frac{1}{\sqrt{x^{3}+x+1}}dx$$

Answer: 0


Question: Let $X_1, X_2, X_3, X_4$ be independent standard normal random variables. Define $Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$. Calculate the probability that $Y \leq 3$. \( \text{(Express your answer as a decimal rounded to four decimal places.)} \)

Answer: 0.4422


Question: Calculate the limit of the sequence: \[ \lim_{n \to \infty } \left( \frac{1}{3n} + \frac{1}{3n+1} + \frac{1}{3n+2} + \dots + \frac{1}{4n} \right). \]

Answer: \ln \frac{4}{3}


Question: Find the limit of the sequence defined by the integral: \[ I_n = \int_{0}^{\pi/2} (\tan x)^{1/n} \, dx \]

Answer: \dfrac{\pi}{2}


Question: Calculate the directional derivative of the function \( f(Y) = A^{-1}(Y - \mu) \) at the vector \( V \), where \( Y \) and \( V \) are vectors of the same dimensions, and \( A \) and \( \mu \) are given constants.

Answer: A^{-1}V


Question: Determine whether the contour integral \( \oint_C f(z)dz \) can be zero if the function \( f(z) \) is not holomorphic at an interior point of the contour \( C \).

Answer: Yes


Question: Calculate the expected payment of the winner in a second-price auction with common values, where two players have valuations $v_1 = v_2 = x_1 + x_2$. The players' values $x_1$ and $x_2$ are independent and uniformly distributed on $[0,1]$. The bidding strategy in the Bayesian Nash Equilibrium is $b_i(x_i) = 2x_i$. What is the expected payment of the winner?

Answer: \dfrac{2}{3}


Question: Let $C[0, 1]$ be the linear space of all continuous functions on the interval $[0, 1]$. Let $K(x, y)$ be a fixed continuous function on $[0, 1] \times [0, 1]$. Define the operator $B$ by \( Bf(y) = \int^{1}_0 K(x, y)f(x) \, dx. \) For $K(x, y) = y$, find the norm \( \|B\|. \)

Answer: 1


Question: Determine the number of elements of order 5 in the symmetric group $Sym(12)$. Provide your answer as a single integer.

Answer: 4809024


Question: Determine whether 5 is a generator of the multiplicative group of integers modulo 81, denoted as \( \mathbb{Z}_{81}^{\times} \).

Answer: Yes


Question: Find the minimum distance between the point \((1,1,1)\) and the surface defined by the equation \(z - xy = 10\).

Answer: 3\sqrt{2}


Question: Determine if the square of an odd Fibonacci number $F(2n+1)$, for $n > 1$, can be expressed as the sum of $F(2n+1)+1$ nonzero integer squares using purely number-theoretic methods.

Answer: Yes


Question: Evaluate \( \lim_{n\to\infty} \frac{g(n)}{10^n} \), where \( g(n) \) denotes the last \( n \) digits of the decimal expansion of \((10^n-1)^{10^n-1}\).

Answer: 1


Question: Let $n = p_1 p_2 \cdots p_k$ be the product of distinct odd primes, and let $X$ denote the set of elements in $\operatorname{Aut}(\mathbb{Z}/n\mathbb{Z})$ with order 1 or 2. Define the mapping $\varphi_\psi : \mathbb{Z}/2\mathbb{Z} \to \operatorname{Aut}(\mathbb{Z}/n\mathbb{Z})$ by $\varphi_\psi (x) = \begin{cases} \operatorname{id} & \text{if } x = 0, \\ \psi & \text{if } x = 1 \end{cases}$ for each $\psi \in X$. This induces a semidirect product $G_\psi = \mathbb{Z}/n\mathbb{Z} \rtimes_{\varphi_\psi} \mathbb{Z}/2\mathbb{Z}$. Are the groups $G_\psi$ pairwise non-isomorphic for different $\psi \in X$? Provide a justification for your answer.

Answer: Yes


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

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_{-\pi/4}^{\pi/4}(\cos(t)+\sqrt{1+t^2}\cos(t)^3\sin (t)^3)dt \).

Answer: \sqrt{2}


Question: Determine the degree of the field extension \([\mathbb{Q}(\mu):\mathbb{Q}(\mu + \mu^{-1})]\), where \(\mu\) is a primitive \(n\)th root of unity.

Answer: 2


Question: For a positive definite matrix $A$, determine if every vector $x$ can be expressed as a linear combination of the eigenvectors of $A$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether every finite group $G$ of order $n$ is isomorphic to a subgroup of the alternating group $A_n$. Provide a true or false answer.

Answer: False


Question: Let E be an incomplete inner product space and H be the completion of E. Can the inner product from E be extended to H such that H becomes a Hilbert space?

Answer: Yes


Question: Given two independent random variables, $U_1$ and $U_2$, both uniformly distributed on the interval [0, 1], let $M = \max(U_1, U_2)$ and $N = \min(U_1, U_2)$. Determine the conditional joint density function of $(U_1, U_2)$ given that $M \leq \frac{1}{2}$.

Answer: 4


Question: Determine the number of solutions to the equation \(1 - \frac{x^2}{2} = \cos x\).

Answer: 1


Question: Determine the range of the function \( f(x) = \int_{1}^{x^2-2x+2} \frac{\log(t)}{t^2} \, dt \), where \( x \in \mathbb{R}_{0}^{+} \).

Answer: [0, 1)


Question: Calculate the sum \( \sum_{k=1}^n \frac{(-1)^k}{k+1} \binom{n}{k} \) using combinatorial techniques.

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


Question: Find the minimum value of the expression \((x-y)^2+ \left(\sqrt{4-x^2}-\frac{16}{y}\right)^2\) for \(0<x<2\) and \(y>0\).

Answer: 36 - 16\sqrt{2}


Question: A frog starts at the center of a unit circle on a plane. For each jump, the frog randomly selects a point on the circle and jumps to the midpoint of the line segment connecting its current position to the selected point. Determine the expected value of the square of the frog's distance from the center after an infinite number of jumps, i.e., find \( \lim\limits_{n\to\infty}E[|r_n|^2] \).

Answer: \dfrac{1}{3}


Question: Solve the recursive system of equations \( \begin{cases} a_{n+1}=-2a_n-4b_n \\ b_{n+1}=4a_n+6b_n\\ a_0=1,b_0=0 \end{cases} \) and find the value of \( \frac{a_{20}}{a_{20}+b_{20}} \).

Answer: -39


Question: Is the integral of an unbounded function \(f: \mathbb{R}^d \rightarrow \mathbb{R}\) always greater than or equal to the integral of a function \(g: \mathbb{R}^d \rightarrow \{0,1\}\)?

Answer: No


Question: Simplify the expression \( \frac{1}{\log_2(100!)} + \frac{1}{\log_3(100!)} + \frac{1}{\log_4(100!)} + \ldots + \frac{1}{\log_{100}(100!)} \) without using logarithms.

Answer: 1


Question: Determine whether $nc_n \to 0$ for every convergent series $\sum c_n = s$. Provide a justification for your answer.

Answer: No


Question: Let $H$ be a Hilbert space, and $A \in L(H)$ be a bounded linear self-adjoint operator on $H$. Assume that the range $R(A)$ is not closed. Determine whether it is true that $R(A)$ is stable under the operation $e^{-A}$, meaning if $y \in R(A)$, then $e^{-A} y \in R(A)$. Provide a justification for your answer.

Answer: Yes


Question: Determine the winding number of the function \( f: S^1 \to \mathbb{C} \setminus \{ 0 \} \) defined by \( f(z) = 8z^4 + 4z^3 + 2z^2 + z^{-1} \) around the origin.

Answer: 4


Question: Determine the number of terms in the sum given by the sequence: \( q^{-N+1} + \ldots + q^{N+1} \).

Answer: 2N + 1


Question: Evaluate the limit: $$\lim_{n\to\infty}\sum_{k=1}^n \frac{\pi k}{2n}\int_0^1 x^{2n}\sin\frac{\pi x}{2}dx$$

Answer: \dfrac{\pi}{8}


Question: Let \( G = \mathbb{Z}_{p^2} \times \mathbb{Z}_{p^2} \), where \( p \) is a prime number. Determine the number of distinct subgroups of order \( p^2 \) in \( G \).

Answer: p^2 + p + 1


Question: Find all continuous functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(f(f(x))) = x \) for all \( x \in \mathbb{R} \).

Answer: f(x) = x


Question: Determine the Euler-Lagrange equation for the functional \( \int_{0.5}^2 (y')^2(1 + y'^2) \,dx \) subject to the boundary conditions \( y(0.5) = 20 \) and \( y(2) = 2 \).

Answer: y'' = 0


Question: Determine the order of the recurrence relation \( T(n) = n + \sum\limits_{k=1}^n [T(n-k) + T(k)] \) with the initial condition \( T(1) = 1 \).

Answer: 1