Question: Is every primitive element of a finite field of characteristic $2$ a generator of the multiplicative group of that field? Specifically, if $\alpha \in \overline{\mathbb{F}_2}$ is such that $\mathbb{F}_2[\alpha]$ is a field of order $2^n$ (where $n > 1$), does $\alpha$ generate the multiplicative group $\mathbb{F}_2[\alpha]^{\times}$, meaning that the multiplicative order of $\alpha$ is $2^n - 1$?

Answer: No


Question: Calculate the fair value of a game where you flip a coin until you get a head. On the nth flip, if you get a head, you receive $2^n. What is the expected value of this game?

Answer: \infty


Question: Consider the sequence of functions \( a_n(x) = \frac{(\sin x)^n}{x^n} \) for \( n \ge 1 \). Determine the pointwise limit of \( a_n(x) \) as \( n \to \infty \) for \( x \in (0, +\infty) \). Then, compute \( \lim_{n \to \infty} \int_0^{+\infty} a_n(x) \, dx \) using the Dominated Convergence Theorem.

Answer: 0


Question: Determine the interval of convergence for the series \( \sum_{n=1}^\infty \frac{n(x-1)^n}{3^n (2n-1)} \).

Answer: (-2, 4)


Question: Let \( \{x_n\} \) be a non-negative sequence defined for \( n \geq 0 \) such that the inequality \( x_{n+1}^3 < 3x_n - 2 \) holds for every \( n \in \mathbb{N} \). Prove that the sequence is convergent and find its limit.

Answer: 1


Question: Let \( G \) be a finite group, \( V \) an irreducible complex representation, and \( \mathbb{1} \) the trivial representation. Determine if there exists an integer \( n > 0 \) such that \( \mathbb{1} \) is a subrepresentation of \( V^{\otimes n} \).

Answer: Yes


Question:
The circle \(x^2 + y^2 +2gx +2fy +c=0\) bisects the circumference of the circle \(x^2 + y^2 +2ax +2by +d=0\). Which of the following conditions is true?

A) \(2a(g-a) +2b(f-b)=c-d\)

B) \(2g(g-a) +2f(f-b)=d-c\)

C) \(2a(g+a) +2b(f+b)=c+d\)

D) \(2g(g+a) +2f(f+b)=c+d\)

Answer: A


Question: Consider the sequence defined by \( a_1 = \sqrt{k} \) and \( a_{n+1} = \sqrt{k \cdot a_n} \) for \( n \geq 1 \), where \( k > 1 \). Determine whether this sequence converges.

Answer: k


Question: Let $F \subseteq E$ be a field extension. Suppose $\alpha \in E$ is a root of an irreducible polynomial $f(x)$ over $F$ with multiplicity $3$. If $\beta$ is any other root of $f(x)$ in $E$, what is the multiplicity of $\beta$?

Answer: 3


Question: Determine whether the integral \( \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx \) can be solved without using complex analysis methods.

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


Question: Let G be a group and H a subgroup such that the number of elements in H is half of the number of elements in G. If x is an element of G with order 3, prove that x is an element of H.

Answer: x \in H


Question: Determine the type of singularity and compute the residue of the function \( \frac{1}{\sin^2(z)} \) at \( z = 0 \).

Answer: 0


Question: Determine the splitting field of the polynomial \(p(x) = x^4 + x^3 + x - 1\) over \(\mathbb{F}_3\).

Answer: \mathbb{F}_{3^2}


Question: Compute \( \lim_{x \to \infty }\sum_{n=1}^\infty \frac{1}{n(n+x)} \).

Answer: 0


Question: Compute the torsion subgroup of the elliptic curve $E(\mathbb{Q})$, where $E: Y^2 = X^3 + 2$, using the Nagell-Lutz theorem.

Answer: \{\mathcal{O}\}


Question: Let $B_t$ be a standard Brownian motion, and let $L(x, t)$ denote the local time at $x$ for time $t$. Define $L_t = L(0, t)$. Determine whether the following holds with probability one: $$\limsup_{t \to 0} \frac{L_t}{\sqrt{t}} = \infty.$$

Answer: \infty


Question:
Consider the regular grammar given by the following productions: \( S \to bS \mid aA \mid \epsilon \) and \( A \to aS \mid bA \). Determine the Myhill-Nerode equivalence classes for the language generated by this grammar. Choose the correct option:

A) \( \{w \in (a + b)^* \mid \#a(w) \text{ is even}\} \) and \( \{w \in (a + b)^* \mid \#a(w) \text{ is odd}\} \)

B) \( \{w \in (a + b)^* \mid \#a(w) \text{ is even}\} \) and \( \{w \in (a + b)^* \mid \#b(w) \text{ is odd}\} \)

C) \( \{w \in (a + b)^* \mid \#a(w) = \#b(w)\} \) and \( \{w \in (a + b)^* \mid \#a(w) \neq \#b(w)\} \)

D) \( \{\epsilon\} \), \( \{wa \mid w \in (a + b)^*\} \) and \( \{wb \mid w \in (a + b)^*\} \)

Answer: A


Question: Calculate the value of $x$ in the given circle if the radius $r = \sqrt{2}$. Provide your answer as a single value for $x$.

Answer: \sqrt{2}


Question: Given the relation for the deviatoric stress tensor \(\pmb{S} = \pmb{T} + p\pmb{I}\) where \(p = -\frac{1}{3}tr\mathbf{T}\), \(\pmb{T}\) is the Cauchy stress tensor, \(\pmb{S}\) is the deviatoric stress tensor, \(\pmb{I}\) is the identity tensor, and \(p\) is the mean stress, derive the differential of the deviatoric stress tensor \(\frac{\partial \mathbf{S}}{\partial p}\). Is the result the zero tensor?

Answer: \mathbf{0}


Question: Given that \( m = 17 \times 23 = 391 \), an exponent \( e = 3 \), and an encrypted word \( c = 21 \), with a decrypting exponent \( d = 235 \), find the original word \( w \) such that \( w \equiv c^d \pmod{m} \).

Answer: 30


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

Answer: 0


Question: Determine the largest number among ten consecutive numbers that can be obtained from an Antimagic arrangement of a 4x4 matrix, where each element is a unique number from 1 to 16. In an Antimagic arrangement, the sums of the rows, columns, and diagonals form an arithmetic sequence of ten consecutive numbers.

Answer: 39


Question: Let $X$ be an $(n-1)$-connected CW complex of dimension $n$ and let $\{\varphi_i : \mathbb{S}^n \to X \mid i \in I\}$ be a generating set of $\pi_n(X)$. If $\dot{X}$ denotes the complex obtained from $X$ by gluing $(n+1)$-cells via the $\varphi_i$'s, is the image of $\pi_{n+1}(X)$ in $\pi_{n+1}(\dot{X})$ trivial?

Answer: Yes


Question:
Given the transition matrix \(P\) for a Markov chain:
\[
P =
\begin{pmatrix}
0 & 1 & 0 & 0 \\
\frac{1}{3} & 0 & \frac{2}{3} & 0\\
0 & \frac{2}{3} & 0 & \frac{1}{3}\\
0 & 0 & 1 & 0\\ \end{pmatrix}
\]
and the limiting matrix \(\Pi = \lim_{n \to \infty} P^n\):
\[
\Pi =
\begin{pmatrix}
0.25 & 0 & 0.75 & 0 \\
0 & 0.75 & 0 & 0.25\\
0.25 & 0 & 0.75 & 0\\
0 & 0.75 & 0 & 0.25\\ \end{pmatrix}
\]
Calculate the long-run portion of time that the first urn is empty, denoted as \(\pi_0\).

Answer: \dfrac{1}{8}


Question: Determine the set of values of \(m\) for which the function \(f(x) = x^m \sin\frac{1}{x}\) is discontinuous at 0.

Answer: (-\infty, 0]


Question: Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be an $M$-Lipschitz function with respect to the vector $L^2$ norm, meaning that for all $x, y \in \mathbb{R}^n$, \( ||f(x)-f(y)||_2 \leq M ||x-y||_2 \). Let $J_f(x)$ denote the Jacobian matrix of $f$ at $x$, which is symmetric for every $x$. Is it true that the spectral norm of the Jacobian satisfies \( ||J_f(x)||_2 \leq M \) for all $x \in \mathbb{R}^n$? Justify your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to\infty} (-x +\sqrt{x^2+1})^{1/\ln x} \]

Answer: \dfrac{1}{e}


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

Answer: \dfrac{19}{6}


Question:
Find the general term of the sequence \(a_n\) defined by the recurrence relation:
\[ a_0 = 1, \]
\[ 2a_{n+1} = \sum_{i=0}^n \binom{n}{i} a_i a_{n-i}. \]

Answer: \dfrac{n!}{2^n}


Question: Determine the radius of convergence of the Taylor series for the function \( g(z) = \frac{\sin z - z + \frac{z^3}{6}}{\cos z - 1} \) centered at 0.

Answer: 2\pi


Question: Use Parseval's identity to compute the integral \( \int_{-\infty}^{\infty} \left(\frac{\sin x}{x}\right)^2 dx \).

Answer: \pi


Question: Determine the value of the parameter \( a \) such that the function \( f(x) = -\frac{4x^{3}+4x^{2}+ax-18}{2x+3} \) has exactly one point of intersection with the \( x \)-axis.

Answer: -15


Question: Determine the smallest constant \( C > 0 \) such that the inequality \( xy^2z^3 \leq C(x+y+z)^6 \) holds for all non-negative real numbers \( x, y, \) and \( z \).

Answer: \dfrac{1}{432}


Question: Find the minimum value of $a + b$ given that $13 \mid (a + 11b)$ and $11 \mid (a + 13b)$, where $a$ and $b$ are positive integers.

Answer: 28


Question: Evaluate the integral \( I_n = \int_0^\infty \left(\operatorname{E}_n(x)e^x - \frac{1}{1+x}\right) dx \) and show that it equals \(-\psi(n)\), where \(\operatorname{E}_n\) denotes the exponential integral and \(\psi\) denotes the polygamma function.

Answer: -\psi(n)


Question: How many trailing zeroes are there in 100!?

Answer: 24


Question: Estimate the degree of the Taylor polynomial required to approximate \( \cos(x) \) for \( |x| \leq \frac{\pi}{4} \) with an error of no greater than \( 10^{-5} \).

Answer: 6


Question: Evaluate the integral \( \int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx \).

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


Question:
Can subsets $A$ and $B$ of the set of real numbers $\mathbb{R}$ be constructed such that they satisfy the following properties?

1. Disjoint: $A \cap B = \emptyset$.
2. Exhaustive: $A \cup B = \mathbb{R}$.
3. Alternate elements: For all $x, y \in A$ with $x < y$, there exists $z \in B$ such that $x < z < y$, and vice versa for $B$.
4. Same cardinality: $|A| = |B|$, meaning there exists a bijection $h : A \rightarrow B$.

Additionally, can such subsets be constructed on any arbitrary interval of $\mathbb{R}$?

Answer: Yes


Question: Given a point and a line segment, both specified using rational coordinates, determine if the reflection of the point across the line segment can be computed using only rational numbers. Answer "yes" or "no".

Answer: yes


Question:
Consider the following optimization problem in the standard primal conic form:

Minimize \( x_3 + x_4 \) subject to:
\[-x_1 - x_3 + x_4 = 1,\]
\[-x_2 + x_3 - x_4 = 1,\]
\[x \geq 1.\]

Which of the following vectors provides a Farkas-type infeasibility certificate for the above problem?

A. \( y = (1, 1) \)
B. \( y = (-1, -1) \)
C. \( y = (2, 2) \)
D. All of the above

Select the correct answer.

Answer: B


Question: What is the name of a linear bijection whose transpose equals its inverse?

Answer: orthogonal transformation


Question: Does the converse of Lusin's theorem hold? Specifically, if for any $\varepsilon>0$ there exists a continuous function $g$ such that $\mu(\{x\in[a,b]:f(x)\neq g(x)\})<\varepsilon$, is the function $f:[a,b]\to\mathbb{C}$ necessarily measurable?

Answer: Yes


Question: Evaluate the limit: \( \lim_{x \to 0} \sqrt{-\exp(i x)} \).

Answer: i


Question: Find the value of \( n \) such that the sum of the first coefficient and the second-to-last coefficient in the expansion of \( \left(\frac{x\sqrt[3]{x}}{2}+\frac{1}{\sqrt[15]{x^{28}}}\right)^n \) is 13.

Answer: 26


Question: Evaluate the triple integral: \[ 16\int_0^\frac{\pi}{4}\int_0^1\int_0^{\sqrt{1-r^2\cos^2(\theta)}}r\,dz\,dr\,d\theta \] and find the value of the integral.

Answer: 16 - 8\sqrt{2}


Question: Let \( f: A \to B \) be a surjective homomorphism of rings. Determine whether every multiplicative subset \( \Gamma \subseteq B \) can be expressed as \( f(\Sigma) \) for some multiplicative subset \( \Sigma \subseteq A \).

Answer: Yes


Question: Evaluate the limit: \[ \lim_{z\to 0}\frac{2z\sin{z^2}-2z^3\cos{z^2}}{\sin^2{z^2}} \] using the Taylor series expansions for sine and cosine.

Answer: 0


Question: A congress will be attended by two representatives from Colombia, three from Chile, four from Argentina, and five from Brazil. Each of the 14 representatives prepared their own speech, but only 6 will be drawn to speak. If the draw rule requires that each of the four countries must have at least one representative speaking, how many different ways can the set of six speeches be composed?

Answer: 1450


Question: Find the maximum value of \( \int_{0}^{y}\sqrt{x^{4}+(y-y^{2})^{2}}\,dx \) for \( 0 \leq y \leq 1 \).

Answer: \dfrac{1}{3}


Question: Determine whether there exist three positive integers \(a, b, c\) with \(a < b < c\) such that the equation \(ax^2 + a = by^2 + b = cz^2 + c\) has infinitely many integer solutions \((x, y, z)\).

Answer: No


Question: Find the number of distinct arrangements of positive integers \(n_1, n_2, n_3, n_4, n_5\) such that \(n_1 < n_2 < n_3 < n_4 < n_5\) and \(n_1 + n_2 + n_3 + n_4 + n_5 = 20\).

Answer: 7


Question: Suppose $D\subset\mathbb{C}$ is a bounded domain and $f$ is a meromorphic function on the exterior domain $D_+=\hat{\mathbb{C}}\setminus\overline{D}$. Given that $f(\infty)=0$ and $f$ has only poles of finite order, does it follow that $f$ is a rational function on $D_+$?

Answer: Yes


Question: Determine all points where the function \( f(z) = \mathbb{Re}(z) \cdot |z| \) is complex differentiable.

Answer: 0


Question: Evaluate the integral: \[ I = \int_{0}^{\infty} J_{0}(x) \, dx \] where \( J_{0}(x) \) is the Bessel function of order zero.

Answer: 1


Question: In triangle $\Delta ABC$, the sides are $AB = \sqrt{17}$, $BC = 5$, and $CA = 4\sqrt{2}$. Points $D$, $E$, and $F$ are located on lines $AB$, $BC$, and $CA$ respectively, forming triangle $\Delta DEF$ inside $\Delta ABC$. The minimum perimeter of $\Delta DEF$ is given by $\frac{q}{p}\sqrt{34}$, where $p$ and $q$ are positive integers with no common factors. What is the value of $p + q$?\n\nA) 35\nB) 36\nC) 37\nD) 38\nE) 39

Answer: C


Question: Evaluate the series \( \sum_{n=1}^\infty \frac{1}{2n-1} \sin((2n-1)x) \) for \( x \in (0, \pi) \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \left(1+\tan^2(\sqrt{x})\right)^{\frac{1}{2x}} \]

Answer: \sqrt{e}


Question: Find the minimum value of \(|z| + |z - e^{ix}| + |z - 5e^{ix}| + |z - 7e^{ix}| + |z - 32e^{ix}|\) where \(z\) is a complex number.

Answer: 38


Question: Find the closed-form expression for the summation: \( \sum\limits_{i=0}^n 2^i(n-i) \)

Answer: 2^{n+1} - n - 2


Question: Compute the line integral of the vector field \( v_1(x_1, x_2, x_3) = \begin{pmatrix} x_1 x_2 \\ -x_1^2 \\ x_3 \end{pmatrix} \) along the curve \( \gamma(t) = (\cos(t), \sin(t), t) \) for \( t \in [0, 2\pi] \).

Answer: 2\pi^2


Question: Determine the dimension of the affine variety $V(I)$, where $I = \langle zw - y^2, xy - z^3 \rangle \subseteq k[x,y,z,w]$, with $k$ being an algebraically closed field.

Answer: 2


Question:
Given that $w$ is a complex cube root of unity with $w \neq 1$, determine the number of non-singular matrices of the form:
\[A=\begin{bmatrix} 1 & a & b\\ w & 1 & c\\ w^2 & w & 1 \end{bmatrix}\]
where $a, b, c$ are elements of the set $S=\{w, w^2\}$. Provide your answer as a single integer.

Answer: 2


Question: Calculate the volume of the region between the surfaces $z=0$ and $z=\frac{1}{(x^2+y^2)^9}$, and outside the cylinder $x^2+y^2=1$, using a triple integral.

Answer: \dfrac{\pi}{8}


Question:
Which of the following binomial expansions has coefficients that are all odd numbers?

\((\text{A}) \ (x+y)^5\)

\((\text{B}) \ (x+y)^7\)

\((\text{C}) \ (x+y)^9\)

\((\text{D}) \ (x+y)^{11}\)

\((\text{E}) \ (x+y)^{13}\)

Answer: B


Question: Let $X \sim \textrm{Erlang}(k,\lambda)$. Use the Central Limit Theorem to evaluate \[ \lim_{k\rightarrow\infty}\mathbb{P}\left(\left|X-\mathbb{E}\left[X\right]\right|>\epsilon \cdot \mathbb{E}\left[X\right]\right) \] where $\epsilon>0$ is small.

Answer: 0


Question: Determine whether the series \( \sum_{n=0}^\infty e^{-nx} \cos{nx} \) is differentiable on the interval \((0, \infty)\).

Answer: Yes


Question: Determine the degree of the field extension \([\mathbb{Q}(a) : \mathbb{Q}(a+1/a)]\), where \(a\) is an algebraic number.

Answer: 2


Question: Evaluate the integral \( I = \int_{0}^{1} \frac{\sqrt{1-x^4}}{1+x^4} \, dx \) and show that it equals \( \frac{\pi}{4} \).

Answer: \dfrac{\pi}{4}


Question: Evaluate the contour integral \( \oint_{\gamma} \tanh(z) \, dz \) where \( \gamma \) is the circle defined by \( |z - \frac{\pi}{4}i| = \frac{1}{2} \).

Answer: 0


Question:
Find an explicit formula for the sequence defined by the recurrence relation:

\[ a_0 = 1, \quad \forall n \ge 1: a_n = n^3a_{n-1} + (n!)^3 \]

Answer: (n + 1)(n!)^3


Question: Calculate \( \lim_{N \to \infty} P(x^2+y^2 \leq N^2) \) where \(x\) and \(y\) are random positive integers less than \(N\). Use the law of total probability to express this limit as \( \lim_{N \to \infty} \sum_{i=1}^N \frac{\sqrt{N^2-i^2}}{N^2} \).

Answer: \dfrac{\pi}{4}


Question: Determine whether there exists a subset \( X \) of \( \mathbb{R}^2 \) with the usual topology such that \( X \) is countable, has more than two points, and is connected. Answer "True" if such a subset exists, or "False" if it does not.

Answer: False


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt[3]{1+x} - \sqrt[3]{1-x}} \] without using L'Hôpital's rule.

Answer: \dfrac{3}{2}


Question: Consider the following statement: Let \( f: [0, 1] \to [0,1]^2 \) be a non-self-intersecting curve with \( f(0) = \langle0,0\rangle \), \( f(1) = \langle1,1\rangle \), and \( f(t) \in (0,1)^2 \) for \( t \in (0,1) \). Then there exists a non-self-intersecting curve \( g: [0, 1] \to [0,1]^2 \) with \( g(0) = \langle1,0\rangle \), \( g(1) = \langle0,1\rangle \), and \( g(t) \in (0,1)^2 \) for \( t \in (0,1) \) such that there are unique \( t_0 \) and \( t_1 \) with \( f(t_0) = g(t_1) \). Determine whether this statement is true or false.

Answer: True


Question:
Determine the value of $k$ such that the following equation holds:
\[ \int \frac{1+x \cos x}{x(1-x^2 e^{2 \sin x})}dx = k \ln \sqrt{\frac{x^2 e^{2 \sin x}}{1-x^2 e^{2 \sin x}}} + C \]

Answer: 1


Question: Find the integer value of $M$ such that $10 \equiv M^5 \pmod{35}$.

Answer: 5


Question: Consider a compact Riemann surface $C$ of genus $g$. The vector space of holomorphic 1-forms on $C$, denoted by $\mathrm{H}^0(C,\mathrm{K}_C)$, has a basis $\eta_1, \ldots, \eta_g$. A holomorphic 1-form on $C^{\times 2}$ is called a bidifferential. A bidifferential $\omega$ is symmetric if $\omega(z_1,z_2) = \omega(z_2,z_1)$ for all $z_i \in C$. Given two 1-forms $\omega_1, \omega_2 \in \mathrm{H}^0(C,\mathrm{K}_C)$, they induce a bidifferential $\omega_1 \boxtimes \omega_2$. Can any symmetric bidifferential be expressed as $\sum_{i,j=1}^g A_{ij} \, \eta_i \boxtimes \eta_j$ where $A_{ij} \in \mathbb{C}$ satisfy $A_{ij}=A_{ji}$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{y\to\+0} \int_{0}^{1} \exp\left(-\frac{\arctan x}{y}\right)\,\mathrm{d}x \]

Answer: 0


Question: Let $M$ be a von Neumann subalgebra of $B(H)$. Let $f$ and $g$ be normal functionals on $B(H)$ and $M$, respectively, such that the restriction of $f$ to $M$ is $g$, i.e., $f_{|_{M}} = g$. Define the positive linear functional $\phi: M \to \mathbb{C}$ by $\phi(x) = |f|(x)$, where $|f|$ is the absolute value of the normal functional $f$. Can we conclude that $|g| \leq \phi$?

Answer: Yes


Question:
A bob with a mass of \(2\,\text{kg}\) is hanging from the ceiling in a room where a flow of air exerts a constant force of \(4\,\text{N}\) against it. The bob makes a \(53^{\circ}\) angle with the vertical. Assuming the wire is ideal, find the modulus of the tension in the wire in newtons when the sphere passes through its lowest point. Choose the correct answer from the following options:

\(\begin{array}{ll}
1.&26.9\,\text{N}\\
2.&29.6\,\text{N}\\
3.&27.5\,\text{N}\\
4.&23.3\,\text{N}\\
5.&21.2\,\text{N}\\
\end{array}\)

Answer: 2


Question: Consider the linear map from \( F_2[x] \) to \( F_2[x] \oplus F_2[x] \) defined by \( p(x) \mapsto [ p(x)(x^2+1) ; p(x)(x^2+x+1)] \). The Hamming norm of a polynomial \( |p(x)| \) is defined as the number of non-zero monomials in \( p(x) \). For a pair of polynomials \( |[p ; g]| = |p| + |g| \). Is it true that the minimal Hamming norm in the image of this map, excluding \([0; 0]\), is equal to 5?

Answer: Yes


Question: Consider the family of distributions with density \( f(x) = \frac{1}{2\theta} \exp\left(-\frac{|x|}{\theta}\right) , x \in (-\infty, \infty) \). Compute the expected value \( E(|x|) \).

Answer: \theta


Question:
What is the remainder when the polynomial \( P(x) = (x+1)^{2015} + (x-1)^{2015} \) is divided by \( x^2 + 1 \)?

A) \(-2^{2015}x\)
B) \(-2^{1008}x\)
C) \(2^{1008}x\)
D) \(2^{2008}\)
E) \(1\)

Answer: B


Question:
For the function \( f(z) = \frac{1}{z^3(z+3)} \), determine which Laurent series expansion should be used to compute the residue at \( z_0 = 0 \) when integrating over the contour \( |z| = 2 \). The two expansions are:

1. \( \sum_0^{\infty}\frac{(-1)^nz^{n-3}}{3^{n+1}} \) valid for \( 0 < |z| < 3 \).
2. \( \sum_1^{\infty}\frac{(-1)^n3^n}{z^{n-4}} \) valid for \( 3 < |z| < \infty \).

Which expansion should be used to compute the residue?

Answer: 1


Question: Calculate the limit: \[ \lim_{x\to 0} \frac{1}{1-\cos(x^2)}\sum_{n=4}^\infty n^5x^n \]

Answer: 2048


Question: Evaluate the integral: \[ \int_0^\pi \frac{\sin\left(n+\frac{1}{2}\right)x}{\sin \frac{x}{2}} \, dx \]

Answer: \pi


Question: Determine the Krull dimension of the ring \( \mathbb{Z}_6[x,y]/\langle x \rangle \).

Answer: 1


Question: Determine whether the following statement is true or false: If $K$ is the algebraic closure of $F$, then as a ring, $K$ is integral over $F$. Provide a justification for your answer.

Answer: True


Question: Given the function \( f(x) = e^{-1/x^2} \) with \( f(0) = 0 \), determine \( f'(0) \).

Answer: 0


Question: Compute the volume bounded by the parabolic cylinders \(x^2=4-4z\) and \(y^2=4-4z\) and the \(xy\)-plane. Provide the correct volume of this region.

Answer: 8


Question: Find the complex number \(z = x + yi\), where \(x\) and \(y\) are positive integers, such that \(z^3 = -74 + ci\) for some integer \(c\).

Answer: 1 + 5i


Question:
Find the limit as \( n \to \infty \) of the series:
\[ x(n) = \sum\limits_{k=1}^n \frac{k^2}{n^3+k^2} \]

Answer: \dfrac{1}{3}


Question: Find the maximum value of the function \( f(x,y) = (x + 2y)^2 + (3x + 4y)^2 \) subject to the constraint \( x^2 + y^2 = 1 \).

Answer: 15 + \sqrt{221}


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

Answer: \dfrac{\pi}{4}


Question: Evaluate the line integral of the function \( g(x,y) = 4x^3 + 10y^4 \) along the straight line segment from the point \((0,0)\) to the point \((1,2)\) in the \(xy\)-plane.

Answer: 33\sqrt{5}


Question: Let \( A \) be a \( 7 \times 7 \) matrix such that \( 2A^2 - A^4 = I \), where \( I \) is the identity matrix. If \( A \) has two distinct eigenvalues, each with geometric multiplicity 3, how many non-zero entries are there in the Jordan canonical form of \( A \)?

Answer: 8


Question: Calculate the limit \( \lim \limits_{n \to \infty} \left( \sqrt{16^n - 4^n} - \sqrt{16^n - 3^n + n} \right) \).

Answer: -\dfrac{1}{2}


Question: Find the 8th derivative at 0 of the function \( f(x) = \ln(4 + x^2) \).

Answer: -\dfrac{315}{8}


Question: Find the smallest possible integer value of \( p \) such that the identity \((x^3+px^{2}+2x-5)^{19}(x^{2}+qx-41)^{8}(x^4-x^3+x-7)^6 = x^{97}+391x^{96}+a_{95}x^{95}+\cdots+a_1x+a_0\) holds, where \(p, q, a_{95}, \ldots, a_0\) are integers.

Answer: 7


Question: Is there an infinite graph $G = (\omega, E)$ on the vertex set $\omega$ such that every vertex has infinite degree, but $G$ has no perfect matching?

Answer: Yes


Question: Given a probability space $(X, A, P)$ and a sequence of events $(A_n)_{n\in \mathbb{N}}$ such that $P(A_n) = \frac{1}{7^n}$ for $n \in \mathbb{N}$, determine the probability of the event $\limsup A_n$.

Answer: 0


Question: For a natural number $n$ with $n \geq 2$, $a_n$ is defined as the solution to the equation $\tan x = n$ within the interval $((n-2)\pi + \frac{\pi}{2}, (n-1)\pi + \frac{\pi}{2})$. Compute the limit: $$\lim_{n\rightarrow\infty}\frac{a_n}{n}$$.

Answer: \pi


Question: A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows: The $k^{th}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{th}$ pirate receive?

Answer: 1925


Question: Let $S^n$ be the $n$-dimensional unit sphere equipped with the uniform measure $\mu$. Given a bounded function $f: S^n \to \mathbb{R}$ with a bound $C$, determine if the following inequality holds: \[ \int_{S^n} f \, d\mu \leq C \mu(S^n) = C. \]

Answer: Yes


Question: Estimate the probability that the sum of the faces showing on one hundred fair dice exceeds 370, using a continuity correction in your analysis.

Answer: 0.115


Question: Let \( f(x, y, z) = x^3 + y^3 + z^3 \). Suppose \( L \) is a linear map from \( \mathbb{R}^3 \) to \( \mathbb{R} \) such that \[ \lim_{(x, y, z) \to (0, 0, 0)} \frac{f(1+x, 1+y, 1+z) - f(1, 1, 1) - L(x, y, z)}{\sqrt{x^2 + y^2 + z^2}} = 0. \] Find the value of \( L(1, 2, 4) \).

Answer: 21


Question:
Find the positive integer \( q \leq 50 \) such that the following expression holds:
\[ \binom{98}{30} + 2\binom{97}{30} + 3\binom{96}{30} + \cdots + 68\binom{31}{30} + 69\binom{30}{30} = \binom{100}{q}. \]

Answer: 32


Question: Let $U, V: X \to \mathbb{R}$ be two utility representations of a preference relation $P$. Is there always a strictly monotonic function $f: \mathbb{R} \to \mathbb{R}$ such that $V = f \circ U$? Provide a justification for your answer.

Answer: Yes


Question: Determine whether the following assertion is true or false: There exists a nonzero function \( f \in C([0,1]) \) such that \( \int_0^1 f(x)x^n \, dx = 0 \) for all \( n \in \mathbb{N} \). Justify your answer. (Hint: Use the Weierstrass approximation theorem.)

Answer: False


Question: Consider the semilinear partial differential equation: \( xu_x + yu_y = 4u \). Given that \( u(x,y) \) lies on the unit circle defined by \( x^2 + y^2 = 1 \), find the value of \( u(2,2) \).

Answer: 64


Question: Find a divisor of the number $4^{1000} - 1$.

Answer: 3


Question: Determine if there exists a rational number $c$ such that \[ c = \sqrt{\frac{p^2 + 2a^2q^2 - a^2}{a^2q^2}} \] where $p$ and $q$ are integers, and $a$ is a rational number.

Answer: Yes


Question: Provide an example of a compact metric space $X$ such that $X$ and $X \times X$ are homeomorphic, given that $|X| > 1$. Justify why $X$ satisfies these conditions.

Answer: C


Question: Calculate \( \lim_{\lambda \to \infty} \frac{A}{B} \), where \( A = \int_\lambda^\infty \frac{x}{\lambda} e^{-x^2/2} \, dx \) and \( B = \int_\lambda^\infty e^{-x^2/2} \, dx \).

Answer: 1


Question: Evaluate the limit of the infinite product \( a_n = \prod_{k=1}^n \sin(ak) \) as \( n \to +\infty \), where \( a \) is a real number.

Answer: 0


Question: Calculate the degree of the field extension \( F(\alpha) \) over \( F\left(\frac{\alpha^4}{4\alpha^3 - 1}\right) \), where \( F \) is a field and \( \alpha \) is a transcendental element.

Answer: 4


Question: Find the limit of the sequence \( X_n = \sum_{k=1}^n \frac{1}{k} - \ln(n) \).

Answer: \gamma


Question:
Let \( U, V, W, Y, Z \) be independent random variables with the following distributions:

- \( U \sim \text{Exp}(1/5) \)
- \( V \sim \text{Exp}(2/3) \)
- \( W \sim \mathcal{U}(1, 5) \) (uniform distribution on \([1, 5]\))
- \( Y \sim \mathcal{N}(-13, 1) \)
- \( Z \sim \mathcal{N}(0, 5) \)

Determine the expected value:
\[ \mathbb{E}[(Z + \min(U+W, V+W)) \cdot (Y+Z)] \]

Hint: Use the properties of independence and linearity of expectation to simplify the computation. You do not need to derive any density or distribution functions by hand; all necessary means can be computed directly via the parameters of the distributions.

Answer: -49


Question:
Given vectors \(a\) and \(b\) such that \(|a|=2\), \(|b|=3\), and \(a\cdot b=0\), determine the vector result of the expression:
\[
 a \times (a \times (a \times (a \times b)))
\]

Answer: 16b


Question: Given that \(x+y+z=1\), \(x^2+y^2+z^2=2\), and \(x^3+y^3+z^3=3\), find the value of \(x^5+y^5+z^5\).

Answer: 6


Question: Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. Define $f(n) = \frac{d(n)}{n^{1/3}}$. There is a unique integer $N$ such that $f(N) > f(n)$ for all $n \neq N$. Find the sum of the digits of $N$. Options: 5, 6, 7, 8, 9.

Answer: 9


Question: Calculate the variance \( \mathrm{Var}(e^{-X}) \) given that the random variable \( X \) has the probability density function \( f_\theta(x) = \exp(-(x-\theta) - \exp(-(x-\theta))) \).

Answer: e^{-2\theta}


Question: Use Green's Theorem to evaluate the line integral of the vector field \( F(x,y) = x^2y\mathbf{i} + xy\mathbf{j} \) around the positively oriented boundary \( C \), where \( C \) is the region bounded by \( y = x^2 \) and \( y = 4x + 5 \).

Answer: \dfrac{648}{5}


Question: Let $x_{0} \in \mathbb{R}$ and let $f$ be a function defined on a deleted neighborhood of $x_0$. Determine whether the following statement is true or false: If $f$ is differentiable at $x_0$, then the limit \( \lim_{h \to 0} \frac{f(x_{0}+h)-f(x_{0}-h)}{2h} \) exists.

Answer: A


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

Answer: 2\sqrt{3}


Question:
Which of the following sets form an ideal in the ring \(C(\mathbb{R})\) of all continuous real-valued functions on \(\mathbb{R}\)?

(a) The set of all \(C^\infty\) functions with compact support.
(b) The set of all continuous functions with compact support.
(c) The set of all continuous functions which vanish at infinity, i.e., functions \(f\) such that \(\lim_{x \to \infty} f(x) = 0\).

Answer: b


Question: Evaluate the integral \( I = \int_{0}^{\infty} \frac{1-e^{-t}}{t} \sin{t} \operatorname{d}\!t \).

Answer: \dfrac{\pi}{4}


Question: Transform the conic equation \(4x^2 - 4xy + y^2 + 20x + 40y = 0\) by finding the appropriate rotation angle to eliminate the \(xy\) term. What is the angle of rotation?

Answer: \arctan(2)


Question: Consider the function \( f(x, y, z) = \alpha^2(x-y) + \alpha(z-y) + (x-z) \) with a constant \( \alpha \in \mathbb{R} \). If \( f(a, b, c) = f(c, a, b) = f(b, c, a) = 0 \) for some \( a, b, c \in \mathbb{R} \), does it necessarily follow that \( a = b = c \)? Justify your answer.

Answer: Yes


Question: Evaluate the infinite series: \[ \sum_{r=1}^{\infty} \arctan\left(\frac{2}{r^2+r+4} \right) \]

Answer: \arctan 2


Question: Determine the values of \( \alpha \) for which the integral \( \int_0^\infty \frac{\ln(x^2+1)}{x^\alpha} \, dx \) converges.

Answer: (1, 3)


Question:
Determine whether the integral of a nonconstant monomial over the unit disk vanishes. Specifically, evaluate the following integral and determine if it equals zero for all positive integers \( n \):
\[
\int_{-1}^{1} \int_{-1}^{1} [a^2+b^2 < 1] (a+bi)^n \;\text{d}a\text{d}b = 0 \;\; \text{for } n \ge 1.
\]

Answer: 0


Question: Determine the class group of the complement of the union of the three lines $L_1: x=0$, $L_2: y=0$, and $L_3: z=0$ in the projective plane $\mathbb{P}^2$. Let $U = \mathbb{P}^2 \setminus (L_1 \cup L_2 \cup L_3)$. What is the class group of $U$?

Answer: 0


Question:
Let $f:X \rightarrow \mathbb{R}$ be a Lipschitz function on a metric space $X$, and let $K < M$ be constants. Define a function $g:X \rightarrow \mathbb{R}$ by:
\[
g(x) = \begin{cases}
K, & \text{if } f(x) < K, \\
f(x), & \text{if } K \leq f(x) \leq M, \\
M, & \text{if } f(x) > M.
\end{cases}
\]
Is the function $g$ Lipschitz?

Answer: Yes


Question: Find the remainder when the determinant \( \begin{vmatrix} { 2014 }^{ 2014 } & { 2015 }^{ 2015 } & { 2016 }^{ 2016 } \\ { 2017 }^{ 2017 } & { 2018 }^{ 2018 } & { 2019 }^{ 2019 } \\ { 2020 }^{ 2020 } & { 2021 }^{ 2021 } & { 2022 }^{ 2022 } \end{vmatrix} \) is divided by 5.

Answer: 4


Question: Linearize the first-order differential equation \(x' = 2x^2 - 8\) about the equilibrium point \(-2\). What is the resulting linearized equation?

Answer: x' = -8(x + 2)


Question: Evaluate the limit: $$\lim_{x\to0} \frac{\ln\left(x+\sqrt{1+x^2}\right)-x}{\tan^3(x)}.$$

Answer: -\dfrac{1}{6}


Question:
Given a $C^{1}$ function $\mathbf{r}:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{R}^2$ with $\mathbf{r}'(t)=(0,0)$ if and only if $t=t_0$, and defined as:
\[\mathbf{r}(t)=\begin{cases} \mathbf{r}_1 (t), & t\in (t_0-\varepsilon, t_0] \\ \mathbf{r}_2 (t), & t\in [t_0, t_0+\varepsilon)\end{cases}\]
where $\mathbf{r}_1\in C^{\infty}((t_0-\varepsilon, t_0])$ and $\mathbf{r}_2\in C^{\infty}([t_0, t_0+\varepsilon))$, determine if there always exists a function $\mathbf{s}:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{R}^2$, $\mathbf{s}\in C^{\infty}((t_0-\varepsilon, t_0+\varepsilon))$, such that $\mathbf{s}((t_0-\varepsilon, t_0+\varepsilon))=\mathbf{r}((t_0-\varepsilon, t_0+\varepsilon))$. Answer with 'Yes' or 'No'.

Answer: No


Question: Suppose $(X,\mathcal{A},\mu)$ is a measure space, $f \in L^{1}(X)$, and $\{f_{n}\}_{n \in \mathbb{N}} \subseteq L^{1}(X)$ with $f_{n} \geq 0$ a.e. for all $n \in \mathbb{N}$ and $f_{n} \to f$ pointwise a.e. If $\{f_{n}\}_{n \in \mathbb{N}}$ is bounded in $L^{1}(X)$, is it true that $f_{n} \to f$ in $L^{1}(X)$ if and only if $\int_{X} f_{n} \, d \mu \to \int_{X} f \, d \mu$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty}(5+n)^{2n-1}((n+1)^{{1}/{n}}-n^{{1}/{n}}). \]

Answer: \infty


Question: Determine if the equation \(4x^2 - 3y^2 - z^2 = 12\) has integer solutions.

Answer: Yes


Question: Let $X$ and $Y$ be independent random variables, both following a Binomial distribution with parameters $n$ and $\theta$. Calculate the conditional expectation $E[X \mid X+Y=m]$. Provide your answer in terms of $n$, $\theta$, and $m$.

Answer: \dfrac{m}{2}


Question: Find the maximum value of the function \( f(x,y) = (x+y)^4 + y^4 \) subject to the constraint \( x^4 + y^4 = 1 \) using the method of Lagrange multipliers.

Answer: \dfrac{17}{2}


Question: Determine the order of the pole of the function \( f(z) = \frac{1}{e^z - 1} \) at \( z = 0 \).

Answer: 1


Question: Given that $x$ is an eigenvector of matrix $A$ corresponding to the eigenvalue $a$, determine the eigenvector of $A^t$ corresponding to the eigenvalue $a$. Provide your answer in terms of $x$.

Answer: x


Question: Determine the number of roots of the function \( f(z) = 1 + 6z^3 + 3z^{10} + z^{11} \) within the annulus defined by \( 1 < |z| < 2 \) using Rouche's theorem.

Answer: 7


Question:
Calculate \( k := \sup\left\{\left\|\theta\right\|_{*} \: |\: \ell^{*}(\theta)<\infty\right\} \) where \( \ell(x,y):=(y-\beta^{t}x)^{2} \). Let \( \beta\in\mathbb{R}^{d} \), and consider the function \( \ell:\mathbb{R}^{d}\times \mathbb{R}\rightarrow \mathbb{R} \) given by
\[\ell(x,y):=(y-\beta^{t}x)^{2}.\]
Calculate
\[k:=\sup\left\{\left\|\theta\right\|_{*} \: |\: \ell^{*}(\theta)<\infty\right\}\]
where \( \theta\in\mathbb{R}^{d}\times \mathbb{R}=\mathbb{R}^{d+1} \),  \( \left\|\theta\right\|_{*} =\sup_{\xi\in \mathbb{R}^{d+1}}\left\langle \theta,\xi\right\rangle \) and
\[\ell^{*}(\theta)=\sup_{\xi\in \mathbb{R}^{d+1}}\left[\left\langle \theta,\xi\right\rangle  - \ell(\xi)\right] .\]
(Recall \(\left\langle \theta,\xi\right\rangle =\theta^{t}\xi\).)

Answer: \infty


Question: Determine the number of integers \( n \) such that the ideal \( \langle n, x \rangle \) in \( \mathbb{Z}[x] \) is a principal ideal.

Answer: 3


Question: Customers arrive at a service facility according to a Poisson process with a rate of \( \lambda = 5 \) per hour. Given that 12 customers arrived during the first two hours of service, calculate the conditional probability that 5 customers arrived during the first hour.

Answer: \dfrac{99}{512}


Question: Determine the minimal polynomial of \(\sqrt[10]{5}\) over \(\mathbb{Q}(e^{2\pi i/10})\).

Answer: x^5 - \sqrt{5}


Question: Find the coordinates of the point of contact of the tangent line $y = 1 - x$ with the parabola $y^2 - y + x = 0$. The answer is given as $(0, 1)$. Verify this result.

Answer: (0, 1)


Question: Find all pairs of prime numbers \(p\) and \(q\) such that the expression \(343p^3 - q^3\) is a prime number.

Answer: (2, 13)


Question: Evaluate the limit: \[ \lim_{n\to\infty} d^{-n}e^{o(n)} \] where \( d > 1 \).

Answer: 0


Question:
Given a set of elements \(N = \{1, 2, \ldots, n\}\) and two arbitrary subsets \(A \subseteq N\) and \(B \subseteq N\), how many of the \(n!\) permutations \(\pi\) from \(N\) to \(N\) satisfy \(\min(\pi(A)) = \min(\pi(B))\), where \(\min(S)\) is the smallest integer in the set of integers \(S\), and \(\pi(S)\) is the set of integers obtained by applying permutation \(\pi\) to each element of \(S\)? Choose the correct option:

(A) \((n - |A \cup B|) |A| |B| \)
(B) \((|A|^2+|B|^2)n^2\)
(C) \(n!(|A\cap B|/|A\cup B|)\)
(D) \((|A\cap B|)^2/(n/|A\cup B|)\)

Answer: C


Question: Determine the value of \( t \) for which the integral \( \int_{0}^{\infty} \left( \frac{x+1}{3x^2-t} - \frac{t}{2x+1} \right) \, dx \) converges.

Answer: \dfrac{2}{3}


Question: Is a complex matrix \( A \) such that \( A^k = I \) (where \( I \) is the \( n \times n \) identity matrix and \( k > 1 \)) diagonalizable, provided that 1 is not an eigenvalue of \( A \)?

Answer: Yes


Question: Find the smallest integer \( n \) such that \( n! > 10^{100} \).

Answer: 70


Question: Find the x-coordinate of the point of intersection of the functions \( y = 2^x \) and \( y = \frac{1}{x} \).

Answer: 0.641


Question: Let \( A \in M_{3 \times 3} \) such that \( A^{6} = I \). Determine whether the minimal polynomial of \( A \) can be of degree 3. Answer with "True" or "False."

Answer: True


Question: Determine if there exists a real number \( N \) such that the double sum \[ C_N\sum_{r_1\in \mathbb{Z}}\sum_{x_1\in \mathbb{Z}}\left(1+\frac{1}{2}\left(|x_1+r_1|+|x_1|\right)\right)^{-N} \] is convergent. If so, find the range of \( N \) for which the sum converges.

Answer: (2, \infty)


Question:
Let \(f\) be a continuous function on \(\mathbb{R}\). Consider the sequence of functions defined recursively as follows:
\[ F_1(x) = \int_0^x xf(t)\,dt \]
\[ F_2(x) = \int_0^{F_1(x)} xf(t)\,dt \]
\[ F_3(x) = \int_0^{F_2(x)} xf(t)\,dt \]
Given that \(\int_0^1 f(t)\,dt = 1\) and \(f(1) = 2\), evaluate \(F_n'(1)\) for each \(n \in \mathbb{N}\). Use the Fundamental Theorem of Calculus I in its most basic form to justify your answer.

Answer: 2^{n+1} - 1


Question: Find the derivative of the function \( u(-x) \), where \( u(x) \) is the unit step function.

Answer: -\delta(x)


Question: Determine the Galois group \( \text{Gal} \left( \frac{\mathbb{Q}(\sqrt[3]{3},\sqrt{-3})}{\mathbb{Q}} \right) \).

Answer: S_3


Question: Evaluate the limit: $$\lim\limits_{x \to 0} \dfrac{8 [1-\cos(x^2/2)-\cos(x^2/4)+\cos(x^2/2)\cos(x^2/4)]}{x^8}.$$

Answer: \dfrac{1}{32}


Question: Are all elements in the sequence 10, 44, 136, 152, 184, 752, 884, 2144, 2272, 2528, 8384, 12224, 17176, 18632, 18904, 32896, 33664, 34688, 49024, 63248, 85936 even numbers?

Answer: Yes


Question: Evaluate the contour integral \( \int_{\gamma} \sqrt{z^2-1} \, dz \) where \( \gamma \) is the circle of radius \( \frac{1}{2} \) centered at the origin in the complex plane.

Answer: 0


Question:
Let \( \{A_n\} \) and \( A \) be bounded operators on a Hilbert space \( H \). Suppose:

1. \( A_n f \to A f \) for any \( f \in H \).
2. \( \|A_n\| \to \|A\| \).

Is it true that \( A_n \to A \) in the operator norm on \( \mathcal{L}(H) \)?

Answer: No


Question: Determine whether the following statement is true or false: If $X$ is a separable metric space, then the space of all continuous functions on $X$ with the supremum metric is also separable.

Answer: B


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

Answer: 0


Question: Calculate the maximum value of \[ \frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy} \] where \( x, y, z > 0 \) and \( x^2 + y^2 + z^2 = 3xyz \).

Answer: \dfrac{3}{2}


Question: Simplify the infinite series: $$\sum_{i=0}^\infty \frac{4^i}{(i!)^2}$$

Answer: I_0(4)


Question: Find the mass of the solid $E$ enclosed by the planes $z = 0$, $z = x + y + 5$, and the cylinders $x^2 + y^2 = 4$, $x^2 + y^2 = 9$. The density at any point is equal to its distance from the $yz$-plane.

Answer: \dfrac{380}{3}


Question: Let \(A_n\subset \mathbb{R}\) for all \(n\). If \(|A_n|=C\) for all \(n\), where \(C\) is the cardinality of the real numbers, and \(A_n\subset A_{n+1}\), is it true that \(| \bigcup_n A_n|=C\)?

Answer: C


Question: Evaluate the integral \( \int_{-2}^{2} \frac{x^{4}}{1+6^{x}} \, dx \).

Answer: \dfrac{32}{5}


Question: Find the height \( m \) of the horizontal line that divides the volume of the solid obtained by rotating the region bounded by \( y = x^2 \), \( y = 9 \), and the \( y \)-axis, around the vertical line \( x = 3 \), into two equal parts.

Answer: 5.18


Question: Calculate the conditional probability \( P(X_2 > 0 \mid X_1 > 0) \) for a Brownian motion \( X_t \) with \( t \geq 0 \).

Answer: \dfrac{3}{4}


Question: Are open disks on the unit sphere $S^2$ homeomorphic to open disks in $\mathbb{R}^2$? Justify your answer.

Answer: Yes


Question: Determine whether the ring \( \mathbb{Z}[\zeta_8]/\langle 1+3\omega \rangle \) is isomorphic to \( \mathbb{Z}_{82} \), where \( \omega = \frac{\sqrt{2} + i\sqrt{2}}{2} \) and \( \zeta_8 \) is a primitive 8th root of unity. Justify your answer.

Answer: \mathbb{Z}_{82}


Question: Find the degree of the splitting field of the polynomial \( f = X^4 + 2X^2 + 2 \) over \( \mathbf{Q} \).

Answer: 8


Question: Find \( \lim_{x \to -\infty} \left( \frac{x^4 \sin \frac{1}{x} + x^2}{1 + |x|^3} \right) \).

Answer: -1


Question: Determine all values of the parameter \(a\) such that the integral \(\int_0^{+\infty} \frac{dx}{1+x^a \sin^2x}\) converges.

Answer: a > 2


Question: Determine the number of passes in a monkey saddle, given that a pass moves from a hollow, through the critical point, into the adjacent hollow. A monkey saddle has three hollows adjacent to the critical point.

Answer: 3


Question: Calculate the residue of the function \( f(z) = \frac{z^2 + \sin z}{\cos z - 1} \) at the pole \( z = 2n\pi \).

Answer: -8n\pi - 2


Question:
Consider the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by
\[
f(x, y) =
\begin{cases}
\frac{xy}{x^2 + y}, & \text{if } y \neq -x^2 \\
0, & \text{if } y = -x^2
\end{cases}
\]
Which of the following statements is correct at the point \((0,0)\)?

A) The directional derivative does not exist at \((0,0)\).

B) \( f \) is continuous at \((0,0)\).

C) \( f \) is differentiable at \((0,0)\).

D) Each directional derivative exists at \((0,0)\) but \( f \) is not continuous.

Answer: D


Question: Three points are chosen independently and uniformly inside the unit square in the plane. Find the expected area of the smallest closed rectangle with sides parallel to the coordinate axes that contains all three points.

Answer: \dfrac{1}{4}


Question: Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x^2 + f(y)) = f(xy) \) for all real numbers \( x \) and \( y \).

Answer: f(x) = c


Question: For any natural number \( n \), does there exist a group \( G \) and a nontrivial normal subgroup \( N \) of \( G \) such that the order of the quotient group \( |G/N| = n \)?

Answer: Yes


Question: Consider the random variable $Z = \int_0^1 B_s^4 \, ds$, where $(B_s, s \in [0,1])$ is a fractional Brownian motion. Does $Z$ have a density? In other words, is the measure induced on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ by $Z$ absolutely continuous?

Answer: Yes


Question: Find a maximal ideal in the ring $R = \mathbb{Z}[\sqrt{-10}] = \{a + b\sqrt{-10} \mid a, b \in \mathbb{Z}\}$. Provide a justification for your answer.

Answer: (2, \sqrt{-10})


Question: In triangle $\triangle ABC$ with inradius $r$, if \( \frac{c + a}{b} + \frac{c + b}{a} = \frac{c}{r} \), determine which angle is $90^\circ$.

Answer: C


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{e^{i\theta}}d\theta \) using contour integration.

Answer: 2\pi


Question: Determine the values of \( p \) for which the double series \( \sum_{i, j=1}^\infty a_{i+j}^p \) converges, given that \( (a_i)_{i=1}^\infty \) is a sequence of positive numbers such that \( \sum_{i=1}^\infty a_i < \infty \). Specifically, investigate the case when \( p = 2 \).

Answer: p \geq 2


Question: Given \( n \) numbers, find the smallest number \( k \) in the range \([a, b]\) such that each number is either divisible by \( k \) or divides \( k \). For example, if \( n = 4 \) and the numbers are 1, 20, 5, 2 with the range \([8, 16]\), the answer is \( k = 10 \). Determine the smallest such \( k \).

Answer: 10


Question:
Find the value of \( a \) such that the four planes given by the equations:
\[
\begin{cases}
α_1 :& x + 2ay + az = a^2 \\
α_2 :& x + ay + az = a \\
α_3 :& x + a^2 z = a^3 \\
α_4 :& ax + ay + a^2 z = a \\
\end{cases}
\]
do not have any point in common.

Answer: a \neq 0, 1, 2


Question: Determine the number of non-negative integer solutions to the equation \(x_{1} + x_{2} + 3x_{3} + x_{4} = k\), given the constraints: \(x_{1} \geq 1\), \(x_{2} \leq 2\), \(x_{3} \leq 1\), and \(x_{4}\) is a multiple of 6.

Answer: k


Question: Determine whether the following statement is true: For any \(\lambda \in (0,1)\) and integer \(k > 0\), if \(X\) is a Poisson random variable with mean \(k \lambda\), then \(\Pr(X < k) \geq e^{-\lambda}\).

Answer: True


Question: What is the graph diameter of the 1-skeleton of the omnitruncated $E_8$ polytope?

Answer: 120


Question:
Determine which of the following functions can be defined at the point \( z = 0 \) such that the extended functions are continuous at \( z = 0 \):

1. \( \frac{\text{Re} \, z}{|z|} \)
2. \( \frac{z}{|z|} \)
3. \( \frac{\text{Re} \, z^2}{|z|^2} \)
4. \( \frac{z \text{Re} \, z}{|z|} \)

Answer: 4


Question: Evaluate the double integral \( \int\int_{T} [35xy^2 + 7e^x y^3] \, dx \, dy \), where \( T \) is the region bounded by \( x = y^2 + 1 \) and \( x = 2 \).

Answer: 16


Question: Evaluate the limit: \[ \lim_{x \to \pi/2} \frac{\sin x - (\sin x)^{\sin x}}{1 - \sin x + \log(\sin x)} \]

Answer: 2


Question: Evaluate the limit: \[ \lim_{x,y \to 0,0} \frac{\sin(x+y)-(x+y)}{\sqrt{x^2 +y^2}} \]

Answer: 0


Question:
Determine whether the following statement is true:

Let \(f:(a,b)\subset\mathbb{R} \rightarrow \mathbb{R}\), such that \(f'\) exists for all \(x \in (a,b)\) and \(f''(x_0)\) exists for some \(a<x_0<b\). Then
\[ \lim_{h \to 0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0). \]

Note: The continuity of \(f'\) in a neighborhood of \(x_0\) is not assumed, but the existence of \(f'\) in a neighborhood of \(x_0\) is necessary for the existence of \(f''(x_0)\).

Answer: True


Question:
Find the radius of convergence for the power series:
\[ \sum_{n = 1}^{\infty} \left(1 + \frac{1}{n^{1/n}} \right)^{n} \sin^{2n}\left(\frac{n \pi}{3}\right) z^{n} \]

Answer: \dfrac{2}{3}


Question: Determine the number of solutions to the equation \([f'(x)]^2 = f(x) \cdot f''(x)\) for a function \(f(x)\) of degree 4. The answer is 0 solutions.

Answer: 0


Question: Determine whether there exists an algorithm to decide if there is a nonzero polynomial $Q(x_1,\dots,x_n) \in \mathbb{R}[x_1,\dots,x_n]$ such that the product $P(x_1,\dots,x_n)Q(x_1,\dots,x_n)$ has nonnegative coefficients, where $P(x_1,\dots,x_n) \in \mathbb{R}[x_1,\dots,x_n]$. (Note: The case $n=1$ is well-known and not difficult.)

Answer: Yes


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

Answer: 0


Question: Two ants start at the same point on a plane. Each minute, they choose to walk either north, east, south, or west. In the first minute, they walk 1 meter. In each subsequent minute, the distance they walk is multiplied by a rational number \( q > 0 \). They meet again after a whole number of minutes, but they have not taken exactly the same route. Determine all possible values of \( q \).

Answer: 1


Question:
Determine the value of \( k \) such that the function \( f(x,y) \) is continuous at \((0,0)\):
\[
f(x,y) =
\begin{cases}
\dfrac {\cos^2\left(\sqrt{x^2+y^2}\right)-1}{x^2+y^2},  & \text{if } (x,y) \ne (0,0) \\
[2ex]
k, & \text{if } (x,y) = (0,0)
\end{cases}
\]

Answer: -1


Question: Provide an example of a decreasing sequence \((f_n)\) of continuous functions on \([0,1)\) that converges to a continuous function, but the convergence is not uniform.

Answer: f_n(x) = x^n


Question: Is the space $BL^1_{loc}$, consisting of locally integrable functions $f: \mathbb{R} \to \mathbb{R}$ such that $|f| = \sup_{x \in \mathbb{R}} \int_x^{x+1} |f(y)| \, dy < \infty$, complete with respect to the norm $|f|$?

Answer: Yes


Question: Given a real $n \times k$ matrix $A$ of rank $k$ and an invertible $k \times k$ matrix $C$, apply the Gram-Schmidt process to the columns of $A$ and $A \cdot C$ to obtain matrices $B$ and $D$ such that $B^* B = I_k = D^* D$. Determine if there exists an orthogonal matrix $E \in O_k$ such that $B = D \cdot E$. Provide a proof or counterexample.

Answer: Yes


Question:
Let \( u(x,t) \) satisfy the initial value problem \( u_{t}=u_{xx} \), where \( x\in \mathbb{R}, t>0 \), and the initial condition is given by:
\[
u(x,0)=\begin{cases}
  1, & x\in[0,1]\\
0, & \text{otherwise}
\end{cases}\]
Determine the value of \( \lim_{t\to 0^+} u(1,t) \). Choose the correct answer from the following options:

- \( e \)
- \( \pi \)
- \( \frac{1}{2} \)
- \( 1 \)

Answer: C


Question:
Given the differential equation \( \frac{{d^2x}}{{dy^2}} \), determine the correct expression from the following options:

(A) \( {\left( \frac{{d^2y}}{{dx^2}} \right)^{-1}} \)

(B) \( -{\left( \frac{{d^2y}}{{dx^2}} \right)^{-1}}{\left( \frac{{dy}}{{dx}} \right)^{-3}} \)

(C) \( \left( \frac{{d^2y}}{{dx^2}} \right){\left( \frac{{dy}}{{dx}} \right)^{-2}} \)

(D) \( -\left( \frac{{d^2y}}{{dx^2}} \right){\left( \frac{{dy}}{{dx}} \right)^{-3}} \)

Select the correct answer.

Answer: D


Question: Let \(a, b, c, d\) be integers such that \(\frac{a}{c} \in \mathbb{Q}^+ \cap \mathbb{Z}^+\) and \(\frac{b}{d} \in \mathbb{Q}^- \cap \mathbb{Z}^-\). Determine the number of solutions to the equation \(|ad - bc| = 1\).

Answer: 0


Question: Let \( f \) be an arbitrary, twice differentiable function for which \( f'' \neq 0 \). The function \( u(x,y) = f(x^2 + axy + y^2) \) satisfies the equation \( U_{xx} - U_{yy} = 0 \). Determine the value of the constant \( a \).

Answer: \pm 2


Question: Determine whether the upper density \( \mu^+\big(\sigma^{-1}(\{k\})\big) = 0 \) for all \( k \in \mathbb{N} \), where \( \sigma(n) \) is the number of divisors of \( n \). If not, find the value of \( \sup\big\{\mu^+\big(\sigma^{-1}(\{k\})\big):k\in\mathbb{N}\big\} \).

Answer: 0


Question: Let \(K\) be a cubic extension of the rational numbers with discriminant \(D\), and let \(\{ 1, \omega_2, \omega_3 \}\) be an integral basis for the ring of integers \(\mathcal{O}_K\) of \(K\). Suppose \(\alpha \in \mathcal{O}_K\) is primitive such that no rational prime divides \(\alpha\), and the norm of \(\alpha\) is equal to \(a^3\), where \(a \in \mathbb{Z}\) and \(a\) is prime to \(D\). Determine whether there always exist rational integers \(\beta_2\) and \(\beta_3\) such that \(\{ a, \beta_2 + \omega_2, \beta_3 + \omega_3 \}\) is an integral basis for the ideal \((a, \alpha)\), even when \(\gcd(\text{disc.}(\alpha), a) \neq 1\).

Answer: Yes


Question: Is the stabilizer of $1$ in the symmetric group $S_n$ a maximal subgroup?

Answer: Yes


Question: Determine the chromatic number \( \chi(G) \) of the graph \( G \), where the vertices are the numbers 1, 2, ..., 64, and two vertices are adjacent if and only if one of the corresponding numbers divides the other.

Answer: 7


Question: Evaluate the limit of the sequence of integrals: \[ \lim _{n \rightarrow \infty} \int_{-\infty}^\infty \frac{dx}{n(e^{x^2}-1) +1/n} \]

Answer: \pi


Question: Find the smallest prime factor of the expression \(4^{52} + 52^{2013} + 2013^{52}\).

Answer: 7


Question: Determine whether there exist four distinct elements of order 4 in extra-special 2-groups.

Answer: Yes


Question:
Given a polynomial \( P(x) \) of degree 7, it is known that:

\[ P(4) = P(5) = P(6) = \cdots = P(10) = 10! \]
\[ P(3) = 11! \]

Determine the constant term of the polynomial \( P(x) \).

Answer: 1201 \times 10!


Question: Given that \( f(x) = x + \int_{0}^{1} t(x+t) f(t) \, dt \), calculate the value of \( \int_{0}^{1} f(x) \, dx \).

Answer: \dfrac{42}{23}


Question: Compute the homotopy group \( \pi_3(S^2) \).

Answer: \mathbb{Z}


Question: Calculate the integral of the pullback of the form \( \omega = \frac{xdy-ydx}{x^2+y^2} \) over the interval \([0, 2\pi]\), where \( g(t) = (e^t \sin(17t), e^{t^2} \cos(17t)) \).

Answer: 34\pi


Question: In a round robin tournament with 20 teams, each team plays every other team exactly once. The number of games won by each team is a perfect square. Additionally, the number of teams winning 9 games is equal to the number of teams winning 16 games. Given that there are no ties, determine how many teams have 16 wins.

Answer: 7


Question:
Given a random variable $X$ with the probability density function
\[
f_{X}(x) = \begin{cases} 1 &\text{ if } 0 < x < 1 \\
0 &\text{ otherwise}
\end{cases}
\]
and a random variable $Y$ with the conditional probability density function
\[
f_{Y|X}(y|x) = \begin{cases}
1/x &\text{ if } 0 < y < x \\
0 &\text{ otherwise}
\end{cases}
\]
find the marginal probability density function for $Y$. Express your answer in terms of $y$ for $0 < y < 1$.

Answer: -\ln(y)


Question: Find the largest number \( l_0 \) such that the operator \( T: X_{l_0} \rightarrow C([0,l_0], \mathbf{R}) \), defined by \((T(f))(x) = \int_0^{x} f(t)dt\), maps the set \( X_{l_0} = \{f \in C([0,l_0], \mathbf{R}) : 0 \leq f(x) \leq 2, \forall x \in [0,l_0]\}\) into itself.

Answer: 1


Question: Let $p$ be a prime and $K$ be a finite Galois extension of $\mathbb{Q}$ with degree $p^r$, where $r > 0$. Let $v$ be a prime of $K$ lying above $p$. Determine the degree of the extension $K_v / \mathbb{Q}_p$, where $K_v$ and $\mathbb{Q}_p$ are the completions of $K$ and $\mathbb{Q}$ with respect to the primes $v$ and $p$, respectively.

Answer: p^r


Question: Determine the weight function with respect to which the eigenfunctions of the differential equation \( x^2y'' + xy' + \lambda y = 0 \) are orthogonal, given the boundary conditions \( y(1) = 0 \) and \( y(b) = 0 \) where \( b > 1 \). Verify the orthogonality of these eigenfunctions by direct integration.

Answer: \dfrac{1}{x}


Question: The graph of the function \( f(x) = x^4 + 4x^3 - 16x^2 + 6x - 5 \) has a common tangent line at \( x = p \) and \( x = q \). Compute the product \( pq \).

Answer: -10


Question: Calculate the probability \( P(4X_1^2 + X_2^2 \le 5) \) where \( X_1 \sim N(0,1) \) and \( X_2 \sim N(0,4) \). Use the transformation \( Y = \frac{X_2}{2} \) to express the probability in terms of standard normal variables and find the numeric value using the CHISQ.DIST function in Excel.

Answer: 0.4647


Question: Determine the number of functions that satisfy the equation: \[ y(x) = 2 + \int_{0}^{x}\frac{t}{y(t)}dt \]

Answer: 1


Question: Compute \( E\left((B_t - 1)^2 \int_0^t (B_s + 1)^2 \, dB_s\right) \), where \((B_t)\) is a standard Brownian motion for \( t \geq 0 \).

Answer: t^2 - 2t


Question: Evaluate the integral \( \int_{0}^{2\pi} \left| \sin\theta + \cos\theta \right| \mathrm{d}\theta \).

Answer: 4\sqrt{2}


Question: Evaluate the integral \( \int_C \frac{\sin(z)}{z^2} \, dz \), where \( C \) is the unit circle in \( \mathbb{C} \). Use Cauchy's integral formula and consider the limit approach for handling the singularity at \( z = 0 \).

Answer: 2\pi i


Question: Calculate the limit: \[ \lim_{x\rightarrow+\infty}\left(\int_0^1\sup_{s>x}\frac{s}{e^{(s\log s)t}}dt\right) \]

Answer: 0


Question: Determine whether there exists a permutation $a$ in $S_6$ such that $a^2 = (123)(456)$. Provide a justification for your answer.

Answer: Yes


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: Let $T: X \times X \to \mathbb{R}$ be a continuous bilinear operator on a normed linear space $X$ such that $T(\alpha x + \beta y, z) = \alpha T(x, z) + \beta T(y, z)$ and $T(x, y) = T(y, x)$. Does there exist a constant $C$ such that $||T(x, y)|| \leq C ||x|| ||y||$ for all $x, y \in X$?

Answer: Yes


Question: Evaluate the expression \[ \frac{\sum_{i=0}^{100} \binom{k}{i} \binom{M-k}{100-i} \frac{k-i}{M-100}}{\binom{M}{100}}. \]

Answer: \dfrac{k}{M}


Question: A check-out counter at a supermarket services one customer per unit time if there is anyone in line. Customers arrive at the line with the following probabilities per unit time: the probability of a single new customer arriving is \(\frac{1}{3}\), the probability of two new customers arriving is \(\frac{1}{3}\), and the probability of no new customer arriving is \(\frac{1}{3}\). Initially, there are three customers in line. What is the probability that the line empties before it reaches ten persons?

Answer: \dfrac{7}{10}


Question: Evaluate the limit: $$\lim_{n\to\infty}n^2\left(\sin(2\pi en!)-\frac{2\pi}{n}\right)$$

Answer: 0


Question: Evaluate the limit: $$\lim_{(x,y) \to (-3, 4)} \frac{2x^3 + 5y^3 + 18x^2 + 54x - 60y^2 + 240y - 266}{\sqrt{x^2 + 6x + 25 + y^2 - 8y}}$$

Answer: 0


Question: For any arbitrary set $A$, does there always exist a set $B$ such that $A$ is a proper subset of $B$?

Answer: Yes


Question: Is every flat compact Riemannian homogeneous manifold a flat torus?

Answer: Yes


Question: Find the maximum possible area of a triangle with side lengths at most 2, 3, and 4.

Answer: 3


Question: Evaluate the limit: \[ \lim_{x \to \infty} x(2\pi - 4\arctan(x)) \]

Answer: 4


Question:
Evaluate the improper integral \( \int_e^{\infty}\frac{\ln x}{x}\ dx \). Choose the correct answer from the following options:

A. \( \dfrac{1}{2} \)

B. \( \dfrac{e^2}{2} \)

C. \( \dfrac{\ln(2e)}{2} \)

D. DNE (Does not exist)

Answer: D


Question: Evaluate the limit: \[ \lim_{x\to 0}\left(\frac {e^x}{x}-\frac {1}{\arctan{x}}\right) \] without using L'Hôpital's rule.

Answer: 1


Question: Find the value of the determinant without expanding it: $$\begin{vmatrix}a^2 & (a+1)^2 & (a+2)^2 & (a+3)^2 \\ b^2 & (b+1)^2 & (b+2)^2 & (b+3)^2 \\ c^2 & (c+1)^2 & (c+2)^2 & (c+3)^2 \\ d^2 & (d+1)^2 & (d+2)^2 & (d+3)^2\end{vmatrix} $$

Answer: 0


Question: Find the largest integer \( n \leq 9,000,000 \) such that the expression \( \sqrt{n+\sqrt{n+\sqrt{n+\cdots}}} \) is rational.

Answer: 8997000


Question: Let \(X_n\) be a sequence of random variables that converges in probability to a negative real number \(a\). Prove or disprove that \(\lim_{n\to\infty} P(X_n > 0) = 0\).

Answer: 0


Question: Given that \(\log_{4n} 40\sqrt{3} = \log_{3n} 45\), find the value of \(n^3\).

Answer: 75


Question: Evaluate the limit: \[ \lim_{x\to\infty} \left( (x+2017)^{1+\frac{1}{x}} - x^{1+\frac{1}{x+2017}} \right) \]

Answer: 2017


Question: What is the probability that the three most common results among 12 equally likely outcomes are selected by at least 11 members of a sample of size 22?

Answer: 0


Question: Compute the conditional expectation \( E[\max(X,Y) | \min(X,Y)] \) where \( X \) and \( Y \) are independent and identically distributed random variables with \( X, Y \sim \exp(1) \).

Answer: \min(X,Y) + 1


Question: Let $X$ be a uniformly distributed random variable on $(0,1)$, $Y$ be a uniformly distributed random variable on $(0,2)$, and $Z$ be a uniformly distributed random variable on $(0,4)$. Calculate the probability that $X + Y > Z$. Express your answer as a single probability value.

Answer: \dfrac{3}{8}


Question: Evaluate the limit: \[ \lim_{n \to \infty} (-1)^n \sin\left(\pi(n^2+0.5n+1)^{\frac{1}{2}}\right)\sin\left(\frac{(n+1)\pi}{4n}\right) \]

Answer: \dfrac{1}{2}


Question: Determine the number of connected components of the set \(\{(x,y) \in \mathbb{R}^2 : (xy)^2 = xy\}\) in \(\mathbb{R} \times \mathbb{R}\).

Answer: 3


Question:
Determine whether the following statement is true or false:

Let $X$ be a normed linear space, $x_k \in X$, $k \in \mathbb{N}$, and $\sum_{k=0}^\infty \lVert x_k\rVert$ convergent. Then $\sum_{k=0}^\infty x_k$ is also convergent.

Answer: B


Question: Given the function \(f(x) = x^3 + 2x + 3\), find \((f^{-1})'(0)\).

Answer: \dfrac{1}{5}


Question: Evaluate the definite integral \( \int_0^{\pi} \frac{dx}{a^2\cos^2x + b^2 \sin^2x} \).

Answer: \dfrac{\pi}{ab}


Question: Compute the degree of the field extension \( \mathbb{Q}(\sqrt{2}, \sqrt[3]{2}, w) \) over \( \mathbb{Q} \), where \( w = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \).

Answer: 12


Question: Evaluate the line integral \( \int_C (x+y)^2dx - (x^2+y^2)dy \) where \( C \) is the positively oriented triangle with vertices at \((1,1)\), \((3,2)\), and \((2,5)\).

Answer: -\dfrac{140}{3}


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

Answer: -2


Question: Let \( X \) be a single observation from the uniform distribution with probability density function \( f(x) = \frac{1}{\theta} \) for \( 0 < x < \theta \) and \( f(x) = 0 \) otherwise. Suppose we use \((X, 1.5X)\) as a confidence interval for \( \theta \). Determine the confidence level of this interval.

Answer: \dfrac{1}{3}


Question: Let $X$ be a set. Determine the number of $\sigma$-algebras of subsets of $X$ that contain exactly 5 elements.

Answer: 0


Question: Suppose we have a polynomial \( P(z) = \sum_{i = 0}^{m} a_{i}z^{i} \) with \( m \) real roots. Determine whether it is true that the polynomial \( P(z+qi) + P(z-qi) \) also has \( m \) real roots.

Answer: True


Question: Let $f:[0,1] \to \mathbb{R}$ be a continuous function on $[0,1]$. Given that $f(1) = 2\int_0^{\frac{1}{2}} x f(x) \, dx$, find the value of $f(1)$.

Answer: 0


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

Answer: \dfrac{4}{e}


Question: Find the maximum value of the function \( f = |x_1x_3 + x_1x_4 + x_2x_3 - x_2x_4| \) on the four-dimensional cube \( \{x \in \mathbb{R}^4 : |x_k| \le 1, 1 \le k \le 4\} \).

Answer: 2


Question: Calculate the covariance \( \text{Cov}[(B(t)-B(s))^2,(B(v)-B(u))^2] \) where \( B(t) \) is a Brownian motion and \( 0<s<t<u<v \).

Answer: 0


Question: Prove that for non-negative real numbers \(a, b, c\) such that \(0 \le a \le b \le c \le 1\), the following inequality holds: \[ \frac{a}{1+bc} + \frac{b}{1+ac} + \frac{c}{1+ab} \le 2. \]

Answer: 2


Question: Determine the set of limit points of the set \( A = \left\{ \frac{\sqrt{m} - \sqrt{n}}{\sqrt{m} + \sqrt{n}} \mid m, n \in \mathbb{N} \right\} \).

Answer: [-1, 1]


Question: Find a group that contains elements $a$ and $b$ such that $|a|=2$, $|b|=11$, and $|ab|=2$. Determine the group and the value of $n$ if the group is either $S_n$ or $D_n$.

Answer: 11


Question: Evaluate the limit: \[ \lim_{t\to\infty}\frac{1}{t}\int_0^t \sin(\alpha x)\cos(\beta x)\,dx \]

Answer: 0


Question: Determine the centralizer of the quotient group $A_4/V_4$ in the symmetric group $S_4$, where $V_4$ is the Klein four-group. Recall that if $G$ is a group and $H$ and $K$ are subgroups of $G$ with $K \unlhd H$, then the centralizer $C_G(H/K)$ is defined as $\{ g \in G \mid [g,h] < K, \forall h \in H \}$.

Answer: A_4


Question: You have a black-box that generates random numbers uniformly between 0 and 1. You continue generating random numbers \(X_1, X_2, X_3, \ldots\) and keep a running sum of these numbers. You stop generating numbers as soon as the sum exceeds 1. What is the expected number of random numbers generated in this process?

Answer: e


Question: Use Stokes' theorem to evaluate the integral \( \int_C (x+2y)dx+(2z+2x)dy+(z+y)dz \), where \( C \) is the intersection of the sphere \( x^2 + y^2 + z^2 = 1 \) and the plane \( y = z \).

Answer: 0


Question: Determine the number of maximal ideals in the quotient ring \( \frac{\mathbb{Z}_5[x]}{\langle (x+1)^2(x+2)^3 \rangle} \).

Answer: 2


Question: The area bounded by the curve \( y = e^{x^2} \), the x-axis, and the lines \( x = 1 \) and \( x = 2 \) is given to be \( a \) square units. Find the area bounded by the curve \( y = \sqrt{\ln x} \), the y-axis, and the lines \( y = e \) and \( y = e^4 \) in terms of \( a \).

Answer: 2e^4 - e - a


Question: Suppose $R$ is an excellent discrete valuation ring and consider a local injection $R \hookrightarrow R^{\prime}$ of discrete valuation rings such that the induced residue field extension is separable and a uniformizer of $R$ is also a uniformizer of $R^{\prime}$. Is $R^{\prime}$ necessarily excellent?

Answer: Yes


Question: Determine whether the homomorphism \( Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to \widehat{\mathbb{Z}}^\times \) induced by the action of \( Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \) on \( \widehat{\mathbb{Z}} \) is the same as the map \( Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to Gal(\overline{\mathbb{Q}}/\mathbb{Q})^{ab} \cong \widehat{\mathbb{Z}}^\times \).

Answer: Yes


Question: Evaluate the limit \( \lim_{n\to \infty}\sum_{j=0}^{n}{j+n-1 \choose j}\left(\frac{1}{2^{n+j}}\right) \) using the central limit theorem.

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true: Given two submodules $U,V \subseteq M$ over a commutative ring $R$, and a flat $R$-module $A$, is it necessarily true that $$(U \cap V) \otimes_R A \cong (U \otimes_R A) \cap (V \otimes_R A) ?$$

Answer: True


Question: Evaluate the contour integral \( \oint_{|z| = 2} \frac{1}{\cos(z)\sin(z)} \, dz \) and determine if the integral evaluates to zero based on the presence of poles within the contour.

Answer: -2\pi i


Question:
Determine the number of real numbers \(x\) that satisfy the equation:
\[
\left(|x^2-12x+20|^{\log{x^2}}\right)^{-1+\log x}=|x^2-12x+20|^{1+\log{\frac{1}{x}}}
\]

Answer: 5


Question: Let $A$ and $B$ be two $C^*$-algebras, and let $\psi: A \rightarrow B$ be a non $*$-homomorphism. Suppose $b$ is a nonzero normal element in $B$. We have a $*$-isometric isomorphism $\phi: C(\sigma_B(b)) \to C^*(b,b^*)$, given by $f \mapsto f(b)$. Does there exist a function $f \in C(\sigma_B(b))$ and an element $a \in A$ such that $\psi(a) = f(b)$ is nonzero?

Answer: No


Question: In a group \(G\) generated by elements \(a\) and \(b\), where \(a^5 = e\), \(aba^{-1} = b^2\), and \(b \neq e\), determine the order of the element \(b\).

Answer: 31


Question: Determine all starting values for which the sequence \((b_n)_n\) defined by the recursion formula \(b_{n+1} = \frac{b_n}{2} - \frac{1}{1+b_n}\) converges.

Answer: \emptyset


Question: Let \( f: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by \( f(x,y) = (e^{2x+y}, e^{x+y}) \). Compute the density function \( \frac{df[\lambda_2]}{d\lambda_2} \) of the pushforward measure \( f[\lambda_2] \), where \( \lambda_2 \) is the Lebesgue measure on \( \mathcal{B}(\mathbb{R}^2) \).

Answer: \dfrac{1}{uv}


Question: Determine whether the sequence $P_n$ is equal to the sequence $G_n$, where $G_0=1$ and $G_{n+1}=F_0G_n+F_1G_{n-1}+\cdots+F_nG_0$, with $F_n$ being the $n$th term of the Fibonacci sequence defined by $F_0=F_1=1$ and $F_{n+1}=F_n+F_{n-1}$. The sequence $P_n$ is defined by $P_0=P_1=1$, $P_2=2$, and $P_{n+1}=2P_n+P_{n-1}$ for $n>1$. Is $P_n=G_n$?

Answer: P_n = G_n


Question: Evaluate the expression \(\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}\), where \(q\) is a complex number such that \(q^7=1\) and \(q\neq 1\).

Answer: -2


Question: Is every finitely generated virtually free group residually finite?

Answer: Yes


Question: Consider an infinite series \(\sum a_n\) where each term \(a_n\) is non-negative and the series is known to diverge. Does it follow that \(\sum a_n = +\infty\)? Justify your answer.

Answer: +\infty


Question: Is every $H_{\infty}$-ring spectrum over Morava E-theory $E$ at height $h$ and prime $p$, whose homotopy ring is isomorphic to $W(F_{p^k})[[v_1, \dots, v_{n-1}]][\beta^{\pm 1}]$, a $K(h)$-local spectrum? Here, $k$ is a positive integer and $\beta$ has degree 2.

Answer: Yes


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), the following equation holds: \[ f\left(x^2+f(y)\right)=f(x)^2+y^4+2f(xy). \]

Answer: f(x) = x^2


Question: Given the premise \( \neg (A \rightarrow B) \), derive the conclusion \( A \land \neg B \).

Answer: A \land \neg B


Question: Evaluate the limit: \[ \lim\limits_{n \to \infty} \int_0^1 \frac{\ln^n (1+x)}{1+x^2}dx \]

Answer: 0


Question:
Determine the function \( \phi(x) \) that satisfies the integral equation:
\[ \phi(x) = f(x) + \int_0^x \sin(x-t)\phi(t)\,dt \]
Choose the correct expression for \( \phi(x) \) from the following options:

1. \( \phi(x) = f(x) + \int_0^x (x-t)\phi(t)\,dt \)
2. \( \phi(x) = f(x) + \int_0^x \sin(x-t)\phi(t)\,dt \)
3. \( \phi(x) = f(x) + \int_0^x \cos(x-t)\phi(t)\,dt \)
4. \( \phi(x) = f(x) - \int_0^x \sin(x-t)\phi(t)\,dt \)

Answer: 2


Question: Let $X_1, X_2, \ldots$ be a sequence of random variables such that $P(X_n = \frac{k}{n}) = \frac{1}{n}$ for $k = 1, 2, \dots, n$. Determine the limit distribution of $X_n$ as $n \to \infty$. Express your answer in terms of a standard probability distribution.

Answer: U(0,1)


Question: Determine the number of intersection points between a circle of radius $R$ centered at the origin in 3D space and a Cassini oval defined by two fixed points $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$ such that the product of the distances from any point on the oval to these two points equals a constant $C$. How many points can this circle and Cassini oval have in common?

Answer: 8


Question: Solve the equation: \[ 4^{x+\sqrt{x^2-2}} - 3 \cdot 2^{x-1+\sqrt{x^2-2}} = 10 \]

Answer: \dfrac{3}{2}


Question: Evaluate the limit using L'Hospital's rule: $$\lim_{x\rightarrow \infty }\left( x^2 - \frac x2 - (x^3 + x+1 ) \ln \left(1+ \frac 1x \right) \right)$$

Answer: -\dfrac{4}{3}


Question: Evaluate the limit: \[ \lim_{x \to 0} \left( \frac{1}{x-\sin x}\int_{0}^{x}t\sin t \,\text{d}t \right) \]

Answer: 2


Question: Evaluate the sum of the series \( \sum_{n=0}^{\infty}\frac{(-1)^n}{2n + 1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots \) by setting \( x = \frac{l}{2} \) in the Fourier series representation of the function \( \phi(x) = x \) on the interval \([-l,l]\).

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral \( \int_0^{\infty} x^3 e^{-x^2} \, dx \).

Answer: \dfrac{1}{2}


Question: Let $G$ be a compact connected real Lie group. Is the set of torsion elements $T$ always dense in $G$?

Answer: Yes


Question: Consider the Cayley graph of $\mathbb{Z}_3 \times \mathbb{Z}_3$ with respect to the generating set $S=\{(0,1),(1,0)\}$. Determine if the graph shown in Figure (2) can be regarded as an automorphism of this Cayley graph.

Answer: Yes


Question: Is it possible to paint a plane using two colors such that any two points separated by 10 cm have different colors?

Answer: No


Question: Evaluate the limit of the expression \(\lim\limits_{n\to \infty} \left(\frac{1}{e}\right)^{\frac{3n+2+\frac{1}{(n-1)!}}{2}}\). Specifically, determine the behavior of the term \(\frac{1}{(n-1)!}\) as \(n\) approaches infinity.

Answer: 0


Question: Given the initial position of a submarine at \((-230, 75)\) and a convoy at \((0, 0)\), with the submarine having a maximum speed of 30 km/h and the convoy traveling southwest at 18.5 km/h, determine the direction the submarine should travel to intercept the convoy most quickly. Assume the units are in kilometers for distance and kilometers per hour for speed.

Answer: 308.6^\circ


Question: Evaluate the integral \( \int_0^\infty \frac{\cos(x)}{x^2+1} \, dx \) using complex analysis techniques.

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


Question: What is the minimum total number of vertices required for two unlinked but interlocked polygons, $P$ and $Q$, where $P$ and $Q$ are rigid and cannot flex or distort? Can this interlocking be achieved with a total of 8 vertices?

Answer: 8


Question: Evaluate the limit: \[ \lim_{x\to \infty} \left( x\log(x^2+x) - x^2\log(x +1) \right). \]

Answer: -\infty


Question: Determine the range of values for \(a\) such that the function \(f(x) = (a+2)x^3 - 3ax^2 + 9ax - 1\) is decreasing for all real values of \(x\).

Answer: (-\infty, -3]


Question: Evaluate the limit: $$\lim_{(x,y,z)\to (1/2,1/2,1/2)}{x\cos(\pi y)\sin\left(\frac{1}{2z-1}\right)}$$ Determine whether the limit exists, and if so, find its value.

Answer: 0


Question: Evaluate the definite integral \( \int_{0}^{\infty}\frac{dx}{(1+x^a)(1+x^2)} \) where \( a > 0 \).

Answer: \dfrac{\pi}{4}


Question: Find the covariance \( \operatorname{Cov}(X, Y^2) \) given that \( X \sim N(0,1) \), \( Y \sim N(0,1) \), and \( \operatorname{Cov}(X, Y) = p \) where \( -1 < p < 1 \).

Answer: 0


Question: Consider a matrix $A \in \text{Mat}(\mathbb{N},\{0,1\})$, where each entry is independently 0 with a probability of 50%. What is the probability that $A$ is nilpotent, meaning that there exists some positive integer $N$ such that $A^N$ is the constant 0-matrix?

Answer: 0


Question: Let \( f: B_1(0) \rightarrow B_1(0) \) be a holomorphic function with at least two fixed points. Prove that \( f \) is the identity function, i.e., \( f(z) = z \) for all \( z \in B_1(0) \).

Answer: f(z) = z


Question: Evaluate the contour integral \( \oint_{C} \left( z + \cfrac{z}{z+\cfrac{z}{\vdots}} \right) dz \), where \( C \) is a contour oriented counterclockwise around (but not including) the interval \([-4, 0]\) in the complex plane.

Answer: 2\pi i


Question: Determine the number of positive integer solutions to the equation \(x^2 - y^2 = 3^8 \cdot 5^6 \cdot 13^4\).

Answer: 157


Question: Find the inverse Z-transform of the function \( \frac{1}{(1-z^{-1})^2} \).

Answer: (n + 1) u[n]


Question: If $f(x)$ is a function satisfying the equation $$f\left(\frac{1}{x}\right) + x^2 f(x) = 0$$ for all non-zero $x$, find the value of the integral $$\int_{\sin\theta}^{\csc\theta} f(x) \, dx.$$

Answer: 0


Question: Can a right circular cylinder in \(\mathbb{R}^3\) have non-trivial holonomy?

Answer: No


Question: Find the maximum value of the integral \( \int_{0}^{1} |\ln|s-t|| \, dt \) as \( s \) varies over the interval \( 0 \leq s \leq 1 \).

Answer: 1 + \ln 2


Question: Given an equilateral triangle $ABC$, a point $D$ is such that $DB=4$, $DA=4\sqrt{3}$, and $DC=8$. Calculate the area of triangle $ABC$.

Answer: 28\sqrt{3}


Question:
Verify if the expression for calculating the nth harmonic number is given by:
\[ H_n = \gamma + \lim_{h\to\infty}\left(\ln(h) - \sum_{k=n+1}^{h}\frac{1}{k} \right) \]

Answer: H_n = \gamma + \lim_{h\to\infty}\left(\ln(h) - \sum_{k=n+1}^{h}\frac{1}{k} \right)


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

Answer: 4e^3


Question: Evaluate the limit: \( \lim_{n \to \infty} n^2 \int_{1/(n+1)}^{1/n} \frac{\tan^{-1}(nx)}{\sin^{-1}(nx)} \, dx \). Choose the correct answer from the following options: (a) 1, (b) 0, (c) -1, (d) \frac{1}{2}.

Answer: D


Question: Does every solvable negative Pell equation, $a^2 - Db^2 = -1$, have infinitely many integer solutions $(a, b)$ where $a$ and $b$ are both positive integers?

Answer: Yes


Question:
Given the function \( f(x) \) defined for each real number \( x \), satisfying the equation:
\[ 6 + f(x) = 2f(-x) + 3x^2 \left( \int_{-1}^{1} f(t) \, dt \right) \]
Find the value of \( \int_{-1}^{1} f(x) \, dx \).

Answer: 4


Question: Evaluate the limit: \[ \lim_{x\to0^+} \frac{(1-x)^{x}-\cos x e^{-\frac{x^2}{2}}}{\arctan(x^2)\sin x +x^3 \ln x} \]

Answer: 0


Question: Suppose 28 objects are placed around a circle at equal distances. In how many ways can you choose 3 objects from among them so that no two of the 3 chosen are adjacent nor diametrically opposite?

Answer: 2268


Question: Are there uncountably many cube-free infinite binary words?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \left(\left(\frac{f(n)}{g(n)}\right)^q \left(\frac{1-f(n)}{1-g(n)}\right)^{(1-q)}\right)^n \] where \( n \in \mathbb{N} \), \( f(n),g(n) \in [0,1] \), \( q\in (0.5,1) \), \( f(n) \leq g(n) \), and both \( f(n) \) and \( g(n) \) converge to some real number in \([0,1]\).

Answer: 0


Question:
Evaluate the integral \( \int \frac{1}{\sqrt{(x^2-4x)^3}} \, dx \) and determine which of the following solutions is correct:

1. \( \frac{\sqrt{(x-2)^2-4}}{x-2} - \ln\left| \frac{x-2}{2}+\frac{\sqrt{(x-2)^2-4}}{2} \right| + C \)

2. \( \frac{2-x}{\sqrt{(x-2)^2-4}} + C \)

Answer: 2


Question: Consider the equation \(x^{2}+y^{2}-3z^{2}-3t^{2}=0\). Determine the total number of integral solutions for this equation where \(1 \leq x, y, z, t \leq 10000\).

Answer: 0


Question:
Calculate the norm \( ||T|| \) of the operator \( T \) defined on \( H = L^2(0,1) \) by:
\[ Tf(s) = \int_0^1 (5s^2t^2 + 2)f(t) \, dt \]
where \( f \in L^2(0,1) \).

Answer: \dfrac{8}{3}


Question: Find the orthogonal projection of the polynomial \( p(x) = x^3 \) onto the subspace \( P_2 \) of polynomials of degree at most 2, using the inner product \( \langle f, g \rangle = \int_{-1}^1 f(t)g(t) \, dt \).

Answer: \dfrac{3}{5}x


Question: Evaluate the integral \( \int_{0}^{2n} \max\{|x|, |x-1|, |x-2|, \ldots, |x-2n|\} \, dx \) where \( n \in \mathbb{N} \).

Answer: 3n^2


Question: Find the limit of the sequence defined by the recurrence relation \(x_{n+1} = x_n + 2\frac{3-e^{x_n}}{3+e^{x_n}}\).

Answer: \ln 3


Question: An ant is moving on the coordinate plane. Initially, it is at (6, 0). Each move consists of a counter-clockwise rotation of 60° about the origin followed by a translation of 7 units in the positive x-direction. After 2020 moves, the ant's position is \((p, q)\). Find the value of \(p^2 + q^2\).

Answer: 57


Question: Determine the number of ordered pairs (x, y) of whole numbers that satisfy the equation \(x \cdot y = 2^2 \cdot 3^4 \cdot 5^7 \cdot (x+y)\).

Answer: 676


Question: Consider the unit sphere $S$ in the Urysohn space $\mathbb{U}$. Determine whether any isometry $S \to S$ can be extended to an isometry $\mathbb{U} \to \mathbb{U}$. Provide a justification for your answer.

Answer: Yes


Question: Given a connected graph with up to 105 nodes, determine the minimum number of nodes that need to be removed to make the graph disconnected, such that there exists at least one node that cannot reach all other nodes. What is this minimum number?

Answer: 1


Question: Prove that if a subgroup $H$ of the symmetric group $S_n$ contains the transpositions $(1,2), (1,3), \ldots, (1,n)$, then $H$ is equal to $S_n$.

Answer: S_n


Question: Evaluate the integral \( \int_0^1 \int_0^1 e^{\max\{x^2,y^2\}} \,\mathrm{d}x\,\mathrm{d}y \).

Answer: e - 1


Question: Determine whether the function \( \sqrt{1-x^2} \) is of bounded variation on the interval \((-1, 1)\).

Answer: Yes


Question: Evaluate the integral \( \frac{1}{2\pi i}\int_\gamma \frac{dz}{z} \), where \( \gamma(t) = i + 2e^{it} \) and \( t \in [-\pi, 3\pi] \).

Answer: 2


Question: Let $R$ be a commutative unital ring with no non-trivial idempotents. Is it true that the additive group $(R,+)$ is not isomorphic to the group of units $(R^\times, \cdot)$?

Answer: Yes


Question:
Simplify the vector expression:
\[ (p^2 + q^2 |\mathbf{P}|^2 + (\mathbf{P} \bullet \mathbf{Q})^2)\mathbf{X} + (\mathbf{Q} \bullet \mathbf{X})(|\mathbf{P}|^2 \mathbf{Q} - (\mathbf{P} \bullet \mathbf{Q})\mathbf{P}) + (\mathbf{P} \bullet \mathbf{X})(|\mathbf{Q}|^2 \mathbf{P} - (\mathbf{P} \bullet \mathbf{Q})\mathbf{Q}) + (\mathbf{X} \bullet (\mathbf{P} \times \mathbf{Q}))(\mathbf{P} \times \mathbf{Q}), \]
where \(p\) and \(q\) are scalars and \(\mathbf{P} , \mathbf{Q} , \mathbf{X} \) are three-dimensional vectors equipped with the standard dot, \(\bullet\), and cross, \(\times\), products. Also, \(|\mathbf{P}|^2 \equiv (\mathbf{P} \bullet \mathbf{P}) \).
Furthermore, we have the condition that \(p^2 + |\mathbf{P}|^2 = q^2 + |\mathbf{Q}|^2 = 1 \).
Show that this expression simplifies to \(\mathbf{X}\).

Answer: \mathbf{X}


Question: The error for the production of a machine is uniformly distributed over the interval \([-0.7, 0.5]\) units. If there are 10 machines working simultaneously, approximate the probability that the total production error exceeds 1 unit in magnitude.

Answer: 0.534


Question: Find the function \( f : \Bbb N \to \Bbb N \) that satisfies the equation \( f\circ f(x) + f(x) = 2x + 15 \) and is injective. Verify that \( f(x) = x + 5 \) is a solution.

Answer: f(x) = x + 5


Question: Find the function \( f(x) \) that satisfies the functional equation \((x-2)f(2x)+xf(x-2)=0\) for all real numbers \( x \).

Answer: 0


Question: Find the degree of the field extension \( \mathbb{Q} (\sqrt{3}) \cap \mathbb{Q} (i) \) over \( \mathbb{C} \).

Answer: 1


Question: Evaluate the limit \( \lim_{t\to1} \frac{\phi(t,t^3-1)-t}{t^2-1} \), where \( \phi(y,z) \) is defined implicitly by \( x^2+y^2+z^3=2 \) and \( \phi_y=-1 \), \( \phi_z=0 \) in a neighborhood of \( (1,1,0) \).

Answer: -1


Question: Does a $*$-homomorphism from a $C^{*}$-algebra $A$ to $B(\mathcal{H})$, where $\mathcal{H}$ is a Hilbert space, always exist? Justify your answer.

Answer: Yes


Question: Find the volume of the solid generated by rotating the curve described by the parametric equations \(x(t) = a(t - \sin(t))\) and \(y(t) = a(1 - \cos(t))\) for \(t\) in the interval \([0, 2\pi]\) around the x-axis.

Answer: 5\pi^2 a^3


Question:
Consider the following diagram of regular local rings:
\[
\begin{matrix}
\hat{A} & \xrightarrow{\quad\hat\varphi\quad} & \hat{B} \\
\ \uparrow\scriptstyle\alpha & \circlearrowleft & \ \uparrow\scriptstyle\beta \\
A & \xrightarrow{\quad\varphi\quad} & B
\end{matrix}
\]
where \(\widehat{\,\dot\,}\) denotes the completion functor. Let \(m\subset A\) and \(n\subset B\) be the respective maximal ideals. Assume that \(\varphi\) is injective and makes \(B\) integral over \(A\), in particular all morphisms are inclusions. Given \(\hat y\in\hat n\setminus \hat n^2\) such that \(\hat y^k=x \in m\setminus m^2\), determine if there exists \(y\in B\) such that \(y^k = x\).

Answer: Yes


Question: Determine the range of values for the parameter \(a \in \mathbb{R}\) such that the inequality \(1 + \log_5(x^2 + 1) \ge \log_5(ax^2 + 4x + a)\) holds for all \(x \in \mathbb{R}\).

Answer: (2, 3]


Question: In a dice game, each player rolls \( k \) dice on their turn. The first player to accumulate a total of \( k \) sixes over all their turns wins the game. What is the expected number of turns needed for a player to win the game?

Answer: 6


Question: Determine the number of group homomorphisms from $\Bbb{Z}_3$ to $S_4$, where $\Bbb{Z}_3$ is the cyclic group of order 3 and $S_4$ is the symmetric group on 4 elements.

Answer: 9


Question: Verify the identity for Legendre polynomials: \( \int_{-1}^{1} (P_{n}^{\prime}(x))^2 \, dx = n(n+1) \) for \( n \geq 1 \).

Answer: n(n+1)


Question: Find the radius of convergence for the series \( \sum_{n=1}^{\infty}\frac{(-3)^{n}+2^{n}}{\sqrt[3]{n+1}}(x-2)^n \).

Answer: \dfrac{1}{3}


Question: Is there a monoidal category \( \mathcal{C} \) with unit object \( I \) such that \( I \otimes A \cong A \cong A \otimes I \) for all objects \( A \in \text{Ob}_{\mathcal{C}} \), and an object \( -1 \) where \( (-1) \otimes (-1) \cong I \) and \( -1 \neq I \)?

Answer: Yes


Question: Given the functions \( f(x) = x^3 + x \) and \( g(x) = x^3 - x \), where \( f^{-1} \) denotes the inverse function of \( f \), find the derivative of the composite function \( g \circ f^{-1} \) at the point \( x = 2 \).

Answer: \dfrac{1}{2}


Question: In a right triangle $\triangle ABC$ with $\angle B = 90^\circ$, an interior bisector $BD$ is drawn, where $I$ is the incenter and $3BI = 4ID$. Find the relationship between the circumradius $R$ and the inradius $r$ of $\triangle ABC$.

Answer: 3


Question: Let $f\colon\mathbb{R}\to\mathbb{R}$ be a continuous function with a weak derivative in $C^1(\mathbb{R})$. Does this imply that $f$ is two times continuously differentiable, i.e., $f \in C^2(\mathbb{R})$? Answer yes or no.

Answer: Yes


Question: A Poisson process has a parameter \( \lambda \). Each time an event occurs, you receive \( K \) dollars. What is the expected total amount you will receive?

Answer: K \lambda


Question:
Given the matrix \( A = \begin{bmatrix}\lambda & 1 & 0 & \ldots & 0 \\
0 & \lambda & 1 & \ldots & 0 \\
\vdots & \vdots & \lambda & \ddots & 1 \\
0 & \ldots & \ldots & \ldots & \lambda \end{bmatrix} \), find the minimal polynomial of \( A \). The characteristic polynomial is given by \((\lambda-x)^n=0\).

Answer: (x - \lambda)^n


Question: Let \( F(x) = \int_{0}^{x} f(t) \, dt \) where \( f \in L^{p}([0,1]) \) for \( 1 < p < \infty \). Prove that \( \lim_{h \to 0} \frac{F(x+h)-F(x)}{h^{1-1/p}} = 0. \)

Answer: 0


Question: For any complex numbers $z$ and $w$, determine the supremum of $\left|z+e^{i\theta}w\right|$ as $\theta$ varies over the interval $[0,2\pi]$. Find $\sup\limits_{\theta\in[0,2\pi]}\left|z+e^{i\theta}w\right|$. Provide your answer in terms of $z$ and $w$.

Answer: |z| + |w|


Question: Let \( f(x):\mathbb{R}\rightarrow \mathbb{R} \) be a continuous function satisfying \( f(x+y)=f(x)+f(y) \) for all \( x,y\in \mathbb{R} \). Prove that \( f \) is a linear transformation.

Answer: f(x) = kx


Question: Evaluate the integral \( \int |z| \, dz \) along the path of the semicircle \( |z| = 1 \) from \( z = -i \) to \( z = i \), where \( -\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{2} \).

Answer: 2i


Question: Determine whether there exists a metric $d$ on $\mathbb{R}$ such that the function $f: (\mathbb{R}, d) \to (\mathbb{R}, d)$ defined by $f(x) = -x$ is not continuous.

Answer: Yes


Question: Find the quadratic variation of the process $X(s) = W_{s+\epsilon} - W_s$, where $W_s$ is a standard Wiener process.

Answer: 2t


Question: Calculate the limit: \[ \lim _{x \to 0}\frac{\int_{0}^{\sin(x)}\,\sqrt{\,\tan(t)\,}\,\mathrm{d}t}{\int_{0}^{\tan(x)}\,\sqrt{\,\sin(t)\,}\, \mathrm{d}t} \]

Answer: 1


Question: Find the greatest integer \(a\) such that the inequality \(x^2 - ax - \ln x + e^{\sin x} - 1 > 0\) holds for every \(x > 0\). Use the approximations \(\sin 1 \approx 0.84\) and \(\ln 2 \approx 0.693\).

Answer: 2


Question: Let $V$ be the vector space $M_{n \times n}(\mathbb{R})$, and let $B : V \times V \to \mathbb{R}$ be the bilinear form defined by $B(X,Y) = \text{tr}(XY^T)$. Calculate the signature of $B$. Here, $\text{tr}$ denotes the trace of a matrix.

Answer: (n^2, 0)


Question: Given that the equation \(x^n + a_{1}x^{n-1} + a_{2}x^{n-2} + \cdots + a_{n} = 5\) with integer coefficients has 4 distinct integer roots, determine the total number of integer roots of the equation \(x^n + a_{1}x^{n-1} + a_{2}x^{n-2} + \cdots + a_{n} = 7\).

Answer: 0


Question: In a non-normal extension of number fields $L/K$, must there exist at least one prime $P$ of $K$ and primes $Q_1, Q_2$ of $L$ lying over $P$ such that $e(Q_1|P) \neq e(Q_2|P)$ or $f(Q_1|P) \neq f(Q_2|P)$?

Answer: Yes


Question: Consider a simply connected closed Riemannian manifold $M$ with no boundary and nonnegative sectional curvature. Suppose the cyclic group ${\bf Z}_n = (g)$, with $n \geq 3$, acts on $M$ isometrically. If $g$ fixes a point $x$, then $g$ acts on the cut locus of $x$. Can $g \cdot x$ belong to the cut locus of $x$ for some $x$?

Answer: No


Question: Given a point P = (0,0,0) and a line L in 3D space defined by L = {P + k*w} where w = (-1, 0, 0), find the scalar q such that the point P' = P + q*w satisfies the condition that the dot product of the unit vector from P' to Q = (0, 1, 0) with the vector v = (0, 1, 0) is as close as possible to the target value T = 1/2. Avoid using expensive calculations such as sin, cosine, or sqrt in your solution.

Answer: \pm \sqrt{3}


Question: Consider a modified random walk \( \{S_n\} \) on the positive integers, where the probability of moving from \( k \) to \( k+1 \) is given by \( P(S_n=k+1|F_n) = \frac{1}{2} - \frac{1}{4k} \). Using the harmonic function \( f(n) = n^2 \), compute the probability that the walk, starting from 50 and absorbed when it hits 0, will ever reach 100.

Answer: \dfrac{1}{4}


Question: Consider a function $u: U \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^n$ and $u \in C^{\infty}(\overline{U})$. Suppose the Lebesgue measure of the boundary of $U$, $m(\partial U)$, is zero, and $U$ has a $C^1$ boundary. Is it true that the Lebesgue measure of the boundary of the level set $\{z: u(z) = 0 \}$ is also zero?

Answer: Yes


Question: Determine whether the function \(-f(z) = -\frac{1}{z}\) is a primitive of the function \(g(z) = \frac{1}{z^2}\) on the domain \(\mathbb{C} \setminus \{0\}\).

Answer: Yes


Question:
Consider the process $X_{t} = \exp\left\{\left(\mu-r-\frac{\sigma^{2}}{2}\right)t+\sigma W_{t}\right\}$, which is the solution to the stochastic differential equation (SDE):
\[dX_{t}=(\mu -r)X_t dt+\sigma X_t d W_{t}, \ X_{0}=1\]
where $W$ is a standard 1-dimensional Brownian motion, $\mu < r$, and $\sigma > 0$. Let $0 < b < 1$ and define $\tau_{b}$ as the hitting time of the level $b$:
\[\tau_{b}:=\inf\{t \geq 0, \ X_{t}=b\}.\]
Determine whether $\mathbb{E}\left(\int_{0}^{\tau_{b}}X_{s}dW_{s}\right)=0$. Provide a justification for your answer.

Answer: 0


Question: Given a $15 \times 15$ matrix $A$ with characteristic polynomial $(x+1)^5(x-1)^3x^7$ and minimal polynomial $(x+1)^3(x-1)^2x^3$, determine the rank of $A$. \( \boxed{12} \)

Answer: 12


Question: Determine whether the sequence \( \{f^{(n)}(0)\} \) is bounded for all positive even integers \( n \), where \( f(x) = \arctan(x) \).

Answer: Yes


Question: Find the coefficient of the term $t^{q^2+q+1}$ in the polynomial \( \left(\frac{1-t^{q+1}}{1-t}\right)^{2q-1} \), where $q$ is a positive integer. Assume $q$ is a prime power and work in the field $\mathbb{F}_q[t]$ if necessary.

Answer: 0


Question: Evaluate the integral \( \int_{0}^{a} \frac{dx}{x + \sqrt{a^{2} - x^{2}}} \) where \( a \) is a positive constant.

Answer: \dfrac{\pi}{4}


Question:
Determine the maximum amount of shuffling for a deck of 52 cards, where shuffling is defined as the average distance each card moves from its original position. Specifically, calculate:
\[ \text{maximum shuffling} = \max \Bigg\{ \sum_{n=1}^{52} \frac{|f(n, 52) - n|}{52} \Bigg\}, \]
where \( f(n, 52) \) is a bijective function mapping the original positions to new positions.

Answer: 26


Question:
In a right triangle \(\triangle ABC\), with \(\angle ABC = 90^\circ\), the altitude \(BH\) is drawn to the hypotenuse \(AC\). In triangle \(BHC\), a cevian \(HM\) is drawn such that \(MC = AB\). Given that \(HC = BH + 2AH\), find the measure of the angle \(\angle MHC\). Choose from the following options:
\[\begin{array}{ll}
1.&\frac{53^{\circ}}{2}\\
2.&\frac{37^{\circ}}{2}\\
3.&53^{\circ}\\
4.&37^{\circ}\\
5.&30^{\circ}\\
\end{array}\]

Answer: 1


Question: Determine whether $T^p - T$ is the greatest common divisor of the set \( \{(T+u)^n - (T+u) : u \in \mathbb{F}_p\} \) in the polynomial ring \( \mathbb{F}_p[T] \), given that \( n > 1 \), \( p \) is an odd prime, \( p-1 \mid n-1 \), and \( p^k - 1 \nmid n-1 \) for any \( k > 1 \).

Answer: T^p - T


Question: Determine the range of values for \( p \in \mathbb{R} \) such that the family of functions \( f_n(x) = \frac{nx^p}{1+n^2x^2} \) for \( n \geq 1 \) is integrable on \((0, \infty)\) with respect to the Lebesgue measure.

Answer: (-1, 1)


Question: Determine the group to which \((\mathbb{Z}\times\mathbb{Z})/\langle (1,2) \rangle\) is isomorphic.

Answer: \mathbb{Z}


Question: Determine the number of nonzero terms with odd exponents in the Taylor series expansion of \( \operatorname{Log}(1+e^z) \) about \( z = 0 \).

Answer: 1


Question: Find the 24th prime number given that \( \pi(x) = 25 \).

Answer: 89


Question: Let \( P(x) \) be a polynomial such that \( P(P(x)) = 16x - 5 \). Find the remainder when \( P(x-3) \) is divided by \( x-5 \).

Answer: 7


Question: Find the limit of the sequence \( Z_n = \left(1+\frac{a+bi}{n}\right)^n \) as \( n \to \infty \).

Answer: e^{a + bi}


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

Answer: k > 1


Question: Determine whether \( \sqrt{a}^{\sqrt{b}} \) can be a rational number if both \( \sqrt{a} \) and \( \sqrt{b} \) are irrational, where \( a \) and \( b \) are rational numbers.

Answer: No


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty}n\int_0^1f(x)x^{2n}dx \] where $f:[0,1]\rightarrow\mathbb R$ is a continuous function such that $f(0)=0$ and $f(1)=1$.

Answer: \dfrac{1}{2}


Question: Find the volume of the solid bounded above by the plane \( z = y \) and below by the paraboloid \( z = x^2 + y^2 \) using cylindrical coordinates.

Answer: \dfrac{\pi}{32}


Question: Evaluate the limit of the infinite product: \[ \lim_{n \to \infty} \prod_{k=1}^{n}\frac{1}{(1 - \frac{1}{2k+1})} \]

Answer: \infty


Question: In a Hausdorff topological group $G$, if $b \in G$, does there exist an open neighborhood $U$ of $b$ such that $U \cap \langle b \rangle$ is finite? Answer with "Yes" or "No".

Answer: No


Question: Evaluate the iterated integral: $$\int_0^4\int_x^4\int_0^y\frac{6}{1 + 48z - z^3}\, dz\, dy\, dx$$.

Answer: \ln 129


Question:
Solve the recurrence relation for the nth term given by:
\[ T(n) = 1 + 2\sum_{i=1}^{n-1} T(i), \quad n > 1 \]
with the initial condition \( T(1) = 1 \).

Answer: 3^{n-1}


Question: Find the limit: \[ \lim_{x \to 0^+} (2\sqrt{x} + x)^{\frac{1}{\ln x}} \]

Answer: \sqrt{e}


Question: Simplify the expression: $$\sqrt[3]{a + \frac{a + 8}{3}\sqrt{\frac{a - 1}{3}}} + \sqrt[3]{a - \frac{a + 8}{3}\sqrt{\frac{a - 1}{3}}}$$

Answer: 2


Question: Calculate the winding number \( n\left ( \gamma ,\frac{1}{3} \right ) \) of the curve \( \gamma :\left [ 0,2\pi  \right ]\rightarrow \mathbb{C}, t \mapsto \sin(2t)+i\sin(3t) \).

Answer: 0


Question: Given two positive real sequences $a_n$ and $b_n$ that both diverge to infinity, determine if it is possible to choose two subsequences $a_{s_n}$ and $b_{t_n}$ such that $\lim_{n \to \infty} \frac{a_{s_n}}{b_{t_n}} = 1$.

Answer: Yes


Question: Given the equation \(|mx+\alpha|+|mx+\beta|=c\), where \(x=2\) and \(x=4\) are solutions, and \(m>0\) with \(\alpha, \beta, c\) as non-zero constants, find the value of \(\left|\frac{\alpha+\beta}{m}\right|\).

Answer: 6


Question: Given that \(x, y, z > 0\) and \(x^2 + y^2 + z^2 = 3\), find the minimal value of \((2-x)(2-y)(2-z)\).

Answer: \dfrac{25}{27}


Question: If $f(x)$ is a polynomial satisfying \( f(x)f(y) = f(x) + f(y) + f(xy) - 2 \) for all real $x$ and $y$, and $f(3) = 10$, find $f(4)$. \( \boxed{} \)

Answer: 17


Question: In a one-dimensional random walk, a person starts at the origin and can move to \((x+1,0)\) or \((x-1,0)\) with equal probability of 0.5. What is the expected number of steps for the person to return to the origin?

Answer: \infty


Question: Evaluate the integral \( \int_{C} \frac{z^{3}}{z^{4} - 3} \, dz \), where \( C \) is the positively oriented circle \( |z| = 3 \).

Answer: 2\pi i


Question: Find the number of solutions in positive integers to the equation \(x_1 + x_2 + x_3 + x_4 = 19\) subject to the conditions \(x_2 \neq 2x_3\) and \(x_1 \neq x_2\).

Answer: 702


Question: Let $A$ be a unique factorization domain (UFD) that is not a principal ideal domain (PID). Does there exist a pair of distinct prime elements $a$ and $b$ in $A$ such that the ideal $(a, b)$ is a proper ideal of $A$?

Answer: Yes


Question: Determine whether the function \( f(x,y) = \frac{x |y|^\alpha}{x^2 + y^2} \) is continuous at the origin, where \( \alpha > 0 \).

Answer: \alpha > 1


Question: Find the last two digits of the number \( n = 2017^{(2018)^{(2019)^{(\cdots)}}} \).

Answer: 41


Question: Five people own a safe with several locks arranged in parallel, meaning all locks must be opened to access the safe. Each lock can have any number of keys. Determine the minimum number of locks required so that any combination of at least 3 people can open the safe.

Answer: 10


Question: Determine whether any finitely generated \( \mathbb{Z}/l^n\mathbb{Z} \)-module has a finite (left) resolution by free finitely generated \( \mathbb{Z}/l^n\mathbb{Z} \)-modules, where \( l \) is a prime number and \( n \in \mathbb{Z} \).

Answer: No


Question: Determine the value of \(a\) such that the function \(f(x) = ax^2 + 1 - \ln(1+x)\), defined for \(x \in (-1, \infty)\), has exactly two extreme points.

Answer: (-\infty, -2)


Question: In a hypothesis test where the null hypothesis is \( H_0: \theta > 0.0625 \) and the alternative hypothesis is \( H_A: \theta \leq 0.0625 \), what value of \( \theta \) should be used to calculate the distribution of the test statistic under the null hypothesis?

Answer: 0.0625


Question: Determine whether the equation \( e^{2x\pi} = \frac{1+x}{1-x} \) has any solution for \( x \in (0,1) \).

Answer: Yes


Question: Evaluate the integral \( \int_a^b (f(x) + f^{-1}(x)) \, dx \) given that \( a \) and \( b \) are fixed points of the function \( f \). Show that the result is \( b^2 - a^2 \).

Answer: b^2 - a^2


Question: Suppose $I$ and $J$ are intervals in $\mathbb{R}$ and $f: I \rightarrow J$ is a continuous bijection. Is $f$ a homeomorphism?

Answer: Yes


Question: Solve the exponential equation: \((\sqrt{2+\sqrt{3}})^x + (\sqrt{2-\sqrt{3}})^x = 2^x\).

Answer: 2


Question: Given a finite measure space \((E, \mathcal{E}, \mu)\) and \(p \ge 1\), let \(\varphi \in L^p(\mu)'\). Suppose \(\varphi(1_B) = \int_B f \, \mathrm{d}\mu\) for all \(B \in \mathcal{E}\). Can we deduce that \(f \in L^p(\mu)\)?

Answer: No


Question: Compute the limit: \[ \lim \limits_{n \to \infty} \frac{\sqrt[4]{n^4+4n}\,-\,\sqrt[3]{n^3+3n}}{\sqrt[5]{n^5+1}\,-\, \sqrt[5]{n^5+n}}\cdot \frac{1}{n^2} \]

Answer: 5


Question: Suppose that $u_1$ and $u_2$ are vectors such that $|u_1| = 2$, $|u_2| = 3$, and $u_1 \cdot u_2 = 5$. Define the vectors $u_3, u_4, u_5, u_6, \ldots$ by $u_3 = \text{proj}_{u_1} u_2$, $u_4 = \text{proj}_{u_2} u_3$, $u_5 = \text{proj}_{u_3} u_4$, and so on. Find the value of the infinite sum $$\sum_{n=1}^\infty |u_n|.$$

Answer: 20


Question: Calculate the limit: \[ \lim_{n\rightarrow\infty}\frac{n^2}{(4+5i)n^2+(3+i)^n} \]

Answer: 0


Question: Let \( X_1, X_2, \ldots, X_n \) be a random sample from a distribution with the following probability mass function (pmf): \( P(X = k_1) = \frac{1 - \theta}{2} \), \( P(X = k_2) = \frac{1}{2} \), and \( P(X = k_3) = \frac{\theta}{2} \), where \( 0 < \theta < 1 \). Find a non-trivial sufficient statistic for \( \theta \).

Answer: (N_1, N_3)


Question: Evaluate the contour integral: \[ \oint_{|z|=2}\frac{dz}{\sqrt{1+z^{10}}}. \]

Answer: 0