Question: Evaluate the Lebesgue integral: $$\int_{\mathbb{R}} 1 \, d\lambda(t).$$

Answer: \infty


Question: Let $(a_{(m,n)})_{m,n \in \mathbb{N}}$ be a double sequence of positive numbers with the property that $\lim_{m \to \infty}\lim_{n \to \infty}a_{(m,n)}=L$. Determine whether there always exists an increasing function $m: \mathbb{N} \to \mathbb{N}$ such that $\lim_{n \to \infty}a_{(m(n),n)}=L$. Provide a proof or counterexample.

Answer: Yes


Question: Evaluate the summation \( \sum_{i=0}^{h-1} 2^i(h-i) \).

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


Question: In a cafeteria, customers enter at a rate of 0.3 clients per minute, and the cafeteria can serve 0.5 clients per minute. Calculate the average number of clients that will have to wait to be served.

Answer: 0.9


Question: Determine the asymptotic complexity of the recurrence relation \(B(n) = 5\cdot B(n/3) + cn^2\) with the initial condition \(B(1) = c\). Express your answer in the form \(O(n^d)\) for some value of \(d\).

Answer: O(n^2)


Question: Solve the equation: \(2^{x/2} + 3^{x/2} = 13^{x/4}\) for \(x\).

Answer: 4


Question: Let $p(z)$ and $q(z)$ be polynomials such that for all complex numbers $z$, the equation \( p(z) \sin^2 z + q(z) \cos^2 z = 2 \) holds. Determine the value of $p(1) + q(1)$. \( \boxed{} \)

Answer: 4


Question:
Find the density function of $X$ given by:
\[ f_X(k) = \int_0^\infty \binom{2}{k}(e^{-y})^{k+1}(1-e^{-y})^{2-k} \, dy \]
where $k = 0, 1, 2$. Evaluate the integral to determine $f_X(k)$. Assume $f_Y(y) = e^{-y}$ and $\mathbf{P}(X=k \mid Y=y) = \binom{2}{k}(e^{-y})^{k}(1-e^{-y})^{2-k}$. Provide the final expression for $f_X(k)$.

Answer: \dfrac{1}{3}


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

Answer: 0


Question: Find the remainder when \(314^{420}\) is divided by 1001.

Answer: 1


Question: Let $f:[0,1]\rightarrow [0,1]$ be a continuous function such that $f(0)=f(1)$. Define the set \( A = \{ (t,s) \in [0,1] \times [0,1] \mid t \neq s \text{ and } f(t)=f(s) \} \). Determine the number of elements in the set $A$. \( \text{(Express your answer as a number or state if it is infinite.)} \)

Answer: \infty


Question: Does every affine $k$-variety have the form $\text{Spec} \ A$, where $A$ is a finitely generated $k$-algebra?

Answer: Yes


Question: A company has $n$ workers. The company is creating subdivisions and distributes the workers among these subdivisions. Each subdivision must have an odd number of workers, and the intersection of workers from any two subdivisions must be even, including the possibility of being zero. What is the maximal number of subdivisions possible?

Answer: n


Question: Let \(a_n\) and \(b_n\) be two sequences such that \(a_n \xrightarrow{n\to\infty} a > 1\) and \(b_n \xrightarrow{n\to\infty} b > 1\). Find \(\lim_{n\to\infty}\sqrt[n]{a_n^n+b_n^n+2018}\).

Answer: \max\{a, b\}


Question: Find all palindrome-prime numbers $p$ such that $6p+1$ is also a palindrome-prime. Verify if $p = 131$ is the only example.

Answer: 131


Question: Determine the order of the pole at $z = 0$ for the function \( \frac{\cos(\pi z)-1}{2z^4 - z^3} \).

Answer: 1


Question: Given the polynomial function \( f(x) = 17x^{12} - 124x^9 + 16x^3 - 129x^2 + x - 1 \), determine \( \frac{d}{dx}(f^{-1})(-1) \) if it exists.

Answer: 1


Question: Calculate the area enclosed by the curve defined by the equation \( \sin^4(x) + (\cos(y) - 3)^2 - 16 = 0 \).

Answer: 0


Question: What is the completion of the metric space \((\mathbb{Q}, |\ \ |)\)?

Answer: \mathbb{R}


Question: Calculate the limit using Lebesgue's dominated convergence theorem: $$ \lim_{k \to \infty} \sum_{n=1}^{\infty} \frac{1}{n^2 + ik} $$

Answer: 0


Question: Find all positive integer solutions \((x, y, z)\) to the equation \(2^x + 2^y + 1 = 3^z\) where \(x > y\).

Answer: (6, 4, 4)


Question: Evaluate the integral \( \int_0^\infty f(x)\delta(x)dx \) using the properties of the Dirac Delta function.

Answer: f(0)


Question: Let $N \triangleleft G$ and $K \triangleleft G$. If $N \cap K = \langle e \rangle$ and $N \vee K = G$, prove that $G/N \cong K$. Use the second theorem of isomorphisms to justify your answer.

Answer: G/N \cong K


Question: Find the integer \(b\) such that \(3^b \equiv 22 \mod 31\).

Answer: 17


Question:
Determine the domain of convergence for the series:
\[ \sum_{n=1}^{\infty} \frac{n}{2^n}(x+2)^n + \sum_{n=1}^{\infty} \frac{n^3}{\sqrt{n!}} \]

Answer: (-4, 0)


Question: Determine the maximum number of arcs in a digraph with $n$ vertices that is unilaterally connected but not strongly connected.

Answer: (n-1)^2


Question: Evaluate the sum of the integrals: \[ \int_{0}^{2} \sqrt{x^3+1} \, dx + \int_{1}^{3} \sqrt[3]{x^2-1} \, dx. \] Consider any potential relationships between the functions that might simplify the evaluation.

Answer: 6


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

Answer: 0


Question: Evaluate the limit: $$\lim\limits_{x \to +\infty} \frac{(x+1)^{10} + (x+2)^{10} + (x+3)^{10} + \dots + (x+100)^{10}}{x^{10}+10^{10}}$$

Answer: 100


Question: A bug is walking on a number line where all numbers are nonnegative. The bug starts at position 1. When it lands on 0, it immediately returns to position 1. From any other position, the bug moves left or right with equal probability. What is the expected number of times the bug will land on 0 before it lands on 4 for the first time?

Answer: 3


Question: Let $C$ be a reduced affine Noetherian scheme of pure dimension 1, and let $p \in C$ be a regular closed point. Suppose there is a morphism $C \backslash p \to \mathbb{P}^n$ that lies in a closed subscheme $Y \subseteq \mathbb{P}^n$. If this morphism extends to $C \to \mathbb{P}^n$, does this extension necessarily factor through $Y$? Justify your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 1^{+}}(\sqrt{x}-1)^{x^2+2x-3} \]

Answer: 1


Question: Find the number of positive integral solutions to the equation \(a_1 + a_2 + a_3 + \ldots + a_6 = 12\) where each variable \(a_i\) can only take the values 1, 2, or 3.

Answer: 141


Question: Evaluate the limit: $$ \lim_{n \rightarrow \infty} \sum_{k=1}^n \left( \sqrt{n^4+k}\ \sin\frac{2k\pi}{n} \right). $$

Answer: -\dfrac{1}{4\pi}


Question: Can the action of $Spin(9)$ on the space of octonionic Hermitian $2 \times 2$ matrices diagonalize any such matrix?

Answer: Yes


Question: Determine the number of inflection points of the function \((x-2)^6(x-3)^9\).

Answer: 3


Question:
Given the complex numbers \(a\) and \(b\), solve the following system of equations to find \(|a|\):

1. \(|2a - b| = 25\)
2. \(|a + 2b| = 5\)
3. \(|a + b| = 2\)

Find the magnitude \(|a|\).

Answer: 9


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

Answer: 1


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that for every \( x, y \in \mathbb{R} \), the following equation holds: \[ f\left(\frac{x+y}{2}\right) = \frac{1}{2}f(x) + \frac{1}{2}f(y). \]

Answer: f(x) = ax + b


Question: Determine the sixth digit after the decimal point in the number \((\sqrt{1978} + \lfloor\sqrt{1978}\rfloor)^{20}\).

Answer: 9


Question: Find all pairs of coprime positive integers \((a, b)\) such that \(b < a\) and \(a^2 + a = b^3 + b\).

Answer: (5, 3)


Question: Determine whether a cubic tree, where every non-leaf vertex has degree 3, can have an even number of edges. Provide a justification for your answer.

Answer: No


Question: Given three admissible heuristics $h_1, h_2, h_3$ for an optimization problem solved using A* search, determine if the heuristic defined by \[ h(n) = \frac{\sqrt{h_1(n)h_2(n)} + 2h_3(n)}{3} \] is admissible for any node $n$ of the search graph.

Answer: Yes


Question: Find the value of \( \dfrac{d z^*}{dz} \), where \( z^* \) denotes the complex conjugate of \( z \) and \( z \in \mathbb{C} \).

Answer: 0


Question: Prove that for a real sequence \( \{a_n\} \), if \( \lim_{n \to \infty} (a_n - a_{n-2}) = 0 \), then \( \lim_{n \to \infty} \frac{a_n - a_{n-1}}{n} = 0 \).

Answer: 0


Question: Suppose $(T_n)$ is a sequence in the space of compactly supported distributions $\mathcal{E}'(\mathbb{R})$ that converges to $T$ in the weak topology $\sigma(\mathcal{E}',\mathcal{E})$. Does this imply that $(T_n)$ also converges to $T$ in the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$?

Answer: Yes


Question: Let \( \{s_n\} \) be a sequence in \([0,1]\) such that \( 2s_n \leq s_{n-1} + s_{n+1} \) for \( n \geq 1 \). Determine the supremum of the set \( \{s_{n+1} - s_n \mid n \geq 1\} \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{i=1}^n \frac{n}{n^2+i^2} \]

Answer: \dfrac{\pi}{4}


Question: Suppose $f \in H^2(\mathbb{R})$ with $\lVert f\rVert_{H^1}<\varepsilon$. Does this imply that $f \in C^1(\mathbb{R})$? Provide a justification for your answer.

Answer: Yes


Question: Evaluate the surface integral \( \iint_S \text{curl} \mathbf{F} \cdot \mathbf{n} \, dS \), where \( \mathbf{F} = \langle xy, yz, zx \rangle \) and \( S \) is the upper half of the ellipsoid \( \frac{x^2}{4} + \frac{y^2}{9} + z^2 = 1 \).

Answer: 0


Question: Does every abelian group have a degree one faithful representation over a field? Provide a justification for your answer.

Answer: No


Question: Find all integer pairs \((m, n)\) such that \(n\cos\frac{\pi}{m} = \sqrt{8+\sqrt{32+\sqrt{768}}}\).

Answer: (24, 4)


Question: Find the limit \( \lim_{n \to \infty} a_n \) given that \( \lim_{n \to \infty} \int_0^{a_n} x^n \, dx = 2 \), where \( \{a_n\} \) is a positive sequence.

Answer: 1


Question: Let $V$ be a finite-dimensional vector space over a field $K$ with dimension $n$, and let $(\phi_1, \ldots, \phi_r)$ be linearly independent elements of the dual space $V^*$. Consider the map $f = (\phi_1, \ldots, \phi_r): V \to K^r$. Is the map $f$ surjective?

Answer: Yes


Question: Let $X_1, X_2, \ldots, X_n$ be iid continuous random variables. What is the probability that $X_n$ is greater than the maximum of $X_1, X_2, \ldots, X_{n-1}$?

Answer: \dfrac{1}{n}


Question: Is the action of $SO(3, \mathbb{R})$ on the sphere $S^2 = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 = 1\}$, defined by $(A, x) \to Ax$, transitive? In other words, for every $x, y \in S^2$, does there exist an $A \in SO(3, \mathbb{R})$ such that $Ax = y$?

Answer: Yes


Question: Determine the values of $a$ for which the integral \( \int_0^{1/e}\frac{x^a}{\ln{x}} \,dx \) converges.

Answer: a > -1


Question: Express $i^{2i}$ in the form $x + iy$, where $x$ and $y$ are real numbers.

Answer: e^{-\pi}


Question: Evaluate the limit: \[ \lim _{n\to \infty }\left( e^{\sqrt{n+1} -\sqrt{n}} -1\right) \times \sqrt{n} \]

Answer: \dfrac{1}{2}


Question: Find all positive integer solutions to the equation: \[ 3x^2 - 7y^2 + 1 = 0 \]

Answer: (3, 2)


Question: Evaluate the limit: \[ \lim _{x\to \infty }\left(1+\sqrt{x}\sin \frac{1}{x}\right)^{\sqrt{x}} \]

Answer: e


Question: Calculate the residue of the function \( \frac{10z^4-10\sin(z)}{z^3} \) at \( z = 0 \).

Answer: 0


Question: Find the limit as \( n \to \infty \) of the following expression: \[ \sum_{r =1}^{n}\frac{6n}{9n^2-r^2} \]

Answer: \ln 2


Question: Evaluate the limit: \[ \lim_{x\rightarrow 1}\frac{(1-x)(1-x^2)(1-x^3)\cdots(1-x^{2n})}{\{(1-x)(1-x^2)(1-x^3)\cdots(1-x^n)\}^2} \]

Answer: \dbinom{2n}{n}


Question: Let $X$ and $Y$ be infinite dimensional Banach spaces, and let $T: X \to Y$ be a compact linear operator. Is it true that the image of $T$ cannot contain any closed subspace of infinite dimension?

Answer: Yes


Question: The heights of certain plants are normally distributed with a mean of 110 cm and a variance of 1050 cm². Two plants are chosen at random. What is the probability that the height of one plant is at least 1.5 times the height of the other plant?

Answer: 0.347


Question:
Given that \( \cot(A-B) = 3 \) and \( \cot(A+B) = 2 \), evaluate the infinite series:
\[ \tan A + \frac{1}{\cot^2 A} + \tan^3 A + \frac{1}{\cot^4 A} + \cdots \]

Answer: \dfrac{\sqrt{2}}{2}


Question: Consider a 2D Gaussian distribution with a mean of zero and a diagonal covariance matrix, having standard deviations \(\sigma_x\) and \(\sigma_y\). An ellipse is drawn to pass through the points \((\pm\sigma_x, 0)\) and \((0, \pm\sigma_y)\). Does this ellipse contain 68% of the volume under the 2D Gaussian curve?

Answer: No


Question: Calculate the conditional probability \( Pr(Y \geq \frac{1}{2} \mid Y \geq 1-2X) \) where \( X \) and \( Y \) are two independent and identically distributed random variables with a uniform distribution on \([0,1]\).

Answer: \dfrac{7}{12}


Question: Construct an explicit isomorphism between the algebraic group $\mathbb{G}_m/\{\pm 1\}$ and $\mathbb{G}_m$.

Answer: x \mapsto x^2


Question: Given two independent random variables $X$ and $Y$ with the following distributions: $F_X(x) = \frac{1}{2}x$ for $0 < x < 2$ and $f_Y(y) = \frac{2}{9}y$ for $0 < y < 3$, find the probability that the product $XY \ge 1$. Provide your answer as a probability value.

Answer: \dfrac{25}{36}


Question: Evaluate the contour integral \( \int_C \frac{dw}{e^w-1} \) over a loop \( C \) contained in the annulus \( 0<|z|<2\pi \).

Answer: 2\pi i


Question: In an algebraically closed field $F$, does every nonzero element have exactly $n$ distinct $n$-th roots for each positive integer $n$? Provide a justification for your answer.

Answer: No


Question: Given the differential equation \( \frac{dy}{dx} = \frac{4+y^2}{1+x^2} \) with the initial condition \( y(1) = 2 \), find the value of \( y(2) \).

Answer: 14


Question: Given the exponential decay model \( N = N_0 \exp(-kt) \), determine the average life expectancy of one person. Express your answer in terms of \( k \).

Answer: \dfrac{1}{k}


Question: Determine the values of \(\alpha\) for which the series \(\sum_{n=1}^{\infty} \frac{n}{\sqrt{4+n^\alpha}}\) converges.

Answer: \alpha > 4


Question: Find the value of \(i^i\).

Answer: e^{-\pi/2}


Question: Calculate \(2^{2^{32}} \mod 11\).

Answer: 9


Question: Given that students are assigned numbers from 1 to \( n \) and three students numbered 1, 3, and 7 are chosen randomly, find the maximum likelihood estimate (MLE) of \( n \) under the condition that \( n \leq 30 \). Assume the students are drawn from a discrete uniform distribution.

Answer: 7


Question: Find the value of $\sin(x)$ that maximizes the expression $(5+3\sin(x))^2 (7-3\sin(x))^3$. What is this value of $\sin(x)$?

Answer: -\dfrac{1}{15}


Question:
Compute $Ext_{\mathbb{Z}}^{1}(S^1, \mathbb{Z})$, where $S^1 = \{e^{2\pi i\theta} \mid 0 \leq \theta < 1\}$ is the circle group under complex multiplication. Consider the short exact sequence of $\mathbb{Z}$-modules:
\[ 0 \to \mathbb{Z} \to \mathbb{R} \xrightarrow{\phi} S^1 \to 0, \]
where $\phi(x) = e^{2\pi i x}$. Determine the structure of $Ext_{\mathbb{Z}}^{1}(S^1, \mathbb{Z})$.

Answer: \mathbb{Z}


Question: Determine whether there exists a non-smooth distribution that satisfies the partial differential equation \( (D_{t}^{2} - D_{x}^{2})u(x,t) = 0 \).

Answer: Yes


Question: Let $V$ be the vector space of all $2 \times 2$ matrices. Given a matrix $B = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$, define the linear transformation $P(A) = AB - BA$ for any $2 \times 2$ matrix $A$. Determine the nullity of $P(A)$. \(\text{Provide your answer as a single integer.}\)

Answer: 2


Question:
Given a finite state automaton $M=(\{0,1,2,3\},\{0,1\},d,0,\{1\})$ with the transition function defined as follows:

- $d(0,0)=2$, $d(1,0)=3$, $d(2,0)=0$, $d(3,0)=1$
- $d(0,1)=1$, $d(1,1)=0$, $d(2,1)=3$, $d(3,1)=2$

where $d(i,j)=k$ means that when in state $i$, the automaton will move to state $k$ after input $j$.

Determine if the language accepted by this automaton consists of all strings that have an odd total number of 1s, such as strings of lengths 1, 3, 5, 7, etc. (i.e., strings of the form $2n+1$).

Answer: No


Question: Suppose \( G \) is a planar graph with 100 vertices such that if each pair of non-adjacent vertices is connected, the resulting graph would be non-planar. What is the number of edges in \( G \)?

Answer: 294


Question: Let $G$ be a finite group acting on a commutative ring $R$ via ring maps. Consider a 1-dimensional free $R$-module $P$ with a $G$-action satisfying $g(r \cdot x) = g(r) \cdot g(x)$ for $r \in R$ and $x \in P$. Is there always some tensor power $P^{\otimes_R n}$ (with diagonal action) that is isomorphic to $R$ as an $R$-module with semilinear $G$-action? Can $n$ be chosen to be $|G|$?

Answer: Yes


Question: Find the degree of the field extension \([\mathbb{Q}(i,2^{\frac{1}{4}}):\mathbb{Q}(2^{\frac{1}{4}})]\).

Answer: 2


Question: Consider a sequence \( \{a_n\} \) with positive terms. Determine \( \lim\limits_{n\to\infty}\left(\frac{a_1}{a_2}+\frac{a_2}{a_3}+\frac{a_3}{a_4}+\ldots+\frac{a_n}{a_1}\right) \) if the sequence is decreasing.

Answer: \infty


Question: Evaluate the contour integral \( \int_{C(0,1)} \frac{z e^z }{\tan^2 z}dz \) over the unit circle centered at the origin with radius 1.

Answer: 2\pi i


Question: Consider a symmetric random walk on a graph constructed from two cycles, $C_{m}$ and $C_{n}$, which share exactly two consecutive common vertices. Let $S$ and $S'$ be the common vertices, $A$ be a vertex of $C_{m}$ not in $C_{n}$, and $B$ be a vertex of $C_{n}$ not in $C_{m}$. Given $m = 2 \times 10^{6} + 1$ with the $SS'A$-arc being a path of $10^{6} + 1$ edges and the other $SA$-arc being a path of $10^{6}$ edges, and $n = 4 \times 10^{6} + 1$ with the $SS'B$-arc being a path of $2 \times 10^{6} + 1$ edges and the other $SB$-arc being a path of $2 \times 10^{6}$ edges, calculate the probability that a symmetric random walk starting from $S$ ends at vertex $A$. Assume the walk ends immediately upon reaching $A$. What is this probability?

Answer: \dfrac{2}{3}


Question: Does there exist a selective ultrafilter \( \mathcal{U} \) and a bijection \( \varphi: \omega \to \omega \) such that \( \varphi(\mathcal{F_1}) \subset \mathcal{U} \) in ZFC and CH?

Answer: Yes


Question: Find the coefficient of the term $x^3y^4z^5$ in the expansion of the polynomial $(x + y + z)^8(x + y + z + 1)^8$. Provide your answer as a single number.

Answer: 1940400


Question:
Determine the number of equivalence relations \( R \) on the set of natural numbers \( \mathbb{N} \) that satisfy the following conditions:

1. If two elements end in the same digit, they are related.
2. The set \( \{(1,2),(101,25),(4,4),(234,20166),(22,7),(153,158),(8,100),(17,27)\} \subseteq R \).
3. The pairs \((1,14),(32,8),(19,10),(309,666)\) are not in \( R \).

How many such equivalence relations exist?

Answer: 4


Question: For positive real numbers $x_1, x_2, \ldots, x_n$ and any $1 \leq r \leq n$, let $A_r$ and $G_r$ be the arithmetic mean and geometric mean of $x_1, x_2, \ldots, x_r$, respectively. Determine if the arithmetic mean of $G_1, G_2, \ldots, G_n$ is never greater than the geometric mean of $A_1, A_2, \ldots, A_n$. Is this statement true for all $n$?

Answer: True


Question: Let $K$ be a number field, $\\mathscr{O}_K$ the ring of algebraic integers of $K$, and $d_K$ the discriminant of $\\mathscr{O}_K$. If a rational prime $p$ divides $d_K$, does $p$ necessarily ramify in $\\mathscr{O}_K$?

Answer: Yes


Question: Suppose \(g(u) = \int_{2}^{u} \sqrt{1 + t^3} \, dt\). Find the value of \((g^{-1})'(0)\) if it exists.

Answer: \dfrac{1}{3}


Question: Suppose that $A$ and $B$ are unitarily equivalent permutation matrices. Are they also permutation similar?

Answer: Yes


Question: The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of $d$, the insurance company reduced the expected claim payment by $10\%$. Calculate the percentage reduction in the variance of the claim payment after the deductible is applied.

Answer: 1\%


Question: Evaluate the definite integral: \[ \int_0^{2\pi} \frac{dx}{\sin^{4}x + \cos^{4}x} \]

Answer: 2\sqrt{2}\pi


Question: Given the polynomial \( P(z) = z^n + a_1z^{n-1} + \ldots + a_n \) with the condition \( |a_1| + |a_2| + \ldots + |a_n| \leq 1 \), find the smallest \( R > 0 \) such that all roots of \( P(z) \) lie within \( |z| \leq R \).

Answer: 1


Question: Compute the surface integral \( \iint_S G \cdot dS \) where \( S \) is the boundary of the solid inside the cylinder \( x^2 + y^2 = 4 \), bounded above by the plane \( z = x \) and below by the \( xy \)-plane. The vector field is given by \( G(x,y,z) = \langle x + \cos z, y + y \sin x, z + \cos y - z \sin x \rangle \). Use the divergence theorem to evaluate the integral.

Answer: 16


Question: Evaluate the integral \( \int{4x^2+y^2} \, dA \) using the substitution \( y+2x=u, \; y-2x=v \) with the limits \( 0<u<2, \; -2<v<0 \). The Jacobian of the transformation is \( \frac{1}{4} \).

Answer: \dfrac{4}{3}


Question: Determine whether the sequence \( a_n = (\sqrt[n]{e} - 1) \cdot n \) converges. If it converges, find its limit.

Answer: 1


Question: Does there exist a continuous, strictly increasing function \( f \) on \([0,1]\) such that \( f'(x) = \infty \) for all \( x \in E \), where \( E \subset [0,1] \) and \( \mu(E) = 0 \)?

Answer: Yes


Question: Let \( x_1, x_2, x_3 \) be the roots of the polynomial \( x^3 + mx + n = 0 \), where \( m, n \in \mathbb{R} \). Consider the matrix \( A = \begin{pmatrix} 1 & 1 & 1 \\ x_1 & x_2 & x_3 \\ x_1^2 & x_2^2 & x_3^2 \end{pmatrix} \). Find the determinant of \( A^2 \).

Answer: -4m^3 - 27n^2


Question: Evaluate the limit without using L'Hôpital's rule: $$\lim_{x\to\infty} x^2\cdot(e^\frac{1}{x-1}-e^\frac{1}{x})$$

Answer: 1


Question: Let $X$ and $Y$ be independent random variables, both following exponential distributions with $E[X] = \frac{1}{2}$ and $E[Y] = 1$. Calculate the density of the conditional distribution $f_{X+Y|X}(z|x)$. Specifically, find this density for $s = 3$. Use the formula $f_{Z|X}(z|x) = \frac{f_{Z,X}(z,x)}{f_X(x)}$ to assist in your calculation.

Answer: e^{-(3 - x)}


Question: Calculate the mean of the normal distribution function \( \frac{1}{2\pi \sigma^2} \exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right] \) by integration.

Answer: \mu


Question: Find the number of solutions to the equation \(x_1 + x_2 + x_3 + \ldots + x_9 = 42\) where each variable satisfies \(0 \le x_n \le 9\) for \(1 \le n \le 9\).

Answer: 44865975


Question: Let $U$ and $V$ be positive integers such that $U = V + 2$. Define the triangular number $T_X = \frac{X(X+1)}{2}$. Can the positive integer $T = T_U - T_V$ be a triangular number if $U$ is a Mersenne prime, i.e., $U = 2^t - 1$? Provide a justification for your answer.

Answer: Yes


Question:
Find the function \( y(x) \) that extremizes the boundary value problem given by the functional:
\[ I[y] = \int_0^1 y'^2(x) \, dx + y^2(0) - 2y^2(1) \]

Answer: 0


Question: Determine whether the following statement is true or false: If \( C \) is a closed subset of a topological space \( X \), then \( C \) is also closed in the one-point compactification of \( X \).

Answer: B


Question: Evaluate the sum \( \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!} \) and determine the function it represents.

Answer: \sinh x


Question: Determine if (n, m) = (18, 7) is the only positive integer solution to the equation \( n^2 + n + 1 = m^3 \).

Answer: (18, 7)


Question: Find the maximum value of the function \( f(x) = \cos(2x) - e^{3x} \).

Answer: 1


Question: Suppose $x$ is an integer such that $3x \equiv 15 \pmod{64}$. If $x$ has a remainder of 2 and a quotient $q$ when divided by 23, determine the remainder when $q$ is divided by 64.

Answer: 53


Question: Evaluate the limit: \[ \lim_{n \to \infty} \cos^2\left(\pi \sqrt[3]{n^3+n^2+2n}\right) \] where \( n \in \mathbb{N} \).

Answer: \dfrac{1}{4}


Question: Let $F$ be a family of analytic functions on the upper half-plane $H = \{z : \mathrm{Im}z > 0\}$ such that $f(i) = 0$ and $|f(z)| \leq 1$ for all $z \in H$. Find $\sup_{f \in F} |f(2i)|$. Use the properties of Möbius transformations to determine the solution.

Answer: \dfrac{1}{3}


Question: Let \( f \) be a \( 2\pi \)-periodic function in \( L^1([0, 2\pi]) \) such that \( \int_{0}^{2\pi} f(x) \, dx = 0 \). Prove or disprove that \( \int_{0}^{2\pi} f(nx) \, dx = 0 \) for all \( n \in \mathbb{N} \).

Answer: 0


Question: Find the limit of the sequence \((a_n)_{n \ge 1}\) given that the sequence \(1 + \dfrac{1}{2} + \dfrac{1}{3} + \ldots + \dfrac{1}{n} - a_n \ln n\) is bounded for \(n \ge 1\).

Answer: 1


Question: In a group of order 16 generated by elements $a$ and $b$, where the order of $a$ is 8, the order of $b$ is 2, and $bab^{-1} = a^{-1}$, determine the number of elements of order 2 in the group.

Answer: 9


Question: Evaluate the contour integral \( \int_\Gamma \frac{dz}{z^2+1} \) using the Cauchy integral formula, where \( \Gamma \) is a contour enclosing the singularities of the integrand.

Answer: 0


Question:
Let $v$ and $w$ be vectors in $\mathbb{R}^{n}$ that are not scalar multiples of each other. Suppose $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a linear transformation such that:
\[ T(-2v-3w) = -2v-3w \quad \text{and} \quad T(v+w) = -2v+3w \]
What is $T(v)$ expressed as a linear combination of $v$ and $w$?

Answer: -8v + 6w


Question: Given that $f(1) = 2013$, find the value of $f(2013)$ for a function $f$ defined on the set of natural numbers such that \[ f(1) + 2^2f(2) + 3^2f(3) + \cdots + n^2f(n) = n^3f(n) \] for all positive integers $n$. Determine $f(2013)$. \( \boxed{} \)

Answer: \dfrac{1}{2013}


Question: Does the equality \(|Tf| = T|f|\) hold for positive operators \(T\) in a finite-dimensional \(\mathrm{C}^*\)-algebra \(A\)? Here, the absolute value is defined as \(|a| := \sqrt{a^*a}\), and an operator \(T\) is positive if \(T(A^+) \subset A^+\), where \(A^+\) is the cone of positive elements in \(A\).

Answer: No


Question: Evaluate the definite integral \( \int_0^\pi \frac{\sin(\frac{21}{2}x)}{\sin(\frac{1}{2}x)} \, dx \).

Answer: \pi


Question: Evaluate the contour integral \( \int_C \frac{1}{z} \, dz \) for any contour \( C \) in the right half-plane from \( z = -3i \) to \( z = 3i \).

Answer: i\pi


Question:
Find the maximum velocity of a point P described by the parametric equations:
\[ x(t) = 2\cos t - \cos(2t) \]
\[ y(t) = 2\sin t - \sin (2t) \]

Answer: 4


Question: If \((A+B)^n\) is binomial for some integer \(n\), does this imply that \(AB = BA\) for matrices \(A\) and \(B\)? A matrix power \((A+B)^n\) is considered binomial if it satisfies the equality \[(A+B)^n = \sum_{j=0}^n \binom{n}{j}A^jB^{n-j}.\] Determine whether the binomial property implies commutativity of \(A\) and \(B\).

Answer: Yes


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

Answer: \dfrac{e}{2}


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

Answer: \dfrac{\pi}{2}


Question: Let $f: \mathbb{R} \to [3,5]$ be a differentiable function such that $\lim_{x \to \infty} (f(x) + f'(x)) = 3$. Find $\lim_{x \to \infty} f(x)$. \( \lim_{x \to \infty} f(x) \) represents the horizontal asymptote of $f(x)$. Determine this limit.

Answer: 3


Question: Determine the number of group homomorphisms from \(\mathbb{Z}_{2p^a}\) to \(D_{p^a}\), where \(p\) is an odd prime and \(a\) is a natural number. Express your answer as a single integer.

Answer: 2p^a


Question: Let \( f = \chi_{\{0\}} \) be the characteristic function of the set \( \{0\} \) on the measure space \((X, \Omega, \mu)\), where \( X = [0,1] \), \( \Omega \) is the Borel subsets of \([0,1]\), and \( \mu(S) = \infty \) if \( 0 \in S \), otherwise \( \mu \) is the Lebesgue measure. Show that \( \|f\|_\infty = 1 \).

Answer: 1


Question:
Is the space $C_\Omega(X)$ of random continuous functions separable? Here, $\Omega$ and $X$ are compact metric spaces, and $P$ is a probability measure on $\Omega$. A random continuous function $f : X \times \Omega \to \mathbb{R}$ satisfies the following conditions:

1. For all $x \in X$, the map $\omega \mapsto f(x, \omega)$ is measurable.
2. For all $\omega \in \Omega$, the map $x \mapsto f(x, \omega)$ is continuous and bounded.
3. The map $\omega \mapsto \sup\{|f (x, \omega)| : x \in X\}$ is integrable with respect to $P$.

Functions $f$ and $g$ are identified if $P(\{\omega : f(\cdot, \omega) \not= g(\cdot, \omega) \}) = 0$. The space $C_\Omega(X)$ is equipped with the norm:
$$
|f|_\infty = \int \sup_{x \in X}|f(x, \omega)|dP(\omega).
$$
Determine whether $C_\Omega(X)$ is separable.

Answer: Yes


Question: Let \( \{f_n\} \) be a sequence of measurable real-valued functions on \([0, 1]\) that is uniformly bounded. Does there exist a subsequence \( n_j \) such that \( \int_A f_{n_j}(x)\,dx \) converges for each Borel subset \( A \) of \([0, 1]\)?

Answer: Yes


Question: Determine whether the limit \( \lim_{(x,y)\rightarrow(0,0)} \frac{7 \sin(2 x) x^2 y}{2 x^3 + 2 x y^2} \) exists. If it does, find its value.

Answer: 0


Question:
Let \((S, ||\cdot||_{\infty})\) be a metric space where \(S\) is the space of bounded and Lipschitz continuous functions. Consider a sequence of functions \(\{\tau_n\}_{n=1}^{\infty}\) mapping \(S\) into itself, each satisfying the Blackwell sufficient conditions, ensuring the existence of a unique fixed point \(s_n\). Suppose \(\{\tau_n\}_{n=1}^{\infty}\) converges to \(\tau_0\) with a fixed point \(s_0\). Prove that for any \(u \in S\), the following holds:
\[\lim_{k\rightarrow \infty} \tau_{k} \circ \tau_{k-1} \circ \ldots \circ  \tau_{1} (u)=s_{0}\]

Answer: s_0


Question: Compute the limit: \[ \lim_{n \to \infty} \sqrt{n} \cdot \left(\sqrt[n]{3} - \sqrt[n]{2}\right) \] using known limits, basic limit properties, and theorems such as the squeeze theorem.

Answer: 0


Question: Find \( \min_{y \in \mathbb{R}} \max_{0 \le x \le 1} |x^2 - xy|. \)

Answer: 3 - 2\sqrt{2}


Question: Evaluate the complex integral \( K = \frac{i}{4-\pi} \int \frac{1}{z\cos(z)}\,dz \) over the circle centered at the origin with radius 4. Determine the value of \( K \).

Answer: 2


Question: Find the remainder when \( 2099^{2017^{13164589}} \) is divided by 99.

Answer: 92


Question: Determine the radius of convergence of the power series representation of the function \( \frac{\tan(\frac{z}{2})}{(z+2)(z+3)} \) around \( z = 0 \).

Answer: 2


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

Answer: n\pi


Question: Determine the number of distinct terms in the binomial expansion of \( \left(x + \frac{1}{x} + x^2 + \frac{1}{x^2}\right)^{15} \).

Answer: 61


Question: Given the sequence \(a_n = (-1)^{n-1}\) and \(s_n = \sum_{i=1}^{n} a_i\), find \( \lim_{n \to \infty} \frac{s_1 + s_2 + \dots + s_n}{n} \).

Answer: \dfrac{1}{2}


Question: Find the indicial equation of the differential equation \(t^2y''+ty'+(t^2-2)y=0\) at \(t=0\).

Answer: r^2 - 2 = 0


Question: Calculate the volume of the region bounded by the surfaces \( z = (x^2 + y^2)^2 \) and \( z = x \).

Answer: \dfrac{\pi}{12}


Question: If A is the sum of the digits of \(5^{10000}\), B is the sum of the digits of A, and C is the sum of the digits of B, what is the value of C?

Answer: 4


Question: Let $H$ be a subgroup of $S_7$ consisting of permutations that send $1,2,3$ to $1,2,3$ in some order, while the remaining values $4,5,6,7$ are permuted among themselves. Is $H$ isomorphic to $S_3 \times S_4$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{ n \to \infty }(1-\tan^2\frac{x}{2})(1-\tan^2\frac{x}{4})(1-\tan^2\frac{x}{8})\cdots(1-\tan^2\frac{x}{2^n}) \]

Answer: x \cot x


Question: If $M$ is a differential manifold, is it always possible to find a complete Riemannian metric on $M$?

Answer: Yes


Question: In a forest consisting of 6 trees with a total of 27 vertices, what is the minimum number of simple paths? Calculate the minimum number of simple paths for this configuration.

Answer: 48


Question: Let $A$ be an $n \times n$ matrix with entries from the set $\{1, 2, \ldots, n\}$. Is it possible to permute the columns of $A$ to obtain a matrix $B = [b_{ij}]$ such that the number of entries $b_{ij} = j$ across all columns is at most $n$?

Answer: Yes


Question: Let $M, N, P$ be $A$-modules, and let $f: M \otimes N \to P$ be an $A$-homomorphism. If $m \otimes n \in \ker f$ implies $m \otimes n = 0$ for all $m \in M, n \in N$, does it follow that $\ker f = 0$? Justify your answer.

Answer: No


Question: Find the closure of the set \( \left\{ \frac{1}{n} : n \in \mathbb{N} \right\} \) in the finite complement topology on \( \mathbb{R} \).

Answer: \mathbb{R}


Question: Determine the values of \( \alpha \in \mathbb{R} \) for which the equation \( e^{\alpha x} - 1 = x \) has a positive solution.

Answer: (0, 1)


Question: Determine the number of partial orders on a set \( S \) where \( |S| = 3 \).

Answer: 19


Question: Find the expected value \( E[\sin(B_t)] \), where \( B_t \) is a standard Brownian motion.

Answer: 0


Question: Determine the radius of convergence for the power series \( \sum_{n=0}^{\infty} \frac{n}{5^{n-1}} z^{\frac{n(n+1)}{2}} \).

Answer: 1


Question: Evaluate the limit: $$\lim_{x \to 0} \frac{\int_0^x \cos(t^2) \, dt}{x}.$$

Answer: 1


Question: Find the value of \( u\left( \frac{1}{4}, \frac{1}{\sqrt{2}} \right) \) where \( u(x,y) \) is the solution to the partial differential equation \( \frac{\partial^{2}u}{\partial x^2} + \frac{\partial^{2}u}{\partial y^2} = 64 \) in the unit disk \( \{(x,y) \mid x^2 + y^2 < 1\} \), with the condition that \( u \) vanishes on the boundary of the disk.

Answer: -7


Question: Given independent random variables $X, Y \sim \exp(1)$ and $U \sim U[0,1]$, find the distribution of the random variable $U(X+Y)$. Provide your answer in terms of a probability density function.

Answer: e^{-z}


Question: Given a stack of 10 cards, where 3 are red, 5 are blue, and 2 are green, what is the expected number of draws, without replacement, before you draw both a red and a blue card? Assume that it does not matter whether the red or blue card is drawn first.

Answer: \dfrac{121}{36}


Question: Evaluate the infinite series \( \sum _{j=0}^{\infty }\binom{-p-1}{j} \left( -\frac {x} {1+x}\right) ^{j} \) in terms of \( p \) and \( x \).

Answer: (1 + x)^{p + 1}


Question:
Consider three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{h}\) in \(\mathbb{R}^n\). Define the function
\[ p(\mathbf{h}) = (\mathbf{a} \cdot \mathbf{h})(\mathbf{b} \cdot \mathbf{h}). \]
Evaluate the limit:
\[ \lim_{\mathbf{h} \to \mathbf{0}} \frac{p(\mathbf{h})}{|\mathbf{h}|}. \]
Does this limit exist, and if so, does it equal zero?

Answer: 0


Question: Let $A$ and $B$ be real symmetric matrices of order $n$, satisfying $A^2B = ABA$. Prove that $AB = BA$.

Answer: AB = BA


Question: Find the value of the sum \( \sum_{r=0}^{2n+1}(-1)^{r}{2n+1\choose r}^2 \).

Answer: 0


Question: Determine the value of the constant \( a \) such that the function \( f(x) = x^4 + ax^3 + \frac{3}{2}x^2 + 1 \) is convex on \( \mathbb{R} \).

Answer: [-2, 2]


Question: Given the equation \(1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{23} = \frac{a}{23!}\), find the value of \(a \mod 13\).

Answer: 7


Question: Let \( f(x) \) be a polynomial such that \( f(f(x)) = f(x-1)f'(x+1) \) and \( f'(0) = 2 \). Find \( f(x) \).

Answer: 2x - 4


Question: Evaluate the integral \( \iiint_K (x^2 -z^2)\, dx\, dy \,dz \), where \( K=\{(x,y,z):x,y,z\ge 0,x+y+z\le 1\} \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to +\infty} \frac{x^{\sqrt{\ln x}}\cdot(\sqrt{\ln x})^x}{(\sqrt{x})^{\ln x}\cdot(\ln x)^{\sqrt{x}}} \]

Answer: +\infty


Question: Given that the function $f$ has continuous derivatives and satisfies $f(\sqrt{2})=2$, and for any real number $x$, \[ f(x)=\lim_{t \to 0}{\dfrac{1}{2t}\int_{x-t}^{x+t}sf'(s)ds}. \] Determine the function $f(x)$. Provide your answer in terms of $x$.

Answer: \sqrt{2}\,x


Question: Evaluate the integral \( \int_{|z|=1} \frac{\tan(z)}{z e^{\frac{1}{z+2}}} \, dz \) for \( z \in \mathbb{C} \).

Answer: 0


Question:
Given the structural causal model with exogenous variables $U_X, U_Y, U_Z$ as independent identically distributed standard normals, and the following equations:
\[\begin{align*}
X &= U_X, \\
Y &= \frac{X}{3} + U_Y, \\
Z &= \frac{Y}{16} + U_Z,
\end{align*}\]
determine the best estimate of $Y$ given that $X=1$ and $Z=3$. Use the hint that for normally distributed variables $X, Y, Z$, the conditional expectation is given by $E[Y|X=x,Z=z]=R_{Y\!X\cdot Z}x+R_{Y\!Z\cdot X}z$. Calculate the necessary covariances and regression coefficients to find the answer.

Answer: \dfrac{400}{771}


Question: Determine the constant \( \alpha \) such that the tangent planes at every intersection point of the spheres \( x^2 + (y - \alpha)^2 + z^2 = 3 \) and \( (x - 1)^2 + y^2 + z^2 = 1 \) are orthogonal.

Answer: \pm \sqrt{3}


Question: Determine whether there exist two functions \( f, g \in C^1(I) \) such that the Wronskian \( W(f, g)(x) = \begin{vmatrix} f(x) & g(x) \\ f'(x) & g'(x) \end{vmatrix} \) is positive for some \( x \in I \) and negative for some other \( x \in I \), where \( I \) is an open interval and \( f, g \) are real-valued.

Answer: Yes


Question: Determine the value of Arg(f(z)) after a complete counter-clockwise rotation of the point z about the origin, starting at z = 2 with the angle there being 0, where \( f(z) = \sqrt{z^2+2z-3} \).

Answer: \pi


Question:
Calculate the value of the expression \( B \) given by:
\[
B = \frac{\left|\begin{matrix}(a-b)&-2\\0&(b-c)\end{matrix}\right| - \left|\begin{matrix}3&3&3\\a&b&c\\bc&ac&ab\end{matrix}\right|}{(b-a)(b-c)}
\]
Which of the following expressions matches the value of \( B \)?
\[
\begin{array}{ll}
1.&c-a-1\\
2.&3b-a+1\\
3.&3c-a-1\\
4.&3c-3a-1\\
\end{array}
\]

Answer: D


Question: Consider an m×n rectangular grid of points in the plane, where m and n are integers greater than 1. Some k of these points are colored red such that no three red points form the vertices of a right-angled triangle with two sides parallel to the sides of the grid. Determine the greatest possible value of k.

Answer: m + n - 2


Question: Evaluate \( \lim_{x\to\infty }x\int_1^{x}\frac{e^t}{t}dt-e^x \).

Answer: \infty


Question: Find the length of the tangent to the curve \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\) that is intercepted between the axes.

Answer: a


Question: Given the joint probability density function \( f(x,y) = 1120x^{3}y^{3} \) for \( 0 \leq x, 0 \leq y, \) and \( x+y \leq 1 \), calculate the correlation coefficient \( \text{Corr}(X,Y) \).

Answer: -\dfrac{4}{5}


Question: Evaluate the integral \( I = \int_2^3 \frac{dx}{\sqrt{x^3 - 3x^2 + 5}} \). Then, find the value of \( \lfloor I + \sqrt{3} \rfloor \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

Answer: 2


Question: The complex number \( \left(\sqrt{\frac{1}{2} + \frac{\sqrt{3}}{2}i}\right)^{1/3} \) can be expressed in polar form as \( r(\cos x + i\sin x) \) with \( r < 0 \). Find the smallest positive value of \( x \) in degrees.

Answer: 10


Question: Given topological spaces $X$ and $Y$, determine whether the product spaces $X \times Y$ and $X_c \times Y$ have the same compact subsets, where $X_c$ is the compactly generated topology on $X$.

Answer: Yes


Question: Calculate the limit: \[ \lim_{x\to1}\left(\frac{1+\cos(\pi x)}{\tan^2(\pi x)}\right)^{\!x^2} \] without using l'Hôpital's rule.

Answer: \dfrac{1}{2}


Question: Determine whether the following holds: \( o\left(\frac{1}{n^{2}}\right) + \cdots + o\left(\frac{n}{n^{2}}\right) = o(1) \).

Answer: Yes


Question: Find the volume of the solid region in the first octant bounded by the coordinate planes, the plane \(y + z = 2\), and the parabolic cylinder \(x = 4 - y^2\).

Answer: \dfrac{20}{3}


Question: Evaluate the limit: \[ \lim_{n\to \infty} \frac{1}{n^4}\left(\frac{1}{a_1} + \frac{2}{a_2} + \frac{3}{a_3} + \cdots + \frac{n}{a_n}\right) \] where the sequence \(a_n\) is defined by \(a_1 = 1\) and \(a_{n+1} = \frac{a_n}{1 + n a_n}\).

Answer: \dfrac{1}{8}


Question: Evaluate the integral \( \int_{0}^{2\pi} f(e^{it}) \cos t \, dt \) where \( f \) is an analytic function.

Answer: \pi f'(0)


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

Answer: -e^2


Question: Evaluate the limit: \( \lim_{x \to \infty} \sin(a + xi) \), where \( a \) is a constant.

Answer: \infty


Question: Let \(\triangle ABC\) be an equilateral triangle with side length 1. Let \(P\) be the midpoint of \(\overline{AB}\) and \(Q\) be a point on \(\overline{AC}\) such that \(\overrightarrow{AQ} = \frac{1}{3}\overrightarrow{AC}\). Define \(T\) as a point such that \(\overrightarrow{CT} = \lambda\overrightarrow{CP}\), where \(\lambda \in [0, 1]\). Determine all values of \(\lambda\) for which \(\measuredangle BTQ = 90^\circ\).

Answer: \dfrac{1}{3}


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

Answer: -\dfrac{1}{2}


Question: Determine whether the following statement is true or false: For a stochastic matrix (where each row sums to 1), if an eigenvalue is 0, then the corresponding eigenfunction is identically 0.

Answer: B


Question: Determine for which of the following primes \( p \), the polynomial \( x^4 + x + 6 \) has a root of multiplicity greater than 1 over a field of characteristic \( p \). Consider the primes \( p = 2, 3, 5, 7 \).

Answer: 3


Question: Evaluate the limit: \( \lim_{x \to 1^+} \frac{x-1}{\sqrt{2x-x^2}-1} \)

Answer: -\infty


Question:
Calculate the seminorm \( \|A\|_S \) for the operator \( A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \) with respect to the operator \( S = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \) on \( (\mathbb{C}^2,\|\cdot\|_2) \). The seminorm is defined as:
\[ \|A\|_S:=\inf\left\{c\geq 0; \quad\sqrt{\langle SAx,Ax\rangle} \leq c  \sqrt{\langle Sx,x\rangle},\;\forall x \in \overline{\text{Im}(S)}\right\}. \]

Answer: \dfrac{1}{2}


Question: Consider a string \( s = a_1 a_2 \ldots a_n \). A period \( p \) is defined such that \( a_i = a_{i+p} \) for all \( i \in [1, n-p] \). Suppose there are two periods \( p \) and \( q \) such that \( p + q \leq n \). Prove that \( \gcd(p, q) \) is also a period of the string \( s \).

Answer: \gcd(p, q)


Question:
Determine the limit of the sequence \(a_n\) defined by the recurrence relation:
\[ a_{n+1} = a_n \sqrt{\frac{n}{n+a_n}}, \quad a_1 > 0. \]

Answer: 0


Question: Find the sum of the coefficients of $x^3$ in the expansion of \((1-\frac{x}{2}+\frac{1}{\sqrt{x}})^8\).

Answer: \dfrac{77}{4}


Question: Determine whether there exists a pair of irrational numbers $a < 0$ and $b > 0$ such that for every $\epsilon > 0$, there is a finite sum $S = m \cdot a + n \cdot b$ (where $m, n$ are non-negative integers) such that $S \in (0, \epsilon)$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether there exists a continuous function \( f: \mathbb{R}^2 \to \mathbb{R} \) that cannot be expressed as a finite sum of the form \( f(x,y) = \sum_i g_i(x)h_i(y) \), where \( g_i \) and \( h_i \) are functions of one variable.

Answer: Yes


Question: Does there exist a diffeomorphism \( f: \mathbb{B}^n \to \mathbb{R}^n \) such that \( \lim_{\|x\| \to 1} \|f(x)\| \neq \infty \)?

Answer: No


Question:
Given a random sample \( X_1, \ldots, X_n \) from the probability density function \( f_{\theta}(x) = \theta e^{-\theta x} \) for \( x > 0 \), where \( \theta > 0 \) is an unknown parameter, determine the uniform minimum variance unbiased estimator (UMVUE) for \( \frac{1}{\theta} \). Choose the correct option:

(A) \( \frac{1}{\bar{X}_n} \)
(B) \( \sum_{i=1}^{n}X_i \)
(C) \( \bar{X}_n \)
(D) \( \frac{1}{\sum_{i=1}^{n}X_i} \)

Answer: C


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

Answer: \dfrac{\pi}{2}


Question: Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that \( f(f(x - y)) = f(x) f(y) - f(x) + f(y) - xy \) for all real numbers \( x \) and \( y \). Find the sum of all possible values of \( f(1) \).

Answer: -1


Question: A player spins a roulette wheel 50 times. The wheel has 5 equal sectors numbered from 1 to 5. Each spin results in a number from one of these sectors, and the player's profit is the sum of these numbers. What is the probability that the player's total profit is at least 170?

Answer: 0.0256


Question: Let $G$ be a group and $H \subseteq G$ a subgroup. Consider an element $a \in G$. Is it possible for the conjugate subgroup $aHa^{-1}$ to be strictly contained in $H$, i.e., $aHa^{-1} \subset H$ but $aHa^{-1} \neq H$? Justify your answer.

Answer: Yes


Question: Prove that \( \lim_{x \to 0} \frac{O(|x|^2)}{1+O(|x|^4)} = 0 \).

Answer: 0


Question: Determine the largest possible size of an $n \times n$ matrix $A$ such that $A^3 = 0$ and $A$ has a Jordan form consisting of exactly 4 blocks.

Answer: 12


Question: Evaluate the double integral \( \iint_{D} xy \, dA \) where \( D \) is the region bounded by the lines \( y=x \), \( y=x+4 \), \( x=0 \), and \( x=2 \). Use the change of variables \( x=2u \) and \( y=2u+v \). Express your answer as a decimal.

Answer: 26.6667


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function. Does there exist a sequence of piecewise linear continuous functions \( f_n \) such that \( f_n \rightarrow f \) pointwise?

Answer: Yes


Question: Can a triangle be equilateral if the distances from its vertices to two given perpendicular lines are whole numbers?

Answer: No


Question: Evaluate \( p(2012)^6 \) where \( p(x) = \sqrt{x + 2 + 3\sqrt{2x-5}} - \sqrt{x - 2 + \sqrt{2x-5}} \).

Answer: 8


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

Answer: (0, 0, 0, 0)


Question: Let $F$ be a global field with ring of integers $o$. Suppose $x \in F$ satisfies $|x|_v = 1$ for all non-archimedean valuations $v$ of $F$. Does this imply that $x \in o^\times$?

Answer: Yes


Question: Find the supremum of the set \( A = \left\{ \frac{5^{3n} \cdot 5^{2m}}{3 \cdot 5^{5n} + 5^n + 2 \cdot 5^{5m}} \mid n, m \in \mathbb{N} \right\} \).

Answer: \dfrac{1}{5}


Question: Determine whether the limit \( \lim\limits_{(x,y)\rightarrow (0,0)}\frac{\sin(xy)}{y} \) exists.

Answer: 0


Question: A machine produces a weighted coin that lands on heads with an unknown probability $p$, where the distribution of $p$ is given by $P(p \leq x) = x^4$. You flip the coin 5 times, and it lands on heads each time. What is the probability that the next flip will also land on heads?

Answer: \dfrac{9}{10}


Question: Given two prime ideals $P$ and $Q$ of a ring $A$ such that $P \subset Q$, determine whether $A_P \subset A_Q$ or $A_Q \subset A_P$, where $A_P$ and $A_Q$ are the localizations of $A$ at $P$ and $Q$, respectively.

Answer: A_Q \subset A_P


Question: Calculate the limit: \[ \lim_{x \to \infty}\frac{x e^{(\ln x)^2}}{(x+1)^{\alpha +1}} \] where \( \alpha > 0 \).

Answer: \infty


Question: Given the expansion \((1+x+x^2)^n = a_{0}+a_{1}x + a_{2}x^{2} +\cdots +a_{2n}x^{2n}\), find the sum of the coefficients indexed by multiples of 3, i.e., find the value of \(a_{0}+a_{3}+a_{6}+\cdots\).

Answer: 3^{n-1}


Question: Find the residue of the function \( \frac{1}{e^z+1} \) at the singular point \( z = \pi i \).

Answer: -1


Question: Verify that \( \int \sec x\ dx = \frac{1}{2} \ln \left|\frac{1+\sin x}{1-\sin x}\right| + C \) by differentiating the right-hand side and showing it equals \( \sec x \).

Answer: \sec x


Question: Let $G$ be a locally compact Hausdorff group and $H \le G$ a closed subgroup with positive Haar measure. Is $H$ open?

Answer: Yes


Question: Solve the equation \(2\tan^{-1}\sqrt{x-x^2} = \tan^{-1}x + \tan^{-1}(1-x)\) for \(x\).

Answer: \dfrac{1}{2}


Question: Find the function \( f: \mathbb{R} \to \mathbb{R} \) that satisfies the equation \( f(-y^2) + f(x^2 + y) - y = (x + y)f(x - y) \) for all real numbers \( x \) and \( y \).

Answer: f(x) = x


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{- \frac{1}{x^2}}}{x^{100}}. \]

Answer: 0


Question:
Determine the values of $k$ such that there exists a constant $c < 1$ for which the following inequality holds for all continuous functions $F$ defined on the interval $[0, r]$, where $r \in \mathbb{R}^+$:
\[ \sup_{x \in [0, r]} \left\{ e^{kx} \int_0^x |F(s)| \, ds \right\} \leq \frac{c}{M} \sup_{s \in [0, r]} \left\{ e^{ks} |F(s)| \right\} \]
where $M > 0$ is fixed.

Answer: k < -M


Question: Solve the equation \(2^{2x} + 2^{2x-1} = 3^{x+0.5} + 3^{x-0.5}\) for \(x\).

Answer: \dfrac{3}{2}


Question: Consider a linearly ordered topological space $X$ with a countable dense subset. Determine whether $X$ must be metrizable. Provide reasoning for your answer.

Answer: Yes


Question: Find the maximum value of the expression \(x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_{2012}x_{2013}\) given that \(x_1, x_2, \ldots, x_{2013} \in [0, \infty)\) and \(x_1 + x_2 + \cdots + x_{2013} = 2013\).

Answer: 671^3


Question: Evaluate the double integral \( \int_{-2}^{0} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \sqrt{x^2+y^2} \, dy \, dx \) by converting it to polar coordinates.

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


Question: What is the probability that you never lose in the following dice game? You start with a four-sided die numbered 1-4. In each round, you roll all your dice. If a die shows 1, you lose that die. If it shows 2, you keep the die. If it shows 3 or 4, you gain an additional identical die. You lose the game if you have no dice left. Calculate the probability that you never lose the game over infinite rounds.

Answer: \dfrac{1}{2}


Question: Consider a field $K$ with characteristic zero and an elliptic curve $E$ over $K$. Let $x \in E(K)$ and $p$ be a prime number different from 2. Suppose $n \in E(\overline{K})$ satisfies $pn = x$ and $n \notin E(K)$. Let $L$ be the smallest algebraic extension of $K$ such that $n \in E(L)$. Is the order of the Galois group $\operatorname{Gal}(L/K)$ greater than 2?

Answer: Yes


Question: Let \( A_1, A_2, A_3, \ldots, A_n \) be subsets of a set \( S \) with \(|S| = 2019\). The union of any three of the \( A_k \) is equal to \( S \), but the union of any two is not equal to \( S \). Determine the maximum possible value of \( n \).

Answer: 64


Question: Determine the function \( f(r) \) such that the divergence of the vector field \( \mathbf{F} = \frac{\mathbf{r}}{r} f(r) \) is zero, where \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) and \( r = \sqrt{x^2 + y^2 + z^2} \).

Answer: \dfrac{C}{r^2}


Question:
Find the value of $m$ that minimizes the mean squared error (MSE) given by the expression:
\[ \frac{2(n-k)\sigma^{4}}{m^{2}} + \left(\frac{(n-k)\sigma^{2}}{m} - \sigma^{2}\right)^{2} \]
Differentiate the expression with respect to $m$ and solve for $m$ to find the minimum.

Answer: n - k + 2


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

Answer: \dfrac{1}{2}


Question: Let \( \{X(t)\}_{t\geq 0} \) be a Poisson process with rate \( \lambda = 2 \). Determine \( E[X(1) \cdot X(2)] \).

Answer: 10


Question: Provide an example of an abelian group $G$ for which the homomorphism group $\operatorname{Hom}(G, \Bbb C \setminus \{0\})$ is not isomorphic to $\operatorname{Hom}(G, \Bbb T)$. Here, $\Bbb C$ is the complex plane and $\Bbb T$ is the unit circle group in the complex plane.

Answer: \mathbb{Z}


Question: In \(\mathbb{R}^3\), consider the 2-form \(\omega = xy \, dx \wedge dy + 2x \, dy \wedge dz + 2y \, dx \wedge dz\). Let \(A = \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2=1, z\geq 0\}\). Show that the integral of \(\omega\) over \(A\) is zero.

Answer: 0


Question: Given $r = \frac{1 + \sqrt{5}}{2}$, if $x, y, z, w$ are integers satisfying \((x + yr)(z + wr) = 1\) and \(x + yr > 0\), prove that there exists an integer $k$ such that \(x + yr = r^k\).

Answer: r^k


Question:
In a convex optimization problem given by:
\[ \min f_0(x) \]
subject to \( f_i(x) \le 0, \) for \( i = 1, \ldots, m, \) and \( Ax = b, \)
which satisfies Slater's constraint qualification, suppose \( x^* \) is an optimal solution with \( f_1(x^*) = -0.2. \) Is it true that for every dual optimal point \( (\lambda^*, \nu^*) \), we have \( \lambda_1^* = 0 \)? Justify your answer.

Answer: Yes


Question: Evaluate the double integral \( \iint_D \frac{\sqrt{x+y}}{\sqrt{x-2y}}~dA \) where \( D \) is the region in \( \mathbb{R}^2 \) enclosed by the lines \( y = \frac{x}{2} \), \( y = 0 \), and \( x+y = 1 \). Use the change of variables \( u = x+y \) and \( v = x-2y \). Calculate the integral in terms of \( u \) and \( v \) and verify if the transformation and the Jacobian \( J = \frac{1}{3} \) are correct.

Answer: \dfrac{1}{3}


Question: Use the Divergence Theorem to compute the flux of the vector field \( F = \langle 0, 0, x+z \rangle \) across the surface \( S \), which is the part of the plane \( x+y+2z=4 \) inside the first octant. Assume the unit normal vector \( n \) to \( S \) has a positive third component. Find the flux of \( F \) across \( S \) by evaluating an appropriate triple integral.

Answer: 16


Question: Find the volume of the solid bounded by the surface given by the equation \(64x^2 - 4y^2 + 16z^2 = 0\) and the plane \(y = 1\).

Answer: \dfrac{\pi}{24}


Question: Find the Laurent series of the function \( f(z) = \frac{1}{z^2} \).

Answer: \frac{1}{z^2}


Question: Find the maximum likelihood estimate of the parameter \(a\) from a single observation with the cumulative distribution function (CDF) given by \(F(x) = \frac{ax^2}{1+ax^2}\) for \(x > 0\).

Answer: \dfrac{1}{x^2}


Question: Compute the limit \( I = \lim_{n \to \infty} \int_{[0,1]^n} \max\{x_1, x_2, \ldots, x_n\} \, dx_1 \, dx_2 \ldots dx_n \).

Answer: 1


Question: Does there exist a finite-dimensional representation of the affine Cremona group $GA_n$ for $n > 1$?

Answer: No


Question: Find the limit: \[ \lim _{n\to \infty }\sum _{k=2}^n\frac{1}{k\cdot \ln(k)} \]

Answer: \infty


Question: In triangle $\Delta ABC$, where $AC > AB$, the internal angle bisector of $\angle A$ meets $BC$ at $D$, and $E$ is the foot of the perpendicular from $B$ onto $AD$. Given that $AB=5$, $BE=4$, and $AE=3$, find the value of $\left(\frac{AC + AB}{AC - AB}\right)ED$. Provide your answer as a single number.

Answer: 3


Question: In a country, 3% of the population has a certain disorder. A test for this disorder has a 97% true positive rate and a 2% false positive rate. If a person takes the test twice independently and receives one positive result and one negative result, what is the probability that they actually have the disorder?

Answer: \dfrac{9}{205}


Question: Given the bivariate probability density function for two independent random variables $X$ and $Y$: \( f(x,y) = 12x^2y^3 \) for \( 0 < x < 1 \) and \( 0 < y < 1 \), and \( f(x,y) = 0 \) otherwise, find the probability that the sample mean of a random sample of size \( n = 50 \) for the random variable $X$ is greater than 0.7.

Answer: 0.9661


Question:
Find the second derivative of the function given by the infinite series:
\[ f(x) = \sum \limits_{n=0}^{\infty} \frac{(-1)^{n+1}2x^{2n+1} }{(2n+1)!} \]

Answer: 2 \sin x


Question: Determine if there exists a regular graph, other than a cycle graph with $n \geq 4$ vertices, such that contracting any edge results in a graph that is still regular.

Answer: Yes


Question: What is the expected value of the area of a rectangle defined by two random points in the unit square?

Answer: \dfrac{1}{9}


Question: Calculate the expected value \( \mathbb{E}(e^{M_n}) \), where \( M_n = \max(X_1, X_2, \ldots, X_n) \) and \( X_1, \ldots, X_n \) are iid random variables each with distribution \( \mathcal{Exp}(1) \). The cumulative distribution function of \( M_n \) is given by \( F_{M_n}(x) = (1 - e^{-x})^n \) for \( x \geq 0 \).

Answer: \infty


Question: Let $f_1, \ldots, f_n \in \mathbb{Q}[x_1, \ldots, x_n]$ be polynomials and let $a = (a_1, \ldots, a_n) \in \mathbb{C}^n$ be a point such that $f_1(a) = \cdots = f_n(a) = 0$, with the Jacobian determinant $J(f_1, \ldots, f_n)(a) \neq 0$. Does this imply that every coordinate $a_i$ of $a$ is algebraic over $\mathbb{Q}$?

Answer: Yes


Question: Let $H$ be a Hilbert space and $x, y$ be self-adjoint compact linear operators acting on $H$. If $x^2 \cdot y^2 = 0$, is it true that $x \cdot y = 0$? Justify your answer.

Answer: 0


Question: Evaluate the integral \( \int f(x) \, \mathrm{d}x \) where \( f(x) = \lim_{n \to \infty} \left[ 2x + 4x^3 + \cdots + 2nx^{2n-1} \right] \) for \( 0 < x < 1 \).

Answer: \dfrac{1}{1 - x^2} + C


Question: Given two positive integers \(a\) and \(b\) that are not multiples of 5, construct a sequence where the first term is 5, and each subsequent term is obtained by multiplying the previous term by \(a\) and adding \(b\). What is the maximum number of prime numbers that can appear in this sequence before the first composite number is encountered?

Answer: 5


Question: Find the volume of the region bounded by the cylinder $x^2 + y^2 = 1$, the planes $x=0$, $z=0$, and $z=y$, within the first octant where $x$, $y$, and $z$ are all positive.

Answer: \dfrac{1}{3}


Question: If both $A - \frac{1}{2}I$ and $A + \frac{1}{2}I$ are orthogonal matrices, which of the following statements is correct?\n\n1. $A$ is orthogonal.\n2. $A$ is a skew-symmetric matrix of even order.\n3. $A^2 = \frac{3}{4}I$.\n\nChoose the correct option.

Answer: 2


Question: Using ropes that burn irregularly, determine the least expensive way to measure exactly 1 minute. A rope that burns in 16 minutes costs 32 rupees, and a rope that burns in 7 minutes costs 14 rupees. You can light the ropes at one end, extinguish them, and relight them as needed. What is the minimum cost to measure exactly 1 minute?

Answer: 46


Question: Evaluate the complex integral \( \int\limits_{\Gamma} |z|^2 \, dz \), where \( \Gamma \) is the boundary of a square with vertices at \( 0, 1, 1+i, i \), traversed anti-clockwise starting at \( 0 \).

Answer: -1 + i


Question: Determine whether the sequence \((x_n)\) defined by \[ x_n = \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} \] converges or diverges.

Answer: \ln 2


Question: Find the limit: \[ \lim_{x \to \infty} \left(\frac{\pi}{2} - \arctan(x)\right)^{\frac{1}{\ln(x)}}. \]

Answer: \dfrac{1}{e}


Question:
In a simulation experiment, two independent observations $X_1$ and $X_2$ are generated from the Poisson distribution with mean 1. The experiment is considered successful if $X_1 + X_2$ is odd. What is the expected value of the number of experiments required to obtain the first success?

(A) 2(1 + $e^{−2}$)
(B) 2/(1 − $e^{-2}$)
(C) 2/(1 − $e^{−4}$)
(D) 2(1 + $e^{−4}$)

Answer: C


Question: If $A$ is a $3 \times 3$ matrix and $A^{T}A$ is diagonal, how many parameters are needed to specify $A$?

Answer: 6


Question:
Let \((a_n),(b_n)\) be sequences of positive real numbers such that \(na_n < b_n < n^2a_n\) for all \(n \geq 2\). If the radius of convergence of \(\sum a_n x^n\) is 4, determine which of the following is true for \(\sum b_n x^n\):

A) converges for all \(|x|<2\)
B) converges for all \(|x|>2\)
C) does not converge for any \(x\) with \(|x|<2\)
D) does not converge for any \(x\) with \(|x|>2\)

Answer: A


Question: Evaluate the integral \( \int_{|z|=1}\frac{dz}{\sin z} \).

Answer: 2\pi i


Question:
Let \( f:(1,4)\to \mathbb{R} \) be a uniformly continuous function and \( \{a_n\} \) be a Cauchy sequence in \((1,2)\). Define the sequences \( x_n = a_n^2 f(a_n^2) \) and \( y_n = \frac{1}{1+a_n^2} f(a_n^2) \). Determine which of the following statements is correct:

(A) Both \( \{x_n\} \) and \( \{y_n\} \) are Cauchy sequences in \( \mathbb{R} \).
(B) \( \{x_n\} \) is a Cauchy sequence in \( \mathbb{R} \) but \( \{y_n\} \) is not a Cauchy sequence in \( \mathbb{R} \).
(C) \( \{y_n\} \) is a Cauchy sequence in \( \mathbb{R} \) but \( \{x_n\} \) is not a Cauchy sequence in \( \mathbb{R} \).
(D) Neither \( \{x_n\} \) nor \( \{y_n\} \) are Cauchy sequences in \( \mathbb{R} \).

Answer: A


Question: Determine if the equation \(2^{x_1} + 2^{x_2} - 2^{x_3} = 2^y\) has any natural number solutions other than the trivial solution where \(x_1 = x_2 = x_3 = y\).

Answer: Yes


Question: In the spectrum of a commutative ring \( A \) with unity, is the closure of the set \( \{p\} \), where \( p \) is a prime ideal in Spec \( A \), equal to \( V(p) \), the set of all prime ideals containing \( p \)?

Answer: V(p)


Question: Find the value of \(a\) such that \[ \lim_{n\rightarrow \infty} \frac{1^a + 2^a + 3^a + \cdots + n^a}{(n+1)^{a-1}\left( (na+1)+(na+2) + \cdots +(na+n)\right)} = \frac{1}{60}. \]

Answer: 7


Question: Given a matrix $A \in \mathbb{R}^{n \times n}$ with spectral radius $\rho(A)<1$, determine if it is possible for the Euclidean norm $||A||$ to be greater than 1.

Answer: Yes


Question: Let $X_t, t \geqslant 0,$ be a Brownian motion. Define the stopping times $T_a := \inf \{t \mid X_t = a\}$. Calculate the probability $\mathbb{P}\{T_{2}< T_{-1} < T_{3}\}$.

Answer: \dfrac{1}{12}


Question: Given a random variable \( X \) with a geometric distribution \( f_{\theta}(x) = (1-\theta) \cdot \theta^x \) for \( x = 0, 1, \dots \), and a prior distribution \( \pi(\theta) = 3\theta^2 \) for \( \theta \in (0,1) \), find the Bayes estimator of \( \theta \) based on the observation \( X=0 \), which is \( E(\Theta|X=0) \).

Answer: \dfrac{3}{5}


Question: Calculate the limit \( \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \left\{\frac{k}{\sqrt{3}}\right\} \), where \( \{\cdot\} \) denotes the fractional part.

Answer: \dfrac{1}{2}


Question: Can there exist an immersion $f: M^n \rightarrow N^m$ if $n > m$, where $M$ and $N$ are manifolds with dimensions $n$ and $m$, respectively?

Answer: No


Question: Evaluate the line integral \( \int_{|z|=2}\frac{\overline{z}}{1-z}\,dz \).

Answer: 0


Question: Evaluate the triple sum: \[ \sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\sum_{p=1}^{\infty}\frac{1}{mnp(m+n+p+1)}. \]

Answer: 6


Question: Let $T: \mathscr{D}(T) \rightarrow H$ be an operator on a complex Hilbert space. If for some $\lambda$, the operator $S(I-\lambda S)^{-1}$ is a bounded inverse for $T-\lambda$, does it follow that $S$ is an inverse for $T$?

Answer: Yes


Question: Determine whether the series \( S_{N}=\sum_{n=0}^{N-1}\frac{c_{n}^{N}}{c_{N}^{N}} \) is convergent, where \( c_{n}^{N} \) is the coefficient of \( x^{n} \) in the Chebyshev polynomial \( T_{N}(x) = \sum_{n=0}^{N}c_{n}^{N}x^n \).

Answer: -1


Question: Evaluate the integral \( I = \int_{0}^{a}\frac{1}{1+f(x)}\,dx \) given that \( f(x)f(a-x)=1 \), where \( a>0 \) and \( f(x) \) is continuous and positive on the interval \([0,a]\).

Answer: \dfrac{a}{2}


Question: A fair die is tossed repeatedly until the last six outcomes form the pattern 131131. What is the expected number of rolls needed to achieve this pattern?

Answer: 46878


Question: Evaluate the limit: $$\lim_{n\rightarrow\infty}\int_0^1\frac{(\lambda_1+\lambda_2x)^n-x^n}{1-x}\,dx,$$ where $\lambda_1+\lambda_2=1$ and both $\lambda_1$ and $\lambda_2$ are positive real numbers.

Answer: -\ln \lambda_2


Question:
Let $\mu$ be a Borel probability measure on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$. Suppose $\mu$ is "approximately uniform" in the sense that there exists a constant $C > 0$ such that
$$
C^{-1} \ \leq \ \frac{\mu(B_r(x))}{\mu(B_r(y))} \ \leq \ C
$$
for all $x,y \in \mathbb{T}$ and all $r>0$, where $B_r(x)$ is the ball of radius $r$ centered at $x$. Does this imply that $\mu$ is absolutely continuous with respect to Lebesgue measure on $\mathbb{T}$? Provide a justification for your answer.

Answer: Yes


Question: How many times must a fair coin be tossed to be at least 99% confident that the percentage of heads will fall between 49.5% and 50.5%?

Answer: 66358


Question: You roll a fair six-sided die repeatedly until you roll a 5. What is the expected value of the maximum number rolled before the first 5 appears? Exclude the roll of 5 from consideration.

Answer: \dfrac{81}{20}


Question: Given a tetrahedron $OABC$ with vectors $\vec{OA} = \vec{a}$, $\vec{OB} = \vec{b}$, and $\vec{OC} = \vec{c}$, let $M$, $N$, and $P$ be the midpoints of segments $AC$, $AB$, and $BC$, respectively. Find the ratio between $\vert \vec{OM} \cdot (\vec{ON} \times \vec{OP}) \vert$ and the volume $V$ of the tetrahedron.

Answer: \dfrac{3}{2}


Question: Given that $P_n(x) = 3x^n + Q(x)$ is the interpolation polynomial of $f$ at the nodes $x_0, \ldots , x_n$, where $Q(x)$ is a polynomial of degree at most $n - 1$, find the divided difference $f[x_0, \ldots , x_n]$. Provide your answer in terms of the given polynomial form.

Answer: 3


Question:
Calculate the circumference of the unit circle, defined by the equation \(x^2 + y^2 = 1\), using the limit-based formula for arc length:
\[
\lim\limits_{h\rightarrow 0} \sum\limits_{r=1}^{\frac{b-a}{h}} \sqrt{\left\{ f\Big(a+rh\Big)-f\Big(a+(r-1)h\Big)\right\}^2 +h^2}
\]

Answer: 2\pi


Question: Determine the number of integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 = 32\) subject to the constraints \(0 \le x_i \le 10\) for each \(i = 1, 2, 3, 4\).

Answer: 165


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

Let \(f_n \colon U\subset\mathbb{C}\longrightarrow \mathbb{C}\) be a sequence of continuous (or holomorphic) functions such that:
- \(\sum f_n\) is absolutely convergent
- \(\sum f_n\) is uniformly convergent

Then \(\sum |f_n|\) is uniformly convergent.

Answer: False


Question: Find all positive integer solutions \((x, y)\) to the equation \(1 + 4^x = 5^y\).

Answer: (1, 1)


Question: Let $G = \langle g_1, g_2 \rangle$ be a group, and let $H = \langle g_1 g_2 g_1^{-1} g_2^{-1} \rangle$ be a subgroup of $G$. Determine the normal subgroup of $G$ generated by $H$.

Answer: [G, G]


Question: Calculate the integral \( \int_1^2 \frac{1}{x^2} \, dx \) using Darboux sums with the partition \( 2^{\frac{i}{n}} \) for \( 0 \leq i \leq n \).

Answer: \dfrac{1}{2}


Question: Calculate the Euler characteristic of a surface of genus $k$ by triangulating it. Use the formula \( \chi = V - E + F \), where \( V \) is the number of vertices, \( E \) is the number of edges, and \( F \) is the number of faces in the triangulation. Assume the surface is a sphere with $k$ tubes sewn in.

Answer: 2 - 2k


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

Answer: 2


Question: In a barn, 100 chicks sit in a circle. Each chick randomly pecks either the chick immediately to its left or right. What is the expected number of chicks that remain unpecked?

Answer: 25


Question: Determine whether the sequence \( \sqrt[n]{n!} \) converges or diverges as \( n \to \infty \).

Answer: \infty


Question: Is it possible to construct an error correction code in the finite field $F_8$ that can correct $n$ errors, where each error is an increment or decrement by 1?

Answer: Yes


Question:
Let \(\omega\) be a complex number such that \(\omega^5 = 1\) and \(\omega \neq 1\). Evaluate the expression:
\[
\frac{\omega}{1 - \omega^2} + \frac{\omega^2}{1 - \omega^4} + \frac{\omega^3}{1 - \omega} + \frac{\omega^4}{1 - \omega^3}.
\]

Answer: 0


Question: Determine the supremum of the set of values $A$ such that \( I_n = \int_{0}^{1}\cos\left(\frac{1}{x}+nx\right)\,dx = O(n^{-A}) \) as \( n \to +\infty \).

Answer: \dfrac{3}{4}


Question: If \(a,b \in \mathbb{R}\) satisfy \(a^2+2ab+2b^2=7,\) find the largest possible value of \(|a-b|\).

Answer: \sqrt{35}


Question: Determine the number of matrices of the form \( \begin{bmatrix} a & a+1 \\ 1-a & -a \end{bmatrix} \) where \( a \) is an integer in the range \([-24, 24]\).

Answer: 49


Question: Determine if $x = 2$ is the only real solution to the equation $a^x + b^x = c^x$ when $(a, b, c)$ is a Pythagorean triplet.

Answer: 2


Question: Find the 3-adic expansion of \( \frac{1}{2} \) in \( \mathbb{Q}_3 \).

Answer: \ldots 1112


Question:
Evaluate the limit:
\[ \lim_{x \to 0}{\frac{-1 + \sqrt{(\tan x - \sin x) + \sqrt{(\tan x - \sin x) + \sqrt{(\tan x - \sin x) + \cdots \infty}}}}{-1 + \sqrt{x^3 + \sqrt{x^3 + \sqrt{x^3 + \cdots \infty}}}}} = \frac{1}{k} \]
Find the value of \( k \).

Answer: 2


Question: Consider a field $k$ with characteristic $\text{char}(k) = p$, where $p$ does not divide the order of a group $G$, $p \nmid |G|$. Are two representations of $G$ over $k$ isomorphic if and only if they have the same Brauer character?

Answer: Yes


Question: Prove that the group of automorphisms of \( \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \) is isomorphic to the symmetric group \( S_3 \).

Answer: S_3


Question: Determine the fundamental group of the space \(\mathbb{CP}^1\) minus a finite set of points \(\{a_1, \ldots, a_n\}\), where \(a_1, \ldots, a_n \in \mathbb{C}\). What is \(\pi_1(\mathbb{CP}^1 \setminus \{a_1, \ldots, a_n\})\)?

Answer: F_{n-1}


Question: Prove that if an even function $f(z)$ is analytic near 0, then there exists an analytic function $h(z)$ such that $f(z) = h(z^2)$. Justify your reasoning.

Answer: h(z^2)


Question: Determine whether there exists a triple of distinct numbers \(a, b, c\) such that \((a-b)^5 + (b-c)^5 + (c-a)^5 = 0\).

Answer: Yes


Question: Determine whether there are infinitely many Fibonacci numbers that are also powers of 2. If not, find the largest Fibonacci number that is a power of 2.

Answer: 8


Question: Let $(\tilde{X},p)$ be a universal covering space of $X$, and let $G$ be a subgroup of $\pi_1(X)$. Consider the canonical map $q : \tilde{X}/G \to X$ corresponding to the injection $G \to \pi_1(X)$. Is $(\tilde{X}/G,q)$ a covering space of $X$?

Answer: Yes


Question: Let \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \) and \( r = \|\vec{r}\| \). Define \( \vec{F} = r^p \vec{r} \). Find all values of \( p \) for which \( \operatorname{div} \vec{F} = 0 \).

Answer: -3


Question: Given that \( \frac{a}{b+c+d} + \frac{b}{a+c+d} + \frac{c}{a+b+d} + \frac{d}{a+b+c} = 1 \), find the value of \( \frac{a^2}{b+c+d} + \frac{b^2}{a+c+d} + \frac{c^2}{a+b+d} + \frac{d^2}{a+b+c} \).

Answer: 0


Question:
Determine the curve on which the point of intersection of tangents drawn to the parabola \((x-1)^2=8(y+1)\), which are perpendicular to the variable line \(y=px-2p^2-p-1\), lies. Choose the correct option:

A) \(y+3=0\)

B) \(y=3\)

C) \(x=3\)

D) \(x+3=0\)

Answer: A


Question: Solve for \(x\) in the equation: \[(x + 1)^{63} + (x + 1)^{62}(x−1) + (x + 1)^{61}(x−1)^{2} + \, . \, . \, . \, + (x−1)^{63} = 0.\]

Answer: 0


Question: Does there exist a continuous function \( f: [0,1] \rightarrow [0,1] \) such that \( f \) takes every value in \([0,1]\) an infinite number of times?

Answer: Yes


Question: Evaluate the limit \( \lim_{n \to \infty} \frac{1}{\sqrt{n^k}} \left(1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}}\right)^k \) for a given integer \( k \) using the Stolz–Cesàro theorem.

Answer: 2^k


Question: Given that $A$, $B$, and $A + B$ are all invertible $3 \times 3$ matrices satisfying $A^{-1} + B^{-1} = (A + B)^{-1}$, and $|AB^{-1}| \in \mathbb{R}$, find the value of $\frac{|A|}{|B|}$.

Answer: 1


Question: Given the equation \( p = \lambda e^{-\lambda} \), where \( p \) and \( e \) are known constants, calculate the value of \( \lambda \).

Answer: -W(-p)


Question: Given that $a, b, c$ are positive real numbers satisfying the equation \( a^{2}(1 + b^{2})+ b^{2}(1 + c^{2}) + c^{2}(1 + a^{2}) = 6abc \), find the value of $a + b + c$.

Answer: 3


Question: Evaluate the limit: \[ \lim_{n\to \infty} \left(1-\frac{1}{n^2-4}\right)^{3n^2+5} \]

Answer: e^{-3}


Question: Find the generator of the cyclic group \( \mathbb{Z}/ m\mathbb{Z} \times \mathbb{Z}/ n\mathbb{Z} \) where \( m \) and \( n \) are relatively prime. Express the generator in the form \((x, y)\).

Answer: (1, 1)


Question: Evaluate the infinite series \( \sum_{i=0}^\infty \frac{i^3}{4^i} \).

Answer: \dfrac{44}{27}


Question: Evaluate the limit \( \lim_{n\rightarrow +\infty }\left ( 2\sqrt[n]{x}-1 \right )^n \) for a real number \( x \geq 1 \).

Answer: x^2


Question: Evaluate the surface integral \( \iint_S \left( \frac{2x}{\pi} + \sin(y^2)\right)\hat{x} + \left(e^z - \frac{y}{\pi}\right)\hat{y} + \left(\frac{2z}{\pi} + \sin^2y\right)\hat{z} \ d\sigma \) over the unit sphere \( S = \{(x,y,z)\in \mathbb{R}^3:x^2+y^2+z^2=1\} \).

Answer: 4


Question: Calculate the Lebesgue integral of the function \( f(x) = \frac{1}{x} \) over the interval \([0, 1]\).

Answer: \infty


Question: Given a natural number \( k \geq 1 \), determine if there always exists an odd natural number \( n > k \) such that for any \( k \) pairwise different boolean vectors \( v_1, v_2, \ldots, v_k \in \mathbb{Z}_2^n \) with Hamming distance \( d(v_i, v_j) < n \) for all \( i \neq j \), there is a vector \( w \in \mathbb{Z}_2^n \) with \( 0 < d(w, v_i) \leq \frac{n-1}{2} \) for all \( i \).

Answer: Yes


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

Answer: \dfrac{1}{e}


Question:
Determine which of the following functions are isomorphisms:

A. The function \( T: U^{2\times 2} \to P_2 \) given by \( T(\begin{bmatrix}a&b\\ 0&c\end{bmatrix})=(a+b)+ct+(a+b)t^2. \)

B. The function \( T: \mathbb{C} \to \mathbb{R}_2 \) given by \( T(a+bi)=\begin{bmatrix}a\\ a+b\end{bmatrix} \)

C. The function \( T: U^{2\times 2} \to P_2 \) given by \( T(\begin{bmatrix}a&b\\ 0&c\end{bmatrix})=a+bc t+ bt^2. \)

D. The function \( T: \mathbb{R}_3 \to P_3 \) given by \( T(\begin{bmatrix}a\\ b\\ c\end{bmatrix})= at+bt^2+ct^3. \)

An isomorphism of vector spaces is a bijective (or invertible) linear transformation. Vector spaces \( V \) and \( W \) are isomorphic if there exists an isomorphism \( V \to W \).

Answer: B


Question: Minimize the expression \( \left( \frac{2}{a^2} + \frac{1}{b^2} \right)(a^2 + b^2) \).

Answer: 3 + 2\sqrt{2}


Question: Determine whether the following statement is true or false: Any set $A \subseteq \mathbb{R}^k$ with positive $k$-dimensional Lebesgue measure contains a bounded subset $K \subseteq A$ with positive measure.

Answer: True


Question: Given that the polynomial \( W(x) \) has a degree of 2015 and satisfies \( W(n) = \frac{1}{n} \) for \( n = 1, 2, \ldots, 2016 \), calculate \( W(2017) \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} f(x)\tan^n(x)\,\mathrm{d}x \] where \( f \in L^1(\mathbb{R}) \).

Answer: 0


Question: Determine the point through which every plane tangent to the surface \( z = xe^{\frac{y}{x}} \) passes.

Answer: (0, 0, 0)


Question: Given 10 voters and 2 candidates, each voter has a probability $p$ of voting for Candidate 1, where $p$ follows a Beta(3,3) distribution. Calculate the probability that exactly 6 voters will vote for Candidate 1.

Answer: \dfrac{20}{143}


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{-\sin^{2}{x}} \cos\left(6x - \frac{\sin{2x}}{2}\right) \, dx \).

Answer: \dfrac{\pi}{24 \sqrt{e}}


Question: Given a vector of rational numbers \(\vec{X} \in \mathbb{Q}^d\), find the smallest integer vector \(\vec{n} \in \mathbb{Z}^d\) such that \(\vec{n} \cdot \vec{X} = 0\) and \(|\vec{n}|\) is minimized. In other words, solve the equation \(n_1 X_1 + n_2 X_2 + \cdots + n_d X_d = 0\) with the condition of minimizing \(n_1^2 + n_2^2 + \cdots + n_d^2\). If there is no unique solution, any one such vector is acceptable.

Answer: \vec{n}


Question: Given the joint probability density function \( f_{X,Y}(x,y) = \begin{cases} 2 e^{-(x+y)} & \text{for } 0 < x < y < \infty \\ 0 & \text{otherwise} \end{cases} \), find the probability \( P(2X < Y) \).

Answer: \dfrac{2}{3}


Question:
Determine whether the expression $P_1 \land P_2$ implies $P_3 \land P_2$ or simply $P_3$, where:

- $P_1$ is the conjunction $$(x_{11} \le x_{21}) \land (x_{12} \le x_{22}) \land \dots \land (x_{1n} \le x_{2n})$$
- $P_2$ is the disjunction $$(x_{11} < x_{21}) \lor (x_{12} < x_{22}) \lor \dots \lor (x_{1n} < x_{2n})$$
- $P_3$ is the conjunction $$\lnot(x_{11} > x_{21}) \land \lnot(x_{12} > x_{22}) \land \cdots \land \lnot(x_{1n} > x_{2n})$$

Answer: P_3 \land P_2


Question:
Let $f(x)$ and $g(x)$ be periodic functions with periods 5 and 10, respectively, such that \( \lim_{x\to0}\frac{f(x)}{x}=\lim_{x\to0}\frac{g(x)}{x}=k \) where \( k>0 \). Determine the value of:
\[ \lim_{n\to\infty}\frac{f(5(4+\sqrt{15})^n)}{g(10(4+\sqrt{14})^n)}. \]

Answer: 0


Question:
Let $X$ be a projective variety and $F \in \overline{NE}(X)$ an extremal face. A morphism $f: X \longrightarrow Y$ is called the contraction of $F$ if the following conditions hold:

- $f(C) = \text{point}$ for an irreducible curve $C \subset X$ if and only if $[C] \in F$.
- $f_*(\mathcal{O}_X) = \mathcal{O}_Z$.

Is the contraction morphism $f$ proper?

Answer: Yes


Question: Let \( H \) be the collection of all absolutely continuous functions \( f: [0,1] \to \mathbb{C} \) such that \( f(0) = 0 \) and \( f' \in L^2(0,1) \). Define the inner product \( \langle f, g \rangle = \int_0^1 f'(t) \overline{g'(t)} \, dt \) for \( f \) and \( g \) in \( H \). Fix \( t \), \( 0 < t \leq 1 \), and define \( L: H \to \mathbb{C} \) by \( L(h) = h(t) \). Find the operator norm \( \|L\| \) and a function \( h_0 \) such that \( L(h) = \langle h, h_0 \rangle \) for every \( h \) in \( H \).

Answer: \sqrt{t}


Question: Is it possible to have an isomorphic copy of an ordinal \(\alpha\) inside an ordinal \(\beta < \alpha\)? Specifically, could there exist an order-preserving and injective function \(f: \alpha \to \beta\) where \(\alpha\) and \(\beta\) are ordinal numbers with \(\alpha > \beta\)?

Answer: No


Question: Calculate \( \lim_{n \rightarrow \infty} \int_0^\infty \frac{x^n}{ x^{(n+3)}+1} \, dx \).

Answer: \dfrac{1}{2}


Question: Determine whether the following statement is true: \( \sum_{k=0}^{n}k\cdot \binom{n}{k}=O\left(2 ^ {n\log _{3}n}\right) \).

Answer: True


Question: Determine whether the statement is true: A compact non-orientable manifold $M^n$ must have $H_{dR}^n(M)=0$. Provide a justification for your answer.

Answer: True


Question: Evaluate the expression \( \sqrt{-1}^{\sqrt{-1}} \).

Answer: e^{-\pi/2}


Question: Find the limit: \[ \lim_{n \rightarrow \infty} \sqrt[n]{ \left( 3^n-2^n+\frac{1}{2^n} \right ) x^n } \]

Answer: 3x


Question: A model for world population assumes a population of 6 billion at reference time 0, with population increasing to a limiting population of 30 billion. The model assumes that the rate of population growth at time \(t \geq 0\) is \(\frac{Ae^t}{(.02A+e^t)^2}\) billion per year, where \(t\) is regarded as a continuous variable. According to this model, at what time will the population reach 10 billion (nearest 0.1)?

Answer: 0.3


Question: Determine the number of distinct real roots of the equation \( P(P(\cdots P(x))\cdots) = 0 \), where the polynomial \( P(x) = x^2 - 1 \) is applied 2018 times.

Answer: 2019


Question: Calculate the volume of the torus formed by rotating the circle defined by \(x^2 + y^2 = 1\) about the line \(y = 2\).

Answer: 4\pi^2


Question: How many distinct necklaces can be made using 11 beads, where each bead can be a circle, square, or triangle? Necklaces are considered identical if they can be rotated into each other, but not if they are mirror images.

Answer: 16107


Question: Let $G$ be a finite abelian group of order $n=p_1^{a_1}\cdot \cdot \cdot p_k^{a_k}$ and $H$ a subgroup of $G$ of order $m=p_1^{b_1}\cdot \cdot \cdot p_k^{b_k}$. Given that $G \cong A_1 \times \cdot \cdot \cdot \times A_k$ where $|A_i|=p_i^{a_i}$, determine if it is possible to write $H \cong B_1\times \cdot \cdot \cdot \times B_k$ where $|B_i|=p_i^{b_i}$ and $B_i \leq A_i$ for each $i$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{\sin(x)\arcsin(x)-x^2}{x^6} \] using L'Hopital's Rule.

Answer: \dfrac{1}{18}


Question: How many permutations of the numbers $1, 2, ..., 10$ exist such that no even number is mapped to itself?

Answer: 2170680


Question: Evaluate the line integral \( C = \oint_K d\mathbf{r} \cdot \mathbf{A} \) where \( \mathbf{A} = \frac{1}{2} \mathbf{n} \times \mathbf{r} \), \( \mathbf{n} \cdot \mathbf{n} = 1 \), and \( K \) is a circle with radius \( R \) in the plane normal to \( \mathbf{n} \). Use Stokes' Theorem to verify your result.

Answer: \pi R^2


Question: Calculate the Euler characteristic of a topological space formed by joining two projective planes, each with two punctures, along their punctures.

Answer: -2


Question: Evaluate the integral \( \int_{-1}^{1} (4t^4 - 4t^2 - 1) e^{-{(t^2 -1)}^2} \, dt \).

Answer: -2


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

Answer: e


Question: Find the minimum value of the expression \( S = \sum \frac{\sqrt{a}}{\sqrt{b} + \sqrt{c} - \sqrt{a}} \), where \( a, b, c \) are the sides of a triangle.

Answer: 3


Question: Determine the number of normal subgroups of the alternating group $A_4$.

Answer: 3


Question: Does there necessarily exist an order embedding from an arbitrary poset \( \mathfrak{A} \) into some complete lattice \( \mathfrak{B} \) that preserves all suprema and infima defined in \( \mathfrak{A} \)?

Answer: Yes


Question: Compute the limit \( \lim_{t \rightarrow 0} (ax^t + by^t)^{\frac{1}{t}} \) given that \(a, b, x, y\) are all positive and \(a + b = 1\).

Answer: x^a y^b


Question: In a number system with an unknown base, the quadratic equation \(3 x^2 - 25 x + 66 = 0\) has roots \(x_1 = 4\) and \(x_2 = 9\). Determine the base of this number system.

Answer: 17


Question: Show that the process \( e^{2B_t - 2t} \) is a martingale with respect to the natural filtration of the Brownian motion \( B_t \).

Answer: e^{2B_t - 2t}


Question:
Determine the value(s) of \( m \) for which the function \( f: \mathbb{Q} \to \mathbb{R} \) defined by \( f(x) = \{3x^2 + mx + n\} \) is injective, where \( n \in \mathbb{R} \). Choose from the following options:

A: \( m = -7 \)
B: \( m = 0 \)
C: \( m = 2 \)
D: \( m = 2 \cdot 3^{1/3} \)
E: \( m = 4, 9 \)

Answer: D


Question: Determine the radius of convergence of the complex series \( \sum_{n=0}^{\infty} \sqrt{n}(4 + (-1)^n)z^n \).

Answer: 1


Question: How many natural numbers \( x \) satisfy \( 3 \cdot 10^{13} < x < 10^{14} \) such that each digit from 1 to 9 appears exactly once in \( x \) and no zeros are adjacent to each other?

Answer: 35562240


Question: Find the maximum value of \(|z|\) for a complex number \(z\) such that \[ \left|\frac{6z - i}{2 + 3iz}\right| \leq 1. \]

Answer: \dfrac{1}{3}


Question: Let $(K, \nu)$ be a valued field and $x$ be transcendental over $K$. Does there exist a henselian extension of $(K, \nu)$ between $(K, \nu)$ and $(K(x), \nu')$, where $\nu'$ is an arbitrary extension of $\nu$?

Answer: No


Question: Evaluate the limit: \[ \lim_{t \to \infty} \frac{1}{t^2}\int_0^t\ln(2e^x+x^2)dx \]

Answer: \dfrac{1}{2}


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function satisfying the equation \( f(x) + f(x+1) = x^2 + 2x + 3 \). Find the value of \( \int_0^2 f(x) \, dx \).

Answer: \dfrac{13}{3}


Question: Find the general solution to the ordinary differential equation \( x\frac{dy}{dx} = y - \frac{1}{y} \).

Answer: y = \pm \sqrt{1 + C x^2}


Question: Let $p$ and $q$ be two distinct odd prime numbers such that $p \equiv 1 \pmod{8}$, $q \equiv 3 \pmod{4}$, and $\left(\frac{p}{q}\right) = 1$. Determine if there exists a prime $t \neq p, q$ such that both $\left(\frac{p}{t}\right) = 1$ and $\left(\frac{q}{t}\right) = 1$. Provide a justification for your answer.

Answer: Yes


Question: In triangle \( ABC \), \( \angle ABC = 20^\circ \) and \( AB = BC \). Points \( D \) and \( E \) are on segments \( AB \) and \( BC \) respectively, such that \( AD = BE = AC \). Determine the measure of angle \( \angle BDE \).

Answer: 20^\circ


Question: Given the equations \(3^{\log_a x} + 3x^{\log_a 3} = 2\) and \(x = y^{-\log_z a}\), find the value of \(y + z\).

Answer: 5


Question: Solve the inequality: \[ \frac{1}{2}\log_{\frac{1}{2}}(x-1) > \log_{\frac{1}{2}}\left(1-\sqrt[3]{2-x}\right) \]

Answer: (2, 10)


Question: If $f: U \rightarrow \mathbb{R}^m$ is a map of class $C^2$ defined on a compact $U \subset \mathbb{R}^n$ with non-empty interior, is the map $Df : U \rightarrow L(\mathbb{R}^n, \mathbb{R}^m)$ Lipschitz? (Here, $L(\mathbb{R}^n, \mathbb{R}^m)$ is a normed vector space with the operator norm.)

Answer: Yes


Question: In a game involving two students, A assigns integer numbers to the faces of 3 fair dice. B then chooses one die, and A selects one of the two remaining dice. They roll their chosen dice, and the winner is the student with the larger number. Determine which student has a better chance of winning.

Answer: A


Question:
Maximize the objective function \( 24x_1 + 22x_2 + 45x_3 \) subject to the constraints:
\[\begin{align*}
2x_1 + x_2 + 3x_3 &\leq 42, \\
2x_1 + x_2 + 2x_3 &\leq 40, \\
x_1 + \frac{1}{2}x_2 + x_3 &\leq 45, \\
x_1, x_2, x_3 &\geq 0.
\end{align*}\]
Solve this problem by inspection without using Gauss-Jordan row operations.

Answer: 882


Question: Consider a measurable space $(\Omega,\mathscr{F})$ with two probability measures $P$ and $Q$, where $Q$ is absolutely continuous with respect to $P$ and the Radon-Nikodym derivative is $\mathrm{d}Q/\mathrm{d}P=f$. Let $E^P$ and $E^Q$ denote the expectations with respect to $P$ and $Q$, respectively. Suppose $X$ is an integrable random variable and $\mathscr{G}\subset\mathscr{F}$ is a sub $\sigma$-algebra. It is known that the equation $$E^Q(X|\mathscr{G})E^P(f|\mathscr{G})=E^P(Xf|\mathscr{G})$$ holds $Q$-almost surely. Determine if this equation also holds $P$-almost surely.

Answer: Yes


Question: Let $F$ be an ordered field and $A$ a convex subring of $F$, which is a valuation ring and thus a local ring with maximal ideal $m_A$. For any subfield $C$ of $F$ contained in $A$ and maximal with respect to these conditions, the residue field $F_A = A / m_A$ is algebraic over the image $C_A$ of $C$ under the residue map $A \rightarrow A / m_A$. In this context, $F_A$ is ordered by $a + m_A < a' + m_A \Longleftrightarrow a < a'$. If $F$ is real closed, then $C_A$ is real closed and $C_A = F_A$. In general, $C_A$ may be a proper subfield of $F_A$. For example, if $F$ is an ultrapower of $\mathbb{Q}$ modulo some free ultrafilter on $\mathbb{N}$ and $A$ is the ring of finite and infinitesimal elements, then $F_A \cong \mathbb{R}$ while any maximal subfield of $A$ may contain no algebraic irrational number. Is $C_A$ always dense in $F_A$?

Answer: Yes


Question: Determine the parameter $b$ for a Bernoulli random variable $Y$ such that the sum $X + Y$ of two Bernoulli random variables $X$ and $Y$ is also a Bernoulli random variable with parameter $p + b$. Given that $X$ is a Bernoulli random variable with parameter $p$, find the value of $b$.

Answer: 1 - p


Question: Determine the number of permutations of length 20 whose longest cycle is of length 11.

Answer: \dfrac{20!}{11}


Question: Determine whether the inclusion \( \mathbb{Z} \subseteq \mathbb{Z}[X]/(X^5-1) \) is an integral extension.

Answer: Yes


Question: Evaluate the limit using the Stolz-Cesàro theorem: \[ \lim_{n\to \infty} \frac{1+2\sqrt{2}+3\sqrt{3}+\ldots+n\sqrt{n}}{n^2 \sqrt{n}}. \]

Answer: \dfrac{2}{5}


Question: Determine the values of \( p \) for which the integral \( \int_{0}^{\pi/2} (\tan x)^p \,{\rm d}x \) converges.

Answer: (-1, 1)


Question: Does the statement that every cover has a subcover that admits a choice set imply the Axiom of Choice?

Answer: Yes


Question: Find the constant \( a \) such that the process \( M(t) = (aB(t) - t) \exp(2B(t) - 2t) \) is a martingale, where \( B(t) \) is a standard Brownian motion.

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \( \lim_{x\rightarrow \infty} \left( x \ln x + 2x \ln \sin \left(\frac{1}{\sqrt{x}} \right) \right) \)

Answer: -\dfrac{1}{3}


Question: Determine the least upper bound of the set of order types of countable unions of finite sets of ordinals in ZFC.

Answer: \omega_1


Question: Given a polynomial \( f(x) = a_{1} x^n + a_{2} x^{n-1} + \cdots + a_{n} \), if we take \(2n\) points from this polynomial and perform polynomial interpolation to find a polynomial of degree \(2n-1\), is the resulting polynomial equal to the original polynomial \( f(x) \)?

Answer: Yes


Question: Evaluate the flux integral \(\iint_S\operatorname{curl}(\vec{F})  \cdot d\vec{S}\) for the vector field \(\vec{F}(x,y,z) = \langle(x^9 + y^7)z^5, x, y \rangle\), where \(S: \frac{x^2 + y^2}{16} + z^8 = 1, \ z\ge 0\) and is oriented upwards, using Stokes' Theorem.

Answer: 16\pi


Question: Two fair 6-sided dice are rolled, and the sum of the outcomes is noted. If the dice are rolled again, in how many ways can the new sum be less than the previous sum?

Answer: 575


Question: Verify Stoke's Theorem for the vector field \( \mathbf{v} = z\mathbf{i} + x\mathbf{j} + y\mathbf{k} \) over the hemispherical surface defined by \( x^2 + y^2 + z^2 = 1 \) with \( z > 0 \). Compute both the line integral \( \int_C \mathbf{v} \cdot d\mathbf{r} \) and the surface integral \( \int \int_S (\nabla \times \mathbf{v}) \cdot \mathbf{n} \, dS \) and verify if they are equal.

Answer: \pi


Question: Determine the values of \(x\) for which the series \(\sum_{n \geq 1} f\left( \frac{x}{n^2}\right)\) converges, given that \(f: \mathbb{R} \to \mathbb{R}\) is differentiable, \(f(0)=0\), and \(f'\) is bounded.

Answer: \mathbb{R}


Question: Does Goldbach's Conjecture imply that every natural number \(\geq 4\) is centered between a pair of primes or is itself prime?

Answer: Yes


Question: Let \( D \subset \mathbb{R}^3 \) be the set \( D = \{(x, y, z) \mid x^2 + y^2 = 1~\text{and}~z = 0\} \). Let \( T :  \mathbb{R}^3 \to \mathbb{R}^3 \) be the linear transformation defined by \( T(x, y, z) = (2x − y + z, x − y + 10z, 3x − 7y + 2z) \). Find the area of \( T(D) \).

Answer: \pi \sqrt{138}


Question:
Calculate the power over a period for a square wave represented by its Fourier series. Given the voltage function:
\[ v(t) = \frac{4V}{\pi} \left( \sin(\omega t) + \frac{1}{3}\sin(3\omega t) + \frac{1}{5}\sin(5\omega t) + \ldots \right) \]
where \( \omega \) and \( V \) are constants, find the power \( P \) over a period \( T \) using the formula:
\[ P = \frac{1}{T} \int_{0}^{T} \frac{v(t)^2}{R} \, dt \]
where \( R \) is a constant resistance.

Answer: \dfrac{V^2}{R}


Question: In a coffeehouse, the number of guests follows a Poisson distribution with parameter 10, i.e., $X \sim \text{Poi}(10)$. Each guest independently drinks a Poisson(1) number of cups of coffee and raises their hand a Poisson(1) number of times. The number of guests, the number of cups of coffee each guest drinks, and the number of hand raises each guest performs are all independent. Given that no cups of coffee have been sold, calculate the expected value of the total number of hand raises.

Answer: \dfrac{10}{e}


Question: Find the coefficient $b_1$ in the expression \( \sin n\theta = \sum_{r=0}^{n} b_r \sin^r\theta \), where \( n \) is an odd integer.

Answer: n


Question: Determine the value of $R > 0$ such that the flux of the vector field $F(x,y,z) = (zx - x \cos(z), -zy + y \cos(z), 4 - x^2 - y^2)$ through the surface $S = \{(x,y,z) : x^2 + y^2 + (z-1)^2 = R^2, z \geq 1\}$ is maximized. The surface $S$ is oriented with the normal vector having a positive third coordinate.

Answer: 2


Question: Given a vector space \( \mathbb{C}^n \) over \( \mathbb{R} \) and a linear operator \( T : \mathbb{C}^n \rightarrow \mathbb{C}^n \), determine the field to which an eigenvalue \( \lambda \) of \( T \) belongs.

Answer: \mathbb{R}


Question: Evaluate the contour integral \( \oint_C \frac{3z^3 + 2}{(z-1)(z^2 + 9)} \, dz \) where \( C \) is the circle \( |z| = 4 \).

Answer: 6\pi i


Question: Find the derivative of the trace of the matrix expression \( \operatorname{Trace}\left[\left(AB\right)^{T}Q\left(AB\right)\right] \) with respect to the matrix \( A \), where \( Q = Q^T > 0 \).

Answer: 2 Q A B B^\top


Question: Determine the order of the permutation defined by the function \( f(x) = \frac{2}{2-x} \) on the set \( A = \mathbb{R} \setminus \{0, 1, 2\} \).

Answer: 4


Question: A battery has a lifetime of 24 hours and is used for a maximum of three days. Each day, the battery is used for a random number of hours, denoted by \( K \), where \( K \) is uniformly distributed over \([0, 24]\) and independent of usage on other days. Calculate the expected remaining lifetime of the battery at the end of the third day.

Answer: 1


Question: If \( \sum a_n \) diverges, does \( \sum \frac{a_n}{\ln n} \) necessarily diverge for \( a_n > 0 \)? Determine whether this statement is true or false.

Answer: B


Question: Find the dimension over \( \mathbb{C} \) of the quotient ring \( \mathbb{C} [X,Y] / ( Y^2 - X^2 , Y^2 + X^2 ) \).

Answer: 4


Question: Let $E$ be an $n$-dimensional vector space over a field $k$, with basis $\{\beta_1, \ldots, \beta_n\}$. Consider $M_n(k)$, the ring of $n \times n$ matrices over $k$. Given the action $M_n(k) \times E \to E$ defined by $(A, v) \mapsto A \cdot v$, determine how many $M_n(k)$-submodules of $E$ exist.

Answer: 2


Question: Suppose \(X, Y\) are bivariate normal random variables with \(E[X] = 40\), \(\mathrm{Var}(X) = 76\), \(E[Y] = 30\), \(\mathrm{Var}(Y) = 32\), and \(\mathrm{Var}(X | Y = 28.5) = 57.\) Calculate \(\mathrm{Var}(Y | X = 25)\).

Answer: 24


Question: Given the linear transformation $T: \mathbb{R}^4 \to \mathbb{R}^4$ with the characteristic polynomial $x^4 - 3x^2 - 5$, determine if $T$ is an isomorphism. Answer with 'Yes' or 'No'.

Answer: Yes


Question: Assume $f : (0,1) \to \mathbb{R}$ is a measurable and integrable function in $(0,1)$. Determine $\lim_{k \to \infty} \int_0^1 x^k f(x) \, dx$. Note that integrability of $f$ in $(0,1)$ means $\int_0^1 f(x) \, dx < \infty$, not the Lebesgue integrability which is $\int_0^1 |f(x)| \, dx < \infty$. Provide your answer as a single value or expression.

Answer: 0


Question: Compute the residue of the function \( f(z) = z^3 e^{\frac{1}{z}} \) at the point \( z = 0 \).

Answer: \dfrac{1}{24}


Question: Let $A \subseteq \mathbf{R}^{n}$ be a closed set and $B$ be a closed unit normal bundle over $A$. For each $a \in A$, there is a closed subset $B_{a} \subseteq \mathbf{S}^{n-1}$, and $B = \{ (a,u) : u \in B_{a} \}$ is a closed set. Consider the set $F$ of functions $f : A \rightarrow \mathbf{S}^{n-1}$ such that $f(a) \in B_{a}$ for every $a \in A$. Is there at least one continuous function in $F$?

Answer: No


Question: Consider a subspace $E$ of a Banach space $F$, and let $T_0$ be the identity operator from $E$ to $E$. If $T$ is a bounded linear operator extending $T_0$ from $F$ to $E$, is it true that the norm of $T$ is bounded by the norm of $T_0$?

Answer: No


Question: Evaluate the integral \( \int_{0}^{2\pi} e^{\cos(\varphi)} \cos(\varphi - \sin(\varphi)) \, d\varphi \).

Answer: 2\pi


Question: Let \( z \) be a complex number such that \( z^2 + z + \frac{1}{z^2} + \frac{1}{z} + 1 = 0 \). If \( n \) is a natural number, find the value of \( z^{2012n} + z^{1006n} + \frac{1}{z^{2012n}} + \frac{1}{z^{1006n}} \).

Answer: -1


Question: Determine the number of ordered pairs \((x, y)\) such that both \(x\) and \(y\) divide \(20^{19}\), but their product \(xy\) does not divide \(20^{19}\).

Answer: 444600


Question: Find the explicit equation of the elliptic curve corresponding to the lattice $\langle 1, i \rangle$.

Answer: y^2 = 4x^3 - 4x


Question: Evaluate the expectation of the Itô stochastic integral: \[ E\left[\int_0^\infty \left|\int_0^t(W(s))^2 \, dW(s)\right|^{\,2} e^{-t} \, dt\right] \]

Answer: 6


Question: Let $X$ be a smooth projective surface over the complex numbers, and let $L$ be a line bundle on $X$ with self-intersection $L \cdot L < 0$. Does this imply that the space of global sections $H^0(X, L)$ is zero?

Answer: No


Question: Calculate the value of $276^{247} \mod 323$. Provide the most efficient method to find the result.

Answer: 13


Question:
Find a continuous function $f$ that satisfies the integral equation:
$$
f(x) = 1 + \frac{1}{x}\int_1^x f(t) \ dt
$$

Answer: 1 + \ln x


Question: Find the splitting field of the polynomial \( f(x) = x^3 + x^2 + 1 \) over the field \( \mathbb{Z}_2 \).

Answer: \mathbb{F}_8


Question: Evaluate the integral \( \int_C \frac{1-i}{\cosh z} \, dz \), where \( C \) is the triangular contour connecting the points \( 3i, -1-i, \) and \( 1-i \). Do not use the residue theorem.

Answer: 2\pi(1 - i)


Question: Evaluate the integral \( \int_{\gamma} \frac{5!\cos(z)}{(z-2\pi )^6} \, dz \), where \( \gamma \) is the circular contour centered at 0 with radius 8, using Cauchy's differentiation formula.

Answer: 0


Question: How many entire functions are equal to \( \frac{1}{z} \) for \( |z| > 1 \)?

Answer: 0


Question: Evaluate the limit: \[ \lim_{(x,y) \rightarrow (0,0)} \frac{\cos(xy) -1}{x^2 y^2} \]

Answer: -\dfrac{1}{2}


Question: Let $E = \mathbb{R} [X]$ be the space of real polynomials equipped with the norm $||p|| = \int_0^1 (|p(t)| + |p'(t)|) \, dt$. Determine if the functional $\psi : E \to \mathbb{R}$ defined by $\psi(p) = p(0)$ is continuous. If it is continuous, find the norm of $\psi$. Provide a justification for your answer.

Answer: 1


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \frac{2x(1+\sin(x))}{1+\cos^2(x)} \, \mathrm{d}x \).

Answer: \pi^2


Question: Compute the arc length of the curve \( y = \sqrt{x-x^2} + \sin^{-1}(\sqrt{x}) \) for \( 0 \leq x \leq 1 \). Use the formula for arc length: \[ L = \int_0^1 \sqrt{1+\left(\frac{dy}{dx}\right)^2} \, dx \] where \( \frac{dy}{dx} = \frac{1-2x}{2\sqrt{x-x^2}} + \frac{1}{2\sqrt{1-x}\sqrt{x}} \).

Answer: 2


Question: Let $G = H \times K$ be a nonabelian group where $o(H) = p^2$ and $o(K) = p^3$, with $p$ being a prime number. Determine the order of the commutator subgroup $G'$. \( \text{(Express your answer as a power of } p \text{.)} \)

Answer: p


Question: Find the smallest natural number \( n \) such that \( \mathbf{Q}(z_{5},z_{7}) = \mathbf{Q}(z_{n}) \), where \( z_{n} = e^{2\pi i /n} \).

Answer: 35


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{\arctan(x)} - e^{\arcsin(x)}}{1 - \cos^3(x)} \] without using L'Hospital's rule or Taylor series.

Answer: 0


Question: Find the first four terms of the Laurent series expansion of the function \( \frac{1}{e^z - 1} \) about the point \( z_0 = 0 \). Also, determine the residue of the function at this point.

Answer: 1


Question: Evaluate the mixed partial derivative \( f_{xy}(x,y) \) at the point \((0,0)\) using the first principle, where \( f(x) = \begin{cases} y\frac{x^2-y^2}{x^2+y^2} , &\text{if } (x,y) \ne (0,0);\\0,& \text{if } (x,y) = (0,0). \end{cases} \)

Answer: 0


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

Answer: \dfrac{1}{2}