Question: In how many ways can the digits \(2, 3, 4, 5, 2, 3, 4, 5\) be permuted such that no identical digits are adjacent?

Answer: 864


Question: Let \(m, n \in \mathbb{N}\). Determine the number of distinct classes \(\overline{y} \in \mathbb{Z}_n\) such that \(\overline{m} \cdot \overline{y} = \overline{0}\) in \(\mathbb{Z}_n\).

Answer: \gcd(m, n)


Question: Given that \( \tan^2\alpha \tan^2\beta +\tan^2\beta \tan^2\gamma + \tan^2\gamma \tan^2\alpha + 2\tan^2\alpha \tan^2\beta \tan^2\gamma = 1 \), find the value of \( \sin^2\alpha + \sin^2\beta +\sin^2\gamma \).

Answer: 1


Question: Find all positive integers \(n\) such that \(n^3 + n + 1\) is a perfect square.

Answer: \emptyset


Question: Determine whether the cardinal characteristic \( \mathfrak{sd} \), defined as the minimum \( \kappa \) such that player I has a winning strategy in the game \( d(\kappa) \), is equal to \( \omega_1 \).

Answer: \omega_1


Question: Let $K \subset L$ be two fields with rings of integers $\mathcal{O}_K$ and $\mathcal{O}_L$. If a prime $p$ is totally ramified in $\mathcal{O}_K$, is it true that $p$ is also ramified in $\mathcal{O}_L$? Provide a justification for your answer.

Answer: Yes


Question: An underground explorer lost in a cave is faced with three potential exit routes. Route 1 will take him to freedom in 2 hours; route 2 will return him to the cave in 4 hours; and route 3 will return him to the cave in 6 hours. Suppose at all times he is equally likely to choose any of the three exits, and let \(T\) be the time it takes the explorer to reach freedom. Define a sequence of iid variables \(X_1,X_2,\cdots\) and a stopping time \(N\) such that \(T=\sum _{i=1}^{N}X_i\). Use Wald's Equation to find \(E(T)\). Compute \(E(N)\) to complete the solution.

Answer: 12


Question:
Let \( P_n(x) \) be the Legendre polynomial of degree \( n \) such that \( P_n(1) = 1 \) for \( n = 1, 2, \dotsc \). If \[ \int_{-1}^1 \left( \sum_{j=1}^{n} \sqrt{j(2j+1)} P_j(x) \right)^2 \, dx = 20, \] what is the value of \( n \)?

(A) 2

(B) 3

(C) 4

(D) 5

Answer: C


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \arccos\left(\ln(1-2x)^{\frac{1}{4x}}\right) \] without using L'Hôpital's rule.

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


Question: Evaluate the limit: \[ \lim_{x\to \infty }x\left(\sqrt[x]{a}-1\right) \] without using L'Hôpital's rule.

Answer: \ln a


Question: Determine the dimension of the space of solutions to the differential equation \( x' = Ax \), where \( A \) is a constant matrix.

Answer: n


Question: Determine which of the two numbers, $300!$ and $100^{300}$, is greater.

Answer: 300!


Question: Determine whether the series \( \sum_{n=1}^{\infty} \frac{n(x+3)^n}{2^n(n^2+1)} \) converges at the endpoints \( x = -5 \) and \( x = -1 \). Include these endpoints in the interval of convergence if the series converges at those points.

Answer: [-5, -1)


Question: Suppose $F : C \rightarrow D$ is a full and faithful functor from category $C$ to category $D$. Does this imply that the function $F : \text{Ob}(C) \rightarrow \text{Ob}(D)$ is injective?

Answer: No


Question: Evaluate the series \( \sum^{n}_{k=0}(-4)^k\binom{n+k}{2k} \).

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


Question: Let \( f(x) = \begin{cases} \dfrac{\sin{x}}{x} & x \neq 0 \\ 1 & x = 0 \end{cases} \). Find the value of \( f^{(100)}(0) \).

Answer: \dfrac{1}{101}


Question: Is there a sequence of uniformly bounded, continuous, and increasing functions \( f_n: \mathbb{R} \rightarrow \mathbb{R} \) that converges pointwise to a continuous function \( f \) but does not converge uniformly?

Answer: Yes


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \frac{1}{1+\cos2x}dx \).

Answer: \infty


Question: Evaluate the integral of the vector field \( F(x,y,z) = (xy, y^2, y^3) \) over the boundary of the unit cube. Use parametrization for each face of the cube and calculate the surface integral to find the total flux through the cube's surface.

Answer: \dfrac{3}{2}


Question: Given independent exponential random variables $X_1, X_2, X_3, X_4$ with rates 1, 2, 3, and 4, respectively, calculate the probability $P(X_1<X_2<X_3<X_4|X_2<X_4)$. Use the formula for conditional probability to find your answer.

Answer: \dfrac{1}{35}


Question: Determine the largest exponent \(d\) such that the volume of the set \(X(r)\), consisting of matrices \(A \in M(n \times m)\) with \(n \leq m\), where the largest singular value of \(A\) is at most 1 and the smallest singular value is less than \(r\), divided by \(r^d\) remains uniformly bounded as \(r\) approaches 0.

Answer: m - n + 1


Question:
If $a$ and $b$ are the roots of the equation $x^2 + x + 1 = 0$, evaluate the expression:
\[ \sum_{n=1}^{1729} (-1)^n (a^n + b^n) \]

Answer: 1


Question: Let \( n \in \mathbb{N} \). Suppose that \( O \subseteq \mathbb{R}^n \) is an open connected subset. Is it true that for every pair of distinct points \( x, y \in O \), there exists an infinite compact subset \( K \) of \( O \) such that \( x, y \in K \)?

Answer: Yes


Question: Compute the operator norm of the integral operator \( T \) defined by \( Tf(x) = \int_0^1 \frac{f(y)}{1+x^2+y^2} \, dy \), where the norm is given by \( \|f\|_\infty = \sup_{x \in [0,1]} |f(x)| \).

Answer: \dfrac{\pi}{4}


Question:
Find all injective functions \( f: \mathbb{N} \to \mathbb{N} \) such that for all natural numbers \( n \) and \( m \), the following equation holds:
\[ f(n + m) + f(n - m) = f(n) - f(m) + f(f(m) + n) \]

Answer: f(n) = n + c


Question: Determine whether the process $(B_t^2)$, where $(B_t)$ is a Brownian motion, is a Markov process. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the improper integral \( \int\limits_0^\infty x\exp (-x-x^2)\,\text{d}x \). Use the hint: \( \int\limits_0^\infty \exp (-x-x^2)\,\text{d}x = 0.4965 \).

Answer: 0.2518


Question: Given an equilateral triangle $\Delta ABC$ and a point $P$ outside the triangle but inside $\angle ABC$, the distances from $P$ to the sides $BC$, $CA$, and $AB$ are $h_1$, $h_2$, and $h_3$ respectively. If $h_1 - h_2 + h_3 = 6$, find the area of $\Delta ABC$. \(\Delta ABC\).

Answer: 12\sqrt{3}


Question:
Given a positive integer $n$, determine if there exist polynomials $p(x) \in \mathbb{R}[x]$ and $q(x) \in \mathbb{R}[x]$ such that $p(x)$ has degree $cn^{c'}$ and $q(x)$ has degree $dn^{d'}$, with fixed positive constants $c, c', d, d'$, satisfying the following conditions:

1. For all integers $m$ with $|2m| < 2^n$, $p(2m) = 0$ and $p(2m+1) \neq 0$.
2. For all integers $m$ with $|2m| < 2^n$, $q(2m) \neq 0$ and $q(2m+1) = 0$.

Answer with 'Yes' if such polynomials exist, otherwise answer 'No'.

Answer: No


Question: Solve the equation $x^x = 10$ for $x$.

Answer: 2.506


Question: In a town of 9,000 people, 20 people have heard a rumor by 8 am. The spread of the rumor follows the logistic equation \(\frac{dP}{dt}=kP\left(1-\frac{P}{C}\right)\). By 12 noon, 500 people have heard the rumor. Determine how long it takes for half the population to have heard the rumor. Provide your answer in hours, rounded to one decimal place.

Answer: 7.5


Question: Find the sum of the digits of the 6-digit number \(DEFABC\) if the 6-digit number \(ABCDEF\) satisfies the property \(6(ABCDEF) = DEFABC\).

Answer: 27


Question:
Consider a metric space $(X,d)$ with four points $w, x, y, z \in X$. Choose comparison points in the model space $(M_\kappa^2, \bar{d})$ for a given $\kappa > 0$. Construct the comparison triangle $\Delta xyz$ with the same side lengths and points $\bar{x}, \bar{y}, \bar{z}$. Additionally, construct comparison triangles $\Delta xyw$, $\Delta yzw$, and $\Delta xzw$, using points $\bar{w_1}, \bar{w_2}, \bar{w_3}$ for each triangle, respectively, while maintaining $\bar{x}, \bar{y}, \bar{z}$ from the first triangle. If the following inequalities hold:

- $d(w,z) < \bar{d}(\bar{w_1}, \bar{z})$,
- $d(w,x) < \bar{d}(\bar{w_2}, \bar{x})$,
- $d(w,y) < \bar{d}(\bar{w_3}, \bar{y})$,

Does this imply that the space $X$ is not CBB($\kappa$)?

Answer: Yes


Question: Consider the theorem: Let $M$ be a square integrable martingale, that is $\sup_{t\geq 0}E[M_t^2]<\infty$. Furthermore, let $H$ be predictable such that \(E[\int_0^\infty H_u^2 d[M,M]_u]<\infty\). Then $H\bullet M$ is a square integrable martingale. Is the theorem still true if we replace the condition \(E[\int_0^\infty H^2 d[M,M]]<\infty\) with \(E[\int_0^\infty H^2 d\langle M,M\rangle]<\infty\)?

Answer: Yes


Question: Let $M$ be a von Neumann algebra in $B(H)$, and assume $\\zeta\\$ is a cyclic vector for $M$. Does the orthogonal projection of $H$ onto the subspace $\\mathbb{C}\\zeta$ belong to $M$?

Answer: No


Question: Find all positive integers \( x \) such that for any positive real numbers \( a, b, c \) that can form the sides of a triangle, the inequality \( x(ab+bc+ca) > 5(a^2+b^2+c^2) \) holds.

Answer: 11


Question: Suppose \( R \) is a Noetherian ring and \( M \) is a finitely generated \( R \)-module. If \( g \in \text{End}_R(M,M) \), does there exist an integer \( k \) such that \( g^k = g^{k+1} = \ldots \)?

Answer: No


Question: Given two numbers \(N\) and \(M\), find the remainder when the number consisting of \(N\) consecutive ones is divided by \(M\). For example, if \(N=5\) and \(M=18\), the remainder is 5. Determine the remainder for the given values of \(N\) and \(M\).

Answer: 5


Question: Evaluate the limit \( I = \lim_{x \to 0} \frac{\int_{x}^{x^2} e^{x\sqrt{1-y^2}} \, dy}{\arctan{x}} \).

Answer: -1


Question: Solve the equation: $$\frac{x + \sqrt{3}}{\sqrt{x} + \sqrt{x + \sqrt{3}}} + \frac{x - \sqrt{3}}{\sqrt{x} - \sqrt{x - \sqrt{3}}} = \sqrt{x}$$ Find the value of \(x\).

Answer: 2


Question: What is the remainder when \( 32^{32^{32}} \) is divided by 9?

Answer: 4


Question: Consider a sequence $(x_n)_{n \in \mathbb{N}} \subset L^2 (\mathbb{R}^3)$ and a function $x \in L^2 (\mathbb{R}^3)$ such that $\| x_n \|_{L^2 (\mathbb{R}^3)} = C$ (a constant) and $\|x\|_{L^2 (\mathbb{R}^3)} > 0$. Suppose $x_n \to x$ weakly in $L^2 (\mathbb{R}^3)$. Is it possible for $\|x\|_{L^2 (\mathbb{R}^3)} < C$? Provide a justification for your answer.

Answer: Yes


Question: Calculate the line integral \( \oint_C \mathbf{F} \cdot d \mathbf{r} \) using Green's theorem, where \( C \) is parametrized by \( \mathbf{r}(t) = \sin(t) \mathbf{i} + \sin(2t) \mathbf{j} \) for \( t \in [0, 2\pi] \), and \( \mathbf{F}(x, y) = ye^{x^2} \mathbf{i} + x^3e^y \mathbf{j} \).

Answer: 0


Question: Is it true that for any number field $k$ and any split simply connected algebraic group $G$ over $k$, the second Tate-Shafarevich group $Ш^2(k,T)$ is trivial for any maximal torus $T$ of $G$?

Answer: Yes


Question:
Given that \((1+\sqrt{2})^{2011} = a + b\sqrt{2}\) for integers \(a\) and \(b\), express \((1-\sqrt{2})^{2010}\) in terms of \(a\) and \(b\). Choose the correct option:

(a) \(a+2b+(a-b)\sqrt{2}\)
(b) \(a-2b+(a-b)\sqrt{2}\)
(c) \(a+2b+(b-a)\sqrt{2}\)
(d) \(2b-a+(b-a)\sqrt{2}\)
(e) \(a+2b-(a+b)\sqrt{2}\)

Answer: D


Question: Evaluate the integral \( \int_V \sin(\sqrt{x^2+y^2}) \, dx \, dy \) where \( V = \{(x,y) : \pi^2 \le x^2+y^2 \le 4\pi^2\} \).

Answer: -6\pi^2


Question: Evaluate the contour integral \(\int _{\Gamma} \frac{1}{z^4 + 16} \, dz\), where \(\Gamma\) is the circle \(|z-i|=\frac{1}{2}\) oriented positively.

Answer: 0


Question: Determine the sign of the integral \( \int_{0}^{2\pi} \frac{\sin x}{x} \, dx \).

Answer: positive


Question: Evaluate the complex integral \( \int_{|z|=3} \frac{\sin(\pi z^2) + \cos(\pi z^2)}{(z+1)(z+2)} \, dz \).

Answer: -4\pi i


Question: Consider a triangle with side lengths \((a, b, c)\) such that \(a^2, b^2, c^2 \in \mathbb{Q}\). Is it necessarily true that this triangle can be placed in \(\mathbb{R}^n\) with all vertices in \(\mathbb{Q}^n\) for some integer \(n \ge 2\)? Justify your answer.

Answer: Yes


Question: If \( f'''(x) \) is continuous everywhere and \( \lim_{x \to 0}(1+x+ \frac{f(x)}{x})^{1/x}=e^3 \), compute \( f''(0) \).

Answer: 4


Question: Let $G$ be a nilpotent finitely generated torsion-free group. Is it true that $\bigcap_{n} G^{p^n} = \{1\}$ for a fixed prime $p$? Justify your answer.

Answer: Yes


Question: Calculate the integral \( \int_{1}^{n+1} \frac {\{x\}^{[x]}}{[x]} \, dx \) for \( n \in \mathbb{N} \), where \( [x] \) is the largest integer less than or equal to \( x \) and \( \{x\} = x - [x] \). Prove that the result is \( \frac{n}{n+1} \).

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


Question: Find the limit of the integral \[ \int_{[0,1]^n}\frac{x_1^5+x_2^5 + \cdots +x_n^5}{x_1^4+x_2^4 + \cdots +x_n^4} \, dx_1 \, dx_2 \cdots dx_n \] as \( n \to \infty \).

Answer: \dfrac{5}{6}


Question: Given the quadratic recurrence relation \(a_n = \frac{a_{n-1}^2+61}{a_{n-2}}\) with initial conditions \(a_1 = 7\) and \(a_2 = 11\), find \(a_{10}\).

Answer: 21515


Question: How many ways are there to select three committees from 10 people, such that each committee serves a different purpose, someone has to be in every committee, and everyone serves in exactly one committee? Use generating functions to find the solution.

Answer: 55980


Question:
Consider the sequence of polynomials \(f_n(x)\) defined recursively as follows:

\[ f_1(x) = (x - 2)^2 \]
\[ f_{n+1}(x) = (f_n(x) - 2)^2 \quad ; n \ge 1 \]

Let \(a_n\) and \(b_n\) denote the constant term and the coefficient of \(x\) in \(f_n(x)\), respectively. Determine the correct option for \(a_n\) and \(b_n\):

(A) \(a_n = 4, b_n = -4^n\)
(B) \(a_n = 4, b_n = -4n^2\)
(C) \(a_n = 4^{(n-1)!}, b_n = -4^n\)
(D) \(a_n = 4^{(n-1)!}, b_n = -4n^2\)

Answer: A


Question: Evaluate the surface integral \( \iint\limits_S \mathbf{F} \cdot dS \) where \( \mathbf{F}(x,y,z) = (x,y,z) \) and \( S \) is the surface of the torus parameterized by \( x = (R + \cos(\phi)) \cdot \cos(\theta) \), \( y = (R + \cos(\phi)) \cdot \sin(\theta) \), \( z = \sin(\phi) \) with \( 0 \leq \theta \leq 2\pi \) and \( 0 \leq \phi \leq 2\pi \). Assume \( S \) is oriented outward using the outward unit normal.

Answer: 6\pi^2 R


Question: Determine whether the following statement is true or false: If $X, Y \in L^1(P)$ and $P(X \le t) < P(Y \le t)$ for all $t \in \mathbb{R}$, then $E(Y) < E(X)$. Provide a proof or counterexample to support your answer.

Answer: True


Question: Determine whether \( \frac{1000!}{(100!)^{10}} \) is an integer.

Answer: Yes


Question: Let $I$ be an ideal of a Banach algebra $A$. Assume $p \in \overline{I}$ and $p^2 = p$. Prove that $p \in I$.

Answer: p \in I


Question: What is the name of the lattice formed by atoms at integer coordinates in $\mathbb{R}^d$ that lie on a $(d-1)$-dimensional hyperplane through the origin, orthogonal to the vector $(1,1,1,\ldots)$?

Answer: A_{d-1}


Question: Find the Lebesgue measure of the simplex \(\{(x_1, x_2, \ldots, x_d) \in \mathbb{R}^d : 0 < x_1 < x_2 < \ldots < x_d < 1\}\).

Answer: \dfrac{1}{d!}


Question: Determine the number of solutions to the equation \( e^{z} - 3z - 1 = 0 \) within the unit disk \( |z| < 1 \).

Answer: 1


Question: Given a geodesic square on a unit sphere where each interior angle is 2 radians, find the area of the square.

Answer: 8 - 2\pi


Question: Determine the fundamental group of the complement of $n$ lines through the origin in $\mathbb{R}^3$. Prove that it is $F_n$, the free group on $n$ generators.

Answer: F_n


Question: Determine whether there exists a nondiagonal matrix $A \in \mathcal{M}_n(\mathbb{R})$ such that $A^{k+1} = I_n$ and $I_n - A$ is invertible, where $k \in \mathbb{N}$. Provide a justification for your answer.

Answer: Yes


Question: Evaluate the integral \( I = \int_{|z|=1} \left( \tan z + \frac{\csc z}{z} \right) \, dz \) without using the residue theorem.

Answer: 0


Question: Find the greatest common divisor of all numbers of the form \(2020^n + 1\) that are less than \(2020^{2020} + 1\), where \(n\) is a positive integer.

Answer: 1


Question: Determine all natural numbers \( n > 1 \) such that there exists a base \( b \geq 5 \) for which any three-digit number \((xyz)_b\) is divisible by \( n \) if and only if \( z + 3y - 4x \) is divisible by \( n \).

Answer: 13


Question: Let $f: \mathbb{A}_{\mathbb{C}}^n \to \mathbb{A}_{\mathbb{C}}^n$ be a dominant morphism. Determine whether there is always a point where the determinant of the Jacobian matrix is not zero.

Answer: Yes


Question: Let $A$ be a $9 \times 4$ matrix and $B$ be a $7 \times 3$ matrix. Determine if there exists a non-zero matrix $X$ such that $AXB = 0$. Provide a justification for your answer.

Answer: Yes


Question: Use Lagrange multipliers to find the maximum and minimum values of the function \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the constraint \( xyz = 1 \).

Answer: 3


Question: Consider the sequence defined by $a_1 = 0$, $a_2 = 1$, and for $n \geq 3$, $a_n = (n-1)(a_{n-1} + a_{n-2})$. Determine the remainder when $a_{2019}$ is divided by 2019.

Answer: 2018


Question: Are there infinitely many primes of the form $P_{2n-1}$ that can be expressed as $3k-1$? Here, $P_{2n-1}$ represents every other prime starting with $2$: $2, 5, 11, 17, 23, 31, \ldots$. The first few primes in this sequence are of the form $3k-1$, but $31$ is not.

Answer: Yes


Question: Show that the set \( F = \{ f \in \mathbb{N}^{\mathbb{N}}: \ \forall n \in \mathbb{N} \ f(2n)=n \} \) has the cardinality of the continuum.

Answer: \mathfrak{c}


Question: Find the volume of the solid enclosed by the paraboloids \(z = 16 - 3x^2 - 3y^2\) and \(z = 4\).

Answer: 24\pi


Question: Determine whether the statement \(5^{\frac{5k+1}{2}} \equiv (5k+1) \pmod{5k+2}\) is true for a prime of the form \(5k+2\).

Answer: True


Question: Determine the rate of convergence with respect to \(\epsilon\) for the norm \(\|f_\epsilon-f\|_{L^1}\), where \(f_\epsilon = \phi_\epsilon * f\) and \(\phi_\epsilon = (1/\epsilon)\phi(x/\epsilon)\) is a mollifier.

Answer: O(\epsilon)


Question: Evaluate the integral: \[ \int_0^\pi \sin^2\left(x^2-\sqrt{\pi^2-x^2}\right)dx \]

Answer: \dfrac{\pi}{2}


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

Answer: \dfrac{27}{e^2}


Question: Let $\mathcal{H}$ be a complex separable Hilbert space. Consider the space \( \frac{U(\mathcal{H})}{U(V) \times U(V^\perp)} \), where $V \subseteq \mathcal{H}$ is an $m$-dimensional linear subspace. Does this space have the same homotopy type as the Grassmannian $Gr_m(\mathbb{C}^\infty)$?

Answer: Yes


Question: Show that $I_1 - I_2 = \frac{1}{8}$, where $I_n = \int_{0}^{1} x^n \sqrt[3]{3x^2 - 2x^3} \, dx$.

Answer: \frac{1}{8}


Question: Determine the algebraic and geometric multiplicity of the eigenvalues of the matrix \( I - uv^T \), where \( u, v \in \mathbb{R}^n \setminus \{0\} \). The eigenvalues are \( \lambda_1 = 1 \) and \( \lambda_2 = 1 + v^T u \). Which eigenvalue has an algebraic multiplicity of \( n-1 \)?

Answer: 1


Question: Evaluate the expression: \[ \log_3(5(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})+2^{64}) \]

Answer: 64


Question: Let $X_1, X_2, \ldots$ be independent random variables with $E(X_i) = a$ and $Var(X_i) = b$ for $i \geq 1$. Let $N \geq 0$ be an integer-valued random variable with $E(N) = c$ and $Var(N) = d^2$, independent of the sequence $\{X_i\}$. Define $Y_N = X_1 + \cdots + X_N$ with $Y_0 = 0$. Compute $Cov(Y_N, N)$. \( \text{Express your answer in terms of } a, b, c, \text{ and } d. \)

Answer: a d^2


Question: Is it true that if \(D\) is a division ring and \(n \geq 1\), then the only left and right ideals of the ring \(M_n(D)\) are the trivial ones?

Answer: No


Question: Given a list of non-intersecting rectangles $W$, where each rectangle $w$ is represented as $(x, y, \text{width}, \text{height})$ and the list is sorted by $y$-coordinates, determine the maximum factor by which an input rectangle $R$ can grow while maintaining its aspect ratio without intersecting any rectangles in $W$. For example, if $W=\{(10, 2, 2, 8), (5, 5, 3, 3), (7, 7, 2, 2)\}$ and $R=(1, 1, 1, 1)$, find the maximum growth factor for $R$. The aspect ratio of $R$ is $\frac{w}{h}=1$. Determine the maximum factor by which $R$ can grow before it intersects with any rectangle in $W$. Output the maximum growth factor.

Answer: 4


Question: Evaluate the line integral \( \int_{\mathbf{C}} (1+2z+3z^2+4z^3) \, \text{d}z \), where \( C = \{ e^{i\theta} : 0 \leq \theta \leq \pi \} \).

Answer: -4


Question: Find the limit of the function \( f(x) = \frac{\ln(1+x^2)-x^2}{\sqrt{1+x^4}-1} \) as \( x \to 0 \) using Taylor series expansions.

Answer: -1


Question: Evaluate the contour integral \( \int_{|z|=10} \frac{\sin(z)}{z(z-1)} \, dz \) using the residue theorem.

Answer: 2\pi i \sin(1)


Question: For any point P on the unit circle, calculate the average distance between P and all other points on the unit circle.

Answer: \dfrac{4}{\pi}


Question: Calculate the variance of the expression \( 2 \cdot p \cdot (1-p) \cdot \beta^2 \) given the following values: \( p = 0.193 \), \( n = 4778 \), \( \beta = 1.94 \times 10^{-1} \), and \( \text{SE}(\beta) = 2.72 \times 10^{-2} \). Use the fact that \( \text{var}(p) = \frac{p(1-p)}{n} \).

Answer: 1.09 \times 10^{-5}


Question: Let $Gap(n)$ be the smallest natural number $k$ such that the sequence $(k, k+1, \ldots, k+n-1)$ contains no prime numbers. For example, $k = (n+1)! + 2$ satisfies this property. Determine whether the limit \( \lim_{n \to \infty} \frac{Gap(n)}{(n+1)! + 2} \) exists and, if so, whether it equals 1.

Answer: 0


Question: The density function of a random variable \( x \) is given by \( f(x) = ke^{-2x^{2}+10x} \). Determine the upper 5% point of the distribution of the means for a random sample of size 25 drawn from this population.

Answer: 2.66


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

Answer: \sqrt{2}


Question: Evaluate the limit: \[ \lim _ { n \rightarrow \infty } \sum _ { k = 0} ^ { n } \mathrm{arctg} \frac { k + 1} { n ^ { 2} } \]

Answer: \dfrac{1}{2}


Question: Compute the value of the surface integral \( \iint_B (x^2 + 2y^2 - 3z^2) \, dS \) over the unit sphere \( B \subset \mathbb{R}^3 \).

Answer: 0


Question: Determine whether there exists a positive integer $n$ such that the expression $n^2 - 79n + 1601$ is composite.

Answer: Yes


Question: Let $k$ be a finite field, and let $x \in k^\times$. Determine whether the following statement is true: $$x^3 \in \left(k^\times\right)^2 \Longleftrightarrow x \in \left(k^\times\right)^2.$$

Answer: True


Question: Determine whether the following statement is true or false: Let $f \in C[a, b]$ be differentiable in $(a, b)$. If $f(a) = f(b) = 0$, then, for any real number $\alpha$, there exists $x \in (a, b)$ such that $f'(x) + \alpha f(x) = 0$. Justify your answer.

Answer: True


Question: Given finitely many distinct real numbers $x_1, x_2, \ldots, x_n$, determine whether there exists a real number $y$ such that all of $y + x_1, y + x_2, \ldots, y + x_n$ are irrational.

Answer: Yes


Question: Compute the limit \( I = \lim_{n \to\infty} \int_{n}^{e^n} xe^{-x^{2016}} \, dx \).

Answer: 0


Question:
Consider an $8 \times 8$ matrix where each element is either $0$, $1$, or $2$. The matrix must satisfy the following conditions:

- In rows 1, 2, and 3, the number 1 must appear exactly once, and the number 2 must not appear.
- In rows 4, 5, and 6, the number 2 must appear exactly once, and the number 1 must not appear.
- In rows 7 and 8, both the numbers 1 and 2 must appear exactly once per row.
- The same conditions apply to the columns as well.

How many such matrices can be constructed?

Answer: 72


Question: There are 25 different types of coupons, each equally likely to be drawn. If you draw 10 coupons, what is the expected number of different types of coupons you will have? Provide your answer to two decimal places.

Answer: 8.38


Question: Evaluate the integral \( \int_{-\pi}^{\pi} \frac{\sin^2 x}{1+a^x} \,\mathrm{d}x \) where \( a > 0 \).

Answer: \dfrac{\pi}{2}


Question: Find the coefficient of \(x^{98}\) in the expansion of the polynomial \((x-1)(x-2)(x-3)\cdots(x-100)\).

Answer: 12582075


Question: Evaluate the limit: \[ \lim_{a \to 0} \int_{0}^{1+a} \frac{1}{1+x^2+a^2} \, dx \]

Answer: \dfrac{\pi}{4}


Question: Determine the values of $k$ such that the function $f(x) = \sqrt{ke^{-k^2x^2 - 2kx - 1}}$ is a probability density function.

Answer: 2\pi


Question: Let $X$ be a vector space with norm $||·||$, and let $V \subseteq X$ be a linear subspace such that $\operatorname{int}(V) \neq \emptyset$. Prove that $V = X$. ($\operatorname{int}(V)$ is the set of interior points in $V$.)

Answer: V = X


Question: Compute the sum \( \sum_{k=0}^n{2n\choose 2k}(-1)^k\sin^{2k}\theta\cos^{2n-2k}\theta \) using Euler's formula \( e^{i \theta} = \cos \theta + i \sin \theta \).

Answer: \cos(2n\theta)


Question: Find the remainder when 25! is divided by 78125.

Answer: 15625


Question:
Given the equations:
\[-29x^3 + 15y^3 = -61209\]
\[-60x - 90y = -3420\]
and the restriction \(x < y\), find the value of:
\[-50x^2 + 70y^2\]

Answer: 18270


Question: Evaluate the integral \( \oint_{|z|=2} \frac{1}{z^5+1} \, dz \) where the contour is a circle of radius 2 centered at the origin.

Answer: 0


Question: Evaluate the limit: $$\lim_{n\to\infty}\left(\frac{n \cdot n!}{e\cdot (-2)^{n+1}}\cdot \left(1-e^2 \sum_{k=0}^{n} \frac{(-2)^k}{k!}\right)\right)$$

Answer: e


Question: Let $E$ be a Banach space over $\mathbb{C}$, and let $E'$ be its dual space with norm $\|\cdot\|_{E'}$. Consider a sequence $(f_n)_{n \in \mathbb{N}} \subset E'$ such that $f_n \rightarrow f$ in the weak-star topology of $E'$, for some $f \in E'$. If there exists a constant $c > 0$ such that $\|f_n\|_{E'} \leq c$ for all $n \in \mathbb{N}$, does it follow that $\|f\|_{E'} \leq c$?

Answer: Yes


Question: Find the largest integer \( n \) such that \( \sum_{k=1}^{n} \left \lfloor{\frac 12 +\log_{10}k}\right \rfloor \leq 1000. \)

Answer: 450


Question: Evaluate the integral \( \iint_{D} x \, dA \) using polar coordinates, where \( D \) is the region inside the circle \( x^2+(y-1)^2=1 \) but outside the circle \( x^2+y^2=1 \).

Answer: 0


Question: Find the Möbius transformation that satisfies the following conditions: it maps $0$ to $\\infty$, $\\infty$ to $0$, and $5$ to $i$. Express the transformation in the form $f(z) = \frac{az + b}{cz + d}$.

Answer: f(z) = \dfrac{5i}{z}


Question: Find the total curvature of the curve $C$ in $\Bbb{R}^2$ given by $(t-\sin t,1-\cos t)$ for $0 \le t \le 2 \pi$. Express your answer as a single numerical value.

Answer: \pi


Question:
Calculate the following limit involving the Gamma function:
\[ \lim_{r \to \infty}\,\sqrt{\,{r \over 2}\,}\,\
{\Gamma\left(\,\left(r - 1\right)/2\,\right) \over
\Gamma\left(\,r/2\,\right)} = 1 \]

Answer: 1


Question: Given a group $G$ with $|G| = 40$ and a set $X$ with $|X| = 67$, where $G$ acts on $X$ resulting in 5 orbits, determine the number of possible lists of orbit lengths. Each orbit length must divide the order of the group $G$, and the sum of the orbit lengths must equal the size of the set $X$. How many such lists are there?

Answer: 6


Question: If \((I-A)^k=0\) for some positive integer \(k\), is the matrix \(A\) invertible?

Answer: Yes


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f'(x) \leq r < 1 \) for all \( x \in \mathbb{R} \). Does \( f \) necessarily have a fixed point?

Answer: Yes


Question: Calculate the flux of the vector field \( \mathbf{F}(x, y, z) = 6x\mathbf{i} + 6y\mathbf{j} + 2\mathbf{k} \) across the surface \( G \), where \( G \) is the surface cut from the bottom of the paraboloid \( z = x^2 + y^2 \) by the plane \( z = 3 \).

Answer: 48\pi


Question: Determine the degree of the field extension $[F(x):F(x^n)]$, where $F(x)$ is the field of fractions of polynomials over a field $F$.

Answer: n


Question: Stocks $A$ and $B$ open on a trading day at the same price. Let $X(t)$ denote the dollar amount by which stock $A$'s price exceeds stock $B$'s price when $100t\%$ of the trading day has elapsed. $X(t)$ for all $t \in [0, 1]$ is modeled as a Brownian motion process with $\mu = 0$ and $\sigma^2 = 0.3695$. After $75\%$ of the trading day has elapsed, stock $A$'s price is $39.75$ and stock $B$'s price is $40.25$. Find the probability that $X(1) \geq 0$. \(\boxed{0.04997}\)

Answer: 0.04997


Question: Evaluate the limit: $$\lim_{x\to\infty} \sum ^\infty_{n=3} \frac{n\cdot x^{n-2}\cdot(-1)^n\cdot(n-1)}{(2n)!}$$

Answer: -\dfrac{1}{12}


Question: Evaluate the limit: \[ \lim_{n\rightarrow\infty} 2^{-2n} \binom{2n}{n} \]

Answer: 0


Question: Let \( \{a_n\} \) be an increasing sequence of positive real numbers such that the series \( \sum\limits_{k=1}^\infty a_k \) is divergent. Define \( s_n = \sum\limits_{k=1}^n a_k \) for \( n=1,2,\dotsc \) and \( t_n = \sum\limits_{k=2}^n \dfrac{a_k}{s_{k-1}s_k} \) for \( n=2,3,\dotsc \). Determine \( \lim\limits_{n\to\infty} t_n \). Choose the correct answer: (a) \( \dfrac{1}{a_1} \), (b) \( 0 \), (c) \( \dfrac{1}{a_1+a_2} \), (d) \( a_1+a_2 \).

Answer: a


Question: Solve the modular equation: \( x^{19} \equiv 36 \mod 97 \).

Answer: 36


Question: For which primes \( P \) is \( \left(\frac{P-1}{2}\right)! \equiv 1 \pmod{P} \)?

Answer: 3


Question: Two polynomials $P$ and $Q$ satisfy the equation $P(-2x + Q(x)) = Q(P(x) - 2x)$. Given that $Q(x) = x^2 - 4$ and $P(x) = ax + b$, determine the value of $2a + b$.

Answer: 3


Question: Determine the points where the function \( f(z) = e^{x^2-y^2} [\cos(2xy) - i\sin(2xy)] \) is differentiable. Use the Cauchy-Riemann equations to find these points.

Answer: 0


Question: Given a metric on \(\mathbb{R}^2\) in polar coordinates of the form \(dr^2 + f(r,\theta)d\theta^2\), and a curve \(\gamma(t) = (t, \theta_0)\) where \(\theta_0\) is a constant, find the covariant derivative \(\nabla_{\gamma'(t)}\gamma'(t)\).

Answer: 0


Question: Determine the domain of the function \( f(x) = (-1)^x \). Consider the behavior of the function for different types of numbers, such as integers and fractions, and decide whether the domain should be restricted to integers.

Answer: \mathbb{Z}


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

Answer: 0


Question: Let $A, B \subset \mathbb{R}^N$ be such that $A \subset B$. Assume $\mathcal{H}^{N-1}(B \setminus A) < \epsilon$ and $\mathcal{H}^{N-1}(\overline{B} \setminus B) < \epsilon$, where $\epsilon > 0$ is a fixed constant and $\mathcal{H}^{N-1}$ is the $(N-1)$-dimensional Hausdorff measure. Additionally, assume $\mathcal{H}^{N-1}(\overline{A} \setminus A) = 0$ and $A$ is compact, i.e., $\mathcal{H}^{N-1}(\overline{A} \setminus A^{\circ}) = 0$. Does it follow that $\mathcal{H}^{N-1}(\overline{B \setminus A}) < 2\epsilon$?

Answer: Yes


Question: Determine whether the following statement is true or false: The image of a nowhere dense set under a continuous mapping is nowhere dense.

Answer: B


Question: Consider two independent standard Gaussian random variables \(X \sim \mathcal{N}(0, 1)\) and \(Y \sim \mathcal{N}(0, 1)\). Let \(Z = X + Y\), which implies \(Z \sim \mathcal{N}(0, 2)\). Derive the expression for \(\mathbb{E}[(Z - aY)^2]\) for a given scalar \(a \in \mathbb{R}\).

Answer: a^2 - 2a + 2


Question: Evaluate the contour integral \( \int_{\gamma}\frac{z^2+1}{z(16z^2+1)} \, dz \), where \( \gamma \) is the positively oriented circular contour of radius \( \frac{1}{4} \) centered at the point 1.

Answer: 0


Question: Find the volume of the largest closed rectangular box in the first octant with three faces on the coordinate planes and a vertex on the plane \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \), where \( a > 0 \), \( 4b > 0 \), and \( c > 0 \).

Answer: \dfrac{abc}{27}


Question: Calculate the probability of having exactly 40 Type A arrivals before either a single Type B arrival occurs or the time reaches \( t = 3 \). Type A customers arrive at a rate given by \( \lambda_a = 0.7\frac{-t^2+8t+84}{10} \), and Type B customers arrive at a constant rate of \( \lambda_b = 6 \). Use the provided information and the concept of super-positioning to determine this probability.

Answer: 0


Question:
Find the determinant of the $n \times n$ matrix given by:
\[
D_n = \left|
 \begin{matrix}
  n & -1 & -3 &  0 &  0 & \cdots & 0 & 0 &  0 \\
  n &  1 &  2 & -3 &  0 & \cdots & 0 & 0 &  0 \\
  n &  0 &  1 &  2 & -3 & \cdots & 0 & 0 &  0 \\
  n &  0 &  0 &  1 &  2 & \cdots & 0 & 0 &  0 \\
  n &  0 &  0 &  0 &  1 & \cdots & 0 & 0 &  0 \\
  \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots \\
  n &  0 &  0 &  0 &  0 & \cdots & 1 & 2 & -3 \\
  n &  0 &  0 &  0 &  0 & \cdots & 0 & 1 &  2 \\
  n &  0 &  0 &  0 &  0 & \cdots & 0 & 0 &  1 \\
 \end{matrix}
\right|
\]

Answer: n^2


Question: Does there exist a subset $X$ of the interval $[0, 1]$ with Lebesgue measure $0 < \epsilon << 1$ such that for any open set $O$, the measure of the intersection $\mu(O \cap X) > 0$?

Answer: Yes


Question: Evaluate the double integral \( \iint_R x e^{-\sqrt{x^2+y^2}} \, dx \, dy \) using polar coordinates, where \( R \) is the region in the first quadrant (\( x \ge 0, y \ge 0 \)).

Answer: 2


Question: Given a real skew-symmetric matrix \( \mathbf{A} \in \mathbb{R}^{m \times m} \), classify the scalar \( \lambda \) in the eigenvalue equation \( \mathbf{A}X = \lambda X \), where \( X \in \mathbb{R}^{m} \) is a nonzero vector. Recall that a matrix \( \mathbf{A} \) is skew-symmetric if \( \mathbf{A}^{T} = -\mathbf{A} \).

Answer: 0


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

Answer: \infty


Question: Given two random variables \(X\) and \(Y\) with means of zero and variances of one, and \(\text{cov}(X,Y) = 0\), find \(\text{cov}(X, XY)\).

Answer: 0


Question: Does there exist a function \( f: \mathbb{R} \to \mathbb{R} \) that is differentiable at every irrational number and discontinuous at every rational number?

Answer: Yes


Question: Find the maximum value of the function \( f(z) = \frac{z^3}{z^2 - 1} \) along the contour \( |z| = 3 \).

Answer: \dfrac{27}{8}


Question: Prove that \( \sum_{j=0}^{k} 2^{k-j}{k + j \choose j} = 4^k \) for all non-negative integers \( k \).

Answer: 4^k


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{2^{\sqrt{\log_2 x}}}{(\log_2 x)^2} \]

Answer: \infty


Question: Determine the values of $m$ and $n$ such that the polynomial $2X^{19} + X^{13} + mX^{11} + X^8 + 2X^6 + nX^2 + 2$ is divisible by $X^4 + X^3 + X^2 + X + 1$. Provide your answer as a pair $(m, n)$.

Answer: (0, 2)


Question: Find the remainder when \(6^{66^{666}}\) is divided by 71.

Answer: 6


Question: Find the limit of \( \frac{\phi(n)}{n} \) as \( n \to \infty \), where \( \phi \) is a permutation of \( \mathbb{N} \).

Answer: 1


Question: Given that \( \sum_{r=1}^{n}T_r=\frac{n(n+1)(n+2)(n+3)}{8} \), find \( \lim_{n\rightarrow \infty}\sum_{r=1}^{n}\frac{1}{T_r} \).

Answer: \dfrac{1}{2}


Question: Determine the total number of square matrices of order $n$ with entries $1$ or $-1$ such that the product of each row and each column is $-1$.

Answer: 2^{(n-1)^2}


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

Answer: 589824


Question: Let $f(x)$ be a polynomial with complex coefficients of degree $k$ and leading coefficient $a_k$. If $f(n)$ is an integer for every integer $n$, is it true that $|a_k| \ge \frac{1}{k!}$?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{y \to +\infty} y \int_0^{+\infty} e^{-x^2} \sin(2xy) \, dx \]

Answer: \dfrac{1}{2}


Question: Determine the sum \( S = \sum_{m \in M} |m| \), where \( M = \{ m \in \mathbb{Z} \mid f(x) = \sqrt[3]{2x^2 - mx + 1} \text{ is differentiable on } \mathbb{R} \} \).

Answer: 6


Question: In triangle $ABC$, let $O$ be the orthocenter. Given that the distance from $O$ to vertex $C$ is equal to the length of side $AB$, determine the measure of angle $\angle ACB$. \( \text{(Express your answer in degrees.)} \)

Answer: 45


Question: Let $V$ and $W$ be vector spaces over a division ring $D$. Given $v \in V$ and $w \in W$, prove that there exists a linear transformation $f: V \to W$ such that $f(v) = w$. Provide a construction or proof for such a transformation.

Answer: f(v) = w


Question: If \( x - \frac{1}{x} = i\sqrt{2} \), compute \( x^{2187} - \frac{1}{x^{2187}} \). Here, \( i = \sqrt{-1} \).

Answer: i\sqrt{2}


Question: Given the matrix differential equation \( e'(t) = e(t)^{-T}g(t) \), where \(-T\) denotes the inverse transpose and \(g(t)\) is a fixed matrix function, determine if the solution \(e(t)\) is uniquely determined by the initial condition \(e(0)\).

Answer: Yes


Question: Find the largest positive integer \( n \) such that the intersection \( A_1 \cap A_2 \cap \cdots \cap A_n \neq \emptyset \), where \( A_k = \{ x \mid 2^k < x^k + x^{k+1} < 2^{k+1} \} \) for \( k = 1, 2, 3, \ldots \).

Answer: 3


Question: Calculate the tangent map of the function \( f: O(n) \rightarrow O(n) \) defined by \( f(M) = M^3 \) at the identity matrix \( I \).

Answer: 3X


Question:
Find the rank of the matrix \( A \) given by:
\[
A=\begin{bmatrix}
a & b & c& d& e& f& g\\
a+1& b-1& 0 & 0 & e& 0 & g \\
a+1& 0 & c-1& 0 & e& f& 0\\
a+1& 0 & 0& d-1& 0& f & g\\
a+1& b& c& 0&  e-1& 0& 0\\
a+1& 0 & c& d& 0& f-1& 0\\
a+1& b& 0 & d& 0 &0 & g-1
\end{bmatrix}.
\]

Answer: 7


Question: Find the determinant of an invertible $6 \times 6$ matrix $A$ such that $A^4 + 2A = 0$. What is the value of $\det(A)$?

Answer: 4


Question: Calculate the expected value of \(\xi^2\eta^2\), where \((\xi,\eta)\) follows a normal distribution with a zero mean vector and covariance matrix \(\begin{pmatrix} 4 & 1 \\ 1 & 1 \end{pmatrix}\).

Answer: 6


Question: Calculate the flux of the vector field \( G(x,y,z) = x(1-z)\hat{k} \) through the cube with vertices at \((0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)\). Assume the normal vector is outward-pointing.

Answer: -\dfrac{1}{2}


Question:
Given that \( x\sin a = y\cos a = \frac{2z\tan a}{1-\tan^2 a} \), find the value of \( 4z^2(x^2+y^2) \).

(a) \((x^2+y^2)^{3}\)
(b) \((x^2-y^2)^3\)
(c) \((x^2-y^2)^2\)
(d) \((x^2+y^2)^2\)

Select the correct option.

Answer: C


Question: Evaluate the double integral \( \iint _R \frac{y^2}{x} \, dx \, dy \), where \( R \) is the region in the xy-plane bounded by the curves \( y = x^2 \), \( y = 4x^2 \), \( xy = 1 \), and \( xy = 5 \).

Answer: 12


Question:
Evaluate the limit:
\[
\lim_{n \to \infty} \frac{1^6 + 2^6 + 3^6 + \ldots + n^6}{(1^2 + 2^2 + 3^2 + \ldots + n^2)(1^3 + 2^3 + 3^3 + \ldots + n^3)}
\]

Answer: \dfrac{12}{7}


Question: Determine whether the sequence of functions \( g_n(t) = \frac{3^{-|t|}}{1+|t-n|} \) converges to zero in \( L_p(\mathbb{R}) \) for \( 1 \leq p < \infty \).

Answer: Yes


Question: Calculate the limit: \[ \lim_{x \rightarrow 0}\left ( x^{-6}\cdot (1-\cos(x)^{\sin(x)})^2 \right ) \]

Answer: \dfrac{1}{4}


Question: Determine whether the following statement is true or false: If matrix \( A \) has an eigenvector \( \vec{a} \) and \( A = C + D \), then \( \vec{a} \) is an eigenvector of both \( C \) and \( D \). Furthermore, if \( A\vec{a} = \lambda\vec{a} \), \( C\vec{a} = \gamma\vec{a} \), and \( D\vec{a} = \beta\vec{a} \), then \( \lambda = \gamma + \beta \).

Answer: False


Question:
Let $f: \mathbb{R}^{2} \to \mathbb{R}$ be a smooth function with a positive definite Hessian at every point. Let $(a,b) \in \mathbb{R}^{2}$ be a critical point of $f$. Determine the nature of the critical point $(a,b)$ from the following options:

(a) $f$ has a global minimum at $(a,b)$
(b) $f$ has a local, but not a global minimum at $(a,b)$
(c) $f$ has a local, but not a global maximum at $(a,b)$
(d) $f$ has a global maximum at $(a,b)$

Answer: a


Question: Given the expansion \((3x^2 + 2x + c)^{12} = \sum_{r=0}^{24} A_r x^r\), if \(\frac{A_{19}}{A_5} = \frac{1}{2^7}\), find the value of \(c\).

Answer: 6


Question: Find the splitting field of the polynomial \( P(X) = X^4 + 1 \) over \( \mathbb{Q} \) and determine the degree of this field over \( \mathbb{Q} \).

Answer: 4


Question:
Find the limit of the sequence as \( x \to \infty \):
\[
\lim_{x\to\infty}\left[\frac{1}{\sin\left(\frac{1}{x+\psi(x)}\right)}-\frac{1}{\sin\left(\frac{1}{x+\phi(x)}\right)}\right]
\]
where:
\[\psi(x)=(1+\frac{1}{x})^x, \quad \phi(x)=\sqrt[x]{x}\]

Answer: e - 1


Question: Given the polynomial \( p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46 \), find the sum of the arctan of all the roots of \( p(x) \).

Answer: 0


Question: Evaluate the limit: \[ \lim _{x\to \infty }\left(x\left(\arctan(2x)-\arccos\left(\frac{1}{x}\right)\right)\right) \]

Answer: \dfrac{1}{2}


Question: Calculate the limit \( \lim_{n \rightarrow \infty} \int_A \dfrac{1+ \dfrac{\cos^2(x^3)}{n} }{x^2+y^2+1} \, dx \, dy \), where \( A=\{(x,y) \in \mathbb{R}^2 : x^2+y^2 < 4\} \). Justify all steps in your solution.

Answer: \pi \ln 5


Question: Show that there exists a constant \( C \) such that \( |\sin z| \leq C |z| \) for all \( |z| \leq 1 \). Estimate the value of \( C \).

Answer: \sinh 1


Question: Let $X$ be a random variable uniformly distributed over the interval $[-1, 1]$, i.e., $X \sim \operatorname{Uniform}[-1,1]$. Define $Y = -\frac{2}{3} + X^2 + V$, where $V$ given $X$ is normally distributed with mean 0 and variance $\sigma^2$, i.e., $V \mid X \sim N(0, \sigma^2)$. Calculate the covariance $\cov(X, Y)$. \[10pt]

Answer: 0


Question: Calculate the volume of the solid that lies below the plane \( z = x + 3y \) and above the region defined by the ellipse \( 25x^2 + 16y^2 = 400 \) in the first quadrant \( (x \geq 0, y \geq 0) \).

Answer: \dfrac{380}{3}


Question: Evaluate the integral \( \int_{C} \frac{|z| e^{z}}{z^2} \, dz \), where \( C \) is the circumference with radius 2.

Answer: 4\pi i


Question: Consider a sequence $(x_n)$ of real numbers defined recursively by $x_{n+1} = f(x_n)$ for all natural numbers $n$, where $f: \mathbb{R} \to \mathbb{R}$. Suppose the sequence $(x_n)$ converges to a real number $x$. Is it necessarily true that $f(x) = x$? Justify your answer.

Answer: No


Question: Determine the value of \( n \) such that the function \( x^{n^x} \) approximates the integral \( \int_{0}^{x} t^t \, dt \) almost perfectly as \( x \) approaches infinity. Specifically, find \( n \) such that the ratio \( \frac{x^{n^x}}{\int_{0}^{x} t^t \, dt} \) approaches 1 as \( x \) approaches infinity.

Answer: 1


Question: Given that \(a + b + c = 5\) and \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5}\), find the maximum value of \(a^3 + b^3 + c^3\) where \(a, b, c\) are real numbers.

Answer: 125


Question:
For any countably infinite set $M$, consider the following sets:

- $X_{M} = \{A \subseteq M : |A| < \aleph_{0}\}$, the set of all finite subsets of $M$.
- $Y_{M} = \{B \subseteq M : B = M \setminus A, A \in X_{M}\}$, the set of all countably infinite subsets of $M$ whose complement is finite.
- $Z_{M} = \{C \subseteq M : |C| = |M \setminus C| = \aleph_{0}\}$, the set of all countably infinite subsets of $M$ whose complement is countably infinite.

If $f$ is a function from $X_{M}$ to $Y_{M}$ defined by $f(T) = M \setminus T$, then $f$ is bijective, implying $|X_{M}| = |Y_{M}| = \aleph_{0}$. Since $X_{M} \cup Y_{M} \cup Z_{M} = P(M)$ and $X_{M} \cap Y_{M} \cap Z_{M} = \varnothing$, it follows that $|Z_{M}| > \aleph_{0}$.

Based on this, can we conclude that the power set $P(M)$ of any countably infinite set $M$ is uncountable due to the existence of $Z_{M}$?

Answer: Yes


Question: Compute the number of intersection points between $\mathbb{R}P^2$ and $\mathbb{C}P^1$ in $\mathbb{C}P^2$. Consider $\mathbb{R}P^2 = \{ [x_0:y_0:z_0] \mid x_i \in \mathbb{R} \}$ and $\mathbb{C}P^1 = \{ [z_0:z_1:z_2] \mid a_0 z_0 + a_1 z_1 + a_2 z_2 = 0 \}$ where $a_i \in \mathbb{C}$. Determine the number of points satisfying $a_0 x_0 + a_1 x_1 + a_2 x_2 = 0$ with $x_i \in \mathbb{R}$. Provide the number of such intersection points.

Answer: 1


Question: Evaluate the integral \( I = \int_{0}^1 \ln \left\lfloor \frac{1}{x} \right\rfloor \, dx \).

Answer: 1 - \gamma


Question: Find a generator of the multiplicative group of integers modulo 23, \((\mathbb{Z}/23\mathbb{Z})^*\).

Answer: 5


Question: Calculate the standard deviation of the number of times a person will participate in a lottery until their first win, given that the probability function of winning is \( P\{X=i\} = \frac{i^2}{2870} \) for \( i=1,2,...,20 \).

Answer: 3.96


Question: Given the probability density function \( f_{X}(x) = \begin{cases} 2x & 0<x<1 \\ 0 & \text{otherwise} \end{cases} \) and the conditional distribution \( Y|X \sim \textsf{U}_{[0,X]} \), find the conditional expectation \( E(X|Y=\tfrac{1}{4}) \).

Answer: \dfrac{5}{8}


Question: Let $G$ and $H$ be magmas, with $G_1 \subset G$ and $H_1 \subset H$ as submagmas. Given that $G \simeq H_1$ and $H \simeq G_1$, is it true that $G \simeq H$?

Answer: No


Question: Evaluate the infinite series \( \sum_{n=1}^\infty \frac{2^{f(n)} + 2^{-f(n)}}{2^n} \), where \( f(n) = \left[ \sqrt{n} + \frac{1}{2} \right] \) and \( [\cdot] \) denotes the greatest integer function.

Answer: 3


Question: Find all injective functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all natural numbers \( m, n \), the equation \( f(f(m) + f(n)) = f(f(m)) + f(n) \) holds.

Answer: f(n) = n + c


Question: Let \( A \) be a 2x2 square matrix with \( |A| \neq 0 \) such that \( |A + |A| \text{adj}(A)| = 0 \). Find the value of \( |A - |A| \text{adj}(A)| \).

Answer: 4


Question: Determine the number of permutations of the digits 1 through 9 such that the blocks 12, 34, and 567 do not appear.

Answer: 283560


Question: Let $H$ be a subgroup of a group $G$. Suppose for all $x \in G$ and for all $a \in H$, the element $x * a$ belongs to $H$. Does it follow that $x \in H$? Provide a justification for your answer.

Answer: Yes


Question: Find the square root of the complex number \(5 - 12i\) using the square root definition. Express the square root in the form \(a + bi\), where \(a\) and \(b\) are real numbers.

Answer: 3 - 2i


Question: Determine the number of distinct non-singular matrices in the set \( S \), where each matrix is of the form \( \begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end{bmatrix} \), with \( \omega \neq 1 \) being a complex cube root of unity, and each of \( a, b, \) and \( c \) is either \( \omega \) or \( \omega^2 \). The matrix is non-singular if \( ac\omega^2 - (a+c)\omega + 1 \neq 0 \).

Answer: 2


Question: If \( f: \mathbb{N} \to \mathbb{N} \) is an injective function, does the sequence \( (f(n)) \) satisfy \( \lim_{n \to \infty} f(n) = +\infty \)? Justify your answer.

Answer: Yes


Question: If a function \( f \) is uniformly continuous on the interval \((0,1)\), can \( f \) be extended to the closed interval \([0,1]\) such that it is right continuous at 0 and left continuous at 1? Justify your answer.

Answer: Yes


Question: Find the value of $x$ that minimizes the integral \( \int_0^1 \left\lvert -x + e^\varphi \right\rvert d\varphi \).

Answer: \sqrt{e}


Question: Evaluate the integral \( \int_\gamma \frac{z^5}{1-z^3}dz \) using the residue theorem, where \( \gamma \) is the circle \( |z|=2 \).

Answer: -2\pi i


Question: Find the value of \( \lim_{n \rightarrow \infty} \left( 1-\frac{1}{\sqrt 2} \right) \cdots \left(1-\frac{1}{\sqrt {n+1}} \right) \).

Answer: 0


Question: Smith, Brown, and Robinson each bought shares with their wives. Mary bought 50 more shares than Mr. Brown, and Mr. Robinson bought 120 more than Jane. Each man paid as many shillings per share as he bought shares, and each wife as many pence per share as she bought shares. Each man spent one guinea more than his wife. Determine Eliza's surname given these conditions.

Answer: Robinson


Question: Evaluate the limit: \[ \lim_{n\to\infty}\prod_{k=1}^n\cos\left(\frac{ka}{n\sqrt{n}}\right) \] where \( a \) is a constant.

Answer: e^{-\frac{a^2}{6}}


Question: Calculate the expected value \( \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right] \) where \( Z \sim N(0, \sigma^2I) \).

Answer: 0


Question: Find the value of $k$ such that the line $y=k$ intersects the curve $y=2x-3x^3$ in the first quadrant, and the areas of the shaded regions formed are equal.

Answer: \dfrac{4}{9}


Question: Compute the limit: \[ \lim_{(x,y) \to (0,0)} \frac{y^3 \sqrt{|x|}}{|x|+y^4} \]

Answer: 0


Question: Evaluate the limit: \[ \lim \limits_{n\to\infty} \sum_{k=1}^{n} \frac{1}{ka+n} \] Choose the correct answer from the following options: \( A. 0 \), \( B. \ln(a) \), \( C. \frac{1}{a} \ln(1+a) \), \( D. \text{None of the above} \).

Answer: C


Question: Evaluate the integral \( \int_{0}^{2000\pi} \frac{1}{1+e^{\sin x}} \, dx \).

Answer: 1000\pi


Question: Find the smallest power \(\alpha > 0\) such that the inequality \((1+2s^2u^2)\le s^{\alpha}(1+2u^2)\) holds for all \(s \ge 1\) and \(u: \mathbb{R} \to \mathbb{R}\).

Answer: 2


Question: Evaluate the limit: \[ \lim_{x \to 0^+} \frac{\log(\cos(x))}{x} \] where the domain of the quotient is \((0, \pi/2)\).

Answer: 0


Question: Let $A$ be a finite module over the ring $\mathbb{Z}[x_1, \ldots, x_d]$. Are the properties of being torsion-free and torsionless equivalent for $A$?

Answer: Yes


Question: Let \( h_{n} = \sum_{k=0}^{n} \binom{n}{k} \frac{2^{k+1}}{k+1} \). If \( S= \sum_{n=0}^{\infty} \frac{h_{n}}{n!} \), find \( \lfloor S \rfloor \).

Answer: 17


Question: Find the volume below the surface \( z = x^2 - y^2 \) and above the region defined by \( x^2 + y^2 \leq 1 \) in the xy-plane.

Answer: 0


Question: Four entities are traveling around a 1280-meter lap. The first entity travels at 1 m/s starting at 1 meter, the second at 2 m/s starting at 2 meters, the third at 3 m/s starting at 8 meters, and the fourth at 4 m/s starting at 16 meters. Determine the number of seconds it will take for all four entities to be at the same point on the track simultaneously.

Answer: 1279


Question: Consider a sequence of 10 independent tosses of a biased coin, where the probability of Heads is \( p = \frac{3}{4} \) and the probability of Tails is \( 1 - p \). A reward of one unit is given at time \( k \), for \( k \in \{1, 2, \ldots, 10\} \), if the toss at time \( k \) resulted in Tails and the toss at time \( k-1 \) resulted in Heads. Otherwise, no reward is given at time \( k \). Let \( R \) be the sum of the rewards collected at times \( 1, 2, \ldots, 10 \). Calculate \( E(R^2) \).

Answer: \dfrac{117}{32}


Question:
Find the sum of all real numbers \(x\) that satisfy the equation:
\[
(\sqrt{x^2-2020x+2021} + \sqrt{x^2-2020x+2019})^{\frac{3x}{2}} + (\sqrt{x^2-2020x+2021} - \sqrt{x^2-2020x+2019})^{\frac{3x}{2}} = 2^{1+\frac{3x}{4}}.
\]

Answer: 2020


Question:
Evaluate the expectation $U(k)$ given by:
\[ U(k) = \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \prod_{i=1}^n \phi(x_i;0,\sigma^2) \left(\sum_{i=1}^n x_i\right)^k \prod_{i=1}^{n} dx_i \]
where $X_1, \ldots, X_n$ are independent and identically distributed Gaussian random variables with mean 0 and variance $\sigma^2$, and $k$ is a positive integer. Determine $U(4)$.

Answer: 3n^2\sigma^4


Question: Let $f : \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a smooth function such that $f(x,y) = 0$ if either $x = 0$ or $y = 0$. Determine whether there exist constants $k > 0$ and $\epsilon > 0$ such that $|f(x,y)| \leq k|x||y|$ for all $(x,y)$ with $|(x,y)| \leq \epsilon$. Provide a justification for your answer.

Answer: Yes


Question: Given the numbers 1 through 5, determine the total number of permutations that respect the order constraints specified by the ordered sets \(s_1 = (1, 2, 3)\) and \(s_2 = (5, 2)\).

Answer: 10


Question: Determine whether there exists a function \( f \in \mathcal{C}^{\infty}([0,1], \mathbb{R}) \) such that \( f^{(k)}(0) = u_k \) and \( f^{(k)}(1) = v_k \) for every \( k \geq 0 \), and \( f \) is increasing on \([0,1]\), given the sequences \((u_k)_{k\geq 0}\) and \((v_k)_{k\geq 0}\) with \( u_0 < v_0 \) and \( u_1 > 0, v_1 > 0 \).

Answer: Yes


Question: Let $p: \mathbb{R}^4 \to \mathbb{R}$ be a homogeneous polynomial of degree 4 that takes both positive and negative values. Assume that $p$ has a non-trivial zero in $\mathbb{R}^4$. Is there a point $x_0 \in \mathbb{R}^4$ such that $p(x_0) = 0$ and the gradient $\nabla_{x_0} p$ is not the zero vector?

Answer: Yes


Question: Determine the number of solutions to the equation \(x^3 = (1\ 2\ 3\ 4)\) in the symmetric group \(S_7\).

Answer: 3


Question: Evaluate the integral \( \int_{0}^{2\pi } \frac{\sin^{2} (x) }{5+4\cos(x)}\,\mathrm{d}x \).

Answer: \dfrac{\pi}{4}


Question: Determine whether the following statement is true or false: If $T^* = T^{2017}$ for a linear operator $T: V \rightarrow V$ over the complex field $\mathbb{C}$, then $T$ is diagonalizable.

Answer: True


Question:
Calculate \( f(0) \) given the following conditions:

1) \( \int_{0}^{\pi} [f(x) + f''(x)]\sin(x)\,dx = 2 \)
2) \( f(\pi) = 1 \)
3) \( f''(x) \) is continuous.

Find \( f(0) \).

Answer: 1


Question: Given the curve $C$ defined by the parametric equations \( \vec{X} = (3\cos t, 3\sin t, 6\cos t) \) for \( 0 \leq t \leq 2\pi \), find the circulation of the vector field \( \vec{f} \) along \( C \). The vector field \( \vec{f} \) is continuously differentiable in \( \mathbb{R}^3 \) and has a curl given by \( \text{rot } \vec{f}(x,y,z) = (0, 2x-y, z) \). Use Stokes' theorem to determine the circulation. Express your answer as a single number.

Answer: 0


Question: Let \( a, b, \) and \( c \) be real numbers such that \( a + b + c = 0 \). Determine the integral value of \[ P = \frac{a^2}{2a^2 + bc} + \frac{b^2}{2b^2 + ca} + \frac{c^2}{2c^2 + ab}. \]

Answer: 1


Question: Determine which of the quadratic extensions \(\mathbb{Q}_2(\sqrt{d})\) for \(d = -10, -5, -2, -1, 2, 5, 10\) are unramified.

Answer: \mathbb{Q}_2(\sqrt{5})


Question: Consider a 2D grid with \( n \) rows and \( m \) columns. You can fill this grid with numbers ranging from 1 to \( k \) (inclusive). The condition is that for each number \( r \) such that \( 1 \leq r \leq k \), no two rows can have exactly the same number of the number \( r \). Determine the number of ways to fill the grid under these conditions.

Answer: 0


Question: Does an isomorphism between two groups preserve the order of elements?

Answer: Yes


Question: Determine the degree of the field extension \(\mathbb{Q}(2^{\frac{1}{3}},\omega)\) over \(\mathbb{Q}(2^{\frac{1}{3}})\).

Answer: 2


Question: Evaluate the limit: \[ \lim\limits_{x\to 1}\left(\frac{\sqrt{x^2+2x+5-8\sqrt{x}}}{\log(x)}\right) \] Find the value of \( \alpha \).

Answer: \sqrt{2}


Question: Find the least positive integer \( k \) such that \( 10^k \equiv 1 \pmod{2003} \).

Answer: 1001


Question: Determine whether the following statement is true or false: A measure space $(X, \mathbb{A}, \mu)$ is $\sigma$-finite if and only if there exists a measurable function $f : (X, \mathbb{A}) \to (\mathbb{R}, B(\mathbb{R}))$ that is strictly positive, piecewise constant, and has a finite integral with respect to $\mu$. Provide a justification for your answer.

Answer: True


Question: Evaluate the complex integral \( \int _{ C } \frac { \cos(iz) }{ { z }^{ 2 }({ z }^{ 2 }+2i) } dz \) where \( C \) is the rectangle with vertices at -3, 3, -3i, and i.

Answer: 0


Question: Given an infinite cardinal \( \kappa \le 2^{\aleph_0} \), consider a collection of functions \( \{f_i \mid i < \kappa\} \) where each function \( f_i: i \rightarrow \omega \) is defined such that the collection is triangle-free. This means there do not exist indices \( i < j < k < \kappa \) such that \( f_j(i) = f_k(i) = f_k(j) \). Determine if it is always possible to add one more function \( f_\kappa: \kappa \rightarrow \omega \) to this collection such that the extended collection \( \{f_i \mid i \le \kappa\} \) remains triangle-free.

Answer: Yes


Question: Is the axiom asserting the existence of weakly compact cardinals consistent with the axioms of ZFC?

Answer: Yes


Question: Calculate the value of the sum \( \sum_{k=1}^{2007} \frac{5^{2008}}{25^k+5^{2008}} \).

Answer: \dfrac{2007}{2}


Question: Find \( \lim_{n \to \infty} \left(1 - \frac{1}{2} + \frac{1}{3} - \cdots - \frac{1}{2n}\right) \).

Answer: \ln 2


Question: Find the last three digits of \(2013^{2012}\).

Answer: 481


Question: Let $X$ and $Y$ be positive integers such that the first few digits of the fractions \( \frac{29}{70}, \frac{70}{169}, \frac{169}{408}, \frac{408}{X}, \frac{X}{Y} \) are all equal to 0.4142. What is \( X + Y \)? Note: The fractions are not equal.

Answer: 3363


Question: Find the sum of the infinite series: \( \sum_{k=0}^{\infty} \frac{k^3}{3^k} \).

Answer: \dfrac{33}{8}


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{x^3 e^{\frac{x^4}{4}} - \sin^{\frac{3}{2}}(x^2)}{x^7} \]

Answer: \dfrac{1}{2}


Question: Evaluate the improper integral \( \int_0^\infty \frac {e^{-x} -  e^{-ax}} {x} \, dx \) by first considering the double integral \( \int_0^\infty \int_1^a e^{-xy} \, dy \, dx \). Change the order of integration to evaluate the integral.

Answer: \ln a


Question: Determine the range of the function \( f(x) = \sec^{-1}(x) + \tan^{-1}(x) \).

Answer: (0, \pi)


Question: Find the product of the roots of the complex equation \( z^{12} = -i \).

Answer: i


Question: Calculate the limit: \[ \lim_{n \to \infty} \left(P\left(\frac{x}{n}\right)\right)^n \] where \( P(x) = 1 + 2x + 7x^2 + 13x^3 \) and \( x \in \mathbb{R} \).

Answer: e^{2x}


Question: Let \( W(t) \) be a Wiener process with \( W(0) = 0 \) and for \( t \le s, (W(s)-W(t)) \) is normally distributed with mean 0 and variance \( s-t \). For \( u, v, w > 0 \), find the expected value \( E[W(u)W(u+v)W(u+v+w)] \).

Answer: 0


Question: Evaluate the sum: \[ \sum_{r=1}^m \frac{(m+1)(r-1)(m^{r-1})}{r\binom{m}{r}} \] where \( \binom{m}{r} \) is the binomial coefficient.

Answer: m^m - 1


Question: A program is divided into 3 blocks that are compiled on 3 parallel computers. Each block takes an exponentially distributed amount of time with an average of 5 minutes, independently of the other blocks. Compute the expected time it takes for the entire program to be compiled.

Answer: \dfrac{55}{6}


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

Answer: 2\pi i


Question: Let $x$ be a uniformly distributed variable across the interval [0, 0.1], where $x$ represents the probability of a particular event occurring during a trial. Given that 528174 trials occur and the event does not occur in any of these trials, find the smallest real number $y$ such that $x < y$ with at least 95% certainty.

Answer: 0.00000567


Question: Find the smallest possible value of \(x^2 + y^2\) given that \(x\) and \(y\) are positive integers such that \(x(x + 1)\mid y(y + 1)\), but neither \(x\) nor \(x + 1\) divides either \(y\) or \(y + 1\).

Answer: 596


Question: Let \((x_n)_{n\ge2}\) be a sequence with \(x_2 > 0\) that satisfies the recurrence relation \(x_{n+1} = \sqrt[n]{1+n x_n} - 1\) for \(n \ge 2\). Compute \(\lim_{n\to\infty} nx_n\).

Answer: 0


Question: Find the minimum value of \(x^2 + 5y^2 + 8z^2\) subject to the constraint \(xy + yz + zx = -1\).

Answer: 4


Question: For each natural number $n$, determine which $n$-th root of unity is closest to $\frac{1}{2}$.

Answer: 1


Question: Find the covariance \( \text{Cov}(X, Y) \) for the joint probability density function \( f(x, y) = e^{-y} \), where \( 0 \leq x \leq y \) and the conditional expectation \( E[Y|X] = x + 1 \).

Answer: 1


Question: If \( \lim_{x\to 0}\frac{1}{x^3}\left(\frac{1}{\sqrt{1+x}}-\frac{1+ax}{1+bx}\right)=l \), find the value of \( \frac{1}{a}-\frac{2}{l}+\frac{3}{b} \).

Answer: 72


Question: Given two lightbulbs, the first has a lifetime $X \sim \exp(1)$ months and the second has a lifetime $Y \sim \exp(3)$ months. The first lightbulb is turned on, and after one month, the second lightbulb is turned on (even if the first one has burnt out by then). Given that the first lightbulb lasts more than one month, calculate the expected time from turning on the first lightbulb until one of the two lightbulbs burns out.

Answer: \dfrac{5}{4}


Question: Let $G$ be a $p$-adic reductive group and $\pi$ an irreducible non-supercuspidal representation. There exists a parabolic subgroup $P=MN$ and a supercuspidal representation $\sigma$ of $M$ such that $\pi$ appears as a subrepresentation of $\operatorname{Ind}_P^G\sigma$. Is it known that $\pi$ appears with multiplicity one in $\operatorname{Ind}_P^G\sigma$?

Answer: Yes


Question: Evaluate the definite integral: \[ \int_{0}^{3} (x^2+1) \, d[x] \] where \([x]\) is the integer part of \(x\).

Answer: 17


Question: Given a parallelepiped described by the vectors \((2,2,3), (2,4,3), (0,1,5)\) and a density function \(\rho(x, y, z) = 2x + 2y\), calculate the mass of the parallelepiped. Assume the vectors originate from the origin.

Answer: 220


Question:
Suppose $l: k \to K$ is a ring morphism and $\bar k$, $\bar K$ are the algebraic closures of $k$ and $K$, respectively. Does there exist a ring morphism $\bar l: \bar k \to \bar K$ such that the following diagram commutes?
$$
\require{AMScd}
\begin{CD}
k @>{l}>> K
\\
@VVV @VVV
\\
{\overline k} @>{\overline l}>> {\overline K}
\end{CD}
$$

Answer: Yes


Question: Let $T(z)=\dfrac{7z+15}{-2z-4}$. Define the sequence $\{z_n\}$ by $z_1=1$ and $z_n=T(z_{n-1})$ for $n\geq 2$. Find $\lim_{n \to \infty} z_n$. \(\text{Express your answer as a complex number.}\)

Answer: -3


Question: Use the Laplace transform to solve the differential equation: \( tx'' + x' + tx = 0 \).

Answer: C J_0(t)


Question: Given that $X|Y=y \sim N(y,1)$ and $Y$ is a continuous random variable with PDF $3y^2$ for $0<y<1$ and $0$ elsewhere, find $\text{Var}(X)$. The result should be $\text{Var}(X) = 1.04$. Use the law of total variance to solve the problem.

Answer: 1.04


Question: Determine the points at which the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \sin x \) if \( x \) is rational, and \( f(x) = 0 \) otherwise, is continuous.

Answer: k\pi


Question: Let \( f \) be a function such that \( f'(x) \) is continuous on \([0,1]\) and satisfies the following conditions: \( f(1) = 0 \), \( \int_0^1 [f'(x)]^2 \, dx = 7 \), and \( \int_0^1 x^2 f(x) \, dx = \frac{1}{3} \). Calculate \( \int_0^1 f(x) \, dx \).

Answer: \dfrac{7}{5}


Question: Find \( g'(0) \) for the function \( g(x) = (x+1)^2 \left(1+\cos{\left(\sqrt{|\tan{(x+1)}|}\right)}\right) + \sqrt{x^4+4x+1}\left(1-\cos{\left(\sqrt{|\tan{(x+1)}|}\right)}\right) \).

Answer: 4


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{x^3-\sin^3x}{(e^x-x-1)\sin^3x} \]

Answer: 1


Question: In an infinite 2D world with an infinite number of planets, each planet sends out one UFO to the nearest planet. Given that the distance between any two planets is unique, determine the maximum number of UFOs that can land on a single planet.

Answer: 5


Question: Determine the polynomial of maximum degree that best approximates the function $f(x) = e^x$ using minimax approximation over the interval $[0,1]$. What is the degree of this polynomial?

Answer: \infty


Question: Given a polynomial with positive integer coefficients, let \( a_n \) be the sum of the digits in the decimal representation of \( f(n) \) for \( n \in \mathbb{Z}^+ \). Is it true that there is a number which occurs infinitely often in the sequence \( a_1, a_2, a_3, \ldots \)?

Answer: Yes


Question:
Find the minimum value of the expression:
\[
\left[\cos ^{2}\left(\dfrac{\theta }{2}\right) + \,\sqrt{\,{1 - \gamma}\,}\,\sin^{2}\left(\dfrac{\theta }{2}\right)\right] ^{2} + \dfrac{\gamma }{4}\,\sin^{2}\left(\theta\right)\quad \text{where}\quad 0 \le \gamma \le 1.
\]

Answer: 1 - \gamma


Question: Find a closed-form expression for the sequence \( f_n \) defined by the recurrence relation \( f_0 = 0, f_1 = 1, f_{n+1} = 4f_n - 4f_{n-1} \) for \( n \geq 1 \). Use the matrix representation \( \begin{pmatrix} f_n \\ f_{n+1} \end{pmatrix} = A^{n-1} \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) where \( A = \begin{pmatrix} 0 & 1 \\ -4 & 4 \end{pmatrix} \).

Answer: n \cdot 2^{n-1}


Question: Determine the smallest positive real number \( \lambda_0 \) such that the series \( \sum_{n=1}^\infty \left( \frac{\sqrt{n+1}}{\zeta(n+1)} - \frac{\sqrt{n}}{\zeta(n)} \right)^\lambda \) converges for all \( \lambda > \lambda_0 \). Justify your answer.

Answer: 2


Question: Show that if $f, g \in \text{Aut}(\mathbb{I})$, and \[ f^{-1}((f(x)+f(y)-1) \vee 0) = g^{-1}((g(x)+g(y)-1) \vee 0) \] for all $x, y \in [0,1]$, then $f = g$. Note: $\text{Aut}(\mathbb{I})$ is the set of all functions $f$ from $[0,1]$ to $[0,1]$ that are one-to-one and onto, and such that $f(x) \leq f(y)$ if and only if $x \leq y$. Prove this statement.

Answer: f = g


Question:
Given the following system of equations, where \(q, r, s,\) and \(t\) are different constant values, find the sum \(q + r + s + t\):
\[
\frac{1}{qa+1} + \frac{1}{qb+1} + \frac{1}{qc+1} + \frac{1}{qd+1} = 1
\]
\[
\frac{1}{ra+1} + \frac{1}{rb+1} + \frac{1}{rc+1} + \frac{1}{rd+1} = 1
\]
\[
\frac{1}{sa+1} + \frac{1}{sb+1} + \frac{1}{sc+1} + \frac{1}{sd+1} = 1
\]
\[
\frac{1}{ta+1} + \frac{1}{tb+1} + \frac{1}{tc+1} + \frac{1}{td+1} = 1
\]

Answer: 0


Question: What is the minimum number of days in a year, N, required so that 1 million aliens can each have a unique birthday, assuming no two aliens share the same birthday?

Answer: 1000000


Question: Determine the number of roots of the equation \( \sum_{i=1}^{2020}\frac{i^2}{x-i}=0 \).

Answer: 2019


Question: Evaluate the sum: \[ \sum_{i=1}^{100} \frac{100 \cdot 99 \cdot \ldots \cdot (100-i+1)}{100^i} \cdot i \]

Answer: 100


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

Answer: (5, \infty)


Question: Find the sum of all values of \( \theta \) for which the expression \( \cos^2x + \cos^2(\theta+x) - \cos x \cdot \cos(\theta+x) \) is independent of \( x \).

Answer: 2\pi


Question: Evaluate the definite integral: \[ \int_{1}^{5}{\frac{\sqrt{x}}{\sqrt{6-x}+\sqrt{x}}\,dx} \]

Answer: 2


Question: Let the point $A$ lie on the exterior of the circle $k(R)$. From $A$, draw the tangents $AB$ and $AC$ to $k$. If the triangle $ABC$ is equilateral, find the side length of $\triangle ABC$ in terms of $R$.

Answer: R\sqrt{3}


Question: Determine whether the set $S \cap T$ is closed and bounded, where $S = \{ x \in \mathbb{R} : x^6 - x^5 \leq 100 \}$ and $T = \{ x^2 - 2x : x \in (0, \infty) \}$. Answer with "True" if it is closed and bounded, or "False" if it is not.

Answer: A


Question: Given a sequence \( \{f_n\} \) of conformal, one-to-one maps from the unit disk \( \mathbb{D} \) onto the right half-plane \( \mathbb{A} = \{ z \in \mathbb{C} : \mathfrak{R} z > 0 \} \), suppose \( \{f_n\} \) converges to \( f \) uniformly on compact subsets of \( \mathbb{D} \) and \( f \) is not one-to-one. Determine the value of \( \mathfrak{R} f(0) \).

Answer: 0


Question: Evaluate the integral \( \int_{37}^{149} f^{-1}(x) \, \mathrm{d}x \), where \( f(x) = x^3 - 2x^2 + 5 \).

Answer: \dfrac{1732}{3}


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

Answer: 0


Question: Is there an element \( c \in \hat{F_2} \) such that the subgroup \( \langle a, b, c \rangle \le \hat{F_2} \) is isomorphic to the 3-generator abstract free group \( F_3 \)?

Answer: Yes


Question: Does there exist a measurable set $A \subseteq \mathbb{R}$ such that for any open interval $I$, both $|A \cap I|$ and $|A^\complement \cap I|$ are uncountable?

Answer: Yes


Question:
Let $f$ be an analytic function such that $$|f^2(z)-1|=|f(z)-1|\,|f(z)+1|<1$$ on a non-empty connected set $U$. Determine which of the following statements is true:

(A) $f$ is constant.

(B) $\text{Im}(f) > 0$ on $U$.

(C) $\text{Re}(f) \neq 0$ on $U$.

(D) $\text{Re}(f)$ is of fixed sign on $U$.

Choose the correct option.

Answer: D


Question: Evaluate the limit: \[ \lim_{n \to \infty} n^2 \int_0^1 \frac{1}{(1+x^2)^n} \, dx. \]

Answer: \infty


Question: Let $L$ and $M$ be lattices in $\mathbb{C}$. Suppose $\gamma,\delta\in\mathbb{C}$ are non-zero and such that $\gamma L\subseteq M$ and $\delta M\subseteq L$. Does it then follow that $\gamma L=M$?

Answer: No


Question: Find all positive integers \( n \) such that \( \left(\frac{-1 + i\sqrt{3}}{2}\right)^n + \left(\frac{-1 - i\sqrt{3}}{2}\right)^n = 2. \)

Answer: 3k


Question:
Evaluate the integral of the function \( f(x,y,z) = z \) over the region defined by the constraints \( y^2 + z^2 = 9 \), \( x = 0 \), \( z = 0 \), and \( y = 3x \). Set up and compute the integral:
\[
\int_0^1 \int_{3x}^3 \int_0^{\sqrt{9-y^2}} z \, dz \, dy \, dx
\]

Answer: \dfrac{27}{8}


Question: Given that $a$ and $b$ are the roots of the quadratic equation $x^2 - 2cx - 5d = 0$, and $c$ and $d$ are the roots of the quadratic equation $x^2 - 2ax - 5b = 0$, where $a, b, c, d$ are distinct real numbers, find the value of $a + b + c + d$. Assume $a = 2c - b$ and $c = 2a - d$. What is $a + b + c + d$?

Answer: 30


Question: Determine the minimum value of \( p \) such that for \( q > p \), the cubic equation \( x^3 - 7x^2 + qx + 16 = 0 \) has only one real root.

Answer: 8


Question: Find the real number \( a \) such that \( \oint_c \frac{dz}{z^2-z+a} = \pi \), where \( c \) is the closed contour \(|z-i|=1\) taken in the counterclockwise direction.

Answer: \dfrac{5}{4}


Question: Consider a continuous and Riemann integrable function \(f:\mathbb{R}\to\mathbb{R}\) and fix \(a\in\mathbb{R}\). Define \(F:\mathbb{R}\to\mathbb{R}\) by \(F(x) = \int_{a}^{x}f(y)dy\). Is it true that \(F\) is differentiable and \(F'(x)=f(x)\) for \(x<a\)?

Answer: Yes


Question: A polynomial $f(x)$ of degree at most $n$ satisfies $f(k) = 2^k$ for $k = 0, 1, \ldots, n$. Determine the value of $f(n+1)$. \( \text{(Express your answer in terms of } n \text{ if necessary.)} \)

Answer: 2^{n+1} - 1


Question:
Find the number of solutions to the system of equations:
\[ y = \frac{1}{3}[\sin x + [\sin x + [\sin x]]] \]
\[ [y + [y]] = 2 \cos x \]
where \([x]\) represents the greatest integer function.

Answer: 0


Question: Find the supremum of the expression \(|x_1 - y_1| + |x_2 - y_2| + |x_3 - y_3|\) subject to the constraints \(x_1^2 + x_2^2 + x_3^2 \leq 1\) and \(y_1^2 + y_2^2 + y_3^2 \leq 1\).

Answer: 2\sqrt{3}


Question: Evaluate \( \lim_{n \to \infty}\sum_{k=2}^{n} \frac{1}{\sqrt[k]{n^k+n+1}+1} \).

Answer: 1


Question: Given the function $f(x) = x^2 - 2ax + a(a+1)$ with the domain $f:[a, \infty) \to [a, \infty)$, one solution to the equation $f(x) = f^{-1}(x)$ is $5049$. Determine the other possible solution.

Answer: 5050


Question: Determine whether the number $2014! + 1$ is prime or composite.

Answer: Composite


Question: Determine the number of real solutions to the equation \( x = 1964\sin{x} - 189 \).

Answer: 1250


Question: If $M$ is a convex-cocompact hyperbolic 3-manifold, and $S$ is a closed surface with genus $\geq 2$, suppose $f:S\to M$ is a minimal immersion such that $f(S)$ is negatively curved. Given that all the closed geodesics in $f(S)$ are closed geodesics in $M$, can you conclude that $f(S)$ is totally geodesic in $M$?

Answer: Yes


Question: Calculate the value of \( \int_{-1}^1 f(x) \, dx + \int_{0}^4 f^{-1}(x) \, dx \) where \( f(x) = x^3 + 3x + 4 \).

Answer: \dfrac{23}{4}


Question: Evaluate the limit: \[ \lim _{(x,y)\to(2,1)}\frac{\arcsin(xy-2)}{\arctan(3xy-6)} \]

Answer: \dfrac{1}{3}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{1 + \sqrt[2]{2} + \sqrt[3]{3} + \cdots + \sqrt[n]{n}}{n} \] Does this limit equal 1?

Answer: 1


Question: Find and draw the flow lines of the velocity vector field \( \vec{F}(x, y) = (-2y, \frac{1}{2}x) \).

Answer: x^2 + 4y^2 = C


Question:
Evaluate the limit:
\[ \lim \limits_{x \to 0^+} \frac{\sin^{-1}(1 - \{x\}) \times \cos^{-1}(1 - \{x\})}{\sqrt{2\{x\}}\times(1 - \{x\})} \]
where \(\{x\}\) represents the fractional part of the number.

Answer: \dfrac{\pi}{2}


Question: Determine whether the number of orbits of the natural action of order-preserving bijections of $\mathbb{Q}$ on $\mathbb{Q}^n$ is finite. An order-preserving map $f : \mathbb{Q} \to \mathbb{Q}$ satisfies $x > y \implies f(x) > f(y)$. Let $\text{Aut}(\mathbb{Q}) = \{ f : \mathbb{Q} \to \mathbb{Q} : f \text{ is bijective and order-preserving} \}$. The group $\text{Aut}(\mathbb{Q})$ acts on $\mathbb{Q}^n$ by $f \cdot (a_1, \ldots, a_n) = (f(a_1), \ldots, f(a_n))$. Is the number of orbits under this action finite?

Answer: Yes


Question: Let $a,b \in \mathbb{R}_+$ be real positive numbers with $\frac{1}{2}<a<1$, and let $I=[0,b]$ be a closed real interval. Define the sequence of functions $f_n(x) : I \to \mathbb{R}$ by \[ f_n(x) = \frac{n}{(2n-1)^{a+x}} - \frac{n}{(2n)^{a+x}} \] and the sequence of functions $g_n(x) : I \to \mathbb{R}$ by \[ g_n(x) = \frac{d}{dx} f_n(x) = \frac{n \cdot \ln(2n)}{(2n)^{a+x}} - \frac{n \cdot \ln(2n-1)}{(2n-1)^{a+x}}. \] Let $h : I \to \mathbb{R}$ be the function \[ h(x) = \sum_{n=1}^\infty \left| \frac{f_n(x)-f_n(0)}{x} - g_n(0) \right|^2. \] Determine whether \[ \lim_{x \to 0^+} h(x) = 0. \]

Answer: 0


Question:
Let \( a, b, \) and \( c \) be real numbers such that \( a + b + c = 0 \). Find the value of the expression:
\[
P = \frac{a^2}{2a^2+bc} + \frac{b^2}{2b^2+ca} + \frac{c^2}{2c^2+ab}.
\]

Answer: 1


Question: If a commutative ring \( R \) with unity is finitely generated over \( \mathbb{Z} \), is the Jacobson radical of \( R \) necessarily nilpotent?

Answer: Yes


Question: Determine the cardinality of the set \( B = \{ f \in \mathbb{R}^\mathbb{N} \mid f(\mathbb{N}) \text{ is a well-ordered subset of } \mathbb{R} \text{ with respect to the standard ordering} \}. \)

Answer: 2^{\aleph_0}


Question: Let $g = (g_1, ..., g_N)$ be a vector field where each component $g_i: \mathbb{R}^N \to \mathbb{R}$ is in $W^{1,\infty}(\mathbb{R}^N)$ and $Q$-periodic. Define $G = \operatorname{div} g = \frac{\partial g_1}{\partial x_1} + ... + \frac{\partial g_N}{\partial x_N}$. Let $Q$ be the unit cube in $\mathbb{R}^N$ centered at the origin. Prove or disprove that \( \int_Q G = 0 \).

Answer: 0


Question: Find all continuous functions \( f: [a,b] \to \mathbb{R} \) such that \( \int_{a}^x f(t) \, dt = \int_{x}^b f(t) \, dt \) for all \( x \in (a,b) \).

Answer: 0


Question: Evaluate the integral \(2 \int_0^{\infty} \frac{1- \cos(tX)}{\pi t^2}~dt\) and explain why the result is \(|X|\).

Answer: |X|


Question: Compute the limit: \[ \lim\limits_{x \to 0_+} \left(\frac{x^2+1}{x+1}\right)^{\frac{1}{\sin^2 x}} \]

Answer: 0


Question: Does there exist a finitely presented infinite group that has only finitely many finite retracts?

Answer: Yes


Question: Find the maximum value of the function \( f(a,b) = \frac{a^3b}{(a+b)^4} \) for \( a, b > 0 \).

Answer: \dfrac{27}{256}


Question: Prove that the sum of the n nth roots of any complex number is equal to zero.

Answer: 0


Question: Consider the region in the first quadrant bounded by the curves \( y = x^2 \) and \( y = 9 \). This region is rotated around the vertical line \( x = 3 \) to form a solid. Determine the height \( m \) of the horizontal line that divides the volume of this solid into two equal parts.

Answer: 5.18


Question: Find the determinant of the symmetric matrix \( \mathbf{G} = a\mathbf{I} + b\boldsymbol{ee}^T \), where \( a \) and \( b \) are constants, \( \boldsymbol{e} \) is a column vector with all elements equal to 1, and \( \mathbf{G} \) is a \( u \times u \) matrix.

Answer: a^{u-1}(a + bu)


Question: Let \( \alpha \) be a root of the polynomial \( X^3 + X + 1 \in \mathbb{Q}[X] \). Determine the minimal polynomial of \( \alpha^{-1} \) over \( \mathbb{Q} \).

Answer: X^3 + X^2 + 1


Question: Evaluate the integral \( \int\limits_{-\pi/2}^{\pi/2} \cos(\tan x)\,dx \).

Answer: \dfrac{\pi}{e}


Question:
Find the set \( S \) of real numbers \( \lambda \) such that the rank of the matrix \( M - \lambda I_3 \) is less than 3, where \( M \) is the matrix:
\[
\begin{bmatrix}
-3 & 5 & -5\\
-4 & 6 & -5\\
-4 & 4 & -3
\end{bmatrix}
\]

Answer: \{-3, 1, 2\}


Question: Let $1, \alpha_1, \alpha_2, \dots, \alpha_{n-1}$ be the roots of the polynomial $X^n - 1 = 0$. Determine the value of the product $\prod_{i=1}^{n-1} (1 - \alpha_i)$. Express your answer in terms of $n$. \( \boxed{n} \)

Answer: n


Question: Determine the number of ideals in the ring \( R = \mathbb{Z} / \langle 300 \rangle \).

Answer: 18


Question: Determine the set of points where the function \( \frac{1}{z \bar{z}} \) is holomorphic.

Answer: \emptyset


Question:
In a non-obtuse triangle \(\triangle ABC\), determine the lower bound for the expression \(\sum_{\text{cyc}}\frac{\sin B+\sin C}{A}\). Choose the correct answer from the following options:

A) \(\frac{6}{\pi}\)

B) 3

C) \(\frac{12}{\pi}\)

D) \(\frac{1}{\pi}\)

Answer: C


Question: Suppose $H$ is a normal subgroup of a group $G$ with $|H| = 4$. If there exists an element $g \in G$ such that the coset $gH$ has order 3 in the quotient group $G/H$, find a subgroup of order 12 in $G$.

Answer: \langle H, g \rangle


Question:
Determine the type of function that serves as the extremal of the functional \( I = \int\limits_0^{x_1} y^2(y')^2 \, dx \) that passes through the points \((0,0)\) and \((x_1,y_1)\). Choose from the following options:

- A constant function
- A linear function of \(x\)
- Part of a parabola
- Part of an ellipse

Answer: C


Question: Determine the number of solutions \((x, y, z)\) to the system of equations \(\begin{cases}x+2y+4z=9\\ 4yz+2xz+xy=13\\ xyz=3\end{cases}\) such that at least two of \(x, y, z\) are integers.

Answer: 5


Question: Find the function \( f: \mathbb{R} \to \mathbb{R} \) that satisfies the equation \( f(x) = x f\left(\frac{1}{x}\right) = 1 + f(x+y) - f(y) \) for all \( x, y \in \mathbb{R} \setminus \{0\} \).

Answer: x + 1


Question: Evaluate the Cauchy Principal Value of the integral: \[ \int_{-\infty}^\infty \frac{\cos kx}{x-a} \, dx \] where \( a \) is a real number and \( k > 0 \).

Answer: -\pi \sin(ka)


Question: Determine whether every integer can be expressed as a mixed sum of three integer squares, where each square can have a positive or negative sign. For example, 28 can be expressed as \( 36 - 9 + 1 \).

Answer: Yes


Question: Calculate the correlation coefficient \( \text{Corr}(N, M) \) where \( N \) and \( M \) are Poisson random variables with parameter 1, defined on the intervals \((0,2)\) and \((1,3)\) respectively.

Answer: \dfrac{1}{2}


Question:
Determine if there exists a positive integer \( n \) such that \( n \geq p+2 \) and the following equation holds for a given positive integer \( p \geq 4 \):
\[ \sum_{m=p+1}^{n-1} \frac{2m-2p-p_{m}+p_{p}}{r^{m^2}} = 0 \] where \( p_m \) is the \( m^{th} \) prime number.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n \to \infty} \left(1+\frac{a}{n^2}\right)^\frac{n}{b} \] where \(a\) and \(b\) are constants.

Answer: 1


Question:
Estimate the value of the integral \( \int_{0}^{1}\sqrt {1 + \frac{1}{3x}} \, dx \). Which of the following is closest to the value of this integral?

(A) 1
(B) 1.2
(C) 1.6
(D) 2
(E) The integral doesn't converge.

Answer: C


Question: Let $R$ be a ring and $S$ a subset of $R$. If the two-sided ideal generated by $S$ together with the Jacobson radical of $R$ is equal to $R$, does it follow that the two-sided ideal generated by $S$ alone is equal to $R$?

Answer: Yes


Question: Evaluate the contour integral \( \int\limits_C \left(xy + \ln(x^2 + 1) \right) dx + \left(4x + e^{y^2} + 3\arctan(y)\right) dy \) using Green's Theorem, where \( C \) is the curve enclosing the region \( D \) in \( \mathbb{R}^2 \) defined by \( x^2 + y^2 \leq 1 \) and \( y \geq 0 \), oriented counterclockwise.

Answer: 2\pi


Question: Suppose \(f(x)\) is differentiable on \((0,+\infty)\) and \(\lim_{x\rightarrow +\infty} \frac{f(x)}{x}=0.\) Prove that there exists a sequence \(x_n\rightarrow\infty\) such that \(\lim_{n\rightarrow\infty}f'(x_n)=0.\)

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \to\infty}\frac{1\cdot2\cdot3+2\cdot3\cdot4+\dots+n(n+1)(n+2)}{\left(1^2+2^2+3^2+\dots+n^2\right)^2} \]

Answer: 0


Question: Find the remainder when \(23^{54^{26}}\) is divided by 45.

Answer: 1


Question: Passengers arrive at a train platform following a Poisson process with an intensity of 30 passengers per minute. The passengers start arriving at 4:00. Calculate the expected value of the total wait time of all the passengers if the train leaves at 4:15.

Answer: 3375


Question: Suppose \([K:F]\) is a Galois extension and \(\alpha \in K\) with \(\alpha \not\in F\). Let \(m_\alpha\) be the minimal polynomial of \(\alpha\) over \(F\). Is it true that all the roots of \(m_\alpha\) are Galois conjugates of \(\alpha\)? In other words, if \(\beta\) is a root of \(m_\alpha\), can we say that \(\beta = \phi(\alpha)\) for some \(\phi \in \text{Gal}(K:F)\)?

Answer: Yes


Question: Evaluate the limit: $$ \lim_{x \to 2}\frac{\sqrt{x^3+1}-\sqrt{4x+1}}{\sqrt{x^3-2x}-\sqrt{x+2}} $$

Answer: \dfrac{16}{27}


Question: For given integers \(N\) and \(K\), compute the sum \(\displaystyle\sum_{}\prod_{K_1}^{K_N}\binom{N}{k_i}\) over all sets of non-negative integers \(\{k_1, k_2, \ldots, k_N\}\) such that \(\displaystyle\sum_{i=1}^N k_i = K\). For example, if \(N=3\) and \(K=5\), the sets of \(k_i\) are \{0,0,5\}, \{0,5,0\}, \{5,0,0\}, \{0,1,4\}, \{0,4,1\}, \{1,0,4\}, \{4,0,1\}, \{4,1,0\}, \{1,4,0\}, \{0,2,3\}, \{0,3,2\}, \{2,0,3\}, \{3,0,2\}, \{2,0,3\}, \{3,0,2\}, \{1,2,2\}, \{2,2,1\}, \{2,1,2\}, \{3,1,1\}, \{1,3,1\}, \{1,1,3\}. Compute the sum of the products of binomial coefficients for these sets.

Answer: 126


Question: Consider the stochastic differential equation \(dZ_t = aZ_t dt + \sigma Z_t dB_t\), where \(B_t\) is a Brownian motion, \(a = 0.03\), and \(\sigma = 0.3\). Given that \(Z_0 = 100\), find the probability \(P(Z_1 > 103.5)\).

Answer: 0.4345


Question: Calculate the number of ways to arrange 4 Americans, 3 Russians, and 5 Chinese into a queue such that no nationality forms a single consecutive block. Use the inclusion-exclusion principle to find the solution.

Answer: 445167360


Question: Calculate \( \lim\limits_{x\to\infty}x\left(\int_0^x te^{-2t}\,dt-\frac{1}{4}\right) \).

Answer: 0


Question:
Consider a finite field \( \mathbb{F}_q \) with characteristic not equal to 2, and let \( f \in \mathbb{F}_q[x] \) be a squarefree polynomial that is not in \( \mathbb{F}_q \). Define the ring \( \mathbb{F}_q[x, \sqrt{f}] \). Let \( g \) be an irreducible polynomial in \( \mathbb{F}_q[x] \) that does not divide \( f \). Determine whether the following statement is true:

If the image of \( f \) in \( \mathbb{F}_q[x]/(g) \) is a square of some element, then there are precisely two maximal ideals \( \mathfrak{p} \) of \( \mathbb{F}_q[x, \sqrt{f}] \) such that \( \mathfrak{p} \supset g\mathbb{F}_q[x, \sqrt{f}] \). For these \( \mathfrak{p} \), the isomorphism \( \mathbb{F}_q[x]/(g) \cong \mathbb{F}_q[x, \sqrt{f}]/\mathfrak{p} \) holds.

Answer: True


Question: Find the pair of positive integers \((m, n)\) that satisfies the equation \(19m + 90 + 8n = 1998\) and has the smallest value for \(m\).

Answer: (4, 229)


Question: Given the function \( f : \mathbb{R} \to \mathbb{R} \) such that \( f(f(x)) = x^2 - x + 1 \), find the value of \( f(0) \).

Answer: 1


Question: Find all functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy the equation $f(x+1) - f(x) = f'(x)$.

Answer: f(x) = ax + b


Question: Let $f(z)$ be a holomorphic function on a disk of radius greater than 1 centered at the origin. Suppose that $|f(z)| \leq 1$ for $z$ with $|z|=1$ and $\text{Re}(z)>0$, and $|f(z)| \leq 2$ for $z$ with $|z|=1$ and $\text{Re}(z)\leq 0$. Prove that $|f(0)| \leq \sqrt{2}$.

Answer: \sqrt{2}


Question:
Determine which of the following subsets of the real line is compact:

(a) \( \left\{ \frac{\sin(n\theta)}{n} \mid n \in \mathbb{N} \right\} \)
(b) \( \left\{ \frac{\cos(n\theta)}{n} \mid n \in \mathbb{N} \right\} \)
(c) \( \left\{ \frac{\tan(n\theta)}{n} \mid n \in \mathbb{N} \right\} \)
(d) None of the above.

Here, \( \theta = \frac{\pi}{2019} \) and \( \mathbb{N} \) denotes the set of positive integers.

Answer: D


Question: Evaluate \( \tan(A) \) where \( A = \sum_{k=1}^{17} \tan^{-1}\left(\frac{1}{k^2+k+1}\right) \).

Answer: \dfrac{17}{19}


Question: Find the gradient of the function \( F(A) = X^T \cdot A \cdot Y \) with respect to the matrix \( A \), where \( X \) and \( Y \) are \( m \times 1 \) vectors and \( A \) is an \( m \times m \) matrix.

Answer: X Y^T


Question: Evaluate the improper integral: \[ \int_0^\infty \frac{1 - \cos x}{x^2} \, \mathrm{d}x \]

Answer: \dfrac{\pi}{2}


Question: If a sequence of functions \( f_n \) converges pointwise to a function \( f \), and \( f_n \) also converges uniformly to some function \( g \), must \( f \) be equal to \( g \)? Provide a justification for your answer.

Answer: Yes


Question: Find the 9th derivative of the function \( f(x) = \frac{\cos(5x^4)-1}{x^7} \) at \( x = 0 \) using the MacLaurin Series.

Answer: 9450000


Question: In a game with 2022 nuts on the table, two players, Alice and Bob, take turns eating nuts. On each turn, a player rolls a regular six-sided die and eats the number of nuts shown on the top face. The player who cannot make a move loses the game. Determine the starting number of nuts, \(a\), for which Bob can force a win.

Answer: 2016


Question: Let \(x\in\mathbb{R}.\) For all \(i,j\in\mathbb{N},\) define \(a_{i0} = \frac{x}{2^i}, a_{ij} = a_{i,j-1}^2 + 2a_{i,j-1}.\) Find \(\lim\limits_{n\to\infty} a_{nn}.\)

Answer: e^x - 1


Question: Evaluate the binomial coefficient \( \binom{-1/2}{2} \).

Answer: \dfrac{3}{8}


Question: Determine the constant coefficient \(m\) such that the expression \(a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)\) can be factorized as \(m(a-b)(b-c)(c-a)\).

Answer: 1


Question: Given an anti-palindromic polynomial of degree $n$ with odd coefficients, determine if it has roots on the unit circle.

Answer: Yes


Question: Find all real solutions to the equation \(2^x + 3^x + 6^x = x^2\).

Answer: -1


Question: Find the minimum value of the function \( f(x) = \int_{0}^{1} |x-t|^3 \, dt \) for \( x \in \mathbb{R} \).

Answer: \dfrac{1}{32}


Question: Calculate the limit: \[ \lim_{(x,y)\rightarrow (\infty,\infty)}(x^2+y^2)e^{-(x+y)} \]

Answer: 0


Question: What is the supremum of all ordinals for which there exists an order-preserving injection into the interval [0,1]? Is it the first uncountable ordinal, the initial ordinal of $2^{\aleph_0}$, or something else?

Answer: \omega_1


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

Answer: 0


Question: Determine the maximum value of the real number \(a\) such that the inequality \(ax < y\) holds, given that \(xy^2 + 4x^2y + 5 = 0\) and \(x > 0\).

Answer: -4


Question: Solve the equation: \( 25^{\log_{10}(x)} = 5 + 4x^{\log_{10}(5)} \) and find the value of \( x \).

Answer: 10


Question: Let $S$ be a subset of $\{0,1\}^n$ such that any two elements of $S$ are at least Hamming distance 5 apart. Find an upper bound on the size of the automorphism group of $S$.

Answer: 2 \cdot n!


Question: Evaluate the expression \( \frac{a^4}{(a-b)(a-c)}+\frac{b^4}{(b-c)(b-a)}+\frac{c^4}{(c-a)(c-b)} \) where \( a = -\sqrt{99}+\sqrt{999}+\sqrt{9999} \), \( b = \sqrt{99}-\sqrt{999}+\sqrt{9999} \), and \( c = \sqrt{99}+\sqrt{999}-\sqrt{9999} \).

Answer: 22194


Question: Find the extremal function $y(x)$ that satisfies the integral \( \int^{3}_{0}(x(y')^3 -3y(y')^2)dx \) with the boundary conditions $y(3) = 1$ and $y(0) = 7$.

Answer: y = -2x + 7


Question: Let \( f:[0,1]\rightarrow \mathbb{R} \) be a continuous function such that \( \int_{0}^{1}x^{n}f(x)dx=0 \) for all \( n=0,1,2,\ldots \). Prove that \( f(x)=0 \) for all \( x \in [0,1] \).

Answer: f(x) = 0


Question: Given the recurrence relation \(a_{n+1}a_{n} = a_{n}^{2} + 1\) with \(a_{0} = 5\), determine \(\left \lfloor{a_{1000}}\right \rfloor\).

Answer: 45


Question: Determine the stable points of the dynamical system described by the equation \( \frac{dx}{dt} = 2\sqrt{1-x^2} \) for \( |x| \leq 1 \).

Answer: 1


Question:
Determine the order of accuracy of the following difference operator as an approximation to \( \frac{d}{dx}\left(a(x) \frac{du}{dx} \right)\bigg|_{x_i} \):
\[ \frac{(a_{i+1} + a_i)(u_{i+1} - u_i) - (a_i + a_{i-1})(u_i - u_{i-1})}{2h^2} \]
where \( h \) is the mesh width, and \( a_i = a(x_i) \) and \( u_i = u(x_i) \).

Answer: 2


Question: Calculate the Littlewood-Richardson coefficient \(C^{(2,1)}_{(1), (1,1)}\).

Answer: 1


Question: Given that \( \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\left(\frac{n}{n-i}\right)^{\gamma} = c \), where \( c \) is a constant, determine if for large \( n \), the expression \( \sum_{i=1}^{n}\left(\frac{n}{n-i}\right)^{\gamma} \) is approximately equal to \( n \times c \).

Answer: Yes


Question: Find the infimum of the sequence \( \frac{(n+1)^2}{2^n} \) for \( n \geq 0 \).

Answer: 0


Question: Given that $\sin(t^2)$ is a solution, find a second linearly independent solution to the differential equation $ty'' - y' + 4t^3y = 0$ for $t > 0$ using the method of reduction of order.

Answer: \cos(t^2)


Question: If $f: \mathbb{R} \to \mathbb{R}$ is a $C^1$ and strictly convex function with superlinear growth, i.e., $\lim_{|p|\to \infty} \frac{f(p)}{|p|} = \infty$, determine whether the Legendre transform $f^*$ is $C^1$.

Answer: Yes


Question: Evaluate the limit: \[ \ell = \lim _{x \rightarrow \frac{\pi}{2}} \frac{(1-\sin x)(1-\sin ^{2} x) \ldots (1-\sin ^{n} x)}{\cos ^{2 n} x} \]

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


Question: Evaluate the limit without using L'Hospital's rule: \[ \lim\limits_{x \to 0}\dfrac{\tan{x}-\sin{x}}{x^3} \]

Answer: \dfrac{1}{2}


Question: Determine the number of topological orderings for the directed acyclic graph with the following edges: 1 → 4, 2 → 5, 3 → 6.

Answer: 90


Question: Find the limit of the sequence defined by the recurrence relation \( x_{n+1} = \frac{x_n + nx_{n-1}}{n+1} \) with initial conditions \( x_0 = 0 \) and \( x_1 = 1 \) as \( n \to \infty \).

Answer: \ln 2


Question: Let $(x_n)$ be a sequence in a metric space $(X,d)$, and let $A$ be a compact subset of $X$. If $d(x_n,A) \to 0$, can we conclude that $(x_n)$ has a cluster point in $A$? Provide a rigorous justification for your answer.

Answer: Yes


Question: Evaluate the integral: \[ \int_0^1 \frac{e^x(1+x) \sin^2(x e^x)}{\sin^2(x e^x)+ \sin^2(e-x e^x)} \,dx. \]

Answer: \dfrac{e}{2}


Question: Find the real value of $a$ for which the definite integral \( f(a) = \int_{0}^{\infty} \frac{x^a}{2x^6 + 4x^5 + 3x^4 + 5x^3 + 3x^2 + 4x + 2} \, dx \) is minimized.

Answer: 2


Question: Determine whether there exists a non-discrete, torsion-free, $\sigma$-compact, locally compact abelian group.

Answer: Yes


Question:
Given the premises:
\[ \forall x \forall y: P(x,y) \implies Q(x) \]
and
\[ \forall x \exists y: P(x,y) \]
can you conclude that
\[ \forall x: Q(x) \]?
If so, provide the rationale for this conclusion.

Answer: \forall x: Q(x)


Question: Is the equation \[ l = \sum_{n=0}^{\infty}\sum_{k_1+\dots+k_m=n}a_{1, k_1}\cdots a_{m, k_m} \] correct for the Cauchy product of $m$ series?

Answer: Yes


Question: In a ring \((R, +, \times, 0, 1)\) with unity, consider a non-zero element \(a \in R\) that is nilpotent. Can \(a\) also be idempotent? Justify your answer.

Answer: No


Question: Let $K$ be an algebraically closed field and $M$ a maximal ideal of the polynomial ring $K[X_1,...,X_n]$. The quotient ring $K[X_1,...,X_n]/M$ is a field and is isomorphic to $K$. Does there exist an isomorphism of fields $\sigma: K[X_1,...,X_n]/M \to K$ such that for all $a \in K$, the image of the equivalence class of $a$ in $K[X_1,...,X_n]/M$ under $\sigma$ is $a$ itself?

Answer: Yes


Question: A triangle has an area of 2. The lengths of its medians are equal to the lengths of the sides of a second triangle. The lengths of the medians of the second triangle are equal to the lengths of the sides of a third triangle. This pattern continues indefinitely, with the medians of the zth triangle having the same lengths as the sides of the (z+1)th triangle. Find the limit of the sum of the areas of all the triangles in this infinite sequence.

Answer: 8


Question: Solve the quadratic equation for $x$: $$ 0 = 0.001 + \frac{-0.0018 x + 0.009 x^2}{\left(\sqrt{0.04 - x^2}\right)^3} $$

Answer: 0.0045


Question:
Consider the functional \( I(y) = \int_0^1 \left(\frac{1}{2}(y^{'})^2 - y\right) \, dx \) with boundary conditions \( y(0) = 0 \) and \( y(1) = 0 \), where \( y \in C^2[0, 1] \). Using the Rayleigh-Ritz method, let \( y_\text{app} \) be a polynomial approximation involving only one coordinate function. If \( y_e(x) \) is the exact extremizing function, at which points do \( y_e \) and \( y_\text{app} \) coincide?

(A) \( x = 0 \) but not at remaining points in \([0, 1]\)
(B) \( x = 1 \) but not at remaining points in \([0, 1]\)
(C) \( x = 0 \) and \( x = 1 \), but not at other points in \([0, 1]\)
(D) All points in \( x \in [0, 1] \)

Answer: D


Question: Determine the number of ordered pairs of integers \((a, b)\) such that \(ab \geq 0\) and \(a^3 + b^3 + 99ab = 33^3\).

Answer: 35


Question:
Let \( L^p([0,1]) = \{f: [0,1] \to \mathbb{R} \mid f \text{ is measurable and } \int_{0}^{1}|f|^p \, dx < + \infty\} \) with the norm \( \| f \|_p = \left(\int_{0}^{1} |f|^p \, dx\right)^{1/p} \).

Consider the subspace \( F = \{f \in L^p([0,1]) \mid f(x) = 0 \text{ for } x > \frac{1}{2} \} \).

Define the operator \( T: L^p([0,1]) \to F \) by:
\[
T(f)(x) = \begin{cases}
  f(2x) C_p, & \text{if } x \leq \frac{1}{2}, \\
  0, & \text{if } x > \frac{1}{2}.
\end{cases}
\]

Find the constant \( C_p \) such that \( \| T(f) \|_p = \| f \|_p \) for all \( f \in L^p([0,1]) \).

Answer: 2^{1/p}


Question: Evaluate the limit: $$\lim\limits_{h\rightarrow 0}{\frac{1}{h} \int\limits_{3}^{3+h}{e^{t^2} dt}}$$

Answer: e^9


Question: Consider a sequence of i.i.d. random variables $X_i$ with density $f(x) = |x|^{-3}$ for $|x| > 1$ and $0$ otherwise. Show that the sum $(n\log n)^{-1/2}\sum_{i=1}^n X_i$ converges in distribution to a normal distribution $N(0, \sigma^2)$, where $\sigma$ is to be determined. Use the fact that the truncated version $\bar{X_k}$ of $X_k$ at $\pm\sqrt{n\log n}$ satisfies Lindeberg's condition and converges to $N(0,1)$. Argue why $(n\log n)^{-1/2}\sum_{i=1}^n X_i$ also converges to the same normal distribution.

Answer: 1


Question: Calculate the limit \( \lim_{x \to + \infty}\int_{\mathbb{R}} \frac{t^2}{1+t^2}e^{-(x-t)^2}dt \).

Answer: \sqrt{\pi}


Question: Let $F$ be a finite field and $K/F$ be a field extension of degree 6. Determine whether the Galois group of $K/F$ is isomorphic to $S_3$ or $C_6$.

Answer: C_6


Question: Let $G$ be a group and $M$ a maximal subgroup of $G$ such that the permutizer $P_{G}(M) = G$. Is there an element $x \in G \setminus M$ such that $G = \langle x \rangle M = M \langle x \rangle$?

Answer: Yes


Question: Evaluate the integral \( \int_{-2k-\Lambda}^{2k+\Lambda} \frac{\sin[(q-2k)r]\cos(2kr) + \cos[(q-2k)r]\sin(2kr)}{q-2k}dq \) in the limit as \( \Lambda \rightarrow \infty \) and for \( kr \gg 1 \). Show that the result is \( \pi \cos(2kr) \).

Answer: \pi \cos(2kr)


Question: Let $K/\mathbb{Q}$ be a quadratic extension. Consider $P(K)$, the group of principal fractional ideals of $\mathcal{O}_K$, and $P^+(K)$, the subgroup of principal fractional ideals with generators having positive norm over $\mathbb{Q}$. Determine the order of the quotient group $P(K)/P^+(K)$. Assume $\mathcal{O}_K$ does not contain a unit of norm $-1$. What is the order of $P(K)/P^+(K)$?

Answer: 2


Question: Evaluate the Legendre symbol \(\left(\frac{10}{41}\right)\) to determine if 10 is a quadratic residue modulo 41 without using Euler's Criterion.

Answer: 1


Question: Determine the number of bases in the vector space \( \mathbf{Z}_5^4 \).

Answer: 116064000000


Question: Let $X$ be a random variable from a continuous uniform distribution over $[1,3]$, and let $Y$ be a random variable from a discrete uniform distribution with $P(Y=1)=P(Y=2)=P(Y=3)=\frac{1}{3}$. Calculate the expected value of $\min(X,Y)$.

Answer: \dfrac{19}{12}


Question: Consider three disjoint line segments of equal length labeled AB, CD, and EF. Determine the size of the group of symmetries of this configuration, where symmetries include exchanging line segments and flipping segments end for end.

Answer: 48


Question: Calculate the limit: \[ \lim_{n\to\infty}\int_{0}^{\infty} \frac{x^{n-2}}{1+ x^{n}}\cos(n\pi x) \, dx \]

Answer: 0


Question: Determine the values of the real number $t$ for which the set \( \{(x,y) \in \mathbb{R}^2: x^2+y^2=txy\} \) is a subspace of \( \mathbb{R}^2 \).

Answer: [-2, 2]


Question: Given that $f(x + y) = f(x)f(y)$, $f(5) = 2$, and $f'(0) = 3$, where $f$ is a differentiable function, find the value of $f'(5)$.

Answer: 6


Question: Evaluate the double integral \( \int \int_D xydA \) where \( D \) is the union of the regions defined by the following bounds: \( \int_{-1}^{1}\int _{0}^{\sqrt{1-x^2}}xydydx + \int_{-1}^{1}\int _{-x/3-1/\sqrt{3}}^{\sqrt{x^2-1}}xydydx \).

Answer: 0


Question: Determine the dimension of the vector space of all $3 \times 3$ real matrices where the sum of the elements in each row and each column is zero.

Answer: 4


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

$$(\forall n\in \mathbb{N})  (\exists \alpha \in \mathbb{R^+}) : n\arctan(\alpha)= \sum\limits_{k=1}^n \arctan(\frac{k}{n}).$$

Answer: True


Question: Determine the largest integer value of \( k \) such that the inequality \[ k \cdot \left(\frac{abc}{a+b+c}\right) \leq \left( a+b \right)^{2} + \left( a+b+4c \right)^{2} \] holds for all positive real numbers \( a, b, \) and \( c \).

Answer: 100


Question: Evaluate the triple integral \( \iiint_G y\,dV \) where \( G \) is the solid enclosed by the line \( z=y \), the \( xy \)-plane, and the parabolic cylinder \( y=1-x^2 \).

Answer: \dfrac{32}{105}


Question: Evaluate the limit: \[ \lim_{n\to \infty} n\left[1-\cos\left(\frac{\theta}{n}\right) -i\sin\left(\frac{\theta}{n}\right)\right] \]

Answer: -i\theta


Question: Determine the field that is isomorphic to the quotient ring \( \frac{\mathbb{C}[x]}{\langle x-a \rangle} \).

Answer: \mathbb{C}


Question: Compute the Lebesgue outer measure of the set \((0,1) \cap \mathbb{Q}^c\), where \(\mathbb{Q}\) is the set of rational numbers.

Answer: 1


Question: Evaluate the triple integral \( \iiint\limits_{V} y \, dx \, dy \, dz \) where \( V \) is the region defined by the sphere \( x^2 + y^2 + z^2 \leq 2y \). Use spherical coordinates to perform the integration.

Answer: \dfrac{4\pi}{3}