Question: Determine whether the inequality \((a_1+...+a_n)(a_1^2+...+ a_n^2) \le n(a_1^3+...+a_n^3)\) holds for all real numbers \(a_1, a_2, \ldots, a_n\).

Answer: No


Question:
If the following two equalities hold:
\[\int \omega(x) \, dG(x) = 1\]
\[\int \omega(x) \Omega(x) \, dG(x) = 1\]
is it necessarily the case that \(\Omega(x) = 1\) for all \(x\)?

Answer: No


Question: Let \( f(x) \) be a monic cubic polynomial. The solutions of the equation \( f\left(\frac{1}{x}\right)=0 \) are \( \alpha, \beta, \gamma \). Given that \( \alpha + \beta + \gamma = 10 \) and \( \alpha \beta \gamma = 15 \), find \( \left \lceil{f(10)-f(-10)}\right \rceil \).

Answer: 2014


Question: How many onto functions are there from a set with 9 elements to a set with 7 elements, such that exactly one element in the domain is mapped to a specific element in the codomain?

Answer: 1723680


Question: Let $G$ and $H$ be groups, and $R$ a commutative ring. Consider the group algebras $RG$ and $RH$, which are $R$-modules with bases $G$ and $H$, respectively. Determine whether the tensor product $RG \otimes_R RH$ has a basis given by the simple tensors $g \otimes h$, where $g \in G$ and $h \in H$. Provide a justification for your answer.

Answer: Yes


Question: Consider two random variables: $X_1 \sim \text{Exponential}(\lambda=2)$ and $X_2 \sim \text{Uniform}([-1,2])$. Define $S = X_1 + X_2$. Determine the range of $S$, which is the smallest set $I$ such that $P(S \in I) = 1$. Assume $X_1$ and $X_2$ are independent.

Answer: [-1, \infty)


Question: Find the minimum value of the function \( \frac{2x^{3}+1}{4y(x-y)} \) given that \( x \ge -\frac{1}{2} \) and \( \frac{x}{y} > 1 \).

Answer: 3


Question: Determine the number of elements of order 6 in the symmetric group $S_7$.

Answer: 1470


Question: Given two points $A$ and $B$ on the same side of a line $l$, with orthogonal projections $A'$ and $B'$ on $l$, where $d(A,A') = 5$, $d(B,B') = 4$, and $d(A',B') = 10$, find a point $C$ on line $l$ such that the sum $d(A,C) + d(B,C)$ is minimized.

Answer: \dfrac{50}{9}


Question: Approximate the integral \( \int_{0}^{1} x^{x} \, dx \) such that the error is less than or equal to \( 10^{-6} \).

Answer: 0.783431


Question: Find the dimensions of a cuboid with a fixed volume \( V \) that minimize its surface area. Use the theorem for extrema of a function with two variables to solve the problem.

Answer: \sqrt[3]{V}


Question:
Let \( T \) be a \( 4 \times 4 \) real matrix such that \( T^4 = 0 \). Define \( k_i = \dim \ker T^i \) for \( 1 \leq i \leq 4 \). Determine which of the following sequences \( k_1 \leq k_2 \leq k_3 \leq k_4 \) is not possible:

(i) \( 3 \leq 4 \leq 4 \leq 4 \)
(ii) \( 1 \leq 3 \leq 4 \leq 4 \)
(iii) \( 2 \leq 4 \leq 4 \leq 4 \)
(iv) \( 2 \leq 3 \leq 4 \leq 4 \)

Answer: ii


Question: Let \( A \) be a \( 2 \times 2 \) matrix with integer entries, \( A \in M(2, \mathbb{Z}) \). Define \( \sin A \) as \( \sin A := \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1} \), which is known to converge absolutely in the Banach space \( \mathcal{L}(\mathbb{R}^2) \) under the operator norm. If \( \sin A \in M(2, \mathbb{Z}) \), is it true that \( A^2 = O \)? Justify your answer.

Answer: Yes


Question: Let $A$ be a set with $\text{card}(A) = a$. Determine the cardinal number of the set of countably infinite subsets of $A$. Express your answer in terms of $a$. \( \text{Hint: Consider the set of injective functions from } \mathbb{N} \rightarrow A. \)

Answer: a^{\aleph_0}


Question: Suppose \(x\) and \(y\) are two vectors in \(l_p\) (where \(1\leq p<\infty\)) such that \(||x||=||y||=1\). Can we find a complex scalar \(\alpha\), with \(|\alpha|<1\) such that \(||x+\alpha y||>1\)?

Answer: Yes


Question: Does there exist a continuous open function \(f: B^n \to B^n\) that is not injective, where \(B^n\) is the open unit ball in \(\mathbb{R}^n\)?

Answer: Yes


Question: Find the value of \( a \) such that the probability \( \Pr(X_1 < 4X_2^2) \geq 0.99 \), where \( X_1 \) and \( X_2 \) are independent random variables with uniform distributions \( 1 \leq X_1 \leq 9 \) and \( 1 \leq X_2 \leq a \), respectively.

Answer: \dfrac{53}{3}


Question: Calculate the surface integral \( \int_B \operatorname{curl}(a) \cdot d\vec{S} \), where \( B = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 = 1, z \geq 0\} \) is the surface of the upper half of a unit sphere centered at the origin, and the vector field \( a(x,y,z) = (-y, x, z) \) in \( \mathbb{R}^3 \).

Answer: 2\pi


Question:
Find the minimum value of \( \left [ s,2 \right ] \) such that the following inequality holds:
\[
n\left [ b,2 \right ]+\left [ 0.00155n\left [ b,2 \right ],2 \right ]+\left \lfloor 0.15 \left [ 0.00155n\left [ b,2 \right ],2 \right ],2 \right \rfloor
\le n\left [ s,2 \right ]-\left [ 0.00155n\left [ s,2 \right ],2 \right ]-\left \lfloor 0.15 \left [ 0.00155n\left [ s,2 \right ],2 \right ],2 \right \rfloor
\]
where \( n \) is a natural number, and \( b, s \) are positive real numbers. Use the example where \( n = 350 \) and \( \left [ b,2 \right ] = 28.62 \) to guide your solution.

Answer: 28.73


Question: A solid lies between planes perpendicular to the $x$-axis at $x=0$ and $x=14$. The cross-sections perpendicular to the axis on the interval $0 \leq x \leq 14$ are squares with diagonals running from the parabola $y=-2\sqrt{x}$ to the parabola $y=2\sqrt{x}$. Find the volume of the solid.

Answer: 784


Question: Find the minimum value of the function \( f(x,y,z) = x + 3y + 5z \) subject to the constraint \( x^2 + y^2 + z^2 = 1 \) using the method of Lagrange multipliers.

Answer: -\sqrt{35}


Question: Calculate the residue of the function \( f(z) = \frac{1}{z^2 \sin z} \) at \( z = 0 \).

Answer: \dfrac{1}{6}


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

Answer: 1


Question: Evaluate the limit \(\lim_{x\to\infty} (\sqrt{x+1} + \sqrt{x-1} - \sqrt{2x})\) using the little o notation.

Answer: \infty


Question: Estimate the number of digits of accuracy that can be expected in the solution \( x \) of the linear system \( Ax = b \), given that the entries of \( A \) have about 6 digits of accuracy, \( b \) has about 4 digits of accuracy, and the condition number of \( A \) is about \( 10^3 \).

Answer: 1


Question: A periodic and twice differentiable function \(f\) with period 4 satisfies \(f(x) = f(2-x)\) and \(f'(0) = 0\). Determine the minimum number of solutions to the equation \(f''(x) = 0\) in the interval \([0,6]\).

Answer: 6


Question:
Determine the number of solutions to the Bernoulli initial value problem (IVP):
\[ y'(x) = \frac{y(x)}{x} + \sqrt{x^3y(x)^3}, \quad y(1) = 1 \]
where \( x, y > 0 \).

Answer: 1


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

Answer: \dfrac{1}{8}


Question: How many 10-digit numbers can be formed using the digits \{5, 6, 7\} such that no two consecutive digits are the same and the first and last digits are the same?

Answer: 510


Question: For each positive integer $n \ge 1$, a sequence is defined by the recursive relation $a_{n+1} = \frac{1}{1 + a_n}$. Given that $a_1 = a_{2012}$, find the sum of the squares of all possible values of $a_1$.

Answer: 3


Question: Let $G$ be a group with no subgroups of index $2, 3, \ldots, n-1$. If $G$ has a subgroup $H$ of index $n$, is $H$ normal in $G$?

Answer: No


Question: Find the values of \( \alpha \) for which the state of the system \( \mathbf{x}(k+1) = A \mathbf{x}(k) + \mathbf{b} u(k) \) can be driven to zero in one step, given that \( \mathbf{x}(0) = \begin{bmatrix} 1 \\ -2 \end{bmatrix} \). Here, \( A = \begin{bmatrix} 1 & 2 \\ 3 & \alpha \end{bmatrix} \) and \( \mathbf{b} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \).

Answer: 3


Question: Calculate the limit superior: \( \limsup_{n \to \infty} \sin(n) \).

Answer: 1


Question: Find a function \( f(k) \) such that \( S_k = \Theta(f(k)) \), where \( S_k = \sum_{n=1}^{k^2-1} \sqrt{n} \).

Answer: k^3


Question: Find all values of \( \alpha \) such that the solution to the differential equation \( y''' + y'' - y' - y = 0 \) with initial conditions \( y(0) = 7 \), \( y'(0) = -3 \), and \( y''(0) = \alpha \) satisfies \( y(x) \to 0 \) as \( x \to \infty \).

Answer: -1


Question: A drunk squirrel is dropped at a location \(i\) on a 1-dimensional branch of an oak tree, where \(i\) is drawn from a Gaussian distribution \(i \sim \mathcal{N}(0, 4)\). After the drop, the squirrel makes a step to the right by a distance \(d\), where \(d \sim \mathcal{N}(0, 1)\). The final location \(f\) of the squirrel is given by \(f = i + d\). If the squirrel ends up at location \(f = 2\), what is the most likely initial location \(i\) that the squirrel landed on?

Answer: \dfrac{8}{5}


Question: Evaluate the line integral \( \int_C \sqrt{x^2 + y^2 + z^2} \, ds \), where \( C \) is the curve defined by the parametric equations \( x = 2y = 3z \) with \( z \) ranging from 1 to 3.

Answer: 49


Question: Let \( f(z) \) be an analytic function in the open unit disc \( D \), continuous in the closed unit disc \( \bar{D} \), with \( |f(z)| \leq C \). Suppose \( f(z)/z^n \) can be extended to be analytic in the unit disc. Find a bound for \( |f(z_0)| \) given \( |z_0| < 1 \).

Answer: C |z_0|^n


Question: Find the power series representation of the function $x$.

Answer: x


Question: Is the expression \( \pi \cdot a^{\frac{m}{n}} \) always irrational for every choice of natural numbers \( m, n, \) and \( a \)?

Answer: Yes


Question: Calculate the area enclosed by the curve parameterized by \((\cos^3 (t), \sin^3 (t))\) for \(0 \leq t \leq 2\pi\).

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


Question: Determine the number of ordered quadruples \((a, b, c, d)\) of integers in \(\mathbb{Z}_5\) such that \(ad - bc = 1\).

Answer: 120


Question: The sequence $x_0, x_1, \dots$ is defined by $x_0 = 3$, $x_1 = 18$, and $x_{n+2} = 6x_{n+1} - 9x_n$ for $n = 0, 1, 2, \dots$. Determine the smallest positive integer $k$ such that $x_k$ is divisible by 2013.

Answer: 670


Question: Given a set $\Omega$, let $A_1, A_2, \ldots, A_n$ be distinct subsets of $\Omega$. Determine the number of sets in the $\sigma$-field generated by $\{A_1, A_2, \ldots, A_n\}$.

Answer: 2^{2^n}


Question: Let K be a finite field and let R be a subgroup of a group P. Given that the irreducible KP-modules have dimensions 1, 4, and 16 over K, consider a KP-module M with dimension at most 5 over K. It is known that M does not have a quotient of dimension 1 over K. Additionally, when M is considered as a module over KR, it contains a 2-dimensional irreducible module. Is M necessarily a 4-dimensional irreducible module over KP?

Answer: Yes


Question:
Solve the equation for the real number \( x \):
\[ 3\sqrt{2+x} - 6\sqrt{2-x} + 4\sqrt{4-x^2} = 10 - 3x \]

Answer: \dfrac{6}{5}


Question: Does there exist a metrizable separable Tychonoff space that is almost strongly zero-dimensional but not strongly zero-dimensional?

Answer: Yes


Question: Is the infinite product space $A = \prod_{i=1}^{\infty} R$ homeomorphic to the space $N \times A$, where $R$ is the space of real numbers with the Euclidean topology and $N$ is the space of natural numbers with the discrete topology, both endowed with the product topology?

Answer: No


Question: Determine the number of zeros of the polynomial \( z^{17} + z - 1 \) that lie within the region \( |z| < 1 \).

Answer: 1


Question: Determine the degree of the field extension $[K(x):K]$, where $K(x)$ is the field of rational functions over a field $K$. What is $[K(x):K]$?

Answer: \infty


Question: Determine whether the surface defined by the equation \( xy - y \log(z) + \sin(xz) = 0 \) can be expressed in the form \( z = f(x, y) \) near the point \((0, 2, 1)\) using the Implicit Function Theorem. Assume \( f \) is a smooth function. Consider the invertibility of the matrix formed by differentiating the function with respect to \( z \) at \((0, 2, 1)\). Is this condition sufficient to show the required representation?

Answer: Yes


Question: Determine the smallest integer \( n \) such that the error in the nth degree Taylor polynomial approximation of \( \ln(1-x^2) \) about \( x=0 \) is less than 0.1 for \( 0 \leq x \leq 0.5 \).

Answer: 2


Question: Calculate the volume of the solid that lies under the paraboloid \( z = 8x^2 + 8y^2 \), above the \( xy \)-plane, and inside the cylinder \( x^2 + y^2 = 2x \).

Answer: 12\pi


Question: Evaluate the integral \( \oint_C \frac{dz}{z-2} \) where \( C \) is the square with vertices at \( \pm 2 \pm 2i \).

Answer: \pi i


Question: Solve for integer values of $x$ in the equation: \[ 3^{3x} - 3^x = (3x)! \]

Answer: 2


Question: Let $f(x)$ be a function that is infinitely differentiable on $\mathbb{R}$. Determine whether the following statement is true: \[ \int_{a}^{b} f'(x) \, dx = f(b) - f(a) \]

Answer: True


Question: Suppose \( f: \mathbb{R} \to \mathbb{R} \) is a continuous and one-to-one function. Does \( f \) map Borel sets to Borel sets?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin(x)^{200}}{x^{199}\sin(4x)}. \]

Answer: \dfrac{1}{4}


Question: Find the last 5 digits of the sum: \(1! + 2\cdot 2! + \cdots + 24 \cdot 24!\).

Answer: 99999


Question: Determine if the expression \( \sqrt{-i \sqrt{5}} \) is equal to \( -i\sqrt[4]{-5} \).

Answer: Equal


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

Answer: \ln 2


Question: Given $m$ points $x_1, \ldots, x_m$ in $\mathbb{R}^d$, determine if there exists a polynomial $p$ in $d$ indeterminates such that $p(x_i) = 0$ for all $i$, and $p(x)$ is negative for any other point $x$. Is it possible to construct such a polynomial?

Answer: Yes


Question: Calculate the area under the curve of the function \( f(x) = \sqrt{x} \) over the interval \([1, 9]\) using the Riemann sum approach. Divide the interval into \( n \) equal parts and express the area as a limit of a sum. Evaluate this limit to find the area.

Answer: \dfrac{52}{3}


Question: Let $K$ be a field, $P$ be an irreducible polynomial of $K[X]$, and $v_P$ be the valuation of $K[X]$ associated with $P$. For two polynomials $Q, R$ in $K[X]$, does the equality $v_P(Q \circ R) = v_P(Q) \times v_P(R)$ hold? Provide a justification for your answer.

Answer: No


Question: Determine the radius of convergence of the power series \( \sum\limits_{n=2}^{\infty} \pi(n) z^{n} \), where \( \pi(n) \) denotes the Euler function.

Answer: 1


Question:
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function such that for any two real numbers \( x \) and \( y \), \( |f(x)-f(y)| \leq 7|x-y|^{201} \). Determine which of the following is true:

(A) \( f(101)=f(202)+8 \)
(B) \( f(101)=f(201)+1 \)
(C) \( f(101)=f(200)+2 \)
(D) None of the above.

Answer: D


Question: Under what conditions is the expression \( \frac{aA+b}{cA+d} \) rational, given that \( a, b, c, \) and \( d \) are rational numbers and \( A \) is irrational?

Answer: ad = bc


Question: Suppose $\Delta$ is an intersecting family of subsets of a set $A$, where $A$ has $n$ elements. Additionally, $\Delta$ is an antichain, meaning for any $A_1, A_2 \in \Delta$ with $A_1 \neq A_2$, neither $A_1 \subset A_2$ nor $A_2 \subset A_1$. Each element of $\Delta$ has size at most $k$, where $k \leq n/2$ is a positive integer. Let $\Gamma$ be the set of all subsets of $A$ with $k$ elements. Determine if there exists a one-to-one function $f$ from $\Delta$ to $\Gamma$ such that for every $x \in \Delta$, $x \subseteq f(x)$. Provide a justification for your answer.

Answer: Yes


Question: Given a matrix $A_{n\times n} = (a_{ij})$, where $a_{ij} = b_{i}^{2} - b_{j}^2$ for $i, j = 1, 2, \ldots, n$ and $b_1, b_2, \ldots, b_n$ are distinct real numbers, find the determinant of $A$ when $n$ is even.

Answer: 0


Question: Determine the convergence rate of the Newton iteration for the polynomial \( f(x) = 8x^4 - 12x^3 + 6x^2 - x \) when started close to the root \( r = 0 \).

Answer: 2


Question: Given the sequence \( \{a_n\} \) defined by \( a_0 = 3^{1/3} \) and \( a_{n+1} = (a_n)^{1/a_n} \), find \( \lim_{n \to \infty} a_n \).

Answer: 1


Question: If \( \frac{1}{N}\sum_{i=1}^N |a_i| \to 0 \) as \( N \to \infty \), does it necessarily follow that \( \frac{1}{N}\sum_{i=1}^N a_i \to 0 \)?

Answer: Yes


Question: Evaluate the limit \( \lim_{r \to 1^-} \sum_{n=1}^\infty r^{n!} \).

Answer: \infty


Question:
Consider a distribution function $F$ with associated density $f$. Define the following sets:

- $S_1=\{s: F(s+\varepsilon)> F(s-\varepsilon)\;\forall\;\varepsilon>0\}$;
- $S_2=\text{cl}(\{s: f(s)>0\})$.

Determine if $S_1 = S_2$. Justify your answer.

Answer: S_1 = S_2


Question: The quadratic equation \(x^2 - 3ax + b = 0\) does not have distinct real roots. Find the least possible value of \(\frac{b}{a-2}\), given that \(a > 2\).

Answer: 18


Question:
Let $A_1, A_2, \ldots, A_{2012}$ and $H_1, H_2, \ldots, H_{2012}$ be the arithmetic and harmonic means between $a$ and $b$, respectively. Given that $A_{1006}H_{1007} = 2013$, find the value of:
\[ \frac{A_5H_{2008}}{A_{10}H_{2003}} \]

Answer: 1


Question:
Consider a method for randomly generating a classical Hilbert propositional calculus formula $F$ with $n$ variables. The formula $F$ is generated as follows:
- $F = x_i$ for each $i \leq n$ with probability $\frac{1}{n+2}$.
- $F = \neg F_1$, where $F_1$ is generated independently using the same method.
- $F = F_1 \to F_2$, where $F_1$ and $F_2$ are generated independently using the same method.

Given that the generation process terminates with probability 1, what is the probability that the formula $F$ is a tautology?

Answer: 0


Question: Calculate the covariance \( \text{cov}(5W_7 + 6W_9, W_7) \) where \( W_t \) is a standard Brownian motion.

Answer: 77


Question: Given two matrices \(A\) and \(P\) of order \(n \times n\), where \(A^2P = AP^2\) and \(A^3 = P^3\), find the value of \(|A^2 + P^2|\).

Answer: 0


Question: Calculate the limit: $$\lim_{k \rightarrow \infty} k(1+\cos(2k \pi x)) \chi_{\left(\frac{-1}{2k},\frac{1}{2k}\right)}$$

Answer: \delta(x)


Question: Let $X$ and $Y$ be Polish spaces, and let $A \subset X$ be a Borel subset. Suppose $f: A \to B \subset Y$ is a homeomorphism. Is $B$ a Borel subset of $Y$?

Answer: Yes


Question: In the free group of rank 2, denoted as $G = F_2$, determine if there exists a constant $c > 0$ such that for every commutator $[u,v] = uvu^{-1}v^{-1}$, where $u, v \in G$ and $|u|, |v| > 0$, the word length $|[u,v]|$ is at least $c(|u| + |v|)$, unless $[u,v] = [u_1,v_1]$ for some $u_1, v_1$ with $|u_1| + |v_1| < |u| + |v|$. Provide a justification for your answer.

Answer: Yes


Question: Prove that \(2^{2^n} + 1 \equiv 2 \pmod{2^{2^m} + 1}\) for integers \(n > m \geq 0\).

Answer: 2


Question: Evaluate the integral \( \int x \, d\mu(x) \) where \( \mu \) is a measure on the subsets of \( \mathbb{N} = \{1, 2, \dots\} \) such that \( \mu(\{n, n+1, \dots\}) = \frac{n}{2^n} \).

Answer: 2


Question: Let $(X,d)$ be a Polish metric space and let $\{\mu_n\}_{n\in\mathbb{N}}$ be a sequence of probability measures such that $\mu_n\rightarrow\mu$ narrowly. If there exists a compact set $K$ that contains the supports of the $\mu_n$'s, does $K$ also contain the support of the narrow limit $\mu$? Provide a justification for your answer.

Answer: Yes


Question: Let $A$ be a $227 \times 227$ matrix with entries in $\mathbb{Z}_{227}$, such that all the eigenvalues are distinct. What is the trace of $A$?

Answer: 0


Question: Suppose $X$ is a topological space with a base for the topology given by $\{B_i\}$, where $i$ runs over some index set $I$. If for every covering of $X$ by base elements, $X=\bigcup_{i\in I}B_i$, there exists a finite subcover $X=\bigcup_{i\in S}B_i$ where $S$ is a finite subset of $I$, does this imply that $X$ is compact?

Answer: Yes


Question: Does there exist a function \( f: \mathbb{R} \to \mathbb{R} \) such that for every \( y \in \mathbb{R} \), there are uncountably many \( x \in \mathbb{R} \) with \( f(x) = y \)?

Answer: Yes


Question: Evaluate the line integral \( \int_C (\ln y) e^{-x} \,dx - \dfrac{e^{-x}}{y}\,dy + z\,dz \) where \( C \) is the curve parametrized by \( r(t)=(t-1)i+e^{t^4}j+(t^2+1)k \) for \( 0\leq t\leq 1 \).

Answer: \dfrac{1}{2}


Question:
Suppose $w, x, y, z$ satisfy the following equations:
\[ w+x+y+z=25 \]
\[ wx+wy+wz+xy+xz+yz=2y+2x+193 \]
The largest possible value of $w$ can be expressed in lowest terms as $\frac{w_1}{w_2}$ for some integers $w_1, w_2 > 0$. Find $w_1 + w_2$. \(\text{(PUMAC Algebra A)}\)

Answer: 27


Question: Calculate the joint conditional distribution of $Y$ and $Z$ given $X = 3$ for the probability distribution $f(x,y,z) = \frac{xyz}{108}$, where $x = 1, 2, 3$, $y = 1, 2, 3$, and $z = 1, 2$. Verify if the conditional distribution is $\frac{zy}{18}$.

Answer: \dfrac{zy}{18}


Question: Evaluate the integral: \[ \int_{0}^{1}\frac{e^{2y}(4y^2-4y+2)-2}{y^2}dy \]

Answer: 4


Question:
Solve for \( x \) in the equation:
\[ 2^{x^{x^{x^{\cdots}}}} = 4 \]

Answer: \sqrt{2}


Question: Evaluate the integral \( \int_{0}^{\pi} f'(x) \cos x\, dx \) given that \( f'(0) = f'(\pi) = 0 \).

Answer: 0


Question: Determine whether every vector in the set $[-1,+1]^d$ can be expressed as a convex combination of vectors in the set $\{-1,+1\}^d$. Provide a proof or counterexample.

Answer: Yes


Question: Determine whether the infinite, binary, directed, connected, non-rooted graph is countable. The graph is similar to an infinite, binary, rooted tree in that two edges lead into every vertex and one out, but it does not have a root and is connected. Is this graph countable?

Answer: Yes


Question: Evaluate the limit: \( \lim_{x \to 0} x^{1/x} \).

Answer: 0


Question: Find the smallest positive integer \( n \) such that \((a+1)^{7^n} = (a+1)\) in the field \( \mathbb{F}_7[X]/(X^2+1) \), where \( a \) is the class of \( X \).

Answer: 2


Question: Determine which is greater: \( \log_9 71 \) or \( \log_8 61 \).

Answer: \log_8 61


Question:
Find the coordinates of the points on the curve \( y = x^3 - 3x^2 - 7x + 6 \) where the tangent line cuts off a segment on the negative x-axis that is half the length of the segment it cuts off on the positive y-axis. Choose from the following options:

(A) \((-1, 9)\)
(B) \((3, -15)\)
(C) \((1, -3)\)
(D) None of the above.

Answer: A


Question: Let $\bar X_1$ and $\bar X_2$ be the means of two independent samples of sizes $n$ and $2n$, respectively, from an infinite population with mean $\mu$ and variance $\sigma^2 > 0$. Determine the value of $w$ such that $(1-w)\bar X_1 + w\bar X_2$ is the minimum variance unbiased estimator of $\mu$. What is the value of $w$?

Answer: \dfrac{2}{3}


Question: Evaluate the integral \( \int_0^1 x \sin^{-1}(x) \, dx \).

Answer: \dfrac{\pi}{8}


Question:
Determine the convergence properties of the series \( \frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}+\ldots \). Choose the correct option:

(A) The series is uniformly convergent for all \( x \).
(B) The series is convergent for all \( x \) but the convergence is not uniform.
(C) The series is convergent only for \(|x|\le \frac{1}{2}\) but the convergence is not uniform.
(D) The series is uniformly convergent on \([-\frac{1}{2},\frac{1}{2}]\).

Answer: D


Question:
Evaluate the limit \( \lim_{n \to \infty} a_n \) where the sequence \( a_n \) is defined as follows for any natural number \( n \):
\[
a_n = \sqrt{4+\sqrt{4+3\sqrt{4+5\sqrt{4+7\sqrt{\cdots\sqrt{4+(2n-1)\sqrt{4+(2n+1)}}}}}}}
\]
Determine the value of this limit.

Answer: 3


Question: Evaluate the limit: $$\lim_{x \to 1} \left(\frac{23}{1-x^{23}}-\frac{11}{1-x^{11}} \right)$$

Answer: 6


Question: Does there exist a semigroup that admits no involution? Specifically, is there a finite semigroup that serves as a counterexample?

Answer: Yes


Question: Find the value of \( n \) such that \[ \lim\limits_{x \to 1}\frac{\pi/4-\tan^{-1}x}{e^{\sin(\ln x)}-x^n} =\frac{1}{8}. \]

Answer: 5


Question: Find all integer values of \( y \) such that \((1 + a)^y = 1 + a^y\) for any real number \( a \).

Answer: 1


Question: Evaluate the integral: $$\int^{x^2}_{0} \frac{dz}{2\sqrt{x^2-z}} \delta(i\sqrt{z}-a)$$ where $a > 0$.

Answer: 0


Question: Use Fubini's Theorem to evaluate the integral \( \int_{0}^{\pi/2}\int_{y}^{\pi/2}y\frac{\sin x}{x}dx \, dy \).

Answer: \dfrac{1}{2}


Question: Can the truth functions on $n$ propositional variables, which satisfy $f(T,T,\ldots,T)=T$, be expressed using only the logical connectives AND (∧) and OR (∨)?

Answer: No


Question: Let $X$ and $Y$ be sets such that the cardinality of their power sets are equal, i.e., $|\mathscr{P}(X)| = |\mathscr{P}(Y)|$. Does it follow that the cardinality of $X$ is equal to the cardinality of $Y$, i.e., $|X| = |Y|$? Justify your answer.

Answer: No


Question: Determine whether the following statement is true or false: If $f:[0,1) \to \mathbb{R}$ is continuous and $f(x) > 0$ for all $x$, then there exists $c > 0$ such that $f(x) \geq c$ for all $x$. Provide a justification for your answer.

Answer: B


Question: Does there exist an infinite non-abelian group in which every non-identity element has infinite order?

Answer: Yes


Question: A school buys 60% of its light bulbs from supplier A. The light bulbs from supplier A have exponentially distributed lifetimes with a mean of 600 hours, and the lifetimes of all bulbs are independent of each other. Using a normal approximation, find the probability that 100 light bulbs from supplier A last more than 50,000 hours in total.

Answer: 0.9522


Question: Simplify the complex number \( \frac{(7+5i\sqrt{3})^{145}}{(92+48i\sqrt{3})^{72}} \) and express your answer in algebraic form.

Answer: 7 + 5i\sqrt{3}


Question: Calculate \( \lim_{n\to\infty}\int_{[1,\infty)} \frac{1}{x^{\frac12+\frac1n}}\left(\sin\frac{\pi}{x}\right)^n \,dx \) using the dominated convergence theorem.

Answer: 0


Question: Evaluate the limit: \[ \lim \limits_{x \to \pi /3} \frac{(1-\cos6x)^{1/2}}{\sqrt{2} (\pi/3 - x)} \]

Answer: 3


Question: Let \( p \) be a prime number. Determine whether \( p \) always divides the binomial coefficient \( {p^2+1 \choose p} \).

Answer: Yes


Question: Given a matrix $A \in \mathbb{R}^{n \times n}$, prove that if $x^{T} A^{T} A x = x^T x$ holds for every vector $x \in \mathbb{R}^n$, then $A^{T}A = I_n$.

Answer: A^T A = I_n


Question: Let $a$ and $b$ be any two real numbers, and let $f$ and $g$ be continuous functions on the interval $[a, b]$. If $\int_a^b f(x) \, dx > 0$ and $\int_a^b g(x) \, dx < 0$, is it true that $\int_a^b f(x)g(x) \, dx < 0$?

Answer: No


Question: Suppose $E$ is an elliptic curve on a K3 surface $X$. Determine whether $E$ must intersect any curve $D \subset X$ of genus $g(D) \geq 3$. Provide a justification for your answer.

Answer: Yes


Question: Let X be a random variable uniformly distributed over the interval [0,2]. Find the expected value of \( \max(X, X^3) \).

Answer: \dfrac{17}{8}


Question:
Evaluate the limit: \( \lim_{r \to 0} I_1 (r) \), where

\[ I_1(r)=\int_{-r}^{r} \int_{-\sqrt{r-x^2}}^{\sqrt{r-x^2}} \frac{1} {(x^2+y^2) \log^2\big(\frac{2}{\sqrt{x^2+y^2}}\big)}\,dy\,dx. \]

Answer: 0


Question: Find the distance from the point \((0,0,0)\) to the curve defined by the equations \(\frac{x^2}{4}+y^2+\frac{z^2}{4}=1\) and \(x+y+z=1\) in \(\mathbb{R}^3\).

Answer: 1


Question: Suppose the function \( f(x) \) is bounded on the interval \([0,1]\) and satisfies the equation \( f(ax) = bf(x) \) for all \( 0 \le x \le \frac{1}{a} \), where \( a, b > 1 \). Determine \( \lim_{x \to 0^+} f(x) \).

Answer: 0


Question: Find the kernel of the homomorphism \( D: F[x] \to F[x] \) defined by \( D(f(x)) = f'(x) \), where \( f(x) = a_0 + a_1x + \cdots + a_nx^n \) and \( f'(x) = a_1 + 2a_2x + \cdots + na_nx^{n-1} \), given that the characteristic of the field \( F \) is \( p \).

Answer: F[x^p]


Question:
Find the volume of the region \( X = \{(x, y, z) \in \mathbb{R}^{3} : x+y+z<1 ; 0<x<1 ; 0<y<1 ; z>-1\} \) using the triple integral form:
\[ \int \left( \int \left( \int dx \right) dy \right) dz. \]

Answer: 1


Question: Determine the orthogonal complement \( F^\perp \) of the set \( F = \{x = (x_k)_{k \in \mathbb{N}^*} \in \ell_2 \mid \sum_{k=1}^{+\infty} x_k = 0 \} \) in the Hilbert space \( \ell_2 \) of complex sequences, where \( \ell_2 = \{x = (x_k)_{k \in \mathbb{N}^*} \mid \sum_{k=1}^{+\infty} |x_k|^2 < \infty \} \) with the inner product \( \langle x, y \rangle = \sum_{k=1}^{+\infty} x_k \overline{y_k} \).

Answer: \{0\}


Question: Find the volume of the wedge sliced from the cylinder \(x^2 + y^2 = 1\) by the planes \(z = a(2-x)\) and \(z = a(x-2)\), where \(a > 0\).

Answer: 4\pi a


Question: Calculate the volume of the solid formed by revolving the triangular region bounded by the lines $y = |x|$ and $y = 1$ around the line $x = -2$.

Answer: 4\pi


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

Answer: \dfrac{1}{2}


Question: Let $G$ be a profinite group. Consider the Grothendieck group $K(G,\mathbb{Z}_\ell)$ of the derived category of finitely generated $\mathbb{Z}_\ell$-modules with continuous $G$-action, and the Grothendieck group $K(G,\mathbb{F}_\ell)$ of the derived category of finitely generated $\mathbb{F}_\ell$-modules with continuous $G$-action. The map $[T] \mapsto [T \otimes_{\mathbb{Z}_\ell}^L \mathbb{F}_\ell]$ defines a group homomorphism $K(G,\mathbb{Z}_\ell) \to K(G,\mathbb{F}_\ell)$, known as the reduction map. If $T$ is a $\mathbb{Z}_\ell$-module with continuous $G$-action such that $T$ is killed by some power of $\ell$, is it true that $[T \otimes^L_{\mathbb{Z}_\ell} \mathbb{F}_\ell] = 0$ in $K(G,\mathbb{F}_\ell)$?

Answer: Yes


Question: Solve the ordinary differential equation \( y' = \frac{(y^2 - 4yt + 6t^2)}{t^2} \) with the initial condition \( y(2) = 4 \).

Answer: 2t


Question: Determine the radius of convergence of the series \( \sum_{k=1}^{\infty}\left(\frac{x}{\sin k}\right)^k \).

Answer: 0


Question: Given a probability space \(\left([0,5], \mathcal{B}([0,5]), \frac{\lambda_{1}}{5}\right)\) and the random variable \(X(\omega)=I_{[0,1]}(\omega)-2 I_{[2,4]}(\omega)\), determine the expected value \(E(X)\).

Answer: -\dfrac{3}{5}


Question: Determine the set of all cluster points of the subsequences of the sequence \( \frac{n}{e} - \left[ \frac{n}{e} \right] \), where \( e \) is Euler's number.

Answer: [0, 1]


Question: Given the recurrence relation $a_{n+1} = \sqrt{a_n^2 + 1}$ for a sequence $a_1, a_2, a_3, \ldots, a_n$ of real numbers, suppose there exists a positive integer $n_0$ such that $a_{2n_0} = 3a_{n_0}$. Find the value of $\left\lfloor \sum_{n=1}^{49} \sqrt{\frac{8}{8a_n^2 + 7}} \right\rfloor$, where $\lfloor \cdot \rfloor$ denotes the greatest integer function.

Answer: 12


Question: Is there a subset $\mathfrak{F}$ of $\omega^\omega$, with cardinality $\aleph_1$, such that no subset $\mathfrak{G}$ of $\mathfrak{F}$ with cardinality less than $\aleph_1$ can generate $\mathfrak{F}$, where generation means every function in $\mathfrak{F}$ is a finite composition of functions in $\mathfrak{G}$?

Answer: Yes


Question: For which values of \(a\) is the matrix \(\begin{pmatrix} 4 & 0 & 0 \\ 4 & 4 & a \\ 4 & 4 & 4 \end{pmatrix}\) diagonalizable over \(\mathbb{R}\)?

Answer: a > 0


Question:
Let $f: B(0,R) \rightarrow \mathbb{R}$ be a continuous function. Determine whether the following limit holds:
\[ \lim_{r \rightarrow 0} \frac{1}{\text{area}(\partial B(0,r))} \int_{\partial B(0,r)} (f(x)-f(0)) \, dS(x) = 0, \]
for $B(0,r) \subset B(0,R)$. Provide a justification for your answer.

Answer: 0


Question: Evaluate the infinite series: $$\sum_{n=2}^\infty \ln\left(1+\frac{(-1)^n}{n}\right)$$

Answer: 0


Question: Find the norm of the linear operator \( (Tf)(x) = \int_{-1}^1 xyf(y)dy \), where \( T:L_{\infty}(-1,1) \rightarrow L_{1}(-1,1) \) and \( f\in L_{\infty}(-1,1) \). The norm \( ||T|| \) is defined as \( \sup_{f\neq 0}\frac{||Tf||}{||f||_{\infty}} \), where \( ||f||_{\infty} \) is the essential supremum of function \( f \) over \((-1,1)\). Determine the value of \( ||T|| \).

Answer: 1


Question: Let $H$ and $V$ be Hilbert spaces where $(.,.)_H$ is the scalar product on $H$, and $V \subset H$ is dense. We identify $H$ with its dual, hence we have $V \subset H \subset V'$. Let $A:V \rightarrow V'$ be a linear continuous operator with $(Au,v)_H = a(u,v) = (u,Av)_H$ and $a(u,u) + \lambda (u,u)_H \geq \alpha |u|^2_V$ for some fixed $\lambda \geq 0, \alpha > 0$ and all $u,v \in V$. Define $\mathcal{D}(A) = \{ u \in V : Au \in H \}$. Is the operator $A: H \supset \mathcal{D}(A) \rightarrow H$ a closed operator?

Answer: Yes


Question: Provide a counterexample to the statement: If \(G\) is a group with operation \(*\) and \(a\) and \(b\) are elements of \(G\), then \((a*b)^2 = a^2*b^2\).

Answer: (a * b)^2 \neq a^2 * b^2


Question: Evaluate the integral \( \int_C \left\vert z \right\vert \, dz \) where \( C \) is the left half of a circle from \(-i\) to \(i\).

Answer: 2i


Question: Evaluate the integral: \[ \int_{-\infty}^{+\infty} \mathrm{sinc}(8t)\mathrm{sinc^2(t)}\cos(8\pi t) \, \mathrm{d}t \] where the sinc function is defined as \( \mathrm{sinc}(t) = \frac{\sin(\pi t)}{\pi t} \) for \( t \neq 0 \) and \( \mathrm{sinc}(0) = 1 \).

Answer: \dfrac{1}{16}


Question: Find the coefficient $a_{-3}$ in the Laurent expansion of the function \( f(z) = \frac{1}{2z^3} - \frac{1}{z^3+i} \) in the region \( 0 < |z| < 1 \).

Answer: \dfrac{1}{2}


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{\log^{k+1}(x)}{x} \]

Answer: 0


Question: Determine whether there exist groups $G, H, I$ and injective group homomorphisms $h : G \rightarrow H$ and $i : G \rightarrow I$ such that $\mathrm{img}(h)$ and $\mathrm{img}(i)$ are normal subgroups of $H$ and $I$ respectively, $H/\mathrm{img}(h) \cong I/\mathrm{img}(i)$, but $H$ and $I$ are not isomorphic.

Answer: Yes


Question: Find the exact value of $\sin 30^\circ$ using the formula: \[ \sin \left(k\cdot 30^\circ\right) = \frac{\sqrt{2}}{4}\;\sqrt{\;4\;-\sqrt{\phi\,(4\phi+0\overline{\phi})}\;+\sqrt{\overline{\phi}\,(0\phi+4\overline{\phi})}\;} \] where $\phi = \frac{\sqrt{5}+1}{2}$ and $\overline{\phi} = \phi^{-1} = \frac{1}{2}(\sqrt{5}-1)$. Substitute the given values and simplify to find $\sin 30^\circ$. \(a = 4, b = 0, c = 0, d = 4, \pm_1 = -, \pm_2 = +\).

Answer: \dfrac{1}{2}


Question: Compute the limit: \[ \lim \limits_{x \rightarrow 0} \dfrac{\sin(\sin(\sin(x)))}{\cos\left(\frac{\pi}{2}\cos(x)\right)}\cdot x^k \]

Answer: \dfrac{4}{\pi}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=0}^n \frac{1}{{n \choose k}} \] for \( n \geq 1 \).

Answer: 2


Question: Let $T$ be a tree with exactly two vertices of degree 7 and exactly 20 vertices of degree 100. What is the minimum possible number of vertices in the tree $T$ that satisfies these restrictions?

Answer: 1994


Question: Calculate the limit: \[ \lim_{x \to 3}\frac{x^2}{x - 3}\int_3^x \frac{\sin t}{t}dt. \]

Answer: 3 \sin 3


Question: Is it possible to construct an order on \(\Bbb R^2\) that turns it into a set with the least-upper-bound property, while ensuring the order is non-degenerate (i.e., it can distinguish distinct elements)?

Answer: Yes


Question: For any positive integer \( N \), is it possible to construct infinitely many dissociated sets of size \( N \) consisting only of even positive integers, such that no subset of any set sums to the same number as any other subset of the same set?

Answer: Yes


Question: Suppose $f: \mathbb{D} \rightarrow \mathbb{C}$ is an analytic function such that $|f| < 2$. Given that $f''(0) = 4i$, find the value of $f(i/2)$. \( \mathbb{D} \) denotes the open unit disk in the complex plane.

Answer: -\dfrac{i}{2}


Question: Evaluate the expression \( \frac{I(100)}{I(10)} \cdot \frac{I(36)}{I(6)} \), where \( I(n) = \int_{0}^{\pi} \ln(1-2n\cos(x)+n^2)\,\text{d}x \).

Answer: 4


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

Answer: 2


Question: Find the value of \(a\) in the interval \((2, \infty)\) such that \[ \int_{0}^{1}\frac{x^8-x^2}{x^{12}+ax^6+1}dx=-\frac{\pi}{24}. \]

Answer: 6


Question: A young baseball fan wants to collect a complete set of 262 baseball cards. The cards are distributed randomly, one per package of chewing gum. The fan buys two packages of chewing gum each day. On average, how many days will it take the fan to collect the complete set of cards?

Answer: 805


Question: Find a nontrivial homomorphism from the group \( \mathbb{Z}_2 \oplus \mathbb{Z}_2 \) to \( \mathbb{Z}_4 \).

Answer: \phi(a, b) = 2a


Question: Let \( f:[-1,1]\to \mathbb{R} \) be a continuous function satisfying the functional equation \( f(2x^2-1)=(x^3+x)f(x) \). Determine the value of \( \lim_{x\to 0}\frac {f(\cos x)}{\sin x} \).

Answer: 0


Question: Determine the number of solutions to the equation $t \sin(2t) = 2$ for $0 < t < 3\pi$. Provide your answer as a single integer.

Answer: 4


Question: Given a list of numbers where every number except one appears twice, find the number that appears only once. For example, in the list $2, 3, 4, 3, 2, 1, 1$, identify the number that does not have a pair.

Answer: 4


Question: Find the continuous function \( f : \mathbb{R} \to \mathbb{R} \) that satisfies the functional equation: \( \sin x + f(x) = \sqrt{2} f\left(x - \frac{\pi}{4}\right) \).

Answer: \cos x


Question: Let $f$ be an entire function such that $24 \le |f'''(z)| \le 30$ and $f(0) = f(1) = f(2) = 3$. Evaluate the integral \( \int_{|z|=5} \frac{zf'(z)}{f(z)} \, dz. \)

Answer: 6\pi i


Question: Suppose that $f(x)$ has a Laurent series $g(x)$ around $x=1$, and that $g(x) = 1 + O(h(x))$ near $x=1$. Does it necessarily follow that \( \lim_{x\rightarrow 1} \frac{d^n}{dx^n} f(x) = \lim_{x\rightarrow 1}\frac{d^n}{dx^n} (1+O(h(x))) \)?

Answer: Yes


Question: A coffee machine makes a cup of coffee with probability 0.9, a cup of hot water with probability 0.09, and breaks forever with probability 0.01. Let $X$ be the number of cups of coffee the machine makes before it breaks, and $Y$ be the number of cups of hot water before it breaks. Determine the smallest $a > 0$ such that $P(X > n) = O(a^n)$. Are the random variables $X$ and $Y$ independent?

Answer: \dfrac{90}{91}


Question: Determine the cardinality of the set \( X \), where \( X \) is the set of all relations \( R \) over \( \mathbb{N} \) such that the transitive closure \( R^{*} = \mathbb{N} \times \mathbb{N} \).

Answer: 2^{\aleph_0}


Question: Calculate the expected value \( E[W_{t_i}W_{t_{i+1}}^2] \), where \( W_{t_i} \) and \( W_{t_{i+1}} \) are Brownian motion variables with \( W_{t_i} \sim N(0,t_i) \).

Answer: 0


Question: Find the value of \( f^{(2016)}(0) \) for the function \( f(x) = \sin(x^2) \).

Answer: 0


Question: Use a Riemann sum to compute the integral \( \int_0^\frac{\pi}{2} \sin(x)\, dx \).

Answer: 1


Question: Let \( f(x) \) be a quadratic function such that \( f(0) = -6 \) and \( \int \frac{f(x)}{x^2(x-3)^8} \, dx \) is a rational function. Determine the value of \( f'(0) \).

Answer: 16


Question: Compute the surface integral \( \iint_\sigma (x+y+z)\,\mathrm{d}S \) over the unit cube defined by \( 0 \le x \le 1, 0 \le y \le 1, 0 \le z \le 1 \).

Answer: 9


Question:
Let $A$ be an $n \times n$ real symmetric non-singular matrix. Suppose there exists $x \in \mathbb{R}^n$ such that $x'Ax < 0$. Determine which of the following statements is true:

1. $\det(A) < 0$
2. $B = -A$ is positive definite.
3. $\exists y \in \mathbb{R}^n: y'A^{-1}y < 0$
4. $\forall y \in \mathbb{R}^n: y'A^{-1}y < 0$

Answer: 3


Question: Evaluate the line integral \( \int_{\vert x \vert + \vert y \vert = 5 } -\frac{y}{x^2 + y^2} \, dx + \frac{x}{x^2 + y^2} \, dy \) for the vector field \( F(x,y)=\left(-\frac{y}{x^2 + y^2} , \frac{x}{x^2+y^2}\right) \).

Answer: 2\pi


Question: Let \( f_1 : [-1, 1] \rightarrow \mathbb{R} \) be a continuously differentiable function with \( f_1(0) = 0 \), and let \( \lambda > 1 \). Consider the sequence of functions defined inductively by \( f_k(x) = \lambda f_{k-1}\left(\frac{x}{\lambda}\right) \) for \( k \geq 2 \) and \( x \in [-1, 1] \). Find the pointwise limit of the sequence of functions \((f_n)\).

Answer: f_1'(0) x


Question:
Given the matrices \( A = \begin{bmatrix} 0 & 1-i \\ -1-i & i \end{bmatrix} \) and \( B = A^T \overline{A} \), determine which of the following statements is true:

(A) An eigenvalue of \( B \) is purely imaginary.
(B) An eigenvalue of \( A \) is zero.
(C) All eigenvalues of \( B \) are real.
(D) \( A \) has a non-zero real eigenvalue.

Select the correct option.

Answer: C


Question:
Given that there exist unique non-negative integers \(n_1 > n_2 > \ldots > n_r\) and unique integers \(a_k\) (where \(1 \le k \le r\)) with each \(a_k\) either 1 or -1, such that:
\[ a_1 3^{n_1} + a_2 3^{n_2} + a_3 3^{n_3} + \ldots + a_r 3^{n_r} = 2022 \]
Find the sum \(n_1 + n_2 + \ldots + n_r\).

Answer: 17


Question: Let $f$ be a function from $\mathbb{R}$ to $\mathbb{R}$ such that $|f(x) - f(y)| \leq \frac{\pi}{2} |x-y|^2$ for all $x, y \in \mathbb{R}$, and $f(0) = 0$. Determine the value of $f(\pi)$. \( \boxed{?} \)

Answer: 0


Question:
Which simple interest rate over six years is closest to being equivalent to the following: an effective rate of discount of 3% for the first year, an effective rate of discount of 6% for the second year, an effective rate of discount of 9% for the third year, and an effective rate of interest of 5% for the fourth, fifth, and sixth years?

A. 6.3%  B. 6.4%  C. 6.5%  D. 6.6%  E. 6.7%

Answer: D


Question: Given that $X$ and $Y$ are independent and uniformly distributed on $[0,1]$, and $Z = \arctan\left(\frac{Y}{X}\right)$ with $Z$ restricted to $[0, 2\pi)$, find the conditional cumulative distribution function $F_{X|A}(t)$, where $A = \{0 \leq Z \leq \pi/4\}$.

Answer: t^2


Question:
Given the sequence \( \{ a_n \} \) of real numbers for \( n \ge 1 \) defined by the recurrence relation:
\[ a_{n+2} = a_{n+1} - a_n \]
and the conditions:
\[ \sum_{n=1}^{999} a_n = 1003 \]
\[ \sum_{n=1}^{1003} a_n = -999 \]
find the value of \( \sum_{n=1}^{2002} a_n \).

Answer: 2002


Question:
Consider the product space $X = \mathbb{C} \times \mathbb{R}$, where $\mathbb{C}$ and $\mathbb{R}$ are the complex and real numbers with their usual topologies. Define the group $G$ generated by the transformations:

- $t_1 : X \rightarrow X , (z,x) \mapsto (z+1,x)$
- $t_2 : X \rightarrow X , (z,x) \mapsto (z+\xi,x)$
- $\alpha : X \rightarrow X , (z,x) \mapsto (\omega z,x+1)$

where $i = \sqrt{-1}$, $\omega = \frac{-1+\sqrt{3}i}{2}$, and $\xi = \frac{1+\sqrt{3}i}{2}$. The group $G$ acts on $X$. Consider the quotient space $M = X/G$. Determine whether $M$ has the structure of a 3-dimensional compact $C^{\infty}$ manifold.

Answer: Yes


Question: Determine whether the following limit holds: \[ \lim_{c\to\infty}\sum_{n\in\mathbb{N}\setminus\{0\}}ce^{-n-\frac{c}{n}}=0. \] Equivalently, determine whether \[ \lim_{c\to\infty}\int_1^\infty ce^{-x-\frac{c}{x}}\mathrm{d}x=0. \]

Answer: 0


Question: Determine the last digit of the number \( \underbrace{7^{7^{7^{7...}}}}_{1001\text{ }7's}.\)

Answer: 3


Question: Find the functional derivative of the differential entropy with respect to the probability density function, i.e., determine \( \frac{\delta F[p]}{\delta p(x)} \), where \( F[p] = \int_X p(x)\ln(p(x))dx \) and \( p(x) \) is a probability density function.

Answer: \ln(p(x)) + 1


Question:
Let $X$ and $Z$ be independent random variables uniformly distributed on $[0,1]$. Compute the conditional variance of $Y = X + Z$ given $X$. Specifically, find $\text{Var}(Y|X)$. Use the fact that the probability density function of $Y$ is given by:
\[ f(y) = \begin{cases} y & \text{for $0 < y < 1$} \\ 2-y & \text{for $1 \le y < 2$} \\ 0 & \text{otherwise.} \end{cases} \]
And that $E[Y|X] = 1/2 + X$. Provide a simplified expression for the conditional variance.

Answer: \dfrac{1}{12}


Question: Let \( A \) be a convex closed subset of \( \mathbb{R}^n \). Determine whether there exists a convex function \( g: \mathbb{R}^n \rightarrow \mathbb{R} \) such that \( A = \{x: g(x) \leq 0\} \).

Answer: Yes


Question:
Determine the points $p$ in the set $L = \{ (x,y,z) \mid f(x,y,z) = c \}$, where $f(x,y,z) = \langle x^2 + y^2 + z^2, (x - \frac{1}{2})^2 + y^2 \rangle$ and $c = (1, a^2)$ for $0 < a < \frac{3}{2}$, such that the rank of the Jacobian matrix $Df(p)$ is strictly less than 2. The Jacobian matrix is given by:
\[Df(p) = \begin{bmatrix} 2x & 2y & 2z \\ 2x-1 & 2y & 0 \end{bmatrix}\]

Answer: (1, 0, 0)


Question: Evaluate the integral \( \int_0^{2\pi} \sqrt{1+\cos(t)} \, \mathrm{d}t \).

Answer: 4\sqrt{2}


Question: Evaluate the limit: $$\lim_{n\to\infty}\frac{e^{1/n^2}-\cos \frac{1}{n}}{\frac{1}{n}\log(\frac{n+1}{n})-(\sin\frac{2}{n})^2}, \quad n \in \mathbb{N}$$

Answer: -\dfrac{1}{2}


Question: Let $X$ and $Y$ be two continuous random variables. If $P(X < Y) = P(X < g(Y))$ for some convex function $g$, is it true that $g$ must always be a linear function? Answer 'true' or 'false'.

Answer: false


Question: Find the total number of non-similar triangles that can be formed such that all the angles of the triangles are integers.

Answer: 2700


Question: Let $A \in M_{10}(\mathbb{R})$ be a real matrix that satisfies the equation $A^2 = A + 2I$. Given that the rank of $A + I$ is 3, find the trace of $A$. \( \text{tr}(A) \).

Answer: -1


Question:
Let $S$ and $T$ be Fredholm operators on a separable complex Hilbert space $\mathcal{H}$ such that the tensor product $S \otimes T$ is also a Fredholm operator on $\mathcal{H} \otimes \mathcal{H}$. Determine the index of $S \otimes T$, given by:
\[ j(S \otimes T) = ? \]
where the index $j(T)$ is defined as $\operatorname{dim}(\ker(T)) - \operatorname{dim}(\ker(T^{*}))$. Provide your answer in terms of $j(S)$ and $j(T)$.

Answer: j(S) \cdot j(T)


Question: Let $B$ be a graded ring and $\rho: \operatorname{Proj}B \to \operatorname{Spec}B$ be the canonical injection, where $\rho(\mathfrak{p}) = \mathfrak{p}$ for all $\mathfrak{p} \in \operatorname{Proj}B$. For any non-nilpotent homogeneous element $f \in B_+$, is $D(f)$ the smallest open set in $\operatorname{Spec}B$ such that $D_+(f) \subset D(f)$?

Answer: Yes


Question: Is it possible to find a selfadjoint compact operator $T$ on a Hilbert space such that there exists a vector $v$ which does not belong to any finite dimensional invariant subspace for $T$?

Answer: Yes


Question: Determine the group to which $Gal(\mathbb{Q}(i+\sqrt{2}):\mathbb{Q})$ is isomorphic. The order of this group is 4. Is it isomorphic to the Klein four-group or the cyclic group of order 4?

Answer: V_4


Question: In a finite metric space \((X, d)\), is every prime ideal of the ring of continuous functions \(C(X, \mathbb{R})\) maximal? Justify your answer.

Answer: Yes


Question: Calculate the limit: \[ \lim_{x\to0}\left(\ln(1+x)+\ln(1+2x)+\ldots+\ln(1+px)\right)^x \]

Answer: 1


Question: In an equilateral triangle $ABC$, a point $M$ is located inside the triangle. Let $D$, $E$, and $F$ be the orthogonal projections of $M$ onto sides $AB$, $BC$, and $AC$, respectively, with $MD = 1$, $ME = 2$, and $MF = 3$. Determine the side length of triangle $ABC$.

Answer: 4\sqrt{3}


Question: Find the sum of all solutions to the equation \( \cos^{-1}x - 2\sin^{-1}x = \cos^{-1}2x \) considering the principal values of the inverse trigonometric functions.

Answer: 0


Question: Calculate the integral \( \int_{\mathbb{R}} x \mu (dx) \), where the measure \( \mu \) is defined on the Borel algebra \( \mathcal{X} \) of \( \mathbb{R} \) as follows: for each \( X \) in \( \mathcal{X} \), \( \mu (X) = \sum_{k\,\in\,\mathbb{N}\,\cap\, X} \frac{e^{-\lambda}\lambda^k}{k!} \), with \( \lambda \) being a positive real number.

Answer: \lambda


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

Answer: e


Question: If $(X, ||\cdot||)$ is a normed space and $K$ is a complete subspace such that the quotient space $X/K$ is complete, is $X$ necessarily complete?

Answer: Yes


Question: Determine whether the function \( f(x) = \int^x_0 \frac{1}{1+e^t}dt \) converges as \( x \to \infty \).

Answer: \ln 2


Question: Evaluate the limit of the complex function as \( z \to \infty \): \[ \lim_{z\rightarrow \infty} \frac{z \left| z \right| - 3 \Im z + i}{z \left| z \right|^2 +2z - 3i}. \]

Answer: 0


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

Answer: -\dfrac{3}{16}


Question: Calculate the sum \( \sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k} \) and prove that it equals the harmonic number \( H_n \).

Answer: H_n


Question: Find the area of the region in the first quadrant of the xy-plane that is bounded by the circle defined by the intersection of the paraboloid \( z = x^2 + y^2 \) and the plane \( z = 4 \).

Answer: \pi


Question:
Use polar coordinates to find the limit:
\[\lim_{(x, y)\rightarrow(0, 0)}\dfrac{3e^{-x^2 - y^2}-3}{x^2+y^2}\]

Answer: -3


Question: Determine the best bound constant \( C \) such that \( \sup_{R^3/D}|u| \leq C \cdot \sup_{\partial D}|g| \) for the Helmholtz equation \( \Delta u + k^2 u = 0 \) in the exterior region \( \mathbb{R}^3 \setminus D \), where \( D \) is an open and bounded region in three-dimensional space. Assume Dirichlet boundary condition \( u = g \) on \( \partial D \) and the usual radiation condition at infinity. Consider \( \partial D \) to be smooth and \( g \) continuous on \( \partial D \).

Answer: 1


Question: A triangle has two sides of lengths 4 cm and 9 cm. The radius of the inscribed circle that touches all three sides of the triangle is 1 cm. Determine the length of the third side of the triangle.

Answer: 6


Question: Let $X_1$ and $X_2$ be independent and identically distributed random variables with an exponential distribution, EXP(1). The joint PDF of $X_1$ and $X_2$ is given by $f(x_1, x_2) = e^{-x_1-x_2}$ for $x_1, x_2 \geq 0$. Find the joint PDF of $T_1 = \min(X_1, X_2)$ and $T_2 = X_1 + X_2$. Express your answer in terms of $t_1$ and $t_2$. Assume $t_1 \leq t_2$.

Answer: 2e^{-t_2}


Question: If a set is both $F_\sigma$ and of first category, is it true that this set is a countable union of closed nowhere dense sets? Provide a justification for your answer.

Answer: Yes


Question: You have 100 boxes, and C of them contain a gift. You can open up to 16 boxes. What is the smallest number of C that will give you a probability greater than 0.5 of finding at least one gift? Assume that once you find a gift, you stop opening boxes.

Answer: 4


Question: Determine the largest 3-digit prime factor of \( \binom{2000}{1000} \).

Answer: 661


Question: Evaluate the integral \( \int_{0}^{a} \frac{f(x)}{f(x)+f(a-x)} \, dx \), given that the denominator never vanishes over the interval \([0, a]\). No assumption is made on the continuity of \(f(x)\) or its derivatives.

Answer: \dfrac{a}{2}


Question: Consider a finite simplicial complex $X$ that satisfies the following conditions: (i) every simplex is a face of an $n$-simplex, (ii) every $(n-1)$-simplex is a face of exactly two $n$-simplices, and (iii) the polyhedron of $X$ is path-connected. Determine whether these conditions imply that for any two $n$-simplices $\sigma$ and $\tau$ in $X$, there exists a sequence of $n$-simplices $\sigma_0 = \sigma, \ldots, \sigma_k = \tau$ such that $\sigma_i \cap \sigma_{i+1}$ is an $(n-1)$-dimensional simplex for each $0 \leq i \leq k-1$. Answer with 'True' if the implication holds, or 'False' if it does not.

Answer: False


Question: Evaluate the line integral \( \int_{\alpha} x \, dy - y \, dx \) where \( \alpha \) is the boundary of the parallelogram with vertices \((0,0), (a,b), (c,d), (a+c,b+d)\) oriented counterclockwise.

Answer: 2(ad - bc)


Question: Evaluate the integral: \[ \int_{-2}^{2} \left(x^3 \cos\left(\frac{x}{2}\right) + \frac{1}{2}\right) \sqrt{4-x^2} \, dx \]

Answer: \pi


Question: Find the greatest common divisor of the numbers \(2^{2^m} + 1\) and \(2^{2^n} + 1\), given that \(m > n\) and \(m\), \(n\) are positive integers with \(m \neq n\). Use Euclid's algorithm to solve the problem.

Answer: 1


Question: Compute the volume of the tetrahedron with vertices at $(1, 0, 0)$, $(0, 0, -1)$, $(-2, 0, 0)$, and $(-1, -1, -1)$ using triple integrals.

Answer: \dfrac{1}{2}


Question: Consider the affine surface $X \subset \mathbb{C}^3$ defined by $z = x^a y^b$, where $(x, y, z)$ are coordinates on $\mathbb{C}^3$. On this surface, the divisor $D = \{z = 0\} = a \cdot \{x = 0\} + b \cdot \{y = 0\}$. Given the map $\phi: \mathbb{C}^3 \to \mathbb{C}^3$, $(x, y, z) \mapsto (x, y, z^p)$, determine the pullback $\phi^*D$. Is it $pD$ or $\frac{1}{p}D$? Justify your answer.

Answer: pD


Question: Compute the path integral \( \int_\gamma z^5 \, dz \) for the path \( \gamma(t) = 2e^{i6t} \) where \( \gamma : [0, \pi] \rightarrow \mathbb{C} \).

Answer: 0


Question: In a planar 2-connected graph $G$, if removing two edges $e_1$ and $e_2$ results in a disconnected graph $G-e_1-e_2$, do $e_1$ and $e_2$ necessarily belong to the same face? Provide a justification for your answer.

Answer: Yes


Question: Determine the distribution given by \( \lim_{n \to \infty} F_n \), where \( F_n \in C^\infty_0 \) is defined by \( F_n(\varphi) = \varphi\left(\frac{1}{n}\right) - \varphi\left(-\frac{1}{n}\right) \).

Answer: 0


Question: In a complete graph with 5 vertices, labeled \( \{1, 2, 3, 4, 5\} \), how many Hamiltonian cycles include both the edges \( \{1, 2\} \) and \( \{3, 4\} \)? Assume that a Hamiltonian cycle is defined solely by the collection of edges it contains, without regard to orientation or starting point.

Answer: 4


Question: Find the shortest distance from the point \((1,0)\) to the parabola defined by the equation \(y^2 = 4x\) using the method of Lagrange multipliers. Verify your answer by substitution.

Answer: 1


Question:
Verify the correctness of the following triple integral calculation in cylindrical coordinates:
\[ \int\limits_{0}^{2\pi}\int\limits_{0}^{1}\int\limits_{r}^1 2zr \, dz \, dr \, d\theta. \]
The calculation proceeds as follows:
\[ \int\limits_{0}^{2\pi}\int\limits_{0}^{1}\left[z^2r\right]_r^1 \, dr \, d\theta = \int\limits_{0}^{2\pi}\int\limits_{0}^{1}\left[r-r^3\right] \, dr \, d\theta = \int\limits_{0}^{2\pi}\left[\dfrac{1}{2}r^2-\dfrac{1}{4}r^4\right]_0^1 \, d\theta = \int\limits_{0}^{2\pi}\left[\dfrac{1}{4}\right] \, d\theta = \left[\dfrac{1}{4}\theta\right]_0^{2\pi} = \dfrac{1}{2}\pi. \]
Compare this result with the expected result of \(\dfrac{2}{3}\pi\) obtained using Cartesian coordinates. Is the cylindrical coordinate calculation correct?

Answer: \dfrac{\pi}{2}


Question: Determine the number of solutions to the equation \(\sqrt[a+bi]{1}\), where \(a\) and \(b\) are real numbers.

Answer: \infty


Question:
How many $4 \times 3$ matrices of rank three are there over a finite field of three elements? Choose the correct answer from the following options:

A. $(3^4 - 3)(3^4 - 3^2)(3^4-3^3)$
B. $(3^4 - 1)(3^4 - 2)(3^4 - 3)$
C. $(3^4-1)(3^4-3)(3^4-3^2)$
D. $3^4(3^4 - 1)(3^4 - 2)$

Answer: C


Question:
Consider a function $f(x)$ such that for every $\epsilon > 0$, there exists $\delta > 0$ satisfying \(|f(x) - f(1)| \geq \epsilon\) whenever \(|x - 1| \geq \delta\). Determine which of the following statements is true:

A. $f$ is discontinuous at $x=1$.
B. $f$ is unbounded.
C. $\int_{0}^{\infty}|f(x)|dx$ exists.
D. $\lim_{|x|\to\infty}|f(x)|=\infty$.

Select the correct option.

Answer: D


Question: If $A$ is a nilpotent matrix over a finite field, are all eigenvalues of $A$ equal to zero?

Answer: Yes


Question: Evaluate the surface integral \( \iint_S y \, dS \), where \( S \) is the hemisphere defined by \( z = \sqrt{R^2 - x^2 - y^2} \).

Answer: 0


Question: Let $f(x)$ be a smooth function on $[-1,1]$ such that $f(x) > 0$ for all $x \in (-1,1)$ and $f(-1) = f(1) = 0$. Consider $\gamma \subset \mathbb{R}^2$ as the graph of $f(x)$. Let $T_a$ be the symmetry with respect to the $x$-axis and $T_c$ be the central symmetry with respect to the origin. Define two domains: $D_a$, bounded by the curves $\gamma$ and $T_a(\gamma)$, and $D_c$, bounded by the curves $\gamma$ and $T_c(\gamma)$. Determine whether $D_a \subset D_c$ or $D_c \subset D_a$. Provide your answer with justification.

Answer: D_c \subset D_a


Question: Let $X, Y, Z$ be independent and identically distributed random variables, each following a uniform distribution on the interval $(0, 1)$. Calculate the probability $P(X + Y + Z > 1)$. Assume $X, Y,$ and $Z$ are mutually independent.

Answer: \dfrac{5}{6}


Question:
Determine the language accepted by the nondeterministic pushdown automaton $M = (\{q_0, q_1, q_2\}, \{a, b\}, \{a, b, z\}, \delta, q_0, z, \{q_2\})$ with the following transitions:

$$\delta (q_0, a, z) = \{ (q_1, a), (q_2, \lambda)\},$$
$$\delta (q_1, b, a) = \{ (q_1, b)\},$$
$$\delta (q_1, b, b) =\{ (q_1, b)\},$$
$$\delta (q_1, a, b) = \{ (q_2, \lambda)\}$$

Choose the correct language from the following options:

1. $L(abb^*a)$
2. $\{a\} \cup L(abb^*a)$
3. $L(ab^*a)$
4. $\{a\} \cup L(ab^*a)$

Answer: 2


Question: A tea merchant has five cubical tin tea boxes, each with a picture on each of its six sides, totaling thirty pictures. However, one picture on box No. 1 is repeated on box No. 4, and two pictures on box No. 4 are repeated on box No. 3, resulting in only twenty-seven different pictures. The merchant always places box No. 1 at one end of the row and never allows boxes No. 3 and No. 5 to be adjacent. Determine the number of ways the boxes can be arranged in a row such that the sequence of pictures on the front side is unique for each arrangement.

Answer: 186624


Question: Evaluate the contour integral of \( f(z) = \frac{z^2}{(z-1)^2(z+2)} \) along the circle of radius 3 centered at the origin.

Answer: 2\pi i


Question: Consider a field \(k\) equipped with a nonarchimedean absolute value \(|\cdot| : k\rightarrow\mathbb{R}\) with respect to which it is locally compact. Is it possible for the image of \(|\cdot|\) to contain a nonempty interval \((a,b)\subset\mathbb{R}\)?

Answer: No


Question: Suppose $x$ is an element of a Banach Algebra. Let $U$ be an open set in $\mathbb{C}$ containing the spectrum of $x$. Determine if it is true that \( \inf \{\|(x-\lambda 1_A)^{-1}\|^{-1} : \lambda \in \mathbb{C} - U \} > 0 \).

Answer: True


Question:
Given the equations:
\[ y = \left\lfloor\frac{ x \mod 790}{10}\right\rfloor + 48 \]
\[ z = (x \mod 790) \mod 10 + 10\left\lfloor\frac{x}{790}\right\rfloor + 48 \]
If \(y = 69\) and \(z = 60\), find the value of \(x\).

Answer: 1002


Question: Find the minimum value of \(\left|1-(ab+bc+ca)\right| + \left|1-abc\right|\) given that \(a, b, c \in \mathbb{R}\) and \(a+b+c=1\).

Answer: \dfrac{44}{27}


Question: Determine whether the absolute Galois group of a proper subfield of \(\mathbb{R}\) can be finite. Specifically, given a proper subfield \(F \subsetneq \mathbb{R}\), with \(\bar{F}\) as an algebraic closure of \(F\), can the Galois group of \(\bar{F}\) over \(F\) be finite?

Answer: Yes


Question: Determine the cardinality of the set of all root paths in an infinite complete infinitary tree where the branching factor k is infinite.

Answer: 2^{\aleph_0}


Question: In a bridge game, what is the probability that the North and South players have all the Queens between them? Express your answer as a fraction.

Answer: \dfrac{46}{833}


Question: Determine the values of \( \phi > 0 \) for which the improper integral \( \int _3^{\infty }\:\frac{\sqrt{x^3+2}}{\left(x^2-3x\right)^\phi} \, dx \) converges.

Answer: \emptyset


Question: Evaluate the limit: \[ \lim_{x \to \frac{\pi}{6}}{(1-2\sin(x))}^{\tan(\frac{\pi}{6}-x)} \]

Answer: 1


Question: How many mutually non-isomorphic Abelian groups of order $3^2 \times 11^4 \times 7$ are there?

Answer: 10


Question: Find a specific value of \( \epsilon \) such that the matrix \( I_n + \epsilon A_n \) is invertible, where \( A_n \) is a matrix of 1's, \( I_n \) is the identity matrix, and \( n \) can be any dimension. Use the Banach Lemma and the infinity norm to determine this value of \( \epsilon \).

Answer: 0


Question: Determine whether the eigenvalues or eigenvectors of a transition matrix, where the matrix elements are between 0 and 1 and either rows or columns sum to one, can have an imaginary part.

Answer: Yes


Question: Find the number of integer solutions to the equation \(x_1 + x_2 - x_3 - x_4 = 0\) where each variable \(x_1, x_2, x_3, x_4\) is an integer between -4 and 4, inclusive.

Answer: 489


Question: Find the value of \( n \) such that \( \sum_{r=1}^n r \cdot 2^r = 2 + 2^{n+10} \). Then, calculate the sum of all digits of \( n \).

Answer: 9


Question: Find an analytic function \( f(z) \) such that \( |f(z)|=e^{\rho^2\cos(2\theta)} \) where \( z=\rho e^{i\theta} \).

Answer: e^{z^2}


Question: Find the mean of the probability density function \( f(x) = \frac{1}{2} x^2 e^{-x} \) for \( x > 0 \).

Answer: 3


Question: Compute the average value of the function \( f(x,y,z) = xyz \) over the region \( W \) defined by \( 0 \le z \le y \le x \le 1 \).

Answer: \dfrac{1}{8}


Question: Given that \(x+y+z=3\), \(x^2 + y^2 + z^2 = 5\), and \(x^3+y^3+z^3=7\), find the value of \(x^4+y^4+z^4\).

Answer: 9


Question: Determine whether every finite subgroup of odd order in \(GL(2, \mathbb{R})\) is cyclic.

Answer: Yes


Question: Determine whether the following statement is true or false: For every prime number \( p \) and every positive integer \( e \), the polynomial \( f_{p}(x^{p^{e-1}}) \) is irreducible in \( \mathbb{Q}[x] \).

Answer: True


Question: Given that \( \lim_{n \to \infty}\left|a_n+ 3\left(\frac{n-2}{n}\right)^n \right|^{\frac{1}{n}}=\frac{3}{5} \), find the value of \( \lim_{n\to \infty} a_n \).

Answer: -\dfrac{3}{e^2}


Question: Determine the number of elements in the set \( S = \{0, 1, 2, \ldots, 5399\} \) that satisfy the congruence \( x^2 \equiv 49 \pmod{5400} \).

Answer: 16


Question: Let \( G = (V, E) \) be a simple connected graph with exactly one cycle. The graph has four vertices of degree 2, five vertices of degree 3, three vertices of degree 4, and one vertex of degree 5. If the maximum degree \( \Delta = 5 \), determine the number of pendant vertices in \( G \).

Answer: 14


Question: In a triangle $\Delta ABC$, express the value of $\frac{a\cos A + b\cos B + c\cos C}{a+b+c}$ in terms of the inradius $r$ and the circumradius $R$. Verify if the expression simplifies to $\frac{r}{R}$.

Answer: \dfrac{r}{R}


Question: Consider 'n' continuous random variables, each uniformly distributed, that sum to 1. Determine the probability density function (PDF) for the ratio of one of these variables to the sum of all 'n' variables. Specifically, find the PDF for \( \frac{X_1}{X_1+X_2+\ldots+X_n} \) when \( n=3 \), where \( X_1, X_2, \) and \( X_3 \) are uniformly distributed random variables.

Answer: 2(1 - x)


Question: Find the extremal of the functional \( J(u) = u(0) + 0.5 u(0)^2 + \int_0^1 [u'(t)]^2 \, dt \) subject to the boundary condition \( u(1) = 2 \). The value of \( u(0) \) is unspecified.

Answer: u(t) = t + 1


Question: In a compact, Hausdorff uniform space \((X, \mathcal{U})\), let \(S\) be a closed subset of \(X\) and \(W\) be an open subset of \(X\) such that \(S \subseteq W\). For each \(U \in \mathcal{U}\), define \(U[x] = \{y : (x, y) \in U\}\) and \(U[S] = \bigcup_{x \in S} U[x]\). Is there a \(U \in \mathcal{U}\) such that \(S \subseteq U[S] \subseteq W\)?

Answer: Yes


Question: Find all positive integer solutions (x, y) to the equation \(x^4 + 4x^3 + 6x^2 + 4x + 16 = y^2\).

Answer: \emptyset


Question: Evaluate the contour integral \( \int_\gamma \frac{1}{z} \, dz \), where \( \gamma \) is the semicircular path in the complex plane defined by \( z = e^{i\theta} \) for \( \theta \in [-\pi/2, \pi/2] \).

Answer: i\pi


Question: Evaluate the integral \( \int_{0}^{\infty} \frac{\ln x}{x^2+x+1} \, dx \).

Answer: 0


Question: Prove that the expression \( b = \left \lfloor (\sqrt[3]{28}-3)^{-n} \right \rfloor \) is never divisible by 6 for any integer \( n \geq 1 \).

Answer: b


Question: Given that $B \in M_{n\times n}(\mathbb{R})$ is an invertible matrix and satisfies the equation $B^2 + B^4 + B^7 = I$, find an expression for $B^{-1}$ in terms of $B$.

Answer: B^{-1} = B + B^3 + B^6


Question: Find the derivative of the function \( f(z) = z \bar{z} \) using the limit definition of a derivative.

Answer: 0


Question: Given the joint density function \( f(x,y) = \frac{1}{2} y^2 e^{-x} \) for \( 0 < y < x < \infty \), calculate the conditional probability \( P(Y < 1 \mid X = 3) \).

Answer: \dfrac{1}{27}


Question: Find the limit of \( \frac{\log(n!)}{n\log(n)} \) as \( n \to \infty \).

Answer: 1


Question: Let $f$ be a continuous function defined on $[0,1]$ such that \( \int_{0}^1f^2(x)dx=\left(\int_{0}^1f(x)dx\right)^2 \). Determine the range of $f$.

Answer: \{c\}


Question: Determine the number of idempotent matrices in the matrix space $M_2(\mathbb{F}_p)$. An idempotent matrix $A$ satisfies $A^2 = A$. Calculate the total number of such matrices in this space.

Answer: p^2 + p + 2


Question:
For a prime number $p > 3$, consider the fraction:
\[
1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{p-1}.
\]
Is the numerator of this fraction divisible by $p^3$?

Answer: No


Question: Determine the number of homomorphisms from the group \(C_2 \times C_3\) to the symmetric group \(S_4\).

Answer: 18


Question: For which values of \( x \) does the series \( \sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2} \) converge?

Answer: \mathbb{R}


Question: A music school has 40 students, with 10 each studying violin, viola, cello, and string bass. The director wants to form 10 string quartets, each consisting of one student from each instrument group. In how many ways can this be done?

Answer: (10!)^3


Question: Let $G$ be a group and $F_G$ the free group on the set $G$. Consider the canonical surjective morphism ${\rm can}: F_G \to G$ defined by ${\rm can} (e_x) = x$ for all $x \in G$, where $(e_x)_{x \in G}$ is a copy of $G$ as a set. Is the set $\{e_x e_y e_{xy}^{-1}\}$ a generating set for ${\rm Ker} ({\rm can})$?

Answer: Yes


Question: Find the minimum value of \( \frac{xy}{z} + \frac{yz}{x} + \frac{xz}{y} \) given that \( x^2 + y^2 + z^2 = 1 \) and \( x, y, z > 0 \).

Answer: \sqrt{3}


Question:
Determine the subspace $X \subseteq \mathbb{R}$ with the usual topology, including $\{0,1\} \subseteq X$, for which there exists a continuous function $f: X \to \{0,1\}$ such that $f(0)=0$ and $f(1)=1$. Consider the following options:

1. $X = [0,1]$
2. $X = [-1,1]$
3. $X = \mathbb{R}$
4. $[0,1] \not\subset X$

Answer: D


Question: A twice differentiable function $f(x)$ satisfies the functional equation \( f(x^2y) = x^2f(y) + yf(x^2) \) for all \( x, y > 0 \), and it is given that \( f'(1) = 1 \). Determine the value of \( f''\left(\frac{1}{7}\right) \).

Answer: 7


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty }\left ( \int_{0}^{2\pi}{\frac{\cos(nx)}{x^{2}+n^{2}}}dx \right ) \]

Answer: 0


Question: Compute the limit: \[ \lim_{x\rightarrow 0} \left( \frac{1}{x}\ln (x!)\right) \]

Answer: -\gamma


Question: For a generic choice of matrix $A$ and vector $b$ in the system of linear equations $Ax = b$, can the elements of the solution vector $x$ be distinct?

Answer: Yes


Question: Given that $90^a = 2$ and $90^b = 5$, evaluate $45^{\frac{1-a-b}{2-2a}}$ without using logarithms or calculators.

Answer: 3


Question: Compute the limit: \[ \lim_{n \rightarrow \infty} \int_{0}^{1} \frac{1+nx^2}{(1+x^2)^n} \, dx \]

Answer: 0


Question: Find the smallest integer \( n > 1 \) such that \( a^n \equiv a \pmod{1309} \) for all integers \( a \) in \( \mathbb{Z}/1309\mathbb{Z} \).

Answer: 241


Question:
Solve the following system of equations for real numbers \(x\) and \(y\):
\[ 2^{x^2+y} + 2^{x+y^2} = 8 \]
\[ \sqrt{x} + \sqrt{y} = 2 \]

Answer: (1, 1)


Question: Given that $X$ and $Y$ are independent and identically distributed random variables, each following an exponential distribution with parameter $\lambda = 1$, compute the expected value $E(\sqrt{X+Y})$. Provide your answer to two decimal places.

Answer: 1.33


Question: Is it possible for an algebraically closed field $K$ to be contained in a division ring $D$ such that $K \subseteq Z(D)$, where $Z(D)$ is the center of $D$?

Answer: Yes


Question: Determine the number of integers \( x \) such that \( 1 \le x \le 5^k \) and \( 5^k \mid (x^4 - 1) \).

Answer: 4


Question: Consider a random walk starting at 1, where each step is determined by independent and identically distributed random variables $X_i$ with $P(X_i = 1) = \frac{1}{2}$, $P(X_i = -1) = \frac{1}{4}$, and $P(X_i = 0) = \frac{1}{4}$. The position of the random walk after $n$ steps is given by $S_n = 1 + X_1 + X_2 + \cdots + X_n$. What is the probability that this random walk ever reaches the position $0$?

Answer: \dfrac{1}{2}


Question: Evaluate the integral \( \int_C \frac{1}{z^2(z^2-16)} \, dz \), where \( C \) is the contour \( |z| = 1 \).

Answer: 0


Question: Let $V$ be a subvariety of $\mathbb{C}^n$ with irreducible components of dimension greater than 0. Determine if $H_{2n-1}(\mathbb{C}^n \setminus V) = 0$. Provide a justification for your answer.

Answer: 0


Question: Determine the number of triplets of positive integers \((a, b, c)\) that satisfy the equation \(2^a - 5^b \cdot 7^c = 1\). Choose from the following options: \(0\), \(1\), \(2\), or infinite.

Answer: 0


Question: In how many ways can 20 distinct students be assigned to four distinct dorms, A, B, C, and D, such that each dorm has at least one student?

Answer: 1085570781624


Question: Solve the system of equations and inequalities: \[ \begin{aligned} 2^{x+2} &= \frac{49}{4}x^2 + 4, \\ 2^{x+2} - 4 &\le x^2(14 - 2^{x+2}) \cdot 2^x \end{aligned} \] Find the values of \( x \) that satisfy both conditions.

Answer: 0


Question: For an arithmetic sequence $a_1, a_2, a_3, \ldots$, where $(a_1)^2 + (a_{11})^2 \leq 100$, determine the maximum value of $S = a_{11} + a_{12} + \ldots + a_{21}$. What is this maximum value?

Answer: 55\sqrt{10}


Question: Consider a Riemannian manifold $M$ that is metrically complete, connected, and has dimension $n$. For every point $p \in M$ and every radius $0 \leq R$, there exists another point $q(p,R) \neq p$ and a radius $R'(p,R) \in \mathbb{R}$ such that the set of points at distance $R$ from $p$ is the same as the set of points at distance $R'(p,R)$ from $q(p,R)$. Is $M$ isometric to an $n$-dimensional sphere?

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to \infty}\int_0^{\pi/2}\sqrt[n]{\sin^n{x}+\cos^n{x}}\,dx. \]

Answer: \sqrt{2}


Question: Evaluate the definite integral \( \int_0^a \frac{2a-x}{a+x} \sqrt{\frac{a-x}{a+x}} \, dx \).

Answer: a(7 - 2\pi)


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

Answer: -1


Question: Evaluate the integral \( I = \int_0^{\infty} \log{(x+1/x)}\,\frac{dx}{1+x^2} \) using the result \( \int_0^{\pi/2} \log\sin x\,\mathrm dx= -\dfrac{\pi}{2} \log 2 \).

Answer: \pi \log 2


Question:
Consider the $n \times n$ matrix $A = (a_{ij})$ where $a_{ij} = a_i a_j$ for real numbers $a_1, \dots, a_n$. Determine which of the following statements is true:

1. It is possible to choose $a_1, \dots, a_n$ such that $A$ is non-singular.
2. Matrix $A$ is positive definite if $(a_1, \dots, a_n)$ is a nonzero vector.
3. Matrix $A$ is positive semi-definite for all $(a_1, \dots, a_n)$.
4. For all $(a_1, \dots, a_n)$, $0$ is an eigenvalue of $A$.

Select the correct statement.

Answer: 3


Question: Given the equation \((x+y)(x^2-xy+y^2)=z^3\), where \(x, y, z\) are non-zero integers, assume that \((x+y)\) and \((x^2-xy+y^2)\) are coprime. Set \(x+y=r^3\) and \(x^2-xy+y^2=s^3\). Can \(z\) be expressed as \(z=rs\) where \(r\) and \(s\) are integers? Justify your answer.

Answer: Yes


Question: Evaluate the surface integral \( \iint_{S} \vec{F} \cdot d\vec{S} \) to find the flux of the vector field \( \vec{F}(x,y,z) = y\mathbf{j} - z\mathbf{k} \) across the surface \( S \). The surface \( S \) consists of the paraboloid \( y = x^2 + z^2, 0 \leq y \leq 1, \) and the disk \( x^2 + z^2 \leq 1, y=1 \). Use the positive (outward) orientation for closed surfaces.

Answer: 0


Question: Is the resolvent set of an operator with compact resolvent in a Hilbert space a dense subset of \( \mathbb{C} \)?

Answer: Yes


Question: Given that \(\{a,b,c\}\subset \mathbb{R}\) with \(a \neq b\), and \(a^2(b+c) = b^2(a+c) = 2010\), find the value of \(c^2(a+b)\).

Answer: 2010


Question: Find all solutions to the equation \(x^x = \frac{1}{256}\).

Answer: -4


Question:
Determine the largest value of $s$ such that the point $x_0 = (0,0)$ and direction $\xi_0 = (0,1)$ do not belong to the $H^s$ wavefront set of the 2D step function $f(x,y)$, which is 1 for $y > 0$ and 0 for $y \leq 0$. Consider a cutoff function $\phi$ that is not zero at $x_0$ and a conic neighborhood $V$ of $\xi_0$. The condition to check is:
\[\int_V |\widehat{\phi f}(\xi)|^2 (1 + |\xi|^2)^s \, d\xi < \infty.\]

Answer: \dfrac{1}{2}


Question: For any given \(\epsilon > 0\), determine whether there always exist natural numbers \(n\) and \(m\) such that \(e^n\) is within \(\epsilon\) of \(m\).

Answer: Yes


Question:
Evaluate the integral \( \int_0^\infty \int_0^\infty \frac{e^{-(x+y)}}{x + y}\,dx\,dy \) and choose the correct option:

(A) infinite
(B) finite, but cannot be evaluated in closed form
(C) \(1\)
(D) \(2\)

Answer: C


Question: Consider a family of compact metric spaces \(X_i\) for each \(i \in [0,1]\). Is the Cartesian product \(\prod_{i \in [0,1]} X_i\) also a compact metric space?

Answer: No


Question: Let $p(x)$ be a polynomial in $\mathbb{R}[x]$ such that $p(0)=1$, $p(x) \ge p(1)$ for all $x$, and $\lim_{x \rightarrow \infty} p''(x)=4$. Determine the value of $p(2)$. \( \text{ } \)

Answer: 1


Question:
Calculate the value of \( M = F(7, 2) + F(7, 3) + F(7, 5) - F(7, 6) \) where \( F(x, y) = \cos^y\left(\frac{\pi}{x}\right) + \cos^y\left(\frac{3\pi}{x}\right) + \cos^y\left(\frac{5\pi}{x}\right) \). Choose the correct answer from the following options:

(a) \( \frac{37}{32} \)
(b) \( \frac{7}{4} \)
(c) \( \frac{19}{16} \)
(d) \( \frac{53}{32} \)
(e) \( \frac{41}{32} \)

Answer: D


Question:
Determine the interval of convergence for the power series:
\[ \sum_{n=1}^\infty \frac{(x+5)^{n^2}}{(n+1)^{n}} \]

Answer: [-6, -4]


Question: Determine the values of \( \gamma > 0 \) for which \( \lim_{n \to \infty} \frac{1}{n^2}\sum_{k=1}^n k^\gamma = 0 \).

Answer: (0, 1)


Question: Determine an upper bound for the smallest Lipschitz constant $L$ of a retraction $r: \ell_\infty[0,1] \to C[0,1]$. Is it possible that $L \le 20$?

Answer: 20


Question: Consider a matrix $A$ of size $100 \times 4$. Let $B = A^T A$, where $B$ is a $4 \times 4$ symmetric matrix. Is it possible for $B$ to be non-invertible? Justify your answer.

Answer: Yes


Question: Evaluate the limit \( \lim_{n \rightarrow \infty} S_n \) for the sequence \( S_n = \frac{x}{x+1} + \frac{x^2}{(x+1)(x^2+1)} + \cdots + \frac{x^{2^n}}{(x+1)(x^2+1)\cdots(x^{2^n}+1)} \), where \( x > 1 \).

Answer: 1


Question: Determine the number of subgroups of order 8 in an abelian group of order 72. Explain why there is only one such subgroup.

Answer: 1


Question: In triangle \(\triangle{ABC}\), the side lengths are \(\overline{AB}=62\), \(\overline{AC}=50\), and \(\overline{BC}=105\). Let \(M\) be the midpoint of segment \(AB\), and \(N\) be the midpoint of the median from \(C\) to \(AB\). Define \(P_1\) as the midpoint of \(CN\) and \(P_2\) as the midpoint of \(NM\). Let \(Q_1\) and \(Q_2\) be the points where lines \(AP_1\) and \(AP_2\) intersect \(BC\), respectively. Find the length of segment \(\overline{Q_1Q_2}\).

Answer: 48


Question:
Consider a finite, simple, unconnected graph $G = (V, E)$. The total graph $T(G)$ of $G$ is defined with vertex set $V(T(G)) = (V \times \{0\}) \cup (E \times \{1\})$ and edge set $E(T(G)) = E_v \cup E_e \cup E_{v+e}$, where:

- $E_v = \{\{(v,0), (w,0)\}: \{v,w\} \in E\}$,
- $E_e = \{\{(e,1), (f,1)\}: (e,f \in E) \land (e \neq f) \land (e \cap f \neq \emptyset)\}$,
- $E_{v+e} = \{\{(v,0), (e,1)\}: v \in e\}$.

Determine if there exists a graph $G$ such that the chromatic number $\chi(T(G))$ of the total graph $T(G)$ is greater than its clique number $\omega(T(G))$. Provide a justification for your answer.

Answer: Yes


Question: Determine whether every valuation ring for a field $K$ can be expressed as $R = \{a \in K : v(a) \ge 0\}$ for some valuation $v$ on $K$. Provide a justification for your answer.

Answer: Yes


Question: Find \(\lim_{x \to \infty} x^2\big(\ln(x\cot^{-1}(x))\).

Answer: -\dfrac{1}{3}


Question: Evaluate the integral \( \int^1_{-1} \int^{ \sqrt {1-x^2}}_{-\sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}} \, dz \, dy \, dx \) by converting it to spherical coordinates.

Answer: \dfrac{5\pi}{3}


Question: Evaluate the limit: \[ \lim_{n \to \infty } \int_0^n \left(1+\frac{(-x)^{27}}{n} \right) e^{-2x} \, dx \]

Answer: \dfrac{1}{2}


Question: Given that \( f(x) \) satisfies the condition \( \int_0^1 (x - f(x)) f(x) \, dx = \frac{1}{12} \), find the greatest integer value of the area bounded by \( y = f(x) \) from \( x = 1 \) to \( x = 3 \).

Answer: 2


Question: Determine the points where the function \( f: \mathbb{C} \to \mathbb{C} \), defined by \( f(z) = z|z| \), is differentiable.

Answer: 0


Question: Find the function $f(n)$ defined for all integers $n$ such that $f(x) + f(y) = f(x + y) - 2xy - 1$ for all integers $x$ and $y$, and $f(1) = 1$. Determine $f(n)$.

Answer: n^2 + n - 1


Question: Let \((X_n)_{n \geq 1}\) be a sequence of i.i.d. real random variables with \(\text{Var}(X_1) = 1\) and \(E(X_1) = 0\). Denote by \(S_n\) the partial sum \(S_n = X_1 + X_2 + \cdots + X_n\). Prove that \(\lim_{n \to \infty} E(|S_n|) = \infty\).

Answer: \infty


Question: Let $X$ be a nonempty perfect Polish space, $Y$ a second countable space, and $f: X \to Y$ be an injective and Baire measurable function. Does there exist a homeomorphic copy of the Cantor set $\mathcal{C} = 2^\mathbb{N}$ contained in $f(X)$?

Answer: Yes


Question:
Let $x, y, z, w > 0$ such that $x + y + z + w = 1$. Determine the minimum value of the expression:
\[
\frac{x^2+y^2+z^2+w}{(x+y+z)^3} + \frac{y^2+z^2+w^2+x}{(y+z+w)^3} + \frac{z^2+w^2+x^2+y}{(z+w+x)^3} + \frac{w^2+x^2+y^2+z}{(w+x+y)^3}.
\]

Answer: \dfrac{112}{27}


Question:
Verify whether the net torque on a surface in Stokes flow is zero, given that the stress tensor \( \sigma \) is symmetric and \( \nabla\cdot\sigma = 0 \). The net torque \( L \) is defined as:
\[ L_i = \epsilon_{ijk}\int_V \sigma_{kj}~dV. \]
Is \( L_i = 0 \) due to the symmetry of \( \sigma \)?

Answer: L_i = 0


Question: Consider the set $st_0$ of all bounded statistically convergent complex sequences. Define a linear functional $g: st_0 \to \mathbb{C}$ by $g(x) = \text{st-lim} \, x_n$ for all $x = (x_n) \in st_0$, where $\text{st-lim} \, x_n$ denotes the statistical limit of the sequence $(x_n)$. The topology on $st_0$ is induced by the sup-norm $\|\cdot\|_\infty$. Is the linear functional $g$ bounded (continuous) on $(st_0, \|\cdot\|_\infty)$?

Answer: Yes


Question: Given the polynomial sequence defined by \( p_0(x) = a_mx^m + a_{m-1}x^{m-1} + \dotsb + a_1x + a_0 \) where \( a_m, \dotsc, a_1, a_0 \in \mathbb{R} \), and the recurrence relation \( p_n(x) = p_{n-1}(x) + p_{n-1}^{\prime}(x) \) for \( n = 1, 2, \dotsc \), prove that there exists a natural number \( N \) such that for all \( n \geq N \), all the roots of \( p_n(x) \) are real.

Answer: N


Question: Let \(X\) be a nonempty compact convex subset of \(\mathbb{R}^n\). Suppose that for every point \(x = (x_1, \dots, x_n) \in X\) and for every pair of indices \(1 \le i < j \le n\), the point \((x_1, \ldots, x_{i - 1}, \frac{x_i + x_j}{2}, x_{i + 1}, \ldots, x_{j - 1}, \frac{x_i + x_j}{2}, x_{j + 1}, \ldots, x_n)\) is also in \(X\). Is it true that there exists some \(\lambda \in \mathbb{R}\) such that \((\lambda, \dots, \lambda) \in X\)?

Answer: Yes


Question: Evaluate the limit: $$\lim _{x\to \infty }\left(\int _{x-1}^{x+1}\frac{\cos \left(t\right)}{\ln \left(t\right)+1}dt\:\right)$$

Answer: 0


Question: Let $W=\{W_t:t\geq0\}$ be a Brownian motion. Find $\operatorname{Var}(W_1^3)$. Provide your answer as a single numerical value.

Answer: 15


Question: Find the coefficient of \(x^{17}\) in the expansion of \((3x^7 + 2x^5 - 1)^{20}\).

Answer: -41040


Question: Let \( X \) be a random variable with a symmetric probability density function \( f_X(x) \) and mean \( \mu \). Does the equation \( E[X | \mu-t \leq X \leq \mu+t] = \mu \) hold true for any \( t \)?

Answer: Yes


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

Answer: 3


Question: Given finite groups $H$ and $G$ with $H < G$, determine if there must exist a positive integer $n$ such that there is a homomorphism $f: G \to S_n$ where $f(G)$ is transitive and $f(H)$ is not transitive.

Answer: Yes


Question: Calculate the limit: \[ \lim_{x \to 0} \frac{x - \sin{x}}{x^3} \] without using L'Hopital's rule.

Answer: \dfrac{1}{6}


Question: If \(x\) and \(y\) are integers such that \(\frac{4x^2-1}{4x^2-y^2}=k\) is an integer, does it imply that \(k=1\)? Justify your answer.

Answer: Yes


Question: Express the smallest natural number that is relatively prime to all integers from 2 to \( n \) as an asymptotic function of \( n \).

Answer: n


Question:
Find the area of the region enclosed by the function \( f(x) = \tan x - x^{1000} \) and the x-axis between the lines \( x = 0 \) and \( x = \pi/2 \). Choose the correct option:

A) \( \ln|\sec x_1| - \dfrac{{x_1}^{1001}}{1001} \)

B) \( \dfrac{{x_1}^{1001}}{1001} - \ln |\sec x_1| \)

C) \( \dfrac{{x_2}^{1001}}{1001} - \dfrac{2{x_1}^{1001}}{1001} + 2\ln{\left|\dfrac{\sec x_1}{\sqrt {\sec x_2}}\right|} \)

D) \( \dfrac{{x_2}^{1001}}{1001} - \dfrac{{x_1}^{1001}}{1001} + \ln {\left|\dfrac{\sec x_1}{\sqrt{\sec x_2}}\right|} \)

Answer: C


Question: Let \( \Omega \) be a smooth bounded subset of \( \mathbb{R}^{n} \), and let \( h \) be an \( L^{\sigma_{\alpha}} \)-function with \( h^{+} \neq 0 \). Given that \( \frac{1}{\sigma_{\alpha}} + \frac{\alpha}{p^*} = 1 \), determine whether there exists a \( C_{0}^{\infty} \)-function \( v \), with \( v > 0 \), such that \( \int_{\Omega} h v^{\alpha} > 0 \).

Answer: Yes


Question: Determine whether the equation $e^A e^B = e^{A+B}$ implies that $AB = BA$ for two matrices $A, B \in M_n(\mathbb{C})$. Provide a justification for your answer.

Answer: Yes


Question: Determine the minimum size of a test bank, \( n \), such that when 20 people each take a challenge consisting of 9 unique questions drawn from this bank, the probability of having exactly 3 repeated questions is at least 95%.

Answer: 177


Question: Let $Z \sim N(0,1)$ be a standard normal random variable. Calculate the expected value $E[\max\{Z-1,0\}]$. Express your answer as a single numerical value.

Answer: 0.0833


Question: Evaluate the integral \( \int_0^\infty \frac{2\left(e^{-t^2} -e^{-t}\right)}{t}\ dt \).

Answer: \gamma


Question: Solve for \( x \) in the equation \( x^{x^{x^{.^{.^.}}}} = 2 \).

Answer: \sqrt{2}


Question: Consider functions $f_1, f_2, \ldots \in L^p$ such that $\|f_n - f\|_p \rightarrow 0$ as $n \rightarrow \infty$. Does there exist a subsequence $f_{n_i}$ that converges to $f$ almost everywhere?

Answer: Yes


Question: Calculate the double integral \( \iint_D (x^2+y)\, \mathrm{d}x\, \mathrm{d}y \) where the region \( D \) is defined by \( \{(x,y)\mid -2 \le x \le 4,\ 5x-1 \le y \le 5x+3\} \).

Answer: 240


Question:
For a positive integer $n > 1$, let $S_n$ be the set of all bijective maps $\pi: \{1, \ldots, n\} \to \{1, \ldots, n\}$. Define the neighboring number $N_n(\pi)$ for $\pi \in S_n$ as:
\[ N_n(\pi) = \min \left(\left\{|\pi(k) - \pi(k+1)| : k \in \{1, \ldots, n-1\}\right\} \cup \left\{|\pi(1) - \pi(n)|\right\}\right). \]
Let $E_n$ be the expected value of the neighboring number for a member of $S_n$. Determine whether $\lim\sup_{n \to \infty} \frac{E_n}{n} > 0$. Provide a justification for your answer.

Answer: No


Question: Suppose $f_n, g_n \in L^p([0,1])$ and that \( f_n \to f \) weakly in $L^p$, \( g_n \to g \) weakly in $L^p$, and \( f_ng_n \to h \) weakly in $L^p$ for all $p \in [1,\infty]$. Is it true that $h = fg$?

Answer: No


Question: Let \( P(T) = \sum_{i=0}^n u_iT^i \) be a polynomial where the coefficients \( u_i \in \mathbb{C}[[z]] \) are analytic power series. If \( f \in \mathbb{C}[[z]] \) is a root of \( P \), is \( f \) necessarily analytic?

Answer: Yes


Question: Determine the asymptotic behavior of the sum \( \sum_{i=1}^{n^2 - 1} \frac{i^2}{\left[\frac{n^3}{3}\right]^2} \) as \( n \to \infty \).

Answer: 3


Question: Given the integral \(X = \int_{0}^{\pi}\frac{\cos x}{(x+2)^2} dx\) and the equation \(\int_{0}^{\pi/2}\frac{\sin y\cos y}{y+1}dy = aX + \frac{b}{\pi+2} + c\), where \(a, b, c\) are constants, find the value of \(100a + 50b + 4c\).

Answer: -24


Question: Determine the number of elements in the group ring $R(G)$, where $R$ is a finite ring with $|R|$ elements, and $G$ is a finite group with $|G|$ elements. Express your answer in terms of $|R|$ and $|G|$. \( \boxed{} \)

Answer: |R|^{|G|}


Question: Determine the value of $b$ such that if $(x^2-5x+4)(y^2+y+1)<2y$ for all real $y$, then $x$ belongs to the interval $(2, b)$. What is the possible value of $b$?

Answer: 3


Question: Let $f$ be a continuous, real-valued function on $[0,1]$ such that there exists a constant $K > 0$ for which \( |f(x)| \le K \int_{0}^{x} |f(t)| \, dt \) for all \( x \in [0,1] \). Does it follow that $f(x) = 0$ for all $x \in [0,1]$?

Answer: 0


Question:
Given the equations:
\[ 2\sin^2 A - 2\tan A \sin A + 2 - \sin^2 A - \cos^2 B = 0 \]
\[ 2\cos^2 B - 2\tan A \cos B + 2 - \sin^2 A - \cos^2 B = 0 \]
If \( \sin A \neq \cos B \), find the value of \( \sin^2 A \).

Answer: \dfrac{2}{3}


Question: Find the integer $x$ such that $6 \equiv 7^x \pmod{71}$. Provide your answer as a single integer.

Answer: 32


Question:
In the interval $(-2\pi, 0)$, how many times does the function $f(x) = \sin (1/x^3)$ change its sign? Choose the correct option:

(A) Never changes sign
(B) Changes sign only once
(C) Changes sign more than once, but a finite number of times
(D) Changes sign an infinite number of times

Answer: D


Question: Is there a countable connected topological space \((X, \tau)\) such that for every point \(x \in X\), the space \(X \setminus \{x\}\) is disconnected with the induced subspace topology?

Answer: Yes


Question: Compute the complex limit \( \lim_{z\rightarrow 0} \frac{\tan(z)}{z} \) where \( z \in \mathbb{C} \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{(x,y)\to (0,0) }\frac{\sin(x^2+9y^2+|x|+|3y|)}{|x|+|3y|} \]

Answer: 1


Question: If a real-valued sequence \(x_n\) satisfies \(x_n \to 0 \pmod{a}\) for every real \(a > 0\), does it follow that \(x_n \to 0\) in the real numbers? Justify your answer.

Answer: Yes


Question: A function $f$ is defined for all positive integers such that $f(1) = 2020$ and \( \sum_{r=1}^{x} f(r) = x^2 f(x) \) for all \( x > 1 \). Find the value of $f(2019)$. \( \)

Answer: \dfrac{2}{2019}


Question: Given the functions $g(x) = ax + c$ and $f(x) = x^b + 3$, and the inverse of their composition $(g \circ f)^{-1}(x) = \left( \frac{x - 7}{2} \right)^{\frac{1}{3}}$, find the value of $a + b + c$. \( \)

Answer: 6


Question: Solve the equation \((x+1)^{2010}+(x+2)^{2010}=2^{-2009}\).

Answer: -\dfrac{3}{2}


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

Answer: 1


Question: Let $C$ be a code with a distance $d = 2t + u + 1$. Determine the maximum number of errors that $C$ can detect if it is used simultaneously to correct $t$ errors.

Answer: t + u


Question: Calculate the limit: \[ \lim_{x\rightarrow 0} \frac{\sin{\pi \sqrt{ \cos (x)}}}{x} \] without using L'Hôpital's rule.

Answer: 0


Question: There are 15 students, with 3 students each from 5 different high schools. Five admission officers, each from a different college, will each select 3 students to attend their college. In how many ways can the officers select the students such that no officer selects all 3 students from the same high school?

Answer: 159252480


Question: Determine the values of \( a \) for which the curve \( y = ax^2 \) is tangent to the curve \( y = \ln(2x) \).

Answer: \dfrac{2}{e}


Question: Given the partial differential equation \( u_t = u_{xx} \) with the initial condition \( u(x,0) = \frac{e^{2x}-1}{e^{2x}+1} \), find \( \lim_{t \to \infty} u(1,t) \).

Answer: 0


Question: How many arrangements of the letters in the word 'BARBARIANISM' contain two identical blocks of 3 letters?

Answer: 201600


Question: Given the joint density function of two random variables $X$ and $Y$, $f_{XY}(x,y) = 2e^{-(x+y)}$ for $0 < x < y$, compute the conditional probability $P(Y < 1 | X = 1)$. Use the conditional probability formula: $P(Y < 1 | X = 1) = \frac{P(X = 1, Y < 1)}{P(X = 1)}$. Determine the appropriate bounds for the integrals involved.

Answer: 0


Question: Consider the function \( f(x,y,z,w)=x^2\sin(4y)+z^3(6x-y)+y^4 \). Use Clairaut's Theorem to find the mixed partial derivative \( f_{yyzzx} \).

Answer: 0


Question: True or False: Let $G$ and $H$ be finite groups. Is it true that any subgroup of $G \times H$ is equal to $A \times B$ for some subgroups $A < G$ and $B < H$? Provide a justification for your answer.

Answer: False


Question: Evaluate the surface integral \( \iint \vec{F} \cdot \hat{n} \, ds \) using Gauss's Theorem, where \( \vec{F} = xy \vec{i} + y^2 \vec{j} + 2yz \vec{k} \). The region is bounded by \( z=0 \), \( z=2 \), and \( x^2+y^2=4 \).

Answer: 0


Question: Let $X$ be a normal random variable with mean $0$ and variance $1$. Let $Y$ be a Bernoulli random variable with $p = \frac{1}{2}$, and suppose that $X$ and $Y$ are independent. Define $Z = (Y + 1)X$. Calculate the conditional probability $P(Y = 1 \mid Z = 3)$. Provide your answer as a decimal rounded to three decimal places.

Answer: 0.936


Question: Can the product of two symmetric positive definite matrices result in a matrix with all zero eigenvalues? Provide a proof or counterexample.

Answer: No


Question: Find the last three digits of \(19^{100}\).

Answer: 001


Question: Evaluate the limit using Maclaurin series: \[ \lim_{x\to 0} \frac{x-\ln(1+x)}{x-\arctan x} \]

Answer: \infty


Question:
Suppose \(a, b, c,\) and \(d\) are constants such that the following holds for all real numbers \(x\) where all denominators are nonzero:
\[
\frac{14}{x(x+14)}+\frac{14}{(x+7)(x+21)}+\frac{14}{(x+14)(x+28)} + \frac{14}{(x+21)(x+35)}+\frac{14}{(x+28)(x+42)} = \frac{a(x^2+42x+147)}{x(x+b)(x+c)(x+d)}.
\]
What is the value of \(a+b+c+d\)?

Answer: 154


Question: Compute the volume of the region bounded by the inequalities \(x^2+y^2 \leq 4x\) and \(|z| \leq x^2+y^2\).

Answer: 48\pi


Question: Let \( f(x) \) be a polynomial in \( x \) of degree greater than 1. Define \( g_i(x) \) by \( g_1(x) = f(x) \), and \( g_{k+1}(x) = f(g_k(x)) \). Let \( r_k \) be the average of the roots of \( g_k(x) \). Given that \( r_{19} = 89 \), determine \( r_{89} \).

Answer: 89


Question: Find the constant $a$ such that \( \log_2(a) + \log_2(a)^2 + \log_2(a)^3 + \cdots + \log_2(a)^n = 3n(n+1) \) for all positive integers \( n \).

Answer: 64


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

Answer: \dfrac{1}{2}


Question: Determine whether for any integer $m$, there exists an elliptic curve over the complex numbers and a distortion map $\phi$ such that $\{ T, \phi(T) \}$ forms a basis of $E[m]$ for some point $T \in E[m]$. Provide a justification for your answer.

Answer: Yes


Question: Let $X \in\{0,1\}$ and $Y \in\{0,1\}$ be two uniformly distributed bits. Let $B$ be an arbitrary random variable such that $I(X:B)=0$, $I(Y:B)=0$, and $I(X \oplus Y:B)=0$. Is it true that $I(X,Y:B)=0$? (Here, $I(X:Y)$ denotes Shannon’s mutual information.)

Answer: Yes


Question: Find the last three digits of the sum \(d(1) + d(2) + d(3) + \cdots + d(2^{99})\), where \(d(n)\) is the largest odd divisor of \(n\).

Answer: 782


Question: Let $f: \Bbb{Z} \times \Bbb{Z} \to G$ be an epimorphism with $\ker f$ generated by $\langle (3,0),(0,5) \rangle$. Determine the type of abelian group $G$.

Answer: \mathbb{Z}_{15}


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

Answer: -\dfrac{1}{2}


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

Answer: \infty


Question: Use Stokes' Theorem to compute the line integral \( \int_C (y \, \mathrm{d}x + z \, \mathrm{d}y + x \, \mathrm{d}z) \), where \( C \) is the curve of intersection of the cylinder \( x^2 + y^2 = 1 \) and the surface \( z = x \cdot y \), oriented counterclockwise around the cylinder.

Answer: -\pi


Question: In an n-dimensional lattice $L$ with Voronoi cell $\mathcal{V}$, a vector $v$ is Voronoi relevant if the hyperplane $\{x \in \mathbb{R}^n : \langle x, v \rangle = |v|^2/2\}$ intersects an $(n-1)$-dimensional face of $\mathcal{V}$. Given a vector $\lambda \in L$ that is not Voronoi relevant, is there always a Voronoi relevant vector $v$ such that $||v|| < ||\lambda||$? Answer yes or no.

Answer: Yes


Question: Determine if \(a=1\) and \(b=2\) are the only positive coprime integers with \(a \neq b\) such that \(a^{p} + b^{p} \mid (a+b)^{p}\) for some odd prime \(p\).

Answer: Yes


Question: Find the minimal polynomial of $2^{1/5}$ over the field $\mathbb{Q}(\sqrt{3})$. Provide your answer as a polynomial with rational coefficients.

Answer: x^5 - 2


Question: Calculate the contour integral \( \int_C \frac{8-z}{z(4-z)} \, dz \), where \( C \) is a circle of radius 7 centered at the origin and negatively oriented. Use the residue theorem to find the value of the integral.

Answer: -2\pi i


Question: Consider a Gaussian copula with an identity matrix as its correlation matrix for $n$ continuous marginal random variables. Does this copula structure imply that these random variables are independent?

Answer: Yes


Question: Given integers A and B, find the sum of (m+n) for all pairs of integers (m, n) such that m ≤ n, gcd(m, n) = B, and lcm(m, n) = A. For example, if A = 72 and B = 3, the possible pairs are (3, 72) and (9, 24). Calculate the sum of m+n for these pairs.

Answer: 108


Question: Let \(a\) be a root of the polynomial \(X^4 + 2X + 1 = 0\). Express \(\frac{a+1}{a^2 - 2a + 2}\) as a polynomial in \(a\) with rational coefficients.

Answer: a^3 + a^2


Question: Given two sequences of real numbers \( x_1 \ge x_2 \ge \cdots \ge x_n \ge 0 \) and \( y_1 \ge y_2 \ge \cdots \ge y_n \ge 0 \), such that for any positive integer \( p \), \( \sum_{i=1}^n x_i^p \geq \sum_{i=1}^n y_i^p \), determine if there exists a constant \( c \ge 1 \) such that the sequence \( cx \) weakly majorizes \( y \) from below. Here, \( cx \) denotes the sequence \( cx_1, cx_2, \ldots, cx_n \), and a sequence \( a \) weakly majorizes another sequence \( b \) from below if for all \( 1 \le k \le n \), \( \sum_{i=1}^k a_i \ge \sum_{i=1}^k b_i \).

Answer: Yes


Question: Evaluate the limit: $$\lim \limits_{n\to \infty }\sum _{k=0}^n\frac{C\left(n,\:k\right)}{k2^n+n}$$

Answer: 0


Question: For any complex-valued, nonconstant, and differentiable function $f$, is it possible that \( \text{Im} \int_0^1 f(x) \, \mathrm{d}x = \text{Im} \int_1^\infty f(x) \, \mathrm{d}x \)? Provide a justification for your answer.

Answer: Yes


Question: Calculate the limit: $$\lim_{n\to \infty} \frac{(2n+3)!} {2^{n+2}\, (n+1)!\, (n+1)! }$$

Answer: \infty


Question: Suppose $X$ is a complex normed space of dimension 2 or 3, and $X$ is isometrically isomorphic to its dual. Determine whether $X$ is necessarily a Hilbert space.

Answer: Yes


Question:
Determine the dimension of the affine algebraic set \(X = V(f_1, f_2, f_3) \subseteq \mathbb{A}^6\), where:
\[f_1 = x_1x_5 - x_4x_2, \quad f_2 = x_1x_6 - x_4x_3, \quad f_3 = x_2x_6 - x_5x_3.\]

Answer: 4


Question: Find a holomorphic bijection from the upper half-plane \( H = \{z \in \mathbb{C} \mid \text{Im}(z) > 0\} \) to the slit plane \( M = \mathbb{C} \setminus (-\infty,0] \).

Answer: -z^2


Question: Find the order of the element \(g^{-2}h^{-2}fh^{2}g^{2}\) in the symmetric group \(S_9\), where \(f=(9\ 6\ 3\ 5\ 1\ 4)(2\ 3\ 4\ 5\ 7\ 8)\), \(g=(4\ 6\ 7\ 3\ 1)(1\ 4\ 2\ 6\ 5\ 7\ 8)\), and \(h=(1\ 3\ 5\ 2)(2\ 3\ 6\ 7)\).

Answer: 12


Question: What is the expected number of coin tosses required to get either $n$ heads in a row or $n$ tails in a row?

Answer: 2^n - 1


Question: Determine whether the following statement is true or false: If \( f(x, y) = 0 \) almost everywhere in \( \mathbb{R}^2 \), then there exist constants \( a \) and \( b \) such that \( f(ta, tb) = 0 \) for almost every \( t \in \mathbb{R} \).

Answer: True


Question: Given the incenter $I$, circumcenter $O$, and the midpoint $M_a$ of side $a$ of a triangle $\triangle ABC$, construct the vertices of the triangle using only a straightedge and compass.

Answer: \triangle ABC


Question: Find the smallest positive integer $x$ such that $x^2$ starts with the digits 2017.

Answer: 4492


Question: Determine whether the following statement is true or false: A random variable $X$ is non-degenerate if and only if there exists some $a \in \mathbb{R}$ such that $\mathrm{P}(X < a) \in (0,1)$. Provide a justification for your answer.

Answer: True


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

Answer: -\dfrac{i}{2}


Question: Use Green's Theorem to find the area enclosed between the x-axis and one arch of the cycloid defined by the parametric equations \(x = t - \sin(t)\) and \(y = 1 - \cos(t)\). Assume \(Q_x - P_y = 1\) and apply Green's Theorem to compute the area.

Answer: 3\pi


Question: Find the argument of the complex number \( \mathcal{Z} = \frac{(2\sqrt{3} + 2i)^8}{(1-i)^6} + \frac{(1-i)^6}{(2\sqrt{3} - 2i)^8} \).

Answer: \dfrac{5\pi}{6}


Question: Evaluate the contour integral \( \oint_{C}\frac{\sin(z)}{e^z z^3} \,dz \), where \( C \) is the circle \( |z - 1| = 3 \).

Answer: -2\pi i


Question: Evaluate the integral \( \int_{-\pi}^{\pi}\frac{dx}{\cos^2(x) + 1} \) using the method of residues.

Answer: \pi \sqrt{2}


Question: Determine the number of continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(f(x)) + f(x) + x = 0 \).

Answer: 0