Question: Find the number of compositions of 50 into exactly 5 parts, where the first and last parts are odd, the other parts are even, and no part is greater than 20. Use generating functions to determine the number of such compositions and simplify your answer to an integer.

Answer: 5875


Question: Let \( f(x) = \arcsin\left(\frac{2x}{x^2+1}\right) + 2\arctan x \) for \( |x| \geq 1 \). Calculate \( f(-5) \).

Answer: -\pi


Question: Consider a graded set $A = (A_0, A_1, \dots)$ where each $A_k$ is finite and consists of elements of size $k$. Is it possible to define an associative operation $*$ that is graded, meaning \( *: A_k \times A_\ell \to A_{k+\ell} \), such that there exists an element of size $0$ which is not a neutral element?

Answer: Yes


Question: Let \(I=(a,b)\) be an interval and let \(v_i:I\to\mathbb{R}^n,\ i=1,\ldots,k\) be continuous curves such that \(v_1(t),\ldots,v_k(t)\) are linearly independent in \(\mathbb{R}^n\) for every \(t\in I\). Can we always find continuous curves \(v_{k+1},\ldots,v_n:I\to\mathbb{R}^n\) such that \(v_1(t),\ldots,v_n(t)\) forms a basis for \(\mathbb{R}^n\) for each \(t\in I\)?

Answer: Yes


Question:
Let $M$ be an $n \times n$ matrix with real entries such that $M^3 = I$. Suppose $Mv \neq v$ for any non-zero vector $v$. Determine which of the following statements is true:

A. $M$ has real eigenvalues.
B. $M + M^{-1}$ has real eigenvalues.

Answer: B


Question: Compute the derivative \( f'(e^{i 2\pi/3}) \) for the function \( f(z) = z^{1/4} \), where the branch is defined in \( \mathbb{C} - (\infty,0] \) and satisfies \( f(1) = -i \).

Answer: -\dfrac{1}{4}


Question: Find the closure of the interval \((0,1)\) in the lower-limit topology on \(\mathbb{R}\), where the topology is generated by the subbasis consisting of all half-open intervals \([a,b)\).

Answer: [0, 1)


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

Answer: 0


Question:
Consider a perfect binary tree $T$ with leaves numbered $1$ to $n$. An algorithm $\operatorname{Sum}()$ computes the sum $1 + 2 + \dotsb + n$ as follows:

- If $T$ is a leaf node $k$, return $k$.
- Otherwise, perform the following steps in parallel:
  - $a \leftarrow \operatorname{Sum}(\text{left subtree of }T)$
  - $b \leftarrow \operatorname{Sum}(\text{right subtree of }T)$
- Return $a + b$.

If the steps are performed in parallel, what is the computational complexity of this algorithm?

Answer: O(\log n)


Question: A researcher surveys $n$ college students and counts how many support, oppose, and are undecided about a federal policy. Let $X_1, X_2, X_3$ denote these counts, modeled as $X \sim \mathsf{Multinomial}(n,p)$. Find the restricted maximum likelihood estimate (MLE) of $\mathbf{p} = (p_1, p_2, p_3)$ under the null hypothesis that the proportions of supporters and opposers are equal. Assume $\mathbf{p} = (a, a, 1 - 2a)$ for $a \in (0,1)$. Calculate the MLE for the data $(X_1, X_2, X_3) = (140, 165, 195)$. What is the value of $a$?

Answer: 0.305


Question: Let $A$ be a square matrix with spectral radius less than 1. Determine whether the following limit holds: $$\lim_{k \to \infty} A^k (A^{T})^k = 0,$$ where $A^{T}$ is the transpose of matrix $A$. Provide a justification for your answer.

Answer: 0


Question: Given integers $n$, $p$, and $q$ greater than 0, and $a, b \in \mathbb{Z}/(n)$, determine the number of $p \times q$ matrices with elements in $\mathbb{Z}/(n)$ such that the sum of entries in each row is equal to $a$ and the sum of entries in each column is equal to $b$. Assume that $pa \equiv qb \pmod{n}$. What is this number?

Answer: n^{(p-1)(q-1)}


Question: Find the exact length of the arc of the curve \( y = 2e^x + \frac{1}{8}e^{-x} \) on the interval \([0, \ln(2)]\).

Answer: \dfrac{33}{16}


Question: Determine if the equation \( \sin\left(\frac{x+y}{\sqrt{2}}\right) = \frac{y-x}{\sqrt{2}} \) can be expressed as a function of \(x\).

Answer: Yes


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

Answer: 2


Question:
Given that \( \{ a_n \} \) is an arithmetic sequence of positive integers and \( a_{10} - a_1 = 225 \), find the limit:
\[ \lim_{n \to \infty} \frac{n a_n^2 + 1}{\sum_{k=1}^{n} \left( 1 + 2 + 3 + \ldots + k \right)} \]

Answer: 3750


Question: Given the expansion \((2x^2+3x+4)^{10} = \sum_{r=0}^{20} a_r x^r\), find the value of \(\frac{a_7}{a_{13}}\).

Answer: 8


Question: Evaluate the integral \( \iint_{R} (x+z) \, dA \) where \( R \) is the region in the first octant bounded by the cylinder \( x^2 + y^2 = 9 \) and between \( z = 0 \) and \( z = 4 \). Use cylindrical coordinates to express and solve the integral.

Answer: 36 + 12\pi


Question: Evaluate the limit: \[ \lim_{x \to \infty} \frac{x!}{m!(x-m)!} 5^{-x} \] where \( 0 < m < x \).

Answer: 0


Question: Find the last four digits of \(2^{27653} - 1\).

Answer: 2991


Question: Let $|G| = p^k m$ where $p$ and $m$ are coprime. It is known that there exists a Sylow-$p$-subgroup $S$ of $G$ with $|S| = p^k$. If $U \le G$ is a subgroup of $G$ with $|U| = p^k \cdot n$, does there exist a Sylow-$p$-subgroup $S$ such that $S \le U$?

Answer: Yes


Question: Determine the number of equivalence classes for statements formed with 5 variable letters.

Answer: 2^{32}


Question: Evaluate the limit: \[ \lim_{x\rightarrow \infty} (e^{2x}+x)^{1/x} \]

Answer: e^2


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

Answer: \infty


Question: Suppose you have 100 distinct video games and 50 distinct friends. You must offer each friend 3 video games from your collection of 100, and each friend will select one of the three games offered to them. Assuming all offers are made simultaneously, you may offer the same game to multiple friends, and each set of 3 games and each game chosen has an equal probability of selection. How many distinct games will be chosen more than once?

Answer: 9


Question: Calculate the limit: \[ \lim _{x\to \infty }x\sin\left(\frac{1}{\lceil{\frac{x}{2}}\rceil}\right)^x \]

Answer: 0


Question: Evaluate the integral of $\log(z)$ over a closed counter-clockwise curve $C$ that encloses the origin exactly once.

Answer: 2\pi i


Question: Assume that $H$ is a separable Hilbert space. Is there a polynomial $p(z) \in \mathbb{C}[x]$ with $\deg(p) > 1$ such that every densely defined operator $A: D(A) \to D(A)$, where $D(A) \subset H$, satisfying $p(A) = 0$ is necessarily a bounded operator on $H$?

Answer: Yes


Question: Determine the number of mutually non-isomorphic Abelian groups of order \( p^2q^4 \), where \( p \) and \( q \) are distinct primes.

Answer: 10


Question:
Convert the following triple integral from Cartesian coordinates to cylindrical or spherical coordinates, choosing the system that simplifies the integral the most:
\[ \int_0^1 \int_0^x \int_0^{ \sqrt{x^2+y^2}} \frac{(x^2+y^2)^{3/2}}{x^2+y^2+z^2}\,dz\,dy\,dx \]

Answer: \dfrac{\pi}{12}


Question: Evaluate the expression \( \left(\frac{-\sqrt{2}}{2}-\frac{1}{\sqrt{-2}}\right)^{2016} \).

Answer: 1


Question: Evaluate the limit \( \lim_{c\to\infty} \frac{E\left[ X e^{-\frac{(X-c)^2}{2}} \right]}{E\left[e^{-\frac{(X-c)^2}{2}}\right]} \) where \(X\) is a bounded random variable with \(|X| \le a\).

Answer: a


Question: Given a sequence of $N$ probability distributions $P$ over an alphabet $A$, calculate the probability that a string $S$ of length $N$, where the $i^{th}$ character is drawn from the $i^{th}$ probability distribution, contains a specific character repeated $T$ times consecutively. For example, if $A = \{a, b\}$ and the distributions are $P_1 = (1.0, 0.0)$, $P_2 = (0.5, 0.5)$, and $P_3 = (1.0, 0.0)$, determine the probability that the string contains the substring $aa$. Provide the probability as a single value.

Answer: 0.5


Question: Given a complex number $x$ such that $x^N = 1$ and $x^2 \neq 1$, determine if the following statement is true: \( \sum_{n=0}^{N-1} x^{2n} = 0 \).

Answer: True


Question: Evaluate the limit: \[ \lim_{n\to\infty}\left(2n\int_{0}^{1}\frac{x^n}{1+x^2}dx\right)^n \]

Answer: e


Question:
Calculate the determinant of the following matrix for a general \(n \times n\) matrix of this form:
\[
\operatorname{det}~\begin{pmatrix} 0 & 1 & 1 & \cdots & 1 \\
1 & 0 & 1 &\cdots & 1\\
1 & 1 & 0 &\cdots & 1\\
\vdots & \vdots & \vdots & \ddots & 1\\
1 & 1 & 1 & \cdots & 0\end{pmatrix}
\]

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


Question: Determine the positive integers \(k\) for which the series \(\sum_{n=1}^\infty \frac{(n!)^2}{(kn)!}\) is convergent.

Answer: k \geq 2


Question: Let \( A \in M_{m \times n}(F) \) be a matrix with \( \operatorname{rank}(A) = m \). Prove that there exists a matrix \( B \in M_{n \times m}(F) \) such that \( AB = I_m \).

Answer: AB = I_m


Question: In a tower of fields $E/F/G$ with $[E:G]<\infty$, is it true that $[F:G]<\infty$? Provide a justification for your answer.

Answer: Yes


Question:
Let $F$ be a field of characteristic zero and let $D$ be the formal polynomial differentiation map defined by
\[D(a_0+a_1x+a_2x^2+\ldots+a_nx^n)=a_1+2a_2x+3a_3x^2+\ldots+na_nx^{n-1}.\]
Find the image of $F[x]$ under the map $D$. What is Im$(D)$?

Answer: F[x]


Question: Determine the equilibrium solutions for the first order differential equation: \[ \frac{dy}{dx}=y^2-r\sqrt{x}, ~~ x\geqslant 0 \]

Answer: y = 0


Question: Determine the value of the parameter \(a\) such that the substitution \(y(x) = z(x)^a\) transforms the differential equation \(2xy' + y = y^2\sqrt{x - x^2y^2}\) into a homogeneous equation.

Answer: -\dfrac{1}{2}


Question: Find the orthogonal complement of the subset \( C = \{ u \in L^2(0,2) : \int_0^2 u(t) \, dt = 1 \} \) in the space \( L^2(0,2) \).

Answer: \{0\}


Question: Suppose \( \lim x_n = a \). It is known that the limits of the arithmetic mean and geometric mean of this sequence are also \( a \), i.e., \( \lim \frac{a_1 + \ldots + a_n}{n} = a \) and \( \lim \sqrt[n]{a_1 \cdot \ldots \cdot a_n} = a \). Determine if the following statement is true: \( \lim \frac{n}{\frac{1}{a_1} + \ldots + \frac{1}{a_n}} = a \).

Answer: True


Question:
Let \( X \) be a metric space and \( Y \) a closed subset of \( X \) such that the distance between any two points in \( Y \) is at most 1. Which of the following statements is correct?

A. \( Y \) is compact.
B. Any continuous function from \( Y \) to \( \mathbb{R} \) is bounded.
C. \( Y \) is not an open subset of \( X \).
D. None of the above.

Choose the correct answer.

Answer: D


Question:
Consider the function $f: [0,1] \to \mathbb{R}$ defined by
\[f(x)=\begin{cases}x^\alpha  \sin(1/x^\beta) &\text{if }0 < x \leq 1\\
0 & \text{if } x = 0 \end{cases}\]
where $\alpha$ and $\beta$ are positive real numbers. If $0 < \alpha - \beta \leq 1$, the derivative of $f$ at 0 is not defined, but $f'$ is well-defined on $(0,1]$ and $\int_0^1 |f'| < \infty$. Is it valid to apply the Fundamental Theorem of Calculus to write
\[
f(x) = \int_0^x f'
\]?

Answer: Yes


Question: Let \( f(x) = x^3 + 3x + 1 \) and \( g(x) \) be the inverse of \( f(x) \). Find the value of \( g''(5) \).

Answer: -\dfrac{1}{36}


Question: Prove that if $f: \mathbb{R} \to \mathbb{R}$ is an odd convex function, then $f(x) = ax$ for some $a \in \mathbb{R}$.

Answer: f(x) = ax


Question: Use Stokes' Theorem to calculate \( \iint_S \text{curl} \mathbf{F} \cdot d\mathbf{S} \) for the surface \( S = \{(x,y,z): x^2+y^2+z^2=2; -1\leq z\leq 1\} \) and the vector field \( \mathbf{F}(x,y,z) = (-x-y+xz^2 , x+z , z+e^{x+y}) \).

Answer: 0


Question: Evaluate the contour integral \( \int_\gamma \frac{z}{z^2-i} \, dz \), where \( \gamma \) is a counterclockwise unit square in the complex plane.

Answer: \pi i


Question: A large vase has a square base with a side length of \(6 \text{ cm}\), and its flat sides slope outwards at an angle of \(120^{\circ}\) with the base. Water flows into the vase at a rate of \(12 \text{ cm}^3/\text{s}\). Calculate, to three significant figures, the rate at which the height of the water is rising when the water has been flowing in for \(3\) seconds.

Answer: 0.246


Question: Find all ordered pairs \((x, y)\) of positive integers such that \(x^2 + 4y^2 = (2xy - 7)^2\).

Answer: (3, 2)


Question: Let $L/K$ and $E_i/K$ be field extensions for $i=1, \\ldots, r$, such that $L \cap E_i = K$ for all $i$. Prove or disprove that $L \cap E = K$, where $E = E_1 \cdots E_r$. Provide a justification for your answer.

Answer: L \cap E \neq K


Question: Evaluate the expression \( \frac{{n \choose 1}(n-1)^3+{n \choose 3}(n-3)^3+\ldots}{n^2(n+3)\cdot 2^n} \) for \( n=10 \).

Answer: \dfrac{1}{16}


Question: Four particles are situated at the corners of a square with side length 'a' and move at a constant speed 'v'. Each particle moves towards the particle at the next corner. The time when all particles will meet is given by \( \frac{2a}{\sqrt{n} v} \). Find the value of \( n \).

Answer: 4


Question: Consider the sequence of prime numbers \( \{P_n\} \), where \( P_1 = 2, P_2 = 3, \ldots \). Define a new sequence \( \{M_n\} \) such that \( M_n \) is the product of the digits of the \( n \)-th prime in its decimal representation. Determine whether there exists a natural number \( N \) such that \( M_n = 0 \) for all \( n \geq N \).

Answer: No


Question: Calculate the value of $5^{0.23} \bmod 4$. Provide your answer as a single number.

Answer: 1


Question: Let $X$ be a complete complex projective variety, and $L$ a nef line bundle over it. Is it true that the first Chern class $c_1(L)$ is a limit of Kähler forms in $H^2(X,\mathbb{Q})$?

Answer: Yes


Question: Determine the automorphism group of the graph \( \Gamma \), where \( \Gamma \) is defined as follows: Let \( S_6 \) be the symmetric group on six letters, and let \( X \) denote the conjugacy class containing \((12)(34)\). The vertex set of \( \Gamma \) is \( X \), and the edges are precisely the 2-element subsets of \( X \) that commute as elements of \( S_6 \).

Answer: S_6


Question: Let \( \{X(t) : t \geq 0\} \) be a Poisson process with rate \( \lambda \). Given that \( X(1) = 2 \), determine the mean of \( S_1 S_2 \), where \( S_1 \) and \( S_2 \) are the first two arrival times.

Answer: \dfrac{1}{4}


Question:
Find a constant $c$ such that the smallest eigenvalue $\lambda_n$ of the matrix $AA^T$ satisfies the inequality:
\[ \lambda_n(AA^T) \ge c \|A\|_F^2 \]
where $A$ is a non-singular matrix and $\|A\|_F$ denotes the Frobenius norm of $A$. The matrix $A$ is not assumed to be symmetric.

Answer: 0


Question: Find the function $g(x)$ that is implicitly defined by the relation $\sqrt{x^2 - y^2} + \cos^{-1}\left(\frac{x}{y}\right) = 0$. Show that $y = x$ is the only function that satisfies this relation.

Answer: x


Question:
Let \( C \) be the disk of radius 1 centered at the origin in \( \mathbb{R}^2 \). Consider the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) given by:
\[ T(x, y) = \left( \frac{5x + 3y}{4}, \frac{3x + 5y}{4} \right) \]
What is the smallest natural number \( n \) such that \( T^n(C) \) contains at least 2019 points \( (a, b) \) where \( a, b \in \mathbb{Z} \)?

Answer: 11


Question:
Given that the random vector $(X, Y, Z)$ follows a multivariate normal distribution with expected values $E[X] = 0$, $E[Y] = 1$, $E[Z] = 1$, and covariance matrix:
\[
\begin{bmatrix}
1 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 2
\end{bmatrix}
\]
calculate the variance $\operatorname{Var}(X(Y+Z))$. Express your answer as a single numerical value.

Answer: 11


Question: Consider the initial value problem: \(16y'' + 24y' + 9y = 0\), with \(y(0) = a\) and \(y'(0) = -1\). Find the critical value of \(a\) that separates solutions that become negative from those that are always positive for \(t > 0\).

Answer: \dfrac{4}{3}


Question: Determine whether there is a real solution to the differential equation \( \frac{d^4y}{dx^4} = -y \).

Answer: Yes


Question: Solve the differential equation \( y'' - y' - 2y = 12 \sin t \, \delta(t-\pi) \) with the initial conditions \( y(0) = 1 \) and \( y'(0) = -1 \).

Answer: e^{-t}


Question: Find \( \lim_{n \to \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}. \)

Answer: 1


Question: Compute the bias of the maximum likelihood estimator \( \hat{x}_{ML} = \frac{\sum_{i=1}^N\frac{z_i}{\sigma_i^2}}{\sum_{i=1}^N\frac{1}{\sigma_i^2}} \), where \( z_i = x + v_i \) and \( v_i \) is Gaussian noise with zero mean and variance \( \sigma_i^2 \).

Answer: 0


Question: Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function such that \( x \leq f(x) \leq x+1 \) for all \( x \in \mathbb{R} \). Determine the set \( f(\mathbb{R}) \).

Answer: \mathbb{R}


Question: Let $D$ be an $n \times n$ diagonal matrix, and $A$ be an $n \times n$ rank-one matrix such that $A = a \cdot b^T$, where $a$ and $b$ are $n \times 1$ vectors. Determine the lower bound for the rank of the matrix $D + A$. Provide your answer with justification.

Answer: 0


Question: Determine whether the series \(\sum_{n=1}^{\infty} n^2 x^2 e^{-n^2|x|}\) is uniformly convergent on \(\mathbb{R}\). Answer with 'Yes' or 'No'.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to5}\left(\dfrac{\log{(x+5)}-\log{(5-x)}}{x-5}\right) \]

Answer: -\infty


Question: Evaluate the integral \( \int_{C} \log(z-z_0) \, dz \) where \( C \) is the unit circle enclosing the origin and \( z_0 > 1 \).

Answer: 0


Question: Evaluate the integral \( \int_0^{\infty} \frac{\sin^2(\tan x)}{x^2} \, \mathrm{d}x \).

Answer: \dfrac{\pi}{2}


Question: Evaluate the limit \( \lim_{x \to 0} \frac{\sin(a+2x) - 2\sin(a+x) + \sin(a)}{x^2} \) without using L'Hôpital's rule.

Answer: -\sin a


Question: Given that \(x, y, z \in \mathbb{R}^+ \cup \{0\}\) and \(x + y + z = 32\), find the maximum value of \(x^3y + y^3z + z^3x\).

Answer: 110592


Question: Determine the average length of threads criss-crossing a hollow sphere of radius $R$. The sphere has a large, random number of holes with a constant surface density, and threads pass randomly from one hole to another. Express the average length as a multiple of $R$.

Answer: \dfrac{4}{3}R


Question: Calculate the number of subgroups of \( A = \mathbb{Z}_{360} \oplus \mathbb{Z}_{150} \oplus \mathbb{Z}_{75} \oplus \mathbb{Z}_{3} \) that are isomorphic to \( \mathbb{Z}_{5} \oplus \mathbb{Z}_{5} \).

Answer: 31


Question: Given a sequence \( \{a_n\} \) of real numbers such that \( \sum_{n=1}^\infty a_n \) converges and \( \sum_{n=1}^\infty |a_n| \) diverges, determine the radius of convergence of the power series \( \sum_{n=1}^\infty a_n x^n \).

Answer: 1


Question: Consider a sequence of random variables $(X_n: n \ge 1)$ such that $|X_{n+1} - X_n| \le 1$ for all $n$. Given that $\mathbb{E}[X_n] = O\left(\frac{1}{\log n}\right)$ and $\text{Var}[X_n] = O\left(\frac{1}{n}\right)$ as $n \to \infty$, determine whether $X_n \to 0$ almost surely.

Answer: Yes


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

Answer: \dfrac{e}{2}


Question: Find the value of the sum: \[ \sum_{n=0}^k \binom{2n}{n} \cdot \binom{2(k-n)}{k-n} \]

Answer: 4^k


Question: Given the autonomous differential equation \( y' = y(1-y)(y+2) \), find \( \lim_{t \to \infty} y(t) \) given that \( y(0) = -1 \).

Answer: -2


Question: Determine whether the set \( V = \left\{ f \in C^1[-1,1] \mid f'(0) + \int_{-1}^1 f(x) \, dx = 0 \right\} \) is a vector space.

Answer: Yes


Question: Find the first derivative of the function \( f(X) = \log \det(XX^TA^{-1}) \) with respect to the matrix \( X \), where \( X \in \mathbb{R}^{n \times r} \) and \( A \in \mathbb{R}^{n \times n} \) with \( A^{-1} \) being the inverse of \( A \).

Answer: 2(XX^T)^{-1}X


Question: Evaluate \( \lim _{n\to \infty }\int _{0}^{1}nx^ne^{x^2}dx \).

Answer: e


Question: Evaluate the expression \( \displaystyle \underset{(i \ne j \ne k)}{\sum_{i=0}^\infty \sum_{j=0}^\infty \sum_{k=0}^\infty \frac{1}{3^i 3^j 3^k}} \).

Answer: \dfrac{81}{208}


Question: Find the minimal value of \( \alpha_3 \) such that the function \( u(x,y,z) = x^\alpha(1-x)y(1-y)z(1-z) \) belongs to the Sobolev space \( H^3(\Omega) \).

Answer: \dfrac{5}{2}


Question: Given the series \( y = \frac{2}{5} + \frac{1 \cdot 3}{2!} \left(\frac{2}{5}\right)^2 + \frac{1 \cdot 3 \cdot 5}{3!} \left(\frac{2}{5}\right)^3 + \ldots \), find the value of \( y^2 + 2y \).

Answer: 4


Question: Maximize the function \( f(a) = a e^{-a\sqrt{2r}} \int_0^\infty \exp\left(-{1 \over 2}x^2 - {{\lambda (a - S_0)^2} \over {\sigma^2}} {1 \over x^2} \right)dx \) over \( a > 0 \), where \( \lambda, \sigma, S_0 > 0 \) are constants.

Answer: S_0


Question: Calculate the Laplacian \( \Delta u = u_{xx} + u_{yy} + u_{zz} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \) for the function \( u(x,y,z) = \frac{1}{\sqrt{x^2 + y^2 + z^2}} \) on \( \mathbb{R}^3 \setminus \{0\} \).

Answer: 0


Question: Given the equations \( \log_x W = 24 \), \( \log_y W = 40 \), and \( \log_{xyz} W = 12 \), compute the value of \( \log_z W \).

Answer: 60


Question: Find the derivative with respect to the time-varying variable $x_t$ of the integral $\int_0^T y(x_s) \, ds$, where $T > t$ and $y(x_t)$ is a function of $x_t$, and $x_t$ is a function of $t$. Express your answer in terms of $x_t$ and $y(x_t)$.

Answer: y'(x_t)


Question: Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that are continuous at \( x_0 = 1 \) and satisfy the equation \( f(9x-8) - 2f(3x-2) + f(x) = 4x - 4 \) for all \( x \in \mathbb{R} \).

Answer: f(x) = x + C


Question: Given a normal surface $S$ and a smooth curve $C$ with a flat morphism $\pi: S \to C$, consider a fiber $F$ of $\pi$ in $S$ over $C$. The fiber $F$ is not integral and has no embedded points, and its genus is $0$. Determine the value of $h^0(F, \mathcal O_F)$. Provide a brief justification for your answer.

Answer: 1


Question: Let \( \mathbb{F} \) be a field such that the map \( a \to a^2 \) is a permutation of \( \mathbb{F} \). Determine the characteristic of \( \mathbb{F} \).

Answer: 2


Question: Is it always possible to construct a vector $v=(a_1,a_2,\ldots,a_n)$ consisting of nonnegative integers such that $a_1=1$ and, if $a_j$'s are nonzero, then $a_j\equiv a_{n-j+2}+j-1 \pmod m\ \forall 1<j\le\frac{n}{2}$, where $m$ is the number of nonzero entries, with the additional constraint that all nonzero $a_i$'s are distinct modulo $m$? Assume that the number $m$ itself appears once as a nonzero number satisfying the above congruence.

Answer: Yes


Question: Given that $x^3 + x - 1 = 0$, find the value of $x^3 + 2x^2$. Express your answer in terms of $x$. \( \boxed{} \)

Answer: 2x^2 - x + 1


Question: Determine whether the following statement is true or false: If \( \frac{n_k}{m_k} \rightarrow \xi \in \mathbb{R} \setminus \mathbb{Q} \), where \(m_k\) and \(n_k\) are integers, then \(m_k \rightarrow \infty\).

Answer: True


Question: Determine whether the function \( f: \mathbb{R}_l \times \mathbb{R}_l \to \mathbb{R}_l \) defined by \( f(x,y) = xy \) is continuous, where \( \mathbb{R}_l \) denotes the reals with the lower limit topology and \( \mathbb{R}_l \times \mathbb{R}_l \) has the product topology. Answer "True" if the function is continuous and "False" otherwise.

Answer: False


Question: Let $K \subset \mathbb{R}^d$ be a convex set and $\varepsilon > 0$. Define $K(\varepsilon) \subset K$ as the set of points in $K$ whose distance to the boundary $\partial K$ is less than $\varepsilon$. Is the set $K \setminus K(\varepsilon)$ convex?

Answer: Yes


Question:
Determine the possible values of $k$ such that the graph of the function $f(x)$ is disconnected, where $f(x) = \sin\left(\frac{1}{x}\right)$ for $x \neq 0$ and $f(0) = k$. Choose the correct option:

(A) $k$ could be $-1$
(B) $k$ must be $0$
(C) $k$ must be $1$
(D) $k$ could be less than $1$ and greater than $-1$
(E) $k$ must be less than $-1$ or greater than $1$

Answer: E


Question: Evaluate the limit: \[ \lim_{n\rightarrow \infty}\frac{n}{\log(n)}(n^{1/n}-1) \]

Answer: 1


Question: Evaluate the limit: $$ \lim_{h\to 0} \frac{\frac{\ln(1+h^4)}{h^4}-1}{h} $$

Answer: 0


Question: Determine whether there exist non-constant polynomials $f \in k[t]$ such that $1-f^2$ is a square in the polynomial ring $k[t]$. If such polynomials exist, provide an example.

Answer: t


Question: Given the power series \(\sum_{n=0}^{\infty} c_{n} (z-a)^{n}\) with a radius of convergence \(R\), prove that the series \(\sum_{n=0}^{\infty} c_{n} n(z-a)^{n-1}\) has the same radius of convergence \(R\).

Answer: R


Question: Evaluate the limit: \[ \lim_{a\to \infty}\int_0^1 a^x x^a \,dx \]

Answer: 1


Question: Consider a subset \( E \subseteq [0, 1] \) with the property that for each \( x \in E \), there exists \( r_x > 0 \) such that the intersection \( B(x, r_x) \cap E = (x-r_x, x+r_x) \cap E \) has Lebesgue measure zero. Is it necessarily true that the set \( E \) itself has Lebesgue measure zero?

Answer: Yes


Question: Find the residue of the function \( \frac{\sin (2z)-2z}{(1-\cos z)^2} \) at \( z_0=0 \).

Answer: -\dfrac{16}{3}


Question: Evaluate the double integral \( \iint_{\{(x,y)\mid (x-1)^2+y^2\leq1\}}\sqrt{x^2+y^2} \, dx \, dy \).

Answer: \dfrac{32}{9}


Question: Calculate the area enclosed by the curve \(x^{2/3} + y^{2/3} = 4\) using Green's Theorem. Use the formula \(A = \frac{1}{2}\int_C x \, dy - y \, dx\) to find the area.

Answer: 24\pi


Question: Does there exist a polynomial with integer coefficients \( p(x) \) such that for any polynomial with integer coefficients \( g(x) \), the composition \( p(g(x)) \) is irreducible in \( \mathbb{Q}[x] \)?

Answer: No


Question: Determine whether the ideal \((2, x, y)\) is a maximal ideal in the ring \(\mathbb{Z}[x, y]\).

Answer: Yes


Question: Evaluate the double integral \( \iint_D (x + y) \,dx\,dy \) over the region \( D \) defined by \( 0 \leq y \leq x \) and \( 0 \leq x \leq 1 \), using the change of variables \( x = u + v \) and \( y = u - v \).

Answer: \dfrac{1}{2}


Question: Determine whether the greatest common divisor (GCD) of any two numbers of the form $2^p + 1$, where $p$ is an odd prime, is always equal to 3.

Answer: 3


Question: Calculate the limit: \[ \lim_{n\to\infty} \int_{[0,n]} e^{-\sqrt{x^2+\frac{1}{n^2}}} \, d\lambda_{1}(x) \]

Answer: 1


Question: Does there exist a Banach space \((C,\|.\|)\) with \(C\subset\mathbb{R}^{\mathbb{Z}}\) such that for each \(\varepsilon>0\), there exists \(k\in\mathbb{N}\) where if \(v=(v_n)_{n\in\mathbb{Z}}\in C\) has \(\|v\|\leq1\), then \(|v_n|\leq\varepsilon\) whenever \(|n|\geq k\)?

Answer: Yes


Question: Determine the value of \( \limsup_{n \geq 2} \frac{m(n)}{n^2} \), where \( m(n) \) is the maximum cardinality of the edge set \( E \) in a square hypergraph \(([n^2], E)\), which is linear and each edge has \( n \) elements.

Answer: 1


Question:
Given the equations:
\[ ax + by = 3, \\
ax^2 + by^2 = 7, \\
ax^3 + by^3 = 16, \\
ax^4 + by^4 = 42, \]
where \(a, b, x,\) and \(y\) are real numbers, find the value of \(ax^5 + by^5\).

Answer: 20


Question:
Let \( A \) and \( B \) be matrices such that \( B^2 + AB + 2I = 0 \), where \( I \) denotes the identity matrix. Determine which of the following matrices must be nonsingular:

(A) \( A + 2I \)
(B) \( B \)
(C) \( B + 2I \)
(D) \( A \)

Answer: B


Question: Determine the totally geodesic submanifolds of $\mathbb{R}P^n$.

Answer: \mathbb{R}P^k


Question: Find the value of \( n \) such that the expression \( 2^{200} - 2^{192} \times 31 + 2^n \) is a perfect square.

Answer: 198


Question: Let $\mathbb{X} = C^1([0,1], \mathbb{R})$ with the norm $\|f\|_{\mathbb{X}} = \|f\|_{\infty} + \|f'\|_{\infty}$. Define the operator $A: \mathbb{X} \rightarrow \mathbb{R}$ by $Af = f'\left(\frac{1}{2}\right)$. Find the operator norm of $A$. \[ \|A\|_{op} = \inf\{c \geq 0: \|Af\| \leq c \|f\|_{\mathbb{X}} \} \]

Answer: 1


Question: Evaluate the double integral \( \iint_D x^6y^6 \, dx \, dy \) where the region \( D \) is defined by \( D = \{ (x,y) : x^{2} \le y \le x^{1/8} \} \).

Answer: \dfrac{5}{441}


Question: Suppose a positive sequence $\{a_n\}_{n=1}^\infty$ has a convergent subsequence with $\lim\limits_{j\to\infty}a_{n_j}=\mu$. If for all $n>0$, $|a_{n+1}-a_n|<\frac{1}{n}$, determine whether $\lim\limits_{n\to\infty}a_{n}=\mu$ is true.

Answer: \mu


Question: Evaluate the limit: \[ \lim_{x\to 0^+} (\sin x)^{(1/\ln(x))} \] without using L'Hospital's rule.

Answer: e


Question: Verify Stokes' Theorem for the upper hemisphere of radius 1 centered at the origin, where the vector field is given by \( \vec{F} = \left(x^3 + \frac{z^4}{4}\right) \hat{i} + 4x \hat{j} + (xz^3 + z^2) \hat{k} \). Compute \( \iint_S (\vec{\nabla} \times \vec{F}) \cdot \mathrm{d}\vec{S} \) and compare it with the line integral of \( \vec{F} \) over the boundary of the hemisphere.

Answer: 4\pi


Question:
Let $G$ be a nonabelian group. Let $a \in G$ have order $4$ and let $b \in G$ have order $3$. Determine the order of the element $ab$ in $G$. Choose the correct option:
A. 6
B. 12
C. 12k, where $k \geq 2$
D. Need not be finite

Answer: D


Question: Find the smallest positive integer $n$ such that $\mu(n) + \mu(n+1) + \mu(n+2) = 3$, where $\mu$ is the Möbius function.

Answer: 33


Question: Consider the set \(Y = \{1/n : n \in \mathbb{N}\}\) as a subset of \(\mathbb{R}\). Is every subset of \(Y\) an open set in the subspace topology of \(Y\) induced by \(\mathbb{R}\)?

Answer: Yes


Question:
Let \( A \) be a *-ring, and consider the following definitions:

1. A projection \( p \) is a self-adjoint idempotent, meaning \( p = p^* = p^2 \).
2. For projections \( p \) and \( q \), we write \( p \leq q \) if \( pq = p \).
3. A projection \( p \) is called strongly finite if there exist at most finitely many projections \( q_1, \ldots, q_n \) such that \( q_j \leq p \) for each \( j \).
4. For a given element \( x \in A \), let \( l(x) \) be the smallest projection such that \( l(x)x = x \).

Assume that \( e \) is a strongly finite projection and \( p \) is a projection. Determine whether \( l(ep) \) is a strongly finite projection.

Answer: Yes


Question: In a triangle $\Delta ABC$, it is given that \( \frac{2\cos A}{a} +\frac{\cos B}{b} +\frac{2\cos C}{c}=\frac{a}{bc} +\frac{b}{ca} \). Find the angle $A$.

Answer: 90^\circ


Question:
Determine the point-wise limit of the sequence of functions \( \{f_n(x)\} \) defined on the interval \([0,1]\) by:

\[
f_n(x) = \begin{cases} 0 & \text{if } x=0 \\ n^2x & \text{if } x \in [0,\frac{1}{n}] \\ -n^2x+n^2 & \text{if } x \in (\frac{1}{n},\frac{2}{n}] \\ 0 & \text{if } x \in [\frac{2}{n},1] \end{cases}
\]

Answer: 0


Question: Evaluate the infinite series \( \sum_{k=1}^\infty \frac{k^2}{(k-1)!} \).

Answer: 5e


Question: Find the product of the highest and lowest integer solutions to the inequality: \[ \sqrt{(5+2\sqrt{6})^{2x}}+\sqrt{(5-2\sqrt{6})^{2x}} \leq 98 \]

Answer: -4


Question:
Consider the determinant map \( \det: \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R} \) defined by \( \det(v, w) = v_1w_2 - w_1v_2 \) for \( v = (v_1, v_2) \) and \( w = (w_1, w_2) \). What is the derivative of the determinant map at \((v, w)\) evaluated on \((H, K)\)?

1. \( \det(H, W) + \det(V, K) \)
2. \( \det(H, K) \)
3. \( \det(H, V) + \det(W, K) \)
4. \( \det(V, W) + \det(K, W) \)

Choose the correct option.

Answer: 1


Question: Let \(f(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n\), where \(a_i\) are non-negative integers. Given that \(f(1) = 21\) and \(f(25) = 78357\), find the value of \(\frac{f(10) + 3}{100}\).

Answer: 51


Question: Calculate the integral \( \int_\Gamma (z-y)dx+(x-z)dy+(y-x)dz \) using Stokes' Theorem, where \( \Gamma \) is the surface obtained from the intersection of \( x^2+y^2=1 \) and \( x+2z=1 \).

Answer: 3\pi


Question: Find the value of \(\displaystyle\lim_{n \rightarrow \infty} \dfrac 1{n^4}\left[1\left(\sum^n_{k=1}k\right)+2\left(\sum^{n-1}_{k=1}k\right)+3\left(\sum^{n-2}_{k=1}k\right)+\dots+n\cdot1\right]\).

Answer: \dfrac{1}{24}


Question:
Let $A$ be a unital $C^*$-algebra. A map $\phi:A\to \mathbb{C}$ is called a character if it is a non-zero homomorphism. A character satisfies the following properties:

1. $\phi(1)=1$.
2. $\phi(x)\in \text{sp}(x):=\{\lambda: \lambda -x \text{ is not invertible}\}$ for all $x\in A$.
3. $\phi$ is bounded with $\|\phi\|\leq 1$.

Using these properties, it can be shown that:

4. $\|\phi\|= 1$.
5. $\phi\geq 0$.

A map $\phi:A\to \mathbb{C}$ is called a state if it is positive, linear, and $\|\phi\|= 1$.

Is it correct that every character is a state?

Answer: Yes


Question: For any finite alphabet $A$ with $|A| \geq 2$ and any natural number $k$, does there always exist a thin base of order $k$?

Answer: Yes


Question: Compute the radius of convergence for the Taylor series of the function \( f(x) = \int_0^1 \log\left(x+\sqrt{t^2+1}\right)dt \) around \( x=0 \).

Answer: 1


Question: Find the value of the infinite product: \[ 2^{\frac{1}{4}} \cdot 4^{\frac{1}{8}} \cdot 8^{\frac{1}{16}} \cdots \]

Answer: 2


Question: Solve the functional equation \( f(x+y) = f(x) + f(y) - f(0) \).

Answer: f(x) = ax + b


Question: Let \(x\) be a real number such that \(x^{2014} - x^{2004}\) and \(x^{2009} - x^{2004}\) are both integers. Must \(x\) also be an integer?

Answer: Yes


Question: For a polynomial $f(x)$ of degree 3 with rational coefficients that touches the x-axis, determine how many of its roots are rational.

Answer: 2


Question: Compute the complex integral \( \int_{|z|=1} \frac{1}{z^2+1} \, dz \).

Answer: 0


Question: Find the value of \( g'(0) \) if \( g(x) \) is the inverse of \( f(x) \), where \( f(x) = \int_{2}^{x} \frac{1}{\sqrt{1+t^4}} \, dt \).

Answer: \sqrt{17}


Question: Find the sum of the series \( \sum_{n=1}^{\infty} \frac{2n-1}{2^n} \).

Answer: 3


Question: Evaluate the contour integral \( \int_{|z| = 3} \frac{\exp(-z)}{z^2} \, dz \) around the circle \(|z| = 3\) in the positive sense using the residue theorem.

Answer: -2\pi i


Question: Let $V = \mathbb{C}^{n \times n}$. Define the linear transformation $T: V \to V$ by $T(u) = (u,v)w$ for fixed $v, w \in V$. Determine the rank of the transformation $T$. \( \operatorname{rank} T = ? \)

Answer: 1


Question: Given a matrix $A \in M_5(\mathbb{R})$ that satisfies the equation $A^2 - 4A - I = 0$, where $I$ is the identity matrix, find the value of \( (a_1 - \frac{1}{a_1}) + (a_2 - \frac{1}{a_2}) + \cdots + (a_5 - \frac{1}{a_5}) \), where $a_1, a_2, a_3, a_4, a_5$ are the eigenvalues of $A$. \( \)

Answer: 20


Question: Evaluate the double integral \( \iint_{[0,1]^2} (1+x^2+y^2)^{-3/2} \, dx \, dy \).

Answer: \dfrac{\pi}{6}


Question: Evaluate the limit: $$\lim_{(x,y) \to (0,0)} \frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$$

Answer: 0


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

Answer: e


Question: If $X$ is an infinite set and there exists an injection $X \to \mathbb{N}$, is there also a bijection between $X$ and $\mathbb{N}$? Provide a rigorous justification for your answer.

Answer: Yes


Question: Find the area bounded by the inverse of the function $f(x) = x^3 + 3x + 2$, the x-axis, and the vertical lines $x = -2$ and $x = 6$. You do not need to explicitly find the inverse function to solve this problem.

Answer: \dfrac{9}{2}


Question: Let \( \{\boldsymbol{x}_i\}_{i=1}^n \) be a set of \( n \) linearly independent vectors in \( \mathbb{R}^n \). Is the matrix \( \sum_{i=1}^n \boldsymbol{x}_i \boldsymbol{x}_i^\top \) always positive definite?

Answer: Yes


Question: Determine the number of ways to express \(2^{2012}\) as the sum of four (not necessarily distinct) positive squares.

Answer: 1


Question: Determine the values of \( p \) for which the series \( \sum_{n=1}^{\infty} \left( \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} \right)^p \) converges.

Answer: p > 2


Question: Determine whether the function \( f(x) = \begin{cases}\dfrac{1}{\lfloor 1/x \rfloor} & \text{for } x \neq 0, \\ 0 & \text{otherwise.}\end{cases} \) is differentiable at \( x = 0 \).

Answer: Yes


Question: Let $a \in \mathbb{R}$ and $E$ be the vector space of functions $f : \mathbb{R} \to \mathbb{R}$ which are $C^{\infty}$. Consider the endomorphism $u$ defined by $u(f) = f' - af$. Determine the image of $u^n$ for a given positive integer $n$.

Answer: E


Question: Matrix $A$ is an $n \times n$ matrix with three distinct eigenvalues $a$, $b$, and $c$. The eigenvalue $a$ has a geometric multiplicity of 2 and an algebraic multiplicity of 4. The eigenvalue $b$ has a geometric multiplicity of 3. The eigenvalue $c$ has equal geometric and algebraic multiplicities. What is the minimum possible value of $n$?

Answer: 8


Question: Evaluate the sum \( \sum_{k=1}^{2500}\left \lfloor{\sqrt{k}}\right \rfloor \).

Answer: 82125


Question: A ball falls from rest at a height $H$ and rebounds with a coefficient of restitution $e = 0.8$. After 5 bounces, the ball is damaged, and the coefficient of restitution changes to $g$. Determine the possible range of values for $g$ such that the total distance traveled by the ball is more than $4H$. Assume the ball rebounds to a height of $e^2H$ before being damaged.

Answer: [0, 1)


Question: In a dice game, you start by rolling a single fair six-sided die. If you roll an even number, you then roll two dice. If the sum of the two dice is even, you roll three dice, and this process continues, increasing the number of dice by one each time, until you roll an odd sum. The first odd sum you roll is your score. Calculate the average score per turn in this game.

Answer: \dfrac{27}{4}


Question:
Identify the original function \( f(x) \) for which the Newton's method iteration is given by:
\[ x_{n+1} = 2x_n - x_n^2y \]

Answer: y - \dfrac{1}{x}


Question: Evaluate the limit: \[ \lim_{\epsilon \to 0} \frac{1}{\epsilon}\int_{-\infty}^{\infty} \rho\left(\frac{x}{\epsilon}\right)f(x)\,dx \] where \( \rho: \mathbb{R} \to \mathbb{R} \) is a continuous function with \( \rho(x) \ge 0 \) and \( \rho(x) = 0 \) if \( |x| \ge 1 \), \( f: \mathbb{R} \to \mathbb{R} \) is a continuous function, and \( \int_{-\infty}^{\infty}\rho(t)\,dt=1 \).

Answer: f(0)


Question: Find the limit: \( \lim_{ n \to \infty }\sin ^2(\pi\sqrt{n^2+n}) \).

Answer: 1


Question: Evaluate the infinite nested radical expression: \[ x = \sqrt{9-5\sqrt{3-\sqrt{9-5\sqrt{3-\sqrt{9-\cdots}}}}} \] Determine the value of \( x \).

Answer: 2


Question: Let \( f(x) \) be a quadratic function such that \( f'(3) = 3 \). Define \( I = \int_{0}^{\frac{\pi}{3}} t \tan(t) \, dt \). If the integral \( \int_{3-\pi}^{3+\pi} f(x) \tan\left(\frac{x-3}{3}\right) \, dx \) is equal to \( kI \), find the value of \( k \).

Answer: 54


Question: Let $A$ be a square matrix such that $A^{10} = 0$ and $A^9 \neq 0$. Let $V$ be the space of all polynomials of $A$. Find the dimension of $V$.

Answer: 10


Question: Let \(V\) be a vector bundle over an algebraic curve \(C\), and assume that \(V \cong V \otimes L\) for some line bundle \(L\). If \(V\) is simple, does this imply that \(L \cong \mathcal{O}_C\)? Answer yes or no and justify your answer.

Answer: Yes


Question:
Consider the system of ordinary differential equations in \( \mathbb{R}^2 \):
\[
\frac{dY}{dt} = AY, \quad Y(0) = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \quad t > 0
\]
where \( A = \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix} \) and \( Y(t) = \begin{pmatrix} y_1(t) \\ y_2(t) \end{pmatrix} \). Determine for which values of \( t \), the functions \( y_1(t) \) and \( y_2(t) \) are monotonic. Choose the correct option:

1. \( y_1(t) \) and \( y_2(t) \) are monotonically increasing for \( t > 0 \).
2. \( y_1(t) \) and \( y_2(t) \) are monotonically increasing for \( t > 1 \).
3. \( y_1(t) \) and \( y_2(t) \) are monotonically decreasing for \( t > 0 \).
4. \( y_1(t) \) and \( y_2(t) \) are monotonically decreasing for \( t > 1 \).

Answer: D


Question: Let \(V\) be a closed subspace of \(L^2[0,1]\) and let \(f,g \in L^2[0,1]\) be given by \(f(x)=x\) and \(g(x)=x^2\). If \(V^\perp = \operatorname{Span} f\) and \(Pg\) is the orthogonal projection of \(g\) on \(V\), find the expression for \((g-Pg)(x)\) for \(x \in [0,1]\).

Answer: \dfrac{3}{4}x


Question: Let \(X\) be a random variable uniformly distributed over \([-2, 2]\) and \(Y\) be a random variable uniformly distributed over \([0, 1]\). Define two new random variables \(A = (X + Y)^2\) and \(B = X^2 - Y^2\). Find the exact numerical value of \(\Pr(A < B)\).

Answer: \dfrac{3}{8}


Question: If $a, b, c$ are in a geometric progression (GP) and $\log_b a, \log_c b, \log_a c$ are in an arithmetic progression (AP), find the common difference of the AP.

Answer: 0


Question:
Calculate the perimeter of the curve given by the polar equation \( r = 1 + \cos \theta \) for \( 0 \le \theta < 2\pi \) by evaluating the integral:
\[ \int_P \sqrt{(dx)^2 + (dy)^2} \]
Show that:
\[ (dx)^2 + (dy)^2 = (dr)^2 + (rd\theta)^2. \]

Answer: 8


Question: Evaluate the line integral \( \int_C \vec{F} \cdot d\vec{r} \) using Stokes' Theorem, where \( C \) is a circle of radius 2 in the plane \( x + y + z = 5 \), centered at \( (4,4,-3) \) and oriented clockwise when viewed from the origin. The vector field is given by \( \vec{F} = 4y\vec{i} - 5\vec{j} + 2(y-x)\vec{k} \).

Answer: 0


Question: Find the maximum value of the expression \( \frac{1}{\sin^2 \theta + 3\sin\theta \cos\theta + 5\cos^2 \theta} \).

Answer: 2


Question: Determine the dimension of the vector space \(\mathbb{C}\) over the field \(\mathbb{Q}\).

Answer: 2^{\aleph_0}


Question: Use Green's Theorem to evaluate the line integral \( \int\limits_C (y-x) \, \mathrm{d}x + (2x-y) \, \mathrm{d}y \) where the path \( C \) is defined by the parametric equations \( x = 2\cos\theta \) and \( y = 2\sin\theta \).

Answer: 4\pi


Question: Does the Banach algebra of jets $J^1_{\mathbb R}[0,1]$ have the classical approximation property?

Answer: Yes


Question: Given \( y(x) = f(\sqrt{|x| + 3}) \) and \( f'(2) = -\frac{1}{3} \), compute \( y'(-1) \).

Answer: \dfrac{1}{12}


Question: Find the positive time $T$, in years, such that the amount in Fund A, which accumulates at a rate of $3\%$ convertible monthly, equals the amount in Fund B, which accumulates with a force of interest given by $\delta_t = t^{20}$ for all $t$. At time $t = 0$, $\$1 is deposited in each fund. Calculate $T$. \( \left(1+\frac{0.03}{12}\right)^{12T} = e^{\int_0^T t^{20} \, dt} \)

Answer: 0.977


Question: In a match between two brothers, the younger brother has a probability of 4/7 of winning each game, with no possibility of a draw, and each game's outcome is independent of the others. The match continues until one brother wins two consecutive games. Calculate the expected number of games played in such a match.

Answer: \dfrac{110}{37}


Question: Evaluate the integral \( \int_C \frac{z}{1-\cos(z)} \, dz \), where \( C \) is the unit circle oriented counterclockwise.

Answer: 4\pi i


Question: Does every maximum independent set in a tree contain a leaf? Provide a justification for your answer.

Answer: Yes


Question: Determine the number of messages that can be sent with a probability of \( \frac{1}{2} \) in at most 15 minutes. Assume that the time to send each message follows an exponential distribution with rate \( \lambda = 8 \) and is independent of other messages. Messages are sent sequentially, one after another.

Answer: 120


Question: Determine the number of rational terms in the expansion of \((\sqrt{2} + \sqrt[3]{3})^8\).

Answer: 2


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

Answer: \sqrt{\pi}


Question: Is a differentiable probability density function with a bounded derivative necessarily bounded? Let $g$ be a Lebesgue probability density function that is differentiable and satisfies $\sup_x|g'(x)| < \infty$. Determine if $g$ is bounded.

Answer: Yes


Question: Find the length of the loop of the parameterized curve given by the equations: \( x = 12t - 4t^3 \) and \( y = 12t^2 \).

Answer: 48\sqrt{3}


Question: Let $V$ be a non-singular affine variety in $\mathbb{C}^n$, regarded as a complex manifold. Consider a point $p = (a_1, \dots, a_n)$ on $V$. The tangent space $T_p$ at $p$ consists of tangent vectors, which are derivations $\mathcal{O}_p \rightarrow \mathbb{C}$. Let $f_1, \dots, f_r$ be defining polynomials for $V$, and $L_i$ be the hyperplane defined by $\sum_k \frac{\partial f_i}{\partial x_k}(p)(x_k - a_k) = 0$. Let $S_p = \bigcap_i L_i$. Are the tangent space $T_p$ and the vector space associated with $S_p$ canonically isomorphic?

Answer: Yes


Question: Calculate the area bounded by the lines $x = y$, $x = 2y$, $x + y = a$, and $x + y = 3a$ where $a > 0$. Use the substitution $\frac{x}{y} = u$ and $x + y = v$ to evaluate the double integral and find the area.

Answer: \dfrac{2}{3}a^2


Question: Evaluate the double integral: $$\int_{-14}^{14} \int_{392}^{1081} e^{-(x^2+9y^2)}\sin(y^{191})\,dy\,dx$$

Answer: 0


Question: Given that $p \equiv 1 \pmod{4}$, determine the number of elements of order 4 in the multiplicative group of the finite field $GF(p^n)^*$.

Answer: 2


Question: A bag contains 16 white balls and 14 black balls. Balls are drawn with replacement. Calculate the expected number of white balls drawn before the 13th black ball is drawn.

Answer: \dfrac{104}{7}


Question: In the acute-angled triangle $ABC$, point $D$ is located on $BC$ such that $BD:CD = 2:3$, and point $E$ is located on $AC$ such that $AE:EC = 3:4$. Point $F$ is the intersection of $AD$ and $BE$. Calculate the value of $\frac{AF}{FD} \cdot \frac{BF}{FE}$.

Answer: \dfrac{35}{12}


Question: Evaluate the integral \( I = \int_0^{2\pi} \sin^4\theta \, d\theta \) using residue calculus.

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


Question: Evaluate the triple integral \( \iiint_\Omega{\frac{z}{y+z}}{dxdydz} \), where \( \Omega = \{(x,y,z)\in\mathbb{R}^3 \mid x \geq 0, y \geq 0, z \geq 0, x \leq y+z, 1 \leq 2(x+y+z) \leq 2\} \).

Answer: \dfrac{7}{128}


Question: Find the limit: \[ \lim_{n \to \infty} 2^n \tan\left(\frac{a}{2^n}\right) \]

Answer: a


Question: Evaluate the double infinite series: \[ \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{n}{(k+n)!} \frac{n}{k!} \]

Answer: e^2


Question: If two analytic functions $f(x)$ and $g(x)$ are equal for all real numbers $x$, can we conclude that $f(z) = g(z)$ for all complex numbers $z$? Assume both functions are analytic without poles.

Answer: Yes


Question: Compute the limit \( \lim_{n \to \infty} \int_0^1 \frac{e^{-nt}}{\sqrt{t}} \, dt \).

Answer: 0


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

Answer: 1


Question: Determine whether there exist continuous random variables $X$ and $Y$ such that $P(X \neq Y)$ is neither 0 nor 1.

Answer: Yes


Question: Determine whether the following statement is true or false: Let $F$ be a field and $n \in \mathbb{N}$. Then there exists an extension $E$ of $F$ with $[E:F]=n$. Justify your answer.

Answer: False


Question: Determine whether the set of all continuous functions from \([0,1]\) to \(\mathbb{R}\), denoted as \(C[0,1]\), has a cardinality larger than or equal to that of the set of real numbers \(\mathbb{R}\).

Answer: Yes


Question:
For even values of \(k\) and \(n \geq k\), determine if there exists a closed formula for the sum:
\[
N = \sum_{j=0}^{k/2}\binom{n}{k-j}\binom{k-j}{j}.
\]

Answer: \sum_{j=0}^{k/2} \binom{n}{k-j} \binom{k-j}{j}


Question: Evaluate the limit \( \lim_{t\rightarrow\infty}\left(te^t\int_t^{\infty}\frac{e^{-s}}{s}\text{d}s\right) \).

Answer: 1


Question: Can any countable set in \(\mathbb{R}\) be extended to a perfect set \(P\) in \(\mathbb{R}\) such that \(\mathbb{R} - P\) is uncountable?

Answer: No


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

Answer: \infty


Question:
Verify whether the sum of the absolute values of the roots of the cubic equation \(x^3-x^2-p^2x+p^2=0\) is equal to \(1+2p\) for all \(p \in (0,1)\) using direct calculation. The roots are given by the trigonometric formula:
\[ S_2 = \frac{4}{3} \sqrt{1+3p^2}\, \cos \left(\frac{1}{3}\arccos\left(\frac{1-9p^2}{(1+3p^2)^{\frac{3}{2}}}\right)-\frac{\pi}{3}\right)+\frac{1}{3} \]
Show that \(S_1 = S_2\) for all \(p \in (0,1)\).

Answer: S_1 = S_2


Question: Determine whether the function \( f(x) = x + 2x^2\sin(1/x) \) for \( x \neq 0 \) and \( f(x) = 0 \) when \( x = 0 \) is increasing or decreasing at \( x = 0 \).

Answer: increasing


Question: Calculate \(\int \int \vec{F} \cdot d\vec{S}\) over the surface defined by \(x = u^2\), \(y = uv\), \(z = \frac{1}{2} v^2\) bounded by the curves \(u = 0\), \(u = 1\), \(v = 0\), and \(v = 3\), for the vector field \(\vec{F} = y \vec{i} - x \vec{j} + xy \vec{k}\).

Answer: \dfrac{111}{8}


Question: Does there exist a quasisimple group $G$ and an odd prime $p$ such that $G$ has cyclic Sylow $p$-subgroups and a weakly real element of $p$-power order?

Answer: Yes


Question: A vertical fence is constructed with its base along the curve \( y = x \sqrt{x} \) from \((0,0)\) to \((1,1)\). The height of the fence above each point \((x,y)\) on the curve is given by \( x^3 - y^2 + 27 \). Calculate the area of this fence.

Answer: 13\sqrt{13} - 8


Question: For which natural numbers \( n \), does there exist a bijection \( f : \mathbb{N} \to \mathbb{Q}^n \) and a norm \( \|\cdot\| \) on \( \mathbb{R}^n \) such that \( \|f(k+1)-f(k)\|=1 \) for all \( k \in \mathbb{N} \)?

Answer: n \geq 2


Question: Solve the equation \((5x-4) \cdot \sqrt{2x-3} - (4x-5) \cdot \sqrt{3x-2} = 2\) for \(x\).

Answer: 6


Question: Is the field $L = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}, \ldots)$ a splitting field over $\mathbb{Q}$? Answer Yes or No.

Answer: No


Question:
Let $A$, $B$, and $C$ be finite subsets of the plane such that $A \cap B$, $B \cap C$, and $C \cap A$ are all empty. Let $S = A \cup B \cup C$. Assume that no three points of $S$ are collinear and that each of $A$, $B$, and $C$ has at least 3 points. Which of the following statements is always true?

(a) There exists a triangle having a vertex from each of $A$, $B$, and $C$ that doesn't contain any point of $S$ in its interior.
(b) There exists a triangle having a vertex from each of $A$, $B$, and $C$ that contains all the remaining points of $S$ in its interior.
(c) Any triangle having a vertex from each of $A$, $B$, and $C$ must contain a point of $S$ in its interior.
(d) There exist 2 triangles both having a vertex from each of $A$, $B$, and $C$ such that the two triangles do not intersect.

Answer: a


Question: Given that \( x^{x^9} = \sqrt{3^{\sqrt{3}}} \) and \( y = x^{\left(\frac{1}{y^{y^x}}\right)} \), determine the value of \( y^{3x} \).

Answer: \sqrt{3}


Question: Let \( g(x) = \int_0^x f(y)(y-x)dy \) be a function that is three times continuously differentiable. Determine the greatest integer \( n \) such that the function \( f \) must be \( n \) times continuously differentiable.

Answer: 1


Question: Consider a real sequence $(a_i)_{i \in \mathbb{N}}$ such that $0 \leq a_i$ and $\sum_{i=1}^N a_i \leq N$. Suppose $\lim_{n \rightarrow \infty} \frac{1}{n} \max_{1 \leq i \leq n} a_i = 0$. Determine if it is true that $\lim_{N \rightarrow \infty} \frac{1}{N^2} \sum_{i=1}^N a_i^2 = 0$. Provide a justification for your answer.

Answer: 0


Question:
Find the area bounded by the following parabolas:
1. \( y = x^2 \)
2. \( 4y = x^2 \)
3. \( y^2 = 2x \)
4. \( y^2 = 3x \)

Answer: 1


Question: Suppose \(E\) is an extension field of \(\mathbb{Z}_5\) and \(E\) has exactly 78125 elements. Find the degree of the minimal polynomial of any element \(a \in E - \mathbb{Z}_5\) over \(\mathbb{Z}_5\).

Answer: 7


Question: In a 4-dimensional world, how many 3-dimensional images should a creature's brain receive to perceive a single 4-dimensional image? Generalize this to determine how many (n-1)-dimensional images are needed for a creature in an n-dimensional world to perceive one n-dimensional image, for n ≥ 2.

Answer: 2


Question: Identify the topological space that corresponds to the quotient $SO(n+1)/O(n)$, where $SO(n)$ and $O(n)$ are considered as topological groups.

Answer: \mathbb{RP}^n


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \( f(a+b) = f(a) + f(b) \) for all real numbers \( a \) and \( b \). Assume that \( \lim_{x \to 0} f(x) = L \) for some real number \( L \). Prove that \( L = 0 \).

Answer: 0


Question: Let $ABCD$ be a parallelogram with $AB = 6$, $BC = 10$, and $AC = 14$. A circle passes through the vertices $C$ and $D$, with $AD$ being tangent to the circle and $BC$ being a secant. Calculate the area of the circle.

Answer: 12\pi


Question: Determine the normalizer of the subgroup \(\langle(1,2,...,p)\rangle\) in the symmetric group \(S_p\).

Answer: AGL(1, p)


Question: Let $f:X \to Y$ be an affine morphism. Determine whether the counit map \( f^* f_* \mathcal{F} \to \mathcal{F} \) is surjective for every coherent sheaf \( \mathcal{F} \).

Answer: Yes


Question: Is the product of perfectly-$T_3$ spaces necessarily perfectly-$T_3$? Provide a justification for your answer.

Answer: No


Question:
Consider the following setup: Let \( \Omega_1, \Omega_2, \Omega_3 \) be non-empty sets with \( \sigma \)-algebras \( \mathcal{A}_1, \mathcal{A}_2, \mathcal{A}_3 \) respectively. Let \( P \) be a stochastic kernel from \( \mathcal{A}_1 \) to \( \mathcal{A}_2 \), and \( Q \) be a stochastic kernel from \( \mathcal{A}_1 \otimes \mathcal{A}_2 \) to \( \mathcal{A}_3 \). For each \( \omega_1 \in \Omega_1 \), define \( P_{\omega_1} \) as a probability measure on \( \mathcal{A}_2 \) and \( Q_{\omega_1} \) as a stochastic kernel from \( \mathcal{A}_2 \) to \( \mathcal{A}_3 \). The indexed product measure is given by \( K(\omega_1, B_{2,3}) = (P_{\omega_1} \otimes Q_{\omega_1})(B_{2,3}) \) for \( B_{2,3} \in \mathcal{A}_2 \otimes \mathcal{A}_3 \). Is \( K \) a stochastic kernel, meaning it satisfies the following conditions?

1. For every \( \omega_1 \in \Omega_1 \), \( K(\omega_1, B_{2,3}) \) is a probability measure on \( \mathcal{A}_2 \otimes \mathcal{A}_3 \).
2. For every \( B_{2,3} \in \mathcal{A}_2 \otimes \mathcal{A}_3 \), \( K(\omega_1, B_{2,3}) \) is \( \mathcal{A}_1/\mathfrak{B} \)-measurable, where \( \mathfrak{B} \) is the Borel field on the real line.

Answer: Yes


Question: Evaluate the limit \( L = \lim\limits_{n\to\infty} \frac{2^{n^{1.1}}}{n!} \).

Answer: \infty


Question: Let $G$ be a group of order 77. Determine the order of the center of the group.

Answer: 77


Question: Calculate the sum of the infinite series \( 1 + \frac{1+2}{2!} + \frac{1+2+3}{3!} + \cdots \).

Answer: \dfrac{3e}{2}


Question: Let $E$ be a vector bundle on a smooth projective variety $X$ with $\mathrm{rk}(E) > \mathrm{dim}(X)$. If $E$ is generically generated by global sections, is a general section of $E$ nowhere vanishing?

Answer: Yes


Question: Find the maxima of the function \( f(x) = \frac{\sin(nx)}{\sin(x)} \) in terms of \( x \), where \( n \) is a positive integer greater than 0.

Answer: n


Question: For which values of \(a \in (0,2)\) must there exist \(x, y \in [0,2]\) such that \(|y - x| = a\) and \(f(x) = f(y)\), given that \(f\) is continuous on \([0,2]\) and \(f(0) = f(2)\)?

Answer: (0,2)


Question: Calculate the commutator \([p^2, x]\), where \(p = -i\hbar\frac{d}{dx}\).

Answer: -2i\hbar p


Question: Find the greatest integer \( c \) such that the function \( f(x) = \frac{x-1}{c-x^2+1} \) does not take any values in the interval \([-1, -\frac{1}{3}]\).

Answer: -1


Question: Determine the number of distinct abelian groups of order 96.

Answer: 7


Question: Calculate the degree of the field extension \([\mathbb{Q} (\sqrt[3]{3} + \sqrt{2}) : \mathbb{Q}]\).

Answer: 6


Question: Given a set \( \{1, \sqrt{2}, \sqrt{3}, 2, e, \pi\} \), choose \( p \) and \( q \) independently with replacement from this set. A circle is drawn with center \((p, q)\). What is the probability that the circle has at most two rational points?

Answer: \dfrac{8}{9}


Question: In how many ways can you distribute 21 fish among 6 bears such that no bear receives more than 6 fish? Note: A bear may receive zero fish.

Answer: 7872


Question: Calculate the line integral \( \int_{\rm C}(e^{x^2}- y{x}^{2}){\rm d}x+ \cos({y}^{2}){\rm d}y \), where \( {\rm C} \) is the unit circle traversed counterclockwise. Use Green's theorem to evaluate the integral.

Answer: \dfrac{\pi}{4}


Question: Is it consistent with ZFC that the cardinality of the power set of the natural numbers, \(|\mathcal{P}(\mathbb{N})|\), is greater than \(\aleph_\omega\)?

Answer: Yes


Question: Solve the expression \( \arccos\left(\frac{11}{14}\right) + \arcsin\left(-\frac{1}{7}\right) \) using complex numbers and the property \( \arg (z_1 \cdot z_2) = \arg(z_1) + \arg(z_2) \).

Answer: \dfrac{\pi}{6}


Question: Let \( f \) be a real function such that \( f''(x) \geq C > 0 \) for some constant \( C \). Determine whether \( \lim_{x \to +\infty} f'(x) e^{-f(x)} = 0 \).

Answer: 0


Question: Determine the number of real solutions to the equation \( x^{x^{2015}} = 2015 \).

Answer: 1


Question:
Determine the value to which the series \( \sum a_n \) converges, where the \( n \)-th term is given by:
\[ a_{n} = \frac{1+3+\cdots+(2n-1)}{n!}. \]

Answer: 2e


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

Answer: 2.506


Question: Let \( G \) be a group of order 231, where \( |G| = 3 \times 7 \times 11 \). Given that all Sylow \( p \) subgroups of \( G \) are normal, determine whether \( G \) is cyclic.

Answer: Yes


Question: Find all pairs of positive integers \((m, k)\) such that \(m \ge 2\), \(k \ge 2\), and \(m^k - k^m = 17\).

Answer: (3, 4)


Question: Evaluate the contour integral \( \int_\gamma \frac{dz}{4z^2-1} \), where \( \gamma:[0,1] \to \mathbb{C} \) is the curve defined by \( \gamma(t) = e^{2\pi i t} \) for \( 0 \le t \le 1 \).

Answer: 0


Question: Solve the congruence equation: \(x^5 \equiv 3 \pmod{19}\).

Answer: 10


Question: Evaluate the limit: \[ \lim_{r\to\infty}\frac{\int_0^{\pi/2}x^{r-1}\cos x\,\mathrm dx}{\int_0^{\pi/2}x^r\cos x\,\mathrm dx} \]

Answer: \dfrac{2}{\pi}


Question: Let $f : S^n \to \mathbb{R}^n$ be a function such that $f(-x) = -f(x)$ for all $x \in S^n$. Prove that there exists a point $y \in S^n$ such that $f(y) = 0$. Use the Borsuk-Ulam theorem to justify your answer.

Answer: 0


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

Answer: -\dfrac{e}{2}


Question: In how many ways can 13 identical balls be distributed into 6 different cells such that no cell contains exactly 3 balls?

Answer: 4092


Question: Find the maximum value of the expression: \[ 7\sin^2x + 5\cos^2x + \sqrt{2}\left(\sin\frac{x}{2} + \cos\frac{x}{2}\right) \]

Answer: 9


Question: Evaluate the integral: $$\int_0^{\infty} x^{k-1} e^{-Ax^k} \, dx$$

Answer: \dfrac{1}{A k}


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

If $C$ is the circle in $\mathbb{C}$ of radius $10$ centered at $z=2i$ with positive orientation and
$$f(z) = \frac{\cos(4z)}{z},$$
then
$$ \oint_{C} f(z) \, \mathrm{d}z = 2\pi i.$$

Answer: A


Question: Determine whether the statement \( \gcd(a_m, a_n) = a_{\gcd(m, n)} \) is true or false for a sequence \((a_0, a_1, a_2, \ldots)\) defined by \(a_0 = 0\) and \(a_{n+1} = P(a_n)\), where \(P\) is a polynomial with positive integer coefficients.

Answer: True


Question: Evaluate the limit: $$\lim_{n\rightarrow \infty } \sqrt[n]{2^{n}\cdot3^{0}+2^{n-1}\cdot3+...+2^{0}\cdot3^{n}}$$

Answer: 3


Question:
A quality control plan involves randomly selecting three items from the daily production of a machine, where the proportion of defectives produced by the machine varies daily and follows a uniform distribution on the interval \((0,1)\). What is the variance of the number of defectives among the three sampled items?

(a) 2.5
(b) 1.25
(c) 3
(d) 2.75
(e) none of the above.

Answer: B


Question: Evaluate the double integral \( \iint_D x^4\tan(x) + 3y^2 \,dA \) over the domain \( D = \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 \le 4, y \ge 0\} \).

Answer: 6\pi


Question: Let $X \subseteq \mathbf{R}^n$ be such that $H^d(X) = \infty$, where $H^d$ is the $d$-dimensional Hausdorff outer measure. Given a constant $c$, determine if there exists a subset $Y \subseteq X$ such that $H^d(Y) \in [c, \infty)$. Provide a justification for your answer.

Answer: Yes


Question: In the construction of Borel measures on the real line, consider the proposition involving intervals of the form $(a_j,b_j]$. If these intervals are replaced with $[a_j,b_j)$, does the proposition remain true?

Answer: Yes


Question: Determine the number of subgroups of order \(p^2\) in the group \(\mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}\).

Answer: p^2 + p + 1


Question: Consider a planar graph where all vertices have degree 3, except for one vertex which has degree 2. Must such a graph contain a face of length 1, 2, 3, or 5? (In other words, a face of length less than or equal to 5 that is not a square.)

Answer: Yes


Question: Evaluate the double integral \( \iint\limits_D \frac{dx\,dy}{(x^2+y^2)^2} \) where \( D = \{(x,y) : x^2+y^2 \le 1,\ x+y \ge 1\} \).

Answer: \dfrac{1}{2}


Question: Find the value of the function $u(x,t)$ at $x=3$ and $t=2$ for the partial differential equation \( \frac{du}{dt} = (1-t)\frac{\partial^2{u}}{\partial^2{x}} \) with the initial condition \( u(x,0) = \frac{1}{1+x^2} \) on the domain $-\infty < x < \infty$, $t \ge 0$. Determine $u(3,2)$. \( \boxed{?} \)

Answer: \dfrac{1}{10}


Question: Given a non-empty set $A$ and a symmetric relation $R \subset A \times A$, if $(R \circ R)$ is a function, determine whether $R$ is necessarily a function.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{t \to \infty} -t^{x}e^{-t} = 0 \]

Answer: 0


Question:
Consider a sequence \(f_n\) of continuous functions on \([0,1]\) such that \(0 \le f_n \le 1\) and \(\lim_{n\to\infty}\int_{0}^{1} f_n(x)\,\mathrm dx=0.\) Determine which of the following statements is true:

(a) The sequence \(f_n(x)\) converges uniformly.
(b) The sequence \(f_n(x)\) converges for some \(x\).
(c) The sequence \(f_n(x)\) might not converge for any \(x\).
(d) The sequence \(f_n(x)\) converges for every \(x\) but not necessarily uniformly.

Select the correct option.

Answer: C


Question: Given that $X \sim U(0,1)$ and the conditional density $f_{Y\mid X}(y\mid x) = 1$ for $x < y < x+1$, find the covariance $\operatorname{Cov}(X,Y)$. Assume $X$ and $Y$ are jointly continuous random variables.

Answer: \dfrac{1}{12}


Question: Use the divergence theorem to compute \( \int \int \limits_S F \cdot n \,dS \) where \( n \) is the outward normal. The surface \( S \) is defined by \( x^2 + y^2 = z^2 \) with \( z \in [0,1] \), and the vector field is \( F(x,y,z) = (x, xy, z) \).

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


Question: Evaluate the recursively defined sequence $T_n$ where $T_n = n + \frac{T_{n+2}}{n+1}$, starting with $T_1$. Express the result as a single number.

Answer: 2\sqrt{e}


Question: Evaluate the limit \( \lim_{x \to \infty} x[1-F(x)] \), where \( F(x) \) is a cumulative distribution function and \( 1-F(x) \) is the tail distribution function.

Answer: 0


Question: Find all functions \( f: \mathbb{R} \to \mathbb{R}^+ \) such that the area under the curve of \( f \) over any interval \([a, b]\) is proportional to the length of the interval. Specifically, solve the equation \( \int_a^b f(x) \, dx = K (b-a) \) for all \( a < b \), where \( f \in \mathcal{C}^1(\mathbb{R}) \).

Answer: K


Question: Determine whether the statement "If $E[z] < \infty$, then $z$ is almost surely finite" is true or false.

Answer: True


Question: Find the coefficient of \(x^{20}\) in the expansion of \((x^3 + x^4 + x^5 + \ldots)^5\).

Answer: 126


Question: Find the number of positive integer solutions to the equation \(x+y+z+w=20\) such that \(x, y, z,\) and \(w\) are all different.

Answer: 552


Question: Does there always exist a field \( \Omega \) which contains both fields \( L \) and \( E \) as \( K \)-subfields?

Answer: Yes


Question: Evaluate the integral \( \int_0^1 x^4\left(\ln\frac{1}{x}\right)^3dx \).

Answer: \dfrac{6}{625}


Question: Given the inequality \(8 \cdot 3^{\sqrt{x}+\sqrt[4]{x}} + 9^{\sqrt[4]{x}+1} \geq 9^{\sqrt{x}}\), find the interval for \(x\) after making the substitution \(t = 3^{\sqrt[4]{x} - \sqrt{x}}\) and solving the resulting inequality \(8t + 9t^2 \geq 1\).

Answer: [0, 16]


Question: Given that \( f(f(x)) = x^2 + \frac{1}{4} \), compute the value of \( f\left(\frac{1}{2}\right) \).

Answer: \dfrac{1}{2}


Question: Determine the number of homomorphisms from the dihedral group $D_5$ of order 10 to the Klein four-group $V_4$. Provide your answer as a single integer.

Answer: 4


Question: Determine the region in the $xy$-plane where the initial-value problem \( y' = \frac{x - y}{2x + 5y} \) is guaranteed to have a unique solution.

Answer: 2x + 5y \neq 0


Question: Compute \( \binom{1000}{326} \mod 13 \).

Answer: 0


Question: Consider the partial order \((P(\omega),\subseteq)\). Let \(L\) be a dense linear suborder. Determine whether \(L\) has a countable dense subset.

Answer: Yes


Question: Is the extension of the functional \( f: \langle t \rangle \to \mathbb{R} \) defined by \( f(\alpha t) = \alpha \) to all of \( C[0,1] \) unique, given that \( ||f||_{\infty} = 1 \) and the extension \( F \) satisfies \( ||F|| = 1 \)?

Answer: Yes


Question: Does there exist a strictly-increasing sequence of positive integers \( A = \{a_n\} \) such that the natural density \( \delta(A) = \lim_{n \to \infty} \frac{A(n)}{n} \) does not exist, where \( A(n) \) is the number of elements of \( A \) not exceeding \( n \)?

Answer: Yes


Question: Find the shortest path between the points \(a = (0,1)\) and \(b = (1,0)\) with respect to the quasi-\(p\)-norm where \(0 < p < 1\).

Answer: 2


Question: Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \), the equation \( f(x) + f(2x) = 0 \) holds.

Answer: 0


Question: Evaluate the infinite product: \(\prod_{n=2}^{+\infty} \frac{n^3-1}{n^3+1}\). Show that it converges to \(\frac{2}{3}\).

Answer: \frac{2}{3}


Question: Evaluate the contour integral \( \int_{|z|=1} \exp(1/z)\sin(1/z)\,dz \) along the circle \(|z|=1\) counterclockwise once.

Answer: 2\pi i


Question: Find the real value of the parameter $C$ for which the improper integral \( \int_0^{\infty} \left( \frac{Cx}{x^2+1} - \frac{1}{2x+1} \right) \, dx \) converges.

Answer: \dfrac{1}{2}


Question: A 3D cake is cut by $n$ planes, resulting in the maximum number of pieces. Given that the maximum number of pieces is defined by the Cake Number $N=\frac{1}{6}(n^3+5n+6)$, determine the number of pieces that have the cake crust as one of their boundaries. Assume the entire surface of the cake is crust.

Answer: n^2 - n + 2


Question: Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \left((\sin(x) + \cos(x))^4 - (\sin(x) - \cos(x))^4\right) \, dx \).

Answer: 4


Question: Determine the radius of convergence of the series \( \sum_{n=1}^{\infty} \sin(\sqrt{n+1} - \sqrt{n})(x-2)^n \).

Answer: 1


Question: Let $X$ be a Polish space and $(\mu_n)_{n \geq 1}$ be a sequence of Borel probability measures on $X$. Assume that for every continuous bounded function $f : X \to \mathbb{R}$, the limit $L(f) := \lim_{n \to \infty} \int_X f \, d\mu_n$ exists. Does it follow that there exists a Borel probability measure $\mu$ such that $L(f) = \int_X f \, d\mu$?

Answer: Yes


Question: Find the integer closest to the integral \( \int_{0}^{6} x^{\{\sqrt{x}\}} \, dx \), where \( \{x\} \) denotes the fractional part of \( x \).

Answer: 10


Question: Solve the transcendental equation for $x$: \[ \frac{a^x}{x} + a^{1/x}x = 2a \] for any given $a$. Verify if $x=1$ is a solution and determine if there are any other solutions.

Answer: 1


Question: Let \(X_1, X_2, \ldots\) be a sequence of independent random variables, each uniformly distributed on \([0, e]\). Define \(R_n = \prod_{k=1}^n X_k\). Determine the probability that the sequence \((R_n)\) converges.

Answer: 0


Question: Let $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}_+$ be a non-negative smooth function such that $f(x,y) = 0$ if and only if $x = y$. Determine whether the following equality holds: \[ \left[\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j}f(x,y)\right]_{x=y} = \left[\frac{\partial}{\partial y_i}\frac{\partial}{\partial y_j}f(x,y)\right]_{y=x}. \]

Answer: Yes


Question:
Consider a sequence of differentiable functions \( f_n \) on \([a,b]\) such that \( f_n(x) \rightarrow f(x) \), where \( f \) is Riemann integrable. Determine which of the following statements is not true:

1. \( f_n \) converges uniformly.
2. \( f_n^{'} \) converges uniformly.
3. \( \int_{a}^{b} f_n \rightarrow \int_{a}^{b} f \).
4. \( f \) is differentiable.

Answer: 4


Question:
Given that \(3\left(a+\frac{1}{a}\right) = 4\left(b+\frac{1}{b}\right) = 5\left(c+\frac{1}{c}\right)\) where \(a, b, c\) are positive numbers and \(ab + bc + ca = 1\), find the value of:
\[
5\left(\frac{1-a^2}{1+a^2} + \frac{1-b^2}{1+b^2} + \frac{1-c^2}{1+c^2}\right)
\]

Answer: 7


Question: Consider a Hilbert space $H$ and let $A$ be a closed unbounded operator, while $B \in B(H)$ is a bounded operator. Define that $A$ and $B$ strong-commute if the partial isometry in the polar decomposition of $A$ commutes with $B$, and all the spectral projections of $|A|$ also commute with $B$. Is it true that if $BA \subset AB$ and $BA^* \subset A^*B$, then $A$ and $B$ strong-commute?

Answer: Yes


Question: Let $p: E \rightarrow B$ be a covering map and $b \in B$. Suppose there exists a neighborhood $U$ of $b$ such that \(p^{-1}(U) = \bigcup V_\alpha\) (disjoint union) and each \(p\restriction_{V_\alpha}: V_\alpha \rightarrow U\) is a homeomorphism. Does it follow that each \(V_\alpha \cap p^{-1}(b) \neq \varnothing\)? Provide a justification for your answer.

Answer: Yes


Question: Suppose J is a Jordan curve in the Euclidean plane E, and there are two perpendicular straight lines in E, each serving as an axis of symmetry for J. Does the intersection point of these two lines necessarily lie within the interior of J?

Answer: Yes


Question: Find the number of involutions in the symmetric group $S_4$. An involution is a permutation that is its own inverse.

Answer: 10


Question: Evaluate the sum \( \sum_{r \ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r} \).

Answer: \dbinom{2n}{n}


Question: Evaluate the limit: $$\lim _{x\to 0}\left(\frac{1-\cos \left(x\right)\sqrt{\cos \left(2x\right)}\cdot \sqrt[3]{\cos \left(3x\right)}}{x^2}\right)$$

Answer: 3


Question: Find the characteristic function of a Laplacian random variable with probability density function \( f(x) = \frac{\alpha}{2} e^{-\alpha |x|} \). Then, use the characteristic function to determine the mean of the Laplacian random variable.

Answer: 0


Question: Find the equation of the plane that passes through the point \((2,2,1)\) and forms a tetrahedron with the smallest volume in the first octant, along with the coordinate planes.

Answer: x + y + 2z = 6


Question: Determine the values of \( k \) for which the second partials test is inconclusive for the function \( f(x, y) \) at the point \((0, 0)\).

Answer: \pm 2


Question: Determine whether the indefinite integral of the Dirac delta function \( \delta(ct-x) \) with respect to \( x \) is \( -\theta(ct-x) \) or \( \theta(x-ct) \), where \( \theta \) is the Heaviside step function, \( x \) and \( t \) are variables, and \( c \) is a constant greater than zero.

Answer: \theta(x - ct)


Question: Determine \( \lim_{t\to \infty}\int_{0}^{\infty}e^{-tx}\frac{\sin x}{x}dx \).

Answer: 0


Question: Determine whether the set \( A = \{ (1,x) \in \mathbb{R}^2 : x \in [2,4] \} \) is bounded, closed, and not compact in the metric space \((\mathbb{R}^2, d_C)\), where the metric \(d_C\) is defined as follows: \[ d_C(x,y) =  \begin{cases} d_E(x,y) & \text{if } x, y, 0 \text{ are collinear} \\ d_E(x,0)+d_E(0,y) & \text{otherwise} \end{cases} \] and \(d_E\) is the Euclidean metric.

Answer: A


Question: Let \(|A| = 5\). How many closed binary operations on \(A\) are commutative? Provide your answer as a power of 5.

Answer: 5^{15}


Question: For which values of \(a \in \mathbb{R}\) does the second-order matrix differential equation \(\frac{d^2}{dt^2}x = \begin{pmatrix}1 & 1 \\ 0 & a\end{pmatrix}x\) have periodic solutions?

Answer: a < 0


Question: Does the embedding functor $J_T$ from the Kleisli category $\operatorname{Kl}(T)$ of a monad $(T,m,e)$ on a category $\mathcal{A}$ into its Eilenberg-Moore category $\operatorname{EM}(T)$ have a right adjoint?

Answer: No


Question: Determine the cardinality of the set \( E = \{a: \exists m \text{ such that } a_n = s_n \forall n \ge m \} \), where \( s = (s_1, s_2, \ldots) \) is an infinite binary word with \( s_p = 1 \) if \( p \) is a prime number.

Answer: \aleph_0


Question:
Find an integer $1 < a < 9991$ such that:
\[
a^{4995} \not\equiv 1 \pmod{9991}, \\
a^{4995} \not\equiv -1 \pmod{9991}, \\
a^{2 \cdot 4995} \not\equiv -1 \pmod{9991}.
\]

Answer: 2


Question:
Consider the set $X = (0,1]$ with two different metrics: $d_1$, the usual metric, and $d_2$, the discrete metric. Determine which of the following statements is true:

- $(X,d_1)$ is compact, but $(X,d_2)$ is not.
- $(X,d_2)$ is compact, but $(X,d_1)$ is not.
- Both $(X,d_1)$ and $(X,d_2)$ are compact.
- Neither $(X,d_1)$ nor $(X,d_2)$ is compact.

Answer: D


Question: Let $(u_n)_n \subset H^1_0(\Omega)$ be a sequence such that $u_n \rightharpoonup 0$ in $H^1_0(\Omega)$, where $\Omega$ is an open bounded subset of $\mathbb{R}^N$. Define $u^+ = \max\{0, u\}$. Determine whether the following statement is true: \[ \int_{\Omega} u^{+}_n \ u_n \, dx \to 0 \quad \text{as} \quad n \to +\infty. \]

Answer: True


Question:
Let \( f \) be a continuous strictly increasing function from \([0,\infty) \rightarrow [0,\infty)\) and \( g = f^{-1} \) (i.e., \( f(x) = y \iff g(y) = x \)). Let \( a, b > 0 \) and \( a \neq b \). Evaluate the expression \( \int_{0}^{a} f(x) \, dx + \int_{0}^{b} g(y) \, dy \) and determine whether it is:

(A) greater than or equal to \( ab \)
(B) less than \( ab \)
(C) always equal to \( ab \)

Answer: A


Question: Determine the minimum number of terms needed in the Taylor series expansion of \( f(x) = \ln(x) \) around \( x = e \) to ensure that the error is less than \( 0.05 \) when \( |x-e| < 1 \). Use the error term \( R_{n+1}(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-e)^{n+1} \) and assume \( x = e-1 \) for the worst-case scenario.

Answer: 4


Question: Evaluate the limit \( \lim_{(x,y)\to(0,2)} (1+x)^{y/x} \).

Answer: e^2


Question: Compute the conjugacy classes of subgroups of order 2 in the general linear group GL_n(\(\mathbb{C}\)), where \(n\) is a positive integer.

Answer: n


Question: Determine the completion of the metric space \(c_{00}\), defined as \(c_{00} = \{(x_n) \in \mathbb{R}: x_n=0, n>n_0\}\), with the metric induced from \(l_{\infty}\).

Answer: c_0


Question: Evaluate whether the integral \( \int_{\mathbb{R}^2} \frac{1}{\sqrt{x^2 + y^4}} \, dx \, dy \) is finite.

Answer: \infty


Question: Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), the equation \( f((x + 1) f(y)) = y (f(x) + 1) \) holds.

Answer: f(x) = x


Question: Given the sequence \(\{x_n\}\) with \(\lim_{n \to \infty} x_n = a\), and the line \(AA_n\) defined by the points \(A = (a, a^2)\) and \(A_n = (x_n, x_n^2)\), let \(a_n\) be the abscissa of the intersection point of \(AA_n\) with the x-axis. Find \(\lim_{n \to \infty} a_n\).

Answer: \dfrac{a}{2}


Question: Find the limit: \[ \lim_{n \to \infty} \frac{(-2)^n n!}{n^n} \]

Answer: 0


Question:
Evaluate the limit of the Lebesgue integrals:
\[ \lim_{n\to \infty} \int_{[0,\infty]} \frac{n\sin\left(\frac{x}{n}\right)}{x(1+x^2)}\,dm \] where \( m \) is the Lebesgue measure on \( \mathbb{R} \).

Answer: \dfrac{\pi}{2}


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

Answer: 1


Question: Evaluate the limit \( \lim_{x\to \infty}\frac{1}{x^2}\int_0^x \frac{1}{3+\cos t} \, dt \).

Answer: 0


Question: Let \((X, \tau)\) be a countable Hausdorff space. Does there exist a topology \(\sigma \subset \tau\) such that \((X, \sigma)\) is a second countable Hausdorff space?

Answer: Yes


Question: Evaluate the limit \( L = \lim_{n\to\infty} \left(n-\sum_{k=1}^n e^{\frac{k}{n^2}}\right) \).

Answer: -\dfrac{1}{2}


Question: Determine the range of \( \alpha \in \mathbb{R} \) for which the following statement holds: If a sequence \( \{ x_n \}_{n=1}^{\infty} \) of real numbers satisfies \( \lim_{n \to \infty}{ n^\alpha \,\, |x_n - x_{n+1}|}=0 \), then it is a convergent sequence.

Answer: (1, \infty)


Question: Let \( P(x) = \prod_{k = 1}^{50} (x - k) \) and \( Q(x) = \prod_{k = 1}^{50} (x + k) \). If \( P(x) Q(x) = \sum_{k = 0}^{100} a_k x^k \), find the value of \( a_{100} - a_{99} - a_{98} - a_{97} \).

Answer: 42926


Question: Let $x_1, x_2, \ldots, x_n$ be independent binomial random variables, each with parameters $p$ and $n$. Determine the probability $P[\max(x_1, x_2, \ldots, x_n) < n]$. Express your answer in terms of $p$ and $n$.

Answer: (1 - p^n)^n


Question: Evaluate the limit: $$\lim_{x\rightarrow 0^+}\frac{\exp \left[-\alpha\left( \frac{1}{x^{\beta}} -1\right) \right] }{x^{1+\beta}}$$ where $ \alpha>0 $ and $ \beta>0 $ are constants.

Answer: 0


Question: Find the splitting field of the polynomial $x^4 + x^3 + 1$ over the finite field $\mathbb{F}_{32}$. Determine the degree of the extension field over $\mathbb{F}_{32}$ that contains all the roots of the polynomial.

Answer: 4


Question: Let $f \in C^1( \mathbb{B}, \mathbb{R}^k)$ be a function such that there exists a positive value $\beta$ with \( \lvert Df(0)h \rvert \ge \beta \lvert h \rvert \) for all \( h \in \mathbb{R}^n \). Define the function \( P: \mathbb{B} \rightarrow \mathbb{R}^k \) by \( P(x) = f(x) - Df(0)x \). Calculate \( DP(0) \).

Answer: 0


Question: Let $A$ be any $n \times n$ diagonalizable matrix. Is $p(A)$ diagonalizable for any polynomial $p(x)$?

Answer: Yes


Question: Compute the residue of the function \( \frac{e^{1/z}}{1-z} \) at the point \( z_0 = 0 \).

Answer: e - 1


Question: Compute the integral \( \int_0^{2\pi} -e^{\cos(t)}[\sin(t)\cdot \cos(\sin(t))+\cos(t)\cdot \sin(\sin(t))] \,dt \).

Answer: 0


Question: Let \((X, \mathcal{T})\) and \((Y, \mathcal{S})\) be completely regular topological spaces. Suppose \((X, \mathcal{T})\) is compact and \(X \cap Y\) is dense in both \(X\) and \(Y\). Can we deduce that \((Y, \mathcal{S})\) is compact?

Answer: No


Question: Let \( f: \mathbb{R} \to (0, \infty) \) be a continuous function such that \( \lim_{x\to \infty}\frac{f(x)}{f(x+1)}=0 \). Calculate \( \lim_{n\to \infty} \frac{1}{n} \int_0^n f(x) \, dx \).

Answer: \infty


Question: Find the radius of convergence of the function \( \frac{z+1}{z^2+z+1} \) about the point \( z=3 \).

Answer: \sqrt{13}


Question: Consider the topology on \(\mathbb{R}\) generated by the basis \(\{[a, b) : a, b \in \mathbb{Q}\}\). What is the closure of the set \((0, \sqrt{2})\) in this topology?

Answer: [0, \sqrt{2}]


Question: A positive integer n does not contain any 9s, has four 8s, three 7s, two 6s, and some other digits. If the sum of the digits of n is 104 and the sum of the digits of 2n is 100, determine how many times the digit 5 appears in n.

Answer: 3


Question: Let $X$ and $Y$ be affine varieties, and let $\varphi : X \to Y$ be a dominant morphism. Is it true that the induced map on coordinate rings $\varphi^* : A(Y) \to A(X)$ is injective?

Answer: Yes


Question: Evaluate the line integral \( \oint\limits_{C}(\sin(y)\,dx + x\cos(y)\,dy) \), where \( C \) consists of the circle \( x^2 + y^2 = 2 \) from \((-1,1)\) to \((1,1)\) and the line segment from \((1,1)\) to \((-1,1)\).

Answer: 0


Question: If the equations \( ax^3+3bx^2+3cx+d=0 \) and \( ax^2 +2bx+c=0 \) have a common root, find the value of \( \frac{(bc-ad)^2}{(ac-b^2)(bd-c^2)} \).

Answer: 4


Question: Let $a$, $b$, and $c$ be positive real numbers such that $a + b + 2c = 10$ and $a^2 + b^2 + 2c^2 = 30$. The largest possible value of $c$ can be expressed as $\frac{k + \sqrt{w}}{f}$, where $k$, $w$, and $f$ are positive integers. Determine the value of $k + w + f$. \(c = d\) is given, but it is redundant since $c$ is already defined.

Answer: 12


Question: Consider a finite extension of local fields $K/F$ where both fields have characteristic 0. Determine whether the quotient group $K^\times / F^\times$ is always compact.

Answer: Yes


Question: Use a double integral to find the volume of the solid bounded by the graphs of the equations: \( z = xy^2 \) where \( z > 0 \), \( x > 0 \), and \( 5x < y < 2 \). Determine the appropriate limits of integration and calculate the volume.

Answer: \dfrac{16}{125}


Question: Compute the limit: \[ \lim\limits_{x \to +\infty}\dfrac{\ln x}{\displaystyle \int_0^x \dfrac{|\sin t|}{t}{\rm d}t}. \]

Answer: \dfrac{\pi}{2}


Question: Find the minimum value of the function \( f(a,b) = (a-b)^2 + \left(\frac{a^2}{20} - \sqrt{(17-b)(b-13)}\right)^2 \), where \( a \in \mathbb{R}^+ \) and \( b \in (13, 17) \).

Answer: 54 - 20\sqrt{2}


Question: Compute the limit: $$\lim_{x \to 0} \frac{\cos x - \sqrt{2 - e^{x^2}}}{\ln{(\cos x) + \frac{1}{2} x \sin x}} \cdot \frac{(x+2)^{2017}}{(x-2)^{2015}}$$

Answer: 10


Question: Evaluate the limit: \[ \lim_{n\to\infty} \int_0^1 \frac{e^{x^{n}}}{1+x^2}\,\mathrm{d}x \]

Answer: \dfrac{\pi}{4}


Question: Assume the Continuum Hypothesis and the Axiom of Choice. Let \( S \) be the smallest uncountable ordinal number. Define \( C(S) \) as the set of all mappings from \( S \) into the continuum such that the set of elements in \( S \) not mapped to zero is at most countable. Is the cardinality of \( C(S) \) equal to the cardinality of \( S \)?

Answer: Yes


Question: Find the smallest natural number \( n \) such that \( 2^n \equiv 111 \pmod{125} \).

Answer: 36


Question: Calculate the expected value \( E(\Phi(X-1)) \) where \( X \) is a normally distributed random variable with mean \( \mu = 1 \) and variance \( \sigma^2 = 1 \).

Answer: \dfrac{1}{2}


Question: Evaluate the double integral \( \iint_\Omega \frac{1}{4}(y^2-x^2) \, dA \) using an appropriate change of variables, where \(\Omega\) is the triangle with vertices \((0,0), (2,2), (1,3)\).

Answer: 1


Question:
Let $f:[0,4]\to \mathbb{R}$ be a three times continuously differentiable function. Calculate the divided difference $f[1,2,3,4]$. Choose the correct option:

(a) $\frac{f'(\xi)}{3}$, for some $\xi \in (0,4)$

(b) $\frac{f'(\xi)}{6}$, for some $\xi \in (0,4)$

(c) $\frac{f''(\xi)}{3}$, for some $\xi \in (0,4)$

(d) $\frac{f''(\xi)}{6}$, for some $\xi \in (0,4)$.

Answer: D


Question: Find an element $m \in U(16)$ such that $|m| = 4$ and $|\langle m \rangle \cap \langle 3 \rangle| = 2$. The operation is defined as $m*n = mn \pmod{16}$. Determine if $m$ is unique.

Answer: 5


Question: What is the remainder when \(3^{2015} + 11^{2014}\) is divided by 61?

Answer: 59


Question: Evaluate the limit \( \lim_{n \to \infty} n^2 \cdot x^n \) where \( 0 < x < 1 \).

Answer: 0


Question: Find a function $g(n)$ such that $f(n) = \Omega(g(n))$, where $f(n) = \frac{\sum_{j=1}^{n/2}\binom{n}{2j}\log(2j)}{2^{n-1}}$ and $n$ is an even number.

Answer: \log n


Question: Determine whether the radius of convergence of the power series \( (1+x)^a = \sum_{n=0}^{\infty} \binom{a}{n} x^n = \sum_{n=0}^{\infty} \frac{a(a-1)(a-2) \cdots (a-n+1)}{n!}x^n \) can be equal to 1. If so, under what conditions does this occur?

Answer: a \notin \mathbb{N} \cup \{0\}


Question: Does there exist a function \( p:2^{\Bbb N}\rightarrow [0,1] \) such that \( p \) is finitely additive, i.e., \( p(A_1\cup\ldots\cup A_n)=p(A_1)+\ldots +p(A_n) \) for all pairwise disjoint subsets \( A_i \) of \( \Bbb N \) and some \( n\in\Bbb N \), but \( p \) is not countably additive? Additionally, \( p \) satisfies \( p(\Bbb N)=1 \) and \( p(\emptyset)=0 \).

Answer: Yes


Question: Find the value of the infinite series: \[ \sum_{k=1}^{\infty}\frac{k^2}{k!} \]

Answer: 2e


Question:
Evaluate the limit: \[ \lim_{n\to\infty}\frac{2^{-n^2}}{ \sum_{k=n+1}^{\infty} \frac{1}{2^{k^2}}} \] Choose the correct answer from the following options:

(a) 0
(b) Some \( c \in (0,1) \)
(c) 1
(d) \( \infty \)

Answer: D


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

Answer: \dfrac{1}{2}


Question: Find the positive integer $x$ such that both of the following congruences are satisfied: \(2^{x+1} - 2 \equiv 0 \pmod{29}\) and \(2^{x+1} - 4 \equiv 0 \pmod{28}\).

Answer: 28


Question: Let \(p(x)\) be a polynomial of degree strictly less than 100 such that it does not have \(x^3-x\) as a factor. If \(\frac{d^{100}}{dx^{100}} \left(\frac{p(x)}{x^3-x}\right) = \frac{f(x)}{g(x)}\) for some polynomials \(f(x)\) and \(g(x)\), find the smallest possible degree of \(f(x)\).

Answer: 200


Question: Determine whether there exist any integral solutions for the equation \(x^2 - 2 = y^p\) where \(p \geq 3\).

Answer: Yes


Question:
Given that \( f \) is a periodic function and \( g \) is a polynomial function such that \( f(g(x)) \) is periodic, with \( g(2) = 3 \) and \( g(4) = 7 \), find \( g(6) \).

A) 13
B) 15
C) 11
D) none of these

Answer: C


Question: Determine whether the equation $a^2 + b^2 + c^2 = d^2$ has solutions in positive integers $(a, b, c, d)$. If so, provide an example of such a solution.

Answer: (1, 2, 2, 3)


Question: Given the system of differential equations \( x' = xy + t^2 \) and \( y' = \frac{-y^2}{2} \), with initial conditions \( x(1) = 3 \) and \( y(1) = 2 \), find \( \frac{d x}{d y_0} \) at \( x_0 = 3 \) and \( y_0 = 2 \).

Answer: 0


Question: Calculate the expected number of dice throws needed to roll a 6, given that all previous rolls resulted in even numbers (2 or 4). Use the concept of fixed point updating to determine the expected value.

Answer: 3


Question: Evaluate the limit: \[ \lim_{z \rightarrow i} \left| \frac{z^3 - i}{z^6 + 1} \right| \]

Answer: \infty


Question: Suppose $f: \mathbb{C} \times (-1,1) \to \mathbb{C}$ is a smooth function satisfying $f(0,t)=1$ for all $t \in (-1,1)$, and for any $k \in \mathbb{N}$, $z \in \mathbb{C}$, and $t \in (-1,1)$, we have \( \frac{\partial^k f}{\partial t^k}(z,0)=z^k. \) Determine whether there exist constants $c, \delta > 0$ such that \( e^{-c|z|} \leq |f(z,t)| \leq e^{c|z|}, \) for all $|t| \leq \delta$ and all $z \in \mathbb{C}$. Provide a justification for your answer.

Answer: No


Question: Find the integer $M$ such that there exists a number $N$ with $N^2 = 10^4M + N$. Here, $N$ is a four-digit number represented as $N = 1000a + 100b + 10c + d$, where $a, b, c, d$ are digits. Determine the value of $M$.

Answer: 8790


Question: Find the maximum value of \( u_x(0) \) for a harmonic function \( u: D \rightarrow [0,1] \), where \( D \subset \mathbb{R}^2 \) is the open unit disk centered at 0.

Answer: \dfrac{2}{\pi}


Question: Suppose $M$ is a random variable following a Poisson distribution with parameter $\lambda$. For each of the $M$ objects, you decide to keep it with probability $p$. Let $N$ be the total number of objects you keep. Given that the expectation of $N$ is $\lambda p$, find the variance of $N$. Assume $M \sim \mathrm{Poisson}(\lambda)$ and each object is kept independently with probability $p$. What is the variance of $N$?

Answer: \lambda p


Question: Given that the causal discrete-time signal $x[n]$ satisfies the equation $x[n] \ast x[n] = u[n]$, where $\ast$ denotes the convolution operator and $u[n]$ is the discrete unit step function, and knowing that $x[0] > 0$, determine the value of $x[3]$. Provide your answer as a single number.

Answer: \dfrac{5}{16}


Question: Compute the limit: $$\lim_{x \to 0^+} \left( 1 + \frac{\ln(x)}{x^2} \right)$$

Answer: -\infty


Question: Evaluate the contour integral \( \int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz \), where \( \varGamma \) is the contour \( |z|=4\pi/3 \).

Answer: -6\pi i


Question: Determine the interval of convergence for the series: \[ x + \frac{1}{2} x^2 + 3x^3 + \frac{1}{4}x^4 + \cdots \]

Answer: (-1, 1)


Question: Find the smallest positive integer \( m \) such that \( \binom{2015}{m} \) is an even number.

Answer: 32


Question: Determine the residue of the function \( f(z) = \frac{\tan z}{z} \) at the point \( z_0 = 0 \).

Answer: 0


Question: Given the function \( w = 5 \cos(xy) - \sin(xz) \) where \( x = \frac{1}{t} \), \( y = t \), and \( z = t^3 \), find \( \frac{dw}{dt} \) using the chain rule.

Answer: -2t \cos(t^2)


Question: Let $p$ be a prime number. Show that if $p^2$ divides $3^{2n+1} + 61$, then $p$ must be equal to 2.

Answer: 2


Question: Find integers \(a\) and \(b\) such that for all integers \(c\) and \(d\), the complex number \(ac-bd+i(ad+bc)\) has both its real and imaginary parts either even or both odd.

Answer: a \equiv b \pmod{2}


Question:
Evaluate the integral to find the volume:
\[ V = 4 \int_0^2 (4-x^2) \sqrt{4-x^2} \, dx \]

Answer: 12\pi


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

Answer: 0


Question: Find the first-variational curve corresponding to the functional \( \int_{-1}^1 t^2 \dot{x}^2 \, dt \) with the boundary conditions \( x(-1) = -1 \) and \( x(1) = 1 \).

Answer: \dfrac{1}{t}


Question: Determine the number of nontrivial cyclic subgroups in the group \( \Bbb Z_8 \oplus \Bbb Z_4 \oplus \Bbb Z_2 \).

Answer: 27


Question: Determine the values of $x$ for which the series \( \sum_{n=1}^\infty \left(1+\frac{1}{2}+ \frac{1}{3}+\cdots+\frac{1}{n}\right) x^n \) converges.

Answer: (-1, 1)


Question: Determine whether the limit \( \lim_{x\to a^+}\frac{\cos{x}\ln(x-a)}{\ln(e^x-e^a)} \) exists.

Answer: \cos a


Question: Given the joint probability density function of \( X \) and \( Y \), \( f(x, y) = \frac{1}{2}e^{-x} \) for \( x \geq 0 \) and \( |y| < x \), and \( f(x, y) = 0 \) elsewhere, calculate the conditional variance \( \mathbb{V}(Y|X = x) \).

Answer: \dfrac{x^2}{3}


Question: A particle encounters a series of N identical obstacles, each with a probability D of being overcome. What is the probability that the particle will successfully overcome all N obstacles in sequence?

Answer: D^N


Question: Is the identity matrix an example of a matrix that is both stochastic and unitary?

Answer: Yes


Question: Find all natural numbers $x$ such that $2^x \equiv 1 \pmod{x}$. Provide a complete solution.

Answer: 1


Question: Evaluate the integral \( \int_0^a \frac{1}{x + \sqrt{a^2-x^2}} \, dx \), where \( a \) is a positive constant.

Answer: \dfrac{\pi}{4}


Question: Evaluate the integral \( \int_{|z|=1} \frac{1}{e^z -1-2z}dz \) using Cauchy's residue theorem.

Answer: -2\pi i


Question: Determine whether the following statement is true or false: If \( 0 \leq x_n \leq y_n \xrightarrow{p} 0,\) then \(x_n \xrightarrow{p} 0.\)

Answer: True


Question:
Consider a sequence of functions $(f_n)$ on a probability space $(X,\mu)$ such that:

1. $f_n \to f$ in $L^p$ for every $p \in [1,\infty)$.
2. $f, f_1, f_2, \ldots \in L^\infty$ and they are uniformly bounded in $L^\infty$ by $\Lambda \in [0,\infty)$, i.e., $\|f\|_\infty, \|f_1\|_\infty, \|f_2\|_\infty, \ldots \le \Lambda$.

Does $f_n \to f$ in $L^\infty$?

Answer: No


Question: Consider a continuous and nonconstant function \( f: \mathbb{R} \to \mathbb{R} \) with a positive real number \( p \) such that \( f(x+p) = f(x) \) for all \( x \in \mathbb{R} \). Determine whether there exists an \( n \in \mathbb{N} \) such that \( \frac{p}{n} = \min\{a > 0 \mid f(x+a) = f(x), \forall x \in \mathbb{R}\} \). Is this statement true?

Answer: True


Question: Determine the number of linear functions from \( \{0, 1\}^n \) to \( \{0, 1\} \). A Boolean function \( F: \{0, 1\}^n \rightarrow \{0, 1\} \) is said to be linear if \( F(x \oplus y) = F(x) \oplus F(y) \) for all \( x, y \in \{0, 1\}^n \), where \( x \oplus y \) denotes the component-wise exclusive-or of \( x \) and \( y \).

Answer: 2^n


Question: Determine the asymptotic behavior of the function \(a(n) = \sum_{i=1}^n d(i)\), where \(d(n)\) is the number of divisors of the positive integer \(n\).

Answer: n \log n + (2\gamma - 1)n


Question: Let $K(\alpha)/K$ be a finite separable extension of fields, and let $\bar{K}$ be the algebraic closure of $K$. Determine the embeddings $\beta:K(\alpha)\hookrightarrow \bar{K}$ that fix $K$, meaning $\beta(k) = k$ for all $k \in K$. How many such embeddings exist?

Answer: [K(\alpha) : K]


Question: Find all functions \( f: \mathbb{N}^+ \to \mathbb{N}^+ \) such that for any positive integers \( x \neq y \), the inequality \( 0 < |f(x) - f(y)| < 2|x-y| \) holds.

Answer: f(x) = x + c


Question:
Evaluate the expression \( \sum_{i,j=0, i \neq j}^n \binom{n}{i} \binom{n}{j} \). Choose the correct answer from the following options:

1. \( 2^{2n} - \binom{2n}{n} \)
2. \( 2^{2n-1} - \binom{2n-1}{n-1} \)
3. \( 2^{2n} - \frac{1}{2}\binom{2n}{n} \)
4. \( 2^{n-1} - \binom{2n-1}{n} \)

Answer: 1


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{\log P(Z_1 + Z_2 + \cdots + Z_n > n)}{n} \] where \((Z_n)_{n \ge 1}\) is a sequence of i.i.d. standard normal random variables.

Answer: -\dfrac{1}{2}


Question: Determine the Zariski closure of the set \( X = \{(x, \sin(x)): x \in \mathbb{A}^{1}\} \) in \( \mathbb{A}^{2} \).

Answer: \mathbb{A}^{2}


Question: Suppose $(P, \leq)$ is a partially ordered set that is interval-isomorphic, meaning for all $a < b \in P$, we have $P \cong [a,b]$, where $[a,b] = \{x \in P : a \leq x \leq b\}$. If there exist $a, b \in P$ with $a < b$, does this imply that $(P, \leq)$ is a lattice? Answer yes or no.

Answer: Yes


Question: Evaluate the value of the series \(\sum_{k=0}^\infty \frac{2^k}{5^{2^k}+1}\).

Answer: \dfrac{1}{4}


Question: Let \( f \) be a holomorphic function such that \( |f(z)| > 1 \) for \( |z| = 1 \). Evaluate the limit: \[ \lim_{n\to \infty}\left|\frac{\prod_{z^n=1}f(z)^n}{\prod_{z^n=-1}f(z)^n}\right|. \]

Answer: 1


Question: Consider a simplex $S \subset \mathbb{R}^n$ formed by unit vectors $v_1, v_2, \ldots, v_{n+1}$, where $v_i \in \mathbb{R}^n$. Fix $v_{n+1}$ and consider the other vectors. The volume of the simplex is given by $\frac{|\text{det}(L)|}{n!}$, where $L=[v_1-v_{n+1}, v_2-v_{n+1}, \ldots, v_n-v_{n+1}]$. Choose $v_i, v_j$ such that $\|(v_i-v_j)\|_2$ is maximum and $v_i, v_j \neq v_{n+1}$. Define $u=0.5(v_i+v_j)$. Construct two new simplexes $S_1$ and $S_2$ from $S$ such that $S_1$ contains $u$ instead of $v_i$ and $S_2$ contains $u$ instead of $v_j$. Determine an upper bound for the ratio $\frac{\text{vol}(S_1)}{\text{vol}(S)}$. Provide your answer as a single expression or value.

Answer: \dfrac{1}{2}


Question: Let $X\subset\mathbb{P}^N$ be a rational smooth projective irreducible non-degenerated variety of dimension $n=\dim(X)$. Consider the linear system of hyperplane sections \(\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^2}|\) that is singular at points $p_1,\dots,p_{l}$. Assume $\dim(\mathcal{H})=n$ and that a general divisor $H\in \mathcal{H}$ is singular along a positive dimensional subvariety passing through these points. Additionally, assume the schematic intersection of all these singularities is singular only at $p_1,\dots,p_l$. Locally, near $p_1$, $H$ can be expressed as the zero locus of \(H_f=a_{i,j}x_ix_j+h(x_1,\dots,x_n)=0\), where $x_1,\dots,x_n$ are local coordinates and $h(x_1,\dots,x_n)$ is a polynomial of degree $\geq3$. The quadric \(Q_{H_f}=a_{i,j} x_ix_j=0\) generally has rank $h\leq n$. Denote by $\mathcal{A}_H$ the vertex of $Q_{H_f}$. Prove or disprove that under these assumptions, $\bigcap_{H\in \mathcal{H}} \mathcal{A}_H=p_1$.

Answer: \bigcap_{H\in \mathcal{H}} \mathcal{A}_H = p_1


Question: Let $A$ and $B$ be Hausdorff spaces. If $f: A \to B$ is a continuous function, is $f: (A, \mathcal{B}_A) \to (B, \mathcal{B}_B)$ necessarily measurable with respect to the Borel algebras $\mathcal{B}_A$ and $\mathcal{B}_B$?

Answer: Yes


Question: Let \( F: \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear transformation defined by \( F(1, 0, 0) = (1, 1, -1) \), \( F(1, 1, 0) = (2, -1, 0) \), and \( F(1, 1, 1) = (3, 0, -1) \). Let \( G: \mathbb{R}^3 \to \mathbb{R}^3 \) be the orthogonal projection onto the line through \((1, 2, a)\) for \( a \in \mathbb{R} \). Find the value of \( a \) for which \( G \circ F = 0 \), meaning \( G \circ F(x, y, z) = 0 \) for all \((x, y, z) \in \mathbb{R}^3\).

Answer: 3


Question: Find the smallest positive integer \( n \) such that \( 2001^n - 1 \) is divisible by \( 2^{2002} \).

Answer: 2^{1998}


Question: Suppose \( A \) is an \( n \times n \) real matrix with \( \det(A) < 0 \). Determine whether there exists a positive number \( \delta \) such that \( \|Ax\| \geq \delta \|x\| \) for all \( x \in \mathbb{R}^n \).

Answer: Yes


Question:
Calculate the variance of the random variable $X$ given the following conditions:

1. $E[X \mid Y=1] = 2 + E[X]$
2. $E[X \mid Y=2] = 3 + E[X]$
3. $E[X \mid Y=3] = 0$
4. $P(Y=1) = 0.5$, $P(Y=2) = 0.3$, and $P(Y=3) = 0.2$.

Answer: \dfrac{91}{4}


Question:
Does there exist a group $G$ with subgroups $H$ and $F$ such that:

1. There is no injection $i: F \to H$ or $i: H \to F$.
2. $N_G(H)/H$ and $N_G(F)/F$ are both abelian.
3. $N_G(F \cap H)/F \cap H$ is not abelian?

Determine whether such a group exists.

Answer: Yes


Question:
Consider the function \( f:[-1,1] \to \mathbb{R} \) defined by
\[
f(x)=\begin{cases} \sin x, & x \in [-1,1] \cap \mathbb{Q} \\
0, & x \notin [-1,1] \cap \mathbb{Q}
\end{cases}
\]
Determine the correct statement about the function \( f \) at \( x=0 \):

(A) \( f \) is continuous and Riemann integrable in every interval \((a,b)\) containing \(0\).

(B) \( f \) is continuous and Riemann integrable in exactly one interval \((a,b)\) containing \(0\).

(C) \( f \) is continuous but not Riemann integrable in any interval containing \(0\).

(D) \( f \) is discontinuous.

Answer: C


Question: Evaluate the integral \( \int_{\gamma(0;2)} \frac{e^{i\pi z/2}}{z^2-1} \, dz \) using residue calculus.

Answer: -2\pi


Question: Find the integer part of the sum: \( \sum_{i=2}^{9999} \frac{1}{\sqrt{i}} \).

Answer: 197


Question: Write the Laurent series expansion of the function \( f(z) = \frac{\cos(z^2)}{z^7} \) in the region \( 0 < |z| < \infty \). Find the residue of this function at \( z = 0 \).

Answer: 0


Question: Evaluate the expression \( \left[ \sum \limits^{2006}_{k=1}\left( -1\right) ^{k}\frac{k^{2}- 3}{\left( k+1\right) ! } \right] -1 \).

Answer: \dfrac{2005}{2007!}


Question: Find all analytic functions \( f: \mathbb{C} - \{0\} \to \mathbb{C} \) such that \( f(z) = f(2z) \) for all \( z \in \mathbb{C} - \{0\} \).

Answer: f(z) = c