Question: Determine the number of ordered quadruples \((a, b, c, d)\) of positive integers such that \(ab + bc + cd + da = 2016\).

Answer: 59508


Question: Find the integer value of $x$ such that \( \sqrt{x+\frac{1}{2}\sqrt{2011}} - \sqrt{x-\frac{1}{2}\sqrt{2011}} = y \), where \( x, y \in \mathbb{Z} \).

Answer: 503


Question: Consider a simply connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Let $Rep(G)$ denote the category of finite-dimensional representations of $G$, and $Rep(\mathfrak{g})$ the category of finite-dimensional representations of $\mathfrak{g}$. The functor $$d: Rep(G) \rightarrow Rep(\mathfrak{g}), \pi \mapsto d_e\pi$$ maps a representation of $G$ to a representation of $\mathfrak{g}$ by taking the derivative at the identity element. On morphisms, $d$ acts as the identity. Is the functor $d$ an isomorphism of categories, meaning it is bijective on both objects and morphisms?

Answer: Yes


Question: Evaluate the integral \( \int_C{\omega} \) where \( \omega=\frac{dx \wedge dy}{x^2+y^2} \) and \( C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - \text{{(0,0)}} \).

Answer: 2\pi \ln 2


Question: Evaluate the line integral \( \oint_{C} \left(-\frac{x^2y}{(x^2+y^2)^2}\,dx + \frac{x^3}{(x^2+y^2)^2}\,dy\right) \) where \( C \) is the boundary of the region defined by \( 4x^2 + y^2 = 1 \). Consider the implications of the singularity at the origin on the validity of Green's Theorem for this problem.

Answer: \pi


Question:
Evaluate the sum \( \sum_{n=-\infty}^\infty f(n) \), where for any integer \( n \), \( k(n) = \frac{n^7}{7} + \frac{n^3}{3} + \frac{11n}{21} + 1 \) and
\[
f(n) = \begin{cases}
0 & \text{if } k(n) \text{ is an integer} \\
\frac{1}{n^2} & \text{if } k(n) \text{ is not an integer}
\end{cases}
\]

Answer: 0


Question: Does there exist an inner product space $X$ such that for some orthonormal sequence $(e_k)$ in $X$, the inequality \( \sum_{k=1}^{\infty} |\langle x,e_k \rangle|^2 < \|x\|^2 \) holds for some $x \in X$?

Answer: Yes


Question: Determine the VC-dimension of the class of parity functions defined on the boolean hypercube $\{0,1\}^N$. The class is given by $H_{N-parity} = \{h_I:I\subseteq \{1,2,..., N\}\}$, where for a binary vector $x = (x_1, x_2, ...,x_N) \in \{0,1\}^N$, the parity function $h_I(x) = \bigg(\sum_{i\in I}x_i\bigg) \mod 2$. Provide your answer as a single integer.

Answer: N


Question: Let $k$ be a positive integer and define the set $$S_{k} = \{x \in [0, 1] \mid \text{a decimal expansion of $x$ has a prime digit at its $k$-th place}\}.$$ Determine the Lebesgue measure of $S_{k}$.

Answer: \dfrac{2}{5}


Question:
Compute the determinant of the following matrix:
\[
\begin{vmatrix}
\sin(2x)&\sin(3x)&\sin(4x)\\
\sin(3x)&\sin(4x)&\sin(5x)\\
\sin(4x)&\sin(5x)&\sin(6x)
\end{vmatrix}
\]

Answer: 0


Question:
Let \(T:\mathbb{R}^4\to\mathbb{R}^4\) be a linear transformation. Determine which of the following statements is true about the eigenvalues of \(T\):

(A) \(T\) must have some real eigenvalues which may be less than 4 in number.
(B) \(T\) may not have any real eigenvalues at all.
(C) \(T\) must have infinitely many real eigenvalues.
(D) \(T\) must have exactly 4 real eigenvalues.

Answer: B


Question: Let $X$ be a locally compact Hausdorff space. Is the dual of the space of continuous functions vanishing at infinity, $C_0(X)$, the same as the dual of the space of continuous functions with compact support, $C_b(X)$?

Answer: No


Question:
Compute the joint cumulative distribution function (CDF) for the random variable $X = Y_1 - Y_2$, given the joint probability density function (PDF):
\[
f(y_1, y_2) =
\begin{cases}
e^{-y_1}, &\quad 0 \leq y_2 \leq y_1 < +\infty \\
0, &\quad \text{otherwise}
\end{cases}
\]

Answer: 1 - e^{-x}


Question: Find the Jacobian factor $J$ when changing variables from $z \in S^2$ to $(a, v^{\perp})$, where $z = \langle z, v \rangle v + \sqrt{1 - \langle z, v \rangle^2} v^{\perp}$, $a = \langle z, v \rangle$, and $v^{\perp} \in S^2$ is orthogonal to $v$. Express $dz$ in terms of $dadv^{\perp}$ as $dz = J(a, v^{\perp}) dadv^{\perp}$. Determine the expression for $J$. \(v \in S^2\) is a fixed vector.

Answer: 1


Question:
Given \( \int f(x)\sin x\cos x\,dx = \frac{1}{2(a^2-b^2)}\ln \left|f(x)\right|+\mathcal{C} \), determine the function \( f(x) \) from the following options:

\(\textbf{Options:}\)
\((a)\;\frac{1}{a^2\sin^2 x+b^2 \cos^2 x} \quad (b)\; \frac{1}{a^2\sin^2 x-b^2 \cos^2 x}\quad(c)\; \frac{1}{a^2\cos^2 x-b^2 \sin^2 x}\)
\((d)\; \frac{1}{a^2\cos^2 x+b^2 \sin^2 x}\)

Answer: D


Question: Find the number of positive integer solutions to the equation \( x_1 + x_2 + x_3 + x_4 + x_5 = 100 \) subject to the constraints \( x_1 \le 30 \), \( x_2 \le 40 \), \( x_3 \le 50 \), \( x_4 \le 60 \), and \( x_5 \le 70 \).

Answer: 2154750


Question: Find a non-smooth function \( f : \mathbb{R} \to \mathbb{R} \) such that \( f^3 \) is smooth.

Answer: f(x) = \sqrt[3]{x}


Question: What is the probability of placing 8 rooks on an 8x8 chessboard such that no two rooks can attack each other, given that one specific square (7th column, 7th row) is unavailable? Provide your answer as a fraction.

Answer: \dfrac{7}{8}


Question: Determine whether the following statement is true or false: If $M$ is a smooth manifold and $p \in M$, then $M - \{p\}$ admits a non-complete global smooth field.

Answer: A


Question: Find the area of the portion of the sphere \( x^2 + y^2 + z^2 = 1 \) that lies between the two parallel planes \( z = a \) and \( z = b \), where \( -1 < a < b < 1 \).

Answer: 2\pi(b - a)


Question: The polynomial \(x^3 + ax^2 + bx + c\) has three roots \(\alpha \leq \beta \leq \gamma\), all of which are positive integers. Given that \(2^2(a) + 2^1(b) + 2^0(c) = -2^{11}\), find the sum of all possible values of \(\gamma\).

Answer: 6458


Question:
If \( n \) is a positive integer such that the sum of all positive integers \( a \) satisfying \( 1 \le a \le n \) and \( \gcd(a,n)=1 \) is equal to \( 240n \), what is the value of \( \phi(n) \)?

\(\textbf{(A) } 120 \quad \textbf{(B) } 124 \quad \textbf{(C) } 240 \quad \textbf{(D) } 480\)

Answer: D


Question: Determine the radius of convergence of the power series \( \sum_{n \ge 1} a_n z^n \), where \( a_n \) is the number of divisors of \( n^{50} \).

Answer: 1


Question: Given the recurrence relation \( a_0 = 2 \), \( a_1 = 16 \), and \( a_{n+1}^2 = a_n a_{n-1} \), find the limit of the sequence \( \{a_n\} \) as \( n \to \infty \).

Answer: 8


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

Answer: -1


Question: Given two row vectors $x, y \in \mathbb{R}^n$ such that the product $x^T \cdot y$ yields a positive-definite $n \times n$ matrix, determine if the inner product $x \cdot y^T$ results in a positive scalar.

Answer: Yes


Question:
Determine which of the following numbers does NOT divide the expression \( A = 2009^n - 1982^n - 1972^n + 1945^n \) when \( n = 2009 \):

- 659
- 1977
- 1998
- 2009

Answer: D


Question: Evaluate the expression \( \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\ldots}}}}} \).

Answer: 3


Question: In \(\triangle ABC\), if \(\tan A\), \(\tan B\), \(\tan C\) are in harmonic progression, find the minimum value of \(\cot \frac{B}{2}\).

Answer: \sqrt{3}


Question: Given that $(X,Y)$ follows a bivariate normal distribution with parameters $\mu_X = \mu_Y = 0$, $\sigma_X^2 = \sigma_Y^2 = 1$, and $\rho = \rho_{X,Y} \neq 0$, find the correlation coefficient $\rho(XY, Y)$. Express your answer in terms of $\rho$. \( \rho(XY, Y) = \frac{\text{Cov}(XY, Y)}{\sigma_{XY} \cdot \sigma_Y} \)

Answer: 0


Question: Given the ellipse described by the equation \(3x^2 + 4xy + 3y^2 = 1\), find the value of \(\frac{3\sqrt{5}}{\pi}A\), where \(A\) is the area of the ellipse.

Answer: 3


Question: Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be the linear transformation represented by the matrix \( \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \). Find the operator norm \( ||T|| \) with respect to the norm \( |(u, v)| = \max\{|u|, |v|\} \) in both the domain and codomain.

Answer: 1 + |a|


Question: What is the minimum number of regular tetrahedra required in a path of stacked regular tetrahedra such that the path self-intersects? A path self-intersects if a pair of distinct tetrahedra share a point strictly interior to both.

Answer: 6


Question: Evaluate the integral: \[ \frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty }\frac{k}{2}\sqrt{\frac{\pi}{2}}e^{-2|k-\pi|}dk \]

Answer: \dfrac{\pi}{4}


Question: Determine the Lebesgue measure of the set \( \mathbb{Q} \cap [0,1] \).

Answer: 0


Question: Consider a system described by the equation \( \dot{x} = f(x) \), where \( x \in \mathbb{R}^1 \) and \( f \) is continuous. Given that there are two asymptotically stable equilibrium points, determine if the statement "the number of all equilibria must be at least three" is true. Justify your answer.

Answer: True


Question: Evaluate the limit \( \lim_{n \to \infty} f_n(x) \) for \( f_n(x) = n^2 x (1-x^2)^n \) where \( 0 < x \leq 1 \).

Answer: 0


Question: Determine whether the class of isomorphism classes of fundamental groups of Lie groups is the same as the class of isomorphism classes of finitely generated abelian groups. Answer with 'True' or 'False'.

Answer: A


Question: Evaluate the integral \( I(t) = \int_0^{\infty} e^{\frac{-t(s-1)^2}{2}} \left( \frac{t(s-1)^3}{3} \right) ds \) and determine its value as \( t \to \infty \).

Answer: 0


Question: If $X$ is a dense subspace of a complete metric space $Y$, is $Y$ necessarily the completion of $X$? Provide a justification for your answer.

Answer: Yes


Question: Let $f$ be a real, even, summable function such that $f(0+)$ and $f(0-)$ exist. Prove that either $f(0-) = f(0+)$ or the Fourier transform $\hat{f}(\gamma)$ changes sign infinitely often as $|\gamma| \to \infty$. Note that $\hat{f}(\gamma)$ is a real function, so its "sign" is well-defined.

Answer: f(0-) = f(0+)


Question: Calculate the volume of the solid in \( \mathbb{R}^3 \) that lies under the paraboloid \( x_1^2 + x_2^2 = x_3 \) and above the square region \([0,1]^2\) in the \(x_1x_2\)-plane.

Answer: \dfrac{2}{3}


Question:
Consider a quadratic equation \(p(x) = ax^2 + bx + c = 0\) with real coefficients that has purely imaginary roots. Determine the nature of the roots of the equation \(p(p(x)) = 0\). Choose the correct option:

(A) only purely imaginary roots
(B) all real roots
(C) two real and two purely imaginary roots
(D) neither real nor purely imaginary roots

Answer: D


Question: Maximize the function \( f(x,y) = x^2 - y^2 \) subject to the constraint \( g(x,y) = 1 - x^2 - y^2 = 0 \).

Answer: 1


Question: How many 10-digit numbers can be formed using only the digits 0, 1, and 2, such that each digit appears at least twice or not at all?

Answer: 29306


Question: Find the extreme values of the function \( f(x, y) = x^2 + xy + y^2 - 4 \ln x - 10 \ln y \).

Answer: 7 - 10 \ln 2


Question: Given real numbers \(a, b, c, d, u_1, u_2\), determine the value of \(\frac{u_1}{u_2}\) if it is known that \(\frac{u_1}{u_2} = \frac{b}{d}\).

Answer: \dfrac{b}{d}


Question: Given an unbounded sequence \((x_j)_{j \in \mathbb{N}}\) in \(\mathbb{R}\), does there exist a positive number \(v\) such that \(x_j \bmod v\) is dense in the interval \([0, v]\)?

Answer: Yes


Question: Consider a connected compact group $K$. Determine whether the probability that two randomly chosen elements commute is non-trivial, i.e., $0 < p < 1$.

Answer: 0


Question: Evaluate the integral \( \int_0^{2\pi} P_l^m(\cos\theta) P_{l-1}^m(\cos\theta) \, d\theta \), where \( P_l^m(\cos\theta) \) are the Associated Legendre Polynomials.

Answer: 0


Question: Assume there is a Taylor series for a function $f(x)$ centered at $x=a$ that converges for all $x$. Does the equation \( f(b) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(b-a)^n \) hold true for all \( b \in \mathbb{R} \)? Justify your answer.

Answer: No


Question: Find the radius of convergence of the power series \(\sum_{n=1}^{\infty} a_n z^{n}\), where \(a_n\) is the number of divisors of \(n^{50}\).

Answer: 1


Question: Design a filter to obtain the minimum mean square error (MMSE) estimator of $X_1$ from the vector $Y = (Y_1, Y_2)^T$, where $Y_1 = X_1 + W_1$ and $Y_2 = X_2 + W_2$. The vector $X = (X_1, X_2)^T$ is a bivariate Gaussian with joint density function \( f(x_1, x_2) = \frac{1}{2\pi \sqrt{0.19}} \exp\left(-\frac{x_1^2 + 1.8x_1x_2 + x_2^2}{0.38}\right) \), and $W_1, W_2$ are independent and identically distributed as $N(0, 0.2)$. Given that the correlation coefficient between $X_1$ and $X_2$ is $\rho = -0.9$ and both $X_1$ and $X_2$ are $N(0,1)$, calculate the mean square error (MSE) of this estimator.

Answer: \dfrac{13}{105}


Question:
Solve for \( F \) in the equation:
\[
\frac{aF - b}{\sqrt{(aF - b)^2 + (cF - d)^2 + (eF - g)^2}} = \frac{hF - i}{\sqrt{(hF - i)^2 + (jF - k)^2 + (mF - n)^2}}
\] where all other variables are known.

Answer: F


Question: Evaluate \( \lim_{R \to \infty} \int_{C_{R}} \frac{e^{iz^{2}}}{1+z^{4}} \, \mathrm{d}z \), where \( C_{R} \) is the portion of the circle \( |z|= R \) in the first quadrant.

Answer: 0


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

Answer: 0


Question: Find the smallest integer $x \ge 2$ such that the expression $1 + 12x^2(1 + x)$ is a perfect square.

Answer: 4


Question: Let $p = 40k + 9$ be a prime number. Determine whether the order of 10 modulo $p$ is always even.

Answer: Yes


Question: Determine the frequency \( \omega(x) \) of the function \( x(t) = \sin(t) + \sin(\pi t) \), where the frequency is defined by \( \omega(x) = \limsup_{T \to \infty} \frac{\pi}{T} \nu(x,T) \), and \( \nu(x,T) \) represents the number of zeros of \( x(t) \), counting with their multiplicities, on the interval \((0,T]\).

Answer: \pi


Question: Consider the equation \((\log_2 x)^2 - 4\log_2 x - (m^2 + 2m + 13) = 0\). Let the real roots of the equation be \(x_1\) and \(x_2\) such that \(x_1 < x_2\). Find the sum of the maximum value of \(x_1\) and the minimum value of \(x_2\).

Answer: \dfrac{257}{4}


Question: Evaluate the limit: \[ \lim_{x\to 0}\frac{x\tan x}{\sqrt{1-x^2}-1} \] without using L'Hôpital's rule.

Answer: -2


Question: Determine the number of roots of the polynomial \(z^4 - 8z + 10\) that lie within the annulus defined by \(1 < |z| < 3\).

Answer: 4


Question: Given a measurable set $A \subset [0,1]$ and a positive number $\varepsilon$, is it possible to find an elementary set $B$ such that $\mu(A \Delta B) < \varepsilon$ and $\mu(B) = \mu(A)$?

Answer: Yes


Question: A standard six-sided die is rolled repeatedly until a 6 appears. Calculate the probability that the sum of all rolls, including the roll that results in a 6, is even.

Answer: \dfrac{4}{7}


Question: Determine whether the interval $(0,1)$ with the usual topology can be equipped with a complete metric. Answer "True" if it can, and "False" if it cannot.

Answer: A


Question:
Let \( p(x) \) be the polynomial given by:
\[
p(x) = [(x-2)^{2016}(x+2016)]+[(x-2)^{2015}(x+2015)]+\cdots+(x-2)(x+1).
\]
What is the sum of the coefficients of \( p(x) \)?

Answer: 1008


Question: Determine whether a set $A$ of natural numbers that contains arbitrarily long arithmetic progressions necessarily has positive Banach upper density.

Answer: No


Question: Find the operator norm of the linear operator \( I: C([0,1],\mathbb{R}) \to \mathbb{R} \) defined by \( I(f) = \int_0^1 f(x)\,dx \), where \( C([0,1],\mathbb{R}) \) is the vector space of continuous real-valued functions on the interval \([0,1]\) with the norm \( \|f\| = \int_0^1 |f(x)|\,dx \).

Answer: 1


Question: Given that $Y$ is uniformly distributed over $(0,6)$ and $X|Y=y$ follows an exponential distribution with rate $1/(3y)$ for $x \geq 0$, calculate the mean of $X$. Express the mean as $E[X] = \int_{0}^{\infty} x f_X(x)\ dx$, where $f_X(x)$ is the marginal distribution of $X$.

Answer: 9


Question:
Let \( A \) be a \( 3 \times 3 \) matrix with \( \det(A) = 2 \). Find the value of \( \det(\text{adj}(\text{adj}(\text{adj}(A^{-1})))) \). Choose the correct answer from the options below:

1. \( \frac{1}{512} \)
2. \( \frac{1}{1024} \)
3. \( \frac{1}{128} \)
4. \( \frac{1}{256} \)

Answer: D


Question: Determine the number of positive integers less than $10^8$ whose digits sum to 24.

Answer: 1708575


Question:
Given a sequence \((a_n)_{n=0}^{\infty}\) of real numbers satisfying \(\sum_{n=1}^{\infty} |a_n - a_{n-1}| < \infty\), determine which of the following statements about the convergence of the series \(\sum a_n x^n\) is necessarily true:

1. The series converges nowhere on \(\mathbb{R}\).
2. The series converges everywhere on \(\mathbb{R}\).
3. The series converges on some interval containing \((-1,1)\).
4. The series converges only on \((-1,1)\).

Answer: 3


Question:
Determine if the following system of Diophantine equations has any nontrivial integer solutions (i.e., none of the unknowns are zero):
\[ A^2 + B^2 = C^2 D^2 \]
\[ 2C^4 + 2D^4 = E^2 + F^2 \]

Answer: Yes


Question: Let \( \phi: G \rightarrow H \) be a homomorphism and let \( g \in G \) such that \( h = \phi(g) \). Suppose \( o(h) = 100 \) and \( g \) has finite order. What are the possible values of \( o(g) \)?

Answer: 100k


Question: Determine whether the following statement about generating sets of a group is true: Let $G$ be a group, $X$ a subset of $G$, and $H$ a subgroup of $G$. If $G = \langle X, H \rangle$, can we write $G = \langle X \rangle H$ as an internal product of subgroups?

Answer: False


Question: Let $a$, $b$, and $c$ be the roots of the polynomial $x^3 - 4x + 1 = 0$. Calculate the value of the expression $\frac{a^2bc}{a^3+1} + \frac{ab^2c}{b^3+1} + \frac{abc^2}{c^3+1}$. \(\text{Express your answer as a single number.}\)

Answer: -\dfrac{3}{4}


Question: Let $G$ be a group of order $pr$, where $p$ and $r$ are distinct primes. Suppose $G$ acts on a set $S$ of order $pr - p - r$. Determine whether there exists a point $s \in S$ such that $gs = s$ for all $g \in G$. Justify your answer.

Answer: Yes


Question: Consider the sequence \( a_1, a_2, a_3, \ldots \) defined by \( \frac{1}{a_{k+1}} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} \) for \( k > 1 \) with \( a_1 = 2^{2009} \). Find the value of \( a_{2011} \).

Answer: 1


Question:
Determine the coefficient $a_1$ in the series expansion of the Bessel function of order $n$, given by $J_n(x) = x^n \sum_{k=0}^{\infty} a_k x^k$, by substituting this series into the differential equation:
\[ y'' + \frac{1}{x}y' + \left(1 - \frac{n^2}{x^2}\right)y = 0 \]
for positive $n$. Use the series expansion to find $a_1$.

Answer: 0


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

Answer: \sqrt{e}


Question: Determine the number of fields, up to isomorphism, of order 6.

Answer: 0


Question:
Find an integer x such that:
\[ x^{17} \equiv 7 \pmod{53} \]
and
\[ x^{17} \equiv 1 \pmod{61} \]

Answer: 123


Question: Determine the Euler characteristic of the manifold of fixed rank matrices, \( \mathcal{M}_r \), which is the set of \( n \times m \) matrices over \( \mathbb{R} \) or \( \mathbb{C} \) with rank \( r \).

Answer: 0


Question: Is the Gram matrix associated with the exponential kernel \( g(x, y) = e^{-|x-y|} \) always positive definite? It is known to be at least positive semidefinite. Determine if it is also positive definite.

Answer: Yes


Question:
Consider a group \( G \) and let \( F(G) \) denote the collection of all subgroups of \( G \). Determine which of the following situations can occur:

1. \( G \) is finite but \( F(G) \) is infinite.
2. \( G \) is infinite but \( F(G) \) is finite.
3. \( G \) is countable but \( F(G) \) is uncountable.
4. \( G \) is uncountable but \( F(G) \) is countable.

Select the correct situation.

Answer: 3


Question: For how many distinct triangles \( \triangle ABC \), with \( AB = 2011 \), are both \( \cos(2\angle{A} + 3\angle{C}) \) and \( \sin(2\angle{B} + \angle{C}) \) integers?

Answer: 5


Question: Determine all values of \(a\) for which the series \(\sum_{n=1}^\infty \frac{\log(n!)}{n^a} \) converges.

Answer: a > 2


Question: Find the smallest value of \( a > 1 \) such that \( \frac{a + \sin x}{a + \sin y} \leq e^{(y-x)} \) holds for all \( x \leq y \).

Answer: \sqrt{2}


Question: Determine if $n = 101$ is the only integer solution greater than 1 to the congruence $2^{n-1} \equiv 203 \pmod{n}$. Consider the conditions that $n$ must be odd and analyze the possible prime factors of $n$.

Answer: 101


Question: Let $f:[0,1] \to [0,1]$ be a homeomorphism with $f(0)=0$ and $f(1)=1$. If $f$ is not the identity map, is it true that $f^n \neq f$ for all integers $n>1$? Here, $f^n$ denotes $f$ iterated $n$ times.

Answer: Yes


Question:
Consider a random variable $X$ that follows a Gamma distribution with parameters $\alpha$ and $\beta = \alpha/\mu$. The probability density function of $X$ is given by:
\[ f_{X}(x) = \frac{(\alpha/\mu)^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\alpha x/\mu}. \]
Determine the limit of the density function as $\alpha \to \infty$:
\[ \lim_{\alpha \to \infty} f_{X}(x). \]

Answer: \delta(x - \mu)


Question: Find the equation of the tangent plane to the surface defined by \( g(x,y,z) = 0 \) at the point \((1, -1, 1)\), where \( g(x,y,z) = f(xyz, x^2 + y^2 + z^2) \) and \( f(u,v) \) is a function such that \( f(-1,3) = 0 \), \( f_u(-1,3) = 2 \), and \( f_v(-1,3) = -3 \).

Answer: x - y + z = 3


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{\int_{0}^{x^2} x^2 e^{-t^2} \, dt}{-1+e^{-x^4}} \]

Answer: -1


Question: Calculate the determinant of the matrix \( A \), where \( A \) is an \( n \times n \) matrix with elements defined by \( a_{ij} = \max(i, j) \) for \( i, j = 1, 2, \dots, n \).

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


Question: Let $X, Y, Z$ be metric spaces equipped with the Borel $\sigma$-algebra. Suppose $f: X \rightarrow Y$ is a homeomorphism and $g: Y \rightarrow Z$ is a measurable map with respect to the Borel $\sigma$-algebra. Is the composition $g \circ f: X \rightarrow Z$ necessarily measurable?

Answer: Yes


Question: Suppose $X$ is a normed linear space and $F(x) = F(y)$ for all bounded linear functionals $F$ on $X$. Prove that $x = y$.

Answer: x = y


Question: Does there exist an enumeration \( \{r_n\} \) of the rationals such that the set \( \bigcup_{n=1}^{\infty}(r_n-\frac{1}{n}, r_n+\frac{1}{n}) \) has a non-empty complement?

Answer: Yes


Question: Evaluate the integral \( \int _0 ^ \infty \frac {dx} {x^3 (e^\frac{\pi}{x} -1)} \) without using Taylor expansion.

Answer: \dfrac{1}{6}


Question: Compute the distribution $x\delta'_0$, where $\delta'_0$ is defined by $\delta'_0(u) = -u'(0)$ for $u \in D\mathbb{R}$.

Answer: -\delta_0


Question: Determine whether the statement is true: For a nowhere dense set $E$ in $\mathbb{R}$, is the Lebesgue measure of $\overline{E} - E$ equal to zero, i.e., $m(\overline{E} - E) = 0$?

Answer: False


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{\sum_{k=1}^n\lfloor kx\rfloor}{n^2} \]

Answer: \dfrac{x}{2}


Question: Given the functions \( f, g: \mathbb{R} \to \mathbb{R} \) such that \( f(x + g(y)) = -x + y + 1 \) for all real numbers \( x \) and \( y \), determine the value of \( g(x + f(y)) \).

Answer: -x + y - 1


Question: Find the value of \( t_0 \in [0, 2\pi] \) such that \( \gamma(t_0) = 0 \), where \( \gamma(t) = \frac{\sqrt{3}}{2} + \frac{i}{2} + e^{it} \).

Answer: \dfrac{7\pi}{6}


Question: Determine the values of $z$ on the unit circle $|z|=1$ for which the series \( \sum_{n=1}^\infty \frac{z^n}{n} \) converges.

Answer: z \neq 1


Question: Let $A$ and $B$ be sets such that $|A| = 8$ and $|B| = 5$. Calculate the number of functions $f: A \to B$ such that no element in $B$ has exactly 3 pre-images from $A$. In other words, for all $b \in B$, $|\{a \in A : f(a) = b\}| \neq 3$. Provide your answer as a single number.

Answer: 154305


Question: Evaluate the limit: $$\lim _{n\to \infty }\left(\frac{\sqrt[n]{n!}}{n}\right)$$

Answer: \dfrac{1}{e}


Question: Evaluate the limit: \[ \lim_{n\to \infty} (e^n-2^n)^{1/n} \]

Answer: e


Question: Find the radius of the circle on which all complex roots of the equation \((z + 1)^5 = 32z^5\) lie when plotted in the complex plane.

Answer: \dfrac{2}{3}


Question: Calculate \( \lim_{n\rightarrow \infty} \int_0^\infty \frac{n^2 \sin(x/n)}{n^3x + x(1 + x^3)} \, d x \).

Answer: 0


Question:
Evaluate the integral \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{3\tan^2x+6\tan x+11}{1+\tan ^2x}dx\) and express it in the form \(\frac{k\pi+\lambda}{6}\). Determine the value of \(k + \lambda\).

\(\text{(A) } 10 \hspace{1cm} \text{(B) } 12 \hspace{1cm} \text{(C) } 14 \hspace{1cm} \text{(D) } 16\)

Answer: D


Question:
Let $E = (C[0,1], \|\cdot\|_\infty)$ be the Banach space of continuous functions on $[0,1]$ with the supremum norm. Define the set
\[ A = \{ u \in E : u(0) = 0, \quad \int_0^1 u(t) \, dt \geq 1 \}. \]
Compute the distance from the zero function to the set $A$, denoted as $d_A(0) = \inf\{\|u\|_\infty : u \in A\}$. Determine $d_A(0)$. \( \boxed{?} \)

Answer: 1


Question: Two beads, each with a mass of 1 kilogram, are attached to the ends of a compressed spring and rest on a smooth parabolic wire described by the equation \( y = x^2 \). The spring has a natural length of 1 meter and a spring constant of 5 Newtons per meter. The system is in static equilibrium, and the acceleration due to gravity is 10 meters per second squared. Determine the distance from the y-axis to each bead, providing your answer to three significant digits.

Answer: 0.167


Question: Determine whether the following statement is true: Given two rings $R$ and $S$, and a principal ideal $((a,b)) = I \in R \times S$ where $(a,b) \in R \times S$, is it true that $(R \times S) / ((a,b)) \cong R/(a) \times S/(b)$?

Answer: True


Question: Let $A$ be a complex $n \times n$ matrix with $n$ distinct eigenvalues. Does there exist a vector $u \in \mathbb{C}^n$ such that the vectors $u, Au, A^2u, \ldots, A^{n-1}u$ are linearly independent?

Answer: Yes


Question: Evaluate the integral of the 1-form \( \omega = \frac{-y \,dx + x\,dy}{x^2 + y^2} \) over the triangle with vertices \( A = (-5, -2) \), \( B = (5, -2) \), and \( C = (0, 3) \).

Answer: 2\pi


Question: Given that \( M^2 \equiv 51 \mod 59 \), find the value of \( M^{12} \mod 59 \).

Answer: 7


Question: Let $X$ and $Y$ be continuous random variables with the joint density function \(f(x,y) = \begin{cases}24xy& \text{for } x>0,\; y>0,\; 0<x+y<1\\ 0 &\text{otherwise} \end{cases}\). Find the conditional probability \(P\left(X < \frac{1}{2} \;\middle|\; Y = \frac{1}{4}\right)\).

Answer: \dfrac{4}{9}


Question: Is there a differentiable function $F: [a, b] \to \mathbb{R}$ such that its derivative $F'$ is not Riemann integrable on the interval $[a, b]$?

Answer: Yes


Question: Determine the point to which the origin should be shifted to eliminate the linear terms in the equation \(3x^2 - 2xy + 4y^2 + 8x - 10y + 8 = 0\).

Answer: (-1, 1)


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

Answer: -\dfrac{3}{2}


Question: Let $n > 1$ and $K = \mathrm{GF}(2^n)$. Let $k$ be coprime to $2^n - 1$. Determine whether there always exists $a \neq 0$ in $K$ such that the curve $y^2 + y = x^k + ax$ has exactly $2^n$ affine solutions.

Answer: Yes


Question: Let \( P(x) = (x-1)(x-2)(x-3) \). Determine the number of polynomials \( Q(x) \) for which there exists a polynomial \( R(x) \) of degree 3 such that \( P(Q(x)) = P(x) \cdot R(x) \).

Answer: 22


Question: Find the value of \( k \in \mathbb{R} \) such that \[ \lim_{x \to 0} \frac{1 - \cos(kx)}{x^2} = \int_0^\infty xe^{-x} \, \mathrm{d}x. \]

Answer: \pm \sqrt{2}


Question: A teacher intended to give a typist a list of nine integers that form a group under multiplication modulo 91. However, one integer was inadvertently left out, resulting in the list: $1, 9, 16, 22, 53, 74, 79, 81$. Determine the missing integer.

Answer: 29


Question: Determine the number of roots for the equation \( \frac{3-2x}{x-1}-\sqrt[4]{x^3}=0 \).

Answer: 1


Question: Evaluate the limit: $$\lim_{x\rightarrow0}(1-e^x)^{\sin (x)}$$

Answer: 1


Question: Given an n-dimensional vector space V with orthonormal basis sets \(\{s_1, s_2, \ldots, s_n\}\) and \(\{e_1, e_2, \ldots, e_n\}\), where the basis transformation matrix \(U\) satisfies \(e_i = \sum_j U_{ij}s_j\), determine the basis transformation matrix for the \(n^2\)-dimensional vector space \(V \otimes V\) with basis sets \(\{s_1s_1, s_1s_2, \ldots, s_ns_n\}\) and \(\{e_1e_1, e_1e_2, \ldots, e_ne_n\}\) in terms of \(U\).

Answer: U \otimes U


Question: Calculate the expected area of a triangle formed by the origin and two points chosen from the set of positive natural numbers \((x, y)\) such that \(x + y = n\).

Answer: \dfrac{n^2}{6}


Question: Given the equation \(abc + a + b + c = ab + bc + ca + 5\) where \(a, b, c \in \mathbb{R}\), find the minimum value of \(a^2 + b^2 + c^2\).

Answer: 6


Question:
Compute \( a(x)b(x) + c(x) \) in \( \mathrm{GF}(2^4) \) using the irreducible generator polynomial \( x^4 + x + 1 \). Given:

\[ a(x) = x^3 + x^2 + 1, \]
\[ b(x) = x^2 + x + 1, \]
\[ c(x) = x^3 + x^2 + x + 1 \]

Find the result of \( a(x)b(x) + c(x) \) modulo \( x^4 + x + 1 \) in \( \mathrm{GF}(2^4) \).

Answer: x^3 + x


Question: Let $1 < k < d$ be integers. Suppose $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ and $A = \bigwedge^k B$ for some complex matrix $B \in \text{End}(\mathbb{C}^d)$. Does there exist a real matrix $M \in \text{End}(\mathbb{R}^d)$ such that $A = \bigwedge^k M$?

Answer: No


Question: If the average of a positive sequence converges to zero, does the average of the squares of the sequence also converge to zero? Specifically, if \( \frac{\sum\limits_{i=1}^n x_i}{n}\rightarrow 0 \) with \( x_i \geq 0 \), does it imply that \( \frac{\sum\limits_{i=1}^n x_i^2}{n}\rightarrow 0 \)?

Answer: No


Question: Determine the maximal cardinality of a family of pairwise disjoint figure eights in the plane. A figure eight is an object shaped like the number 8 or the infinity symbol (∞), including all angle variations. What is the maximal cardinality of such a family?

Answer: \aleph_0


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

Answer: 2


Question: Let $\mathcal{A}$ be a $C^{\ast}$-algebra and suppose that $\{ x_1, \ldots, x_n \}$ are positive elements of $\mathcal{A}$ such that $\left\| \sum_{i=1}^{n} x_i \right\| \leq \lambda$, for some $\lambda \in \mathbb{R}$. Is it true that each $\|x_i\| \leq \lambda$?

Answer: Yes


Question: Is it possible to extend the Lebesgue measure on $\mathbb{R}$ to a measure on $[-\infty,\infty]$, defined on the Borel subsets of $[-\infty,\infty]$, such that $[-\infty,\infty]\backslash \mathbb{R}$ is a null set?

Answer: Yes


Question: Evaluate the integral \( \int_C \frac{z^2 \, dz}{z^2+4} \) where \( C \) is the square with vertices at \( 2, -2, 2+4i, -2+4i \). Use the Cauchy integral formula to find the value of the integral.

Answer: -2\pi


Question: Determine the tens digit of the number $T_{1000}$, where $T_1=7^7$, $T_2=7^{7^7}$, $T_3=7^{7^{7^7}}$, and so on.

Answer: 4


Question: Minimize the distance between the points $\mathbf{x} = (x, y)$ and $\mathbf{u} = (u,v)$ in $\mathbb{R^2}$, where $\mathbf{x}$ lies on the plane $\{\mathbf{x} \in \mathbb{R^2} : \mathbf{x} \cdot (1,2) = -10\}$ and $\mathbf{u}$ lies on the parabola $\{\mathbf{u} \in \mathbb{R^2} : v = u^2\}$. Use the method of Lagrange multipliers to find the minimum distance.

Answer: \dfrac{79\sqrt{5}}{40}


Question: Let $X \sim N(0, \sigma^2)$ and $Y \sim N(0, \sigma^2)$ be independent random variables. Calculate the expectation $E \left(\frac{X}{\sqrt{X^2+Y^2}} \right)$. Provide your answer in terms of known constants or functions.

Answer: 0


Question: Given a normal distribution of cheese weights X with mean 10 kg and standard deviation 2 kg, determine the weight threshold h such that the probability of a cheese weighing more than h is at least 0.991. Calculate the value of h.

Answer: 5.27


Question: Determine the value of \( r \) such that the series \( K(x) = \sum_{n=1}^{\infty}\frac{1}{n^r}\cos(nx) \) converges uniformly for \( x \in \mathbb{R} \).

Answer: r > 1


Question: Let $p$ be an odd prime of the form $p=2^m+1$, for some positive integer $m$. Let $g$ be a primitive root mod $p$ with $1 \le g \le p$. How many such $g$ are there?

Answer: 2^{m-1}


Question: Determine whether the vector bundle $Q(\operatorname{Gr}[p+2,2])$ is $U(p+2)$-equivariant.

Answer: Yes


Question: Let \( A \in M_3(\mathbb{C}) \) be a matrix such that \( A^{\star}A = AA^{\star} \). Determine whether the following statement is true or false: If \((1,0,1)^{T}\) and \((1,1,0)^{T}\) are eigenvectors of \( A \) with eigenvalues \( \alpha \) and \( \beta \), respectively, does it imply that \( \alpha = \beta \)?

Answer: True


Question: Does there exist a $C^{\infty}$ function $f: \mathbb{R}^2 \to \mathbb{R}$ such that $\int dx \int dy \,f(x,y)$ diverges to infinity but $\int dy \int dx\, f(x,y)$ converges?

Answer: Yes


Question: Find the last three digits of \(23^{320}\).

Answer: 201


Question: Let \( f(x) \) be a differentiable function satisfying the functional equation \( f(x+y) = f(x)e^y + f(y)e^x + 2xye^xe^y \) for all \( x, y \in \mathbb{R} \). Given that \( f'(x) = 0 \), determine the number of solutions to the equation \( f(x) = 0 \).

Answer: 1


Question: Given that $f(x) \xrightarrow{x\to a+} 0$ and $f'(x) \ne 0$ for all $x \in (a,b)$, where $a,b \in \mathbb{R}$ with $a < b$ and $f:(a,b) \to \mathbb{R}$ is differentiable, is there a $\delta \in (0,b-a]$ such that $f(x) \ne 0$ for all $x \in (a,a+\delta)$?

Answer: Yes


Question: Is the direct limit of a directed set of commutative local rings a local ring? Let \( \{R_i\}_{i\in A} \) be a directed set of commutative local rings with a directed index set \( A \), and let \( R \) be the direct limit of this set. Determine if \( R \) is a local ring.

Answer: Yes


Question: For a 2-parameter Weibull distribution with shape parameter \( k \) and scale parameter \( \lambda \), the mean is given by \( E(X) = \lambda \Gamma(1+\frac{1}{k}) \) and the variance is given by \( \text{Var}(X) = \lambda^2 [\Gamma(1+\frac{2}{k})-(\Gamma(1+\frac{1}{k}))^2] \). Find the value of \( k \) for which the mean equals the variance.

Answer: 1


Question: Calculate \( \lim_{n\to\infty}\frac{1}{9^n}\int_{1}^{2}\big((e^x)(x^2-5x+7)\big)^n \, dx \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{n \rightarrow \infty} \sum_{k=0}^n \left(\frac{1}{\sqrt{(n+k)(n+k+1)}}\right) \]

Answer: \ln 2


Question: Compute the induced group homomorphism \((f_n)_* : \pi_1(S^1,1) \to \pi_1(S^1,1)\) for the map \(f_n : S^1 \to S^1\) defined by \(f_n(z) = z^n\), where \(\pi_1(S^1,1)\) is identified with \(\mathbb{Z}\).

Answer: n


Question: Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \) and \( (1 + x^2)f'(x) \geq 1 + f^2(x) \) for all \( x \in \mathbb{R} \). Determine \( \lim_{x \to \infty} f(x) \).

Answer: \infty


Question:
Consider a random walker starting at $S_0 = 2$ on the integers. The walker moves according to the following rules:

- If the walker is on the $n$th prime number $p_n$, she moves to either $p_n + 1$ or $p_{n+1}$ with equal probability.
- If the walker is on a composite number $x$, she moves to one of the prime factors of $x$, each with probability $1/\omega(x)$, where $\omega(n)$ denotes the number of distinct prime factors of $n$.

Determine whether the probability $\mathbb{P}(\sup_{n\ge 0} S_n = \infty)$ is 1 or less than 1.

Answer: 1


Question:
Given that D is 1.3 meters, C is 0.6 meters, and the angle \( a \) is 30 degrees with a rate of change \( a' = 150 \) rad/s, calculate the rate of change of M. Use the following relationships:

\[ M = \cos(30^\circ) \cdot 0.6 + \cos(13.3424) \cdot 1.2 \]

\[ M' = -\left(\sin(30^\circ) \cdot 0.6 \cdot a' + \sin(13.3424) \cdot 1.3 \cdot B'\right) \]

where \( B' \) is calculated using:

\[ B' = \frac{\cos(30) \cdot 0.6 \cdot 150}{\cos(13.3424) \cdot 1.3} \]

Determine \( M' \) in rad/s or m/s.

Answer: -63.5


Question: Consider a sequence of independent random variables \(\{X_k\}_{k=1}^\infty\) with probability density functions \(f_1(x), \dots, f_M(x)\) that are distinct but have the same expected value \(\mu\). Let \(S=\{S_k\}_{k=1}^\infty\) be a sequence of indices such that \(S_k \in \{1, \dots, M\}\) and \(X_k \sim f_{S_k}(\cdot)\). Determine if the following statement is true: \[ \lim_{k \rightarrow \infty} \sup_{S} \mathbb{P}_S \left( \frac{\sum_{i=1}^k X_{i}}{k} > \mu + \epsilon \right) \rightarrow 0 \] where \(\mathbb{P}_S\) denotes the probability measure specified by \(\{S_k\}_{k=1}^\infty\) and the supremum is over all possible sequences \(\{S_k\}_{k=1}^\infty\).

Answer: True


Question: Find the function $f(x)$ if the arc length of the curve $y=f(t)$ on the interval $0 \le t \le x$ is given by \( L(x) = \ln(\sec x + \tan x) \) for every $x$ on $0 \le x \le 1$, and the curve passes through the origin.

Answer: -\ln(\cos x)


Question: Let $K$ be a finite extension of $\mathbb{Q}$, and let $p$ be a prime number in $\mathbb{Q}$. If $p\mathcal{O}_K = \mathfrak{P}_1\ldots\mathfrak{P}_n$ is the prime decomposition of $p$ in the ring of integers $\mathcal{O}_K$, is it true that any prime ideal in $\mathcal{O}_K$ lying above $p$ is a factor in this decomposition?

Answer: Yes


Question: Find the greatest area of a triangle with its base as the major axis of the ellipse given by the equation \(3x^2 + 2y^2 = 6\), and one of its vertices lying on the circumference of the ellipse \(\frac{x^2}{2} + \frac{y^2}{3} = 1\).

Answer: \sqrt{6}


Question: Determine whether the sequence of normal distributions \( \mu_n \sim N(0,\frac{1}{n^2}) \) converges weakly to \( \mu \sim N(0,0) \).

Answer: Yes


Question: Evaluate the integral \( \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin^{1/3} x}{\sin^{1/3} x + \cos^{1/3} x} \, dx \).

Answer: \dfrac{\pi}{12}


Question: Two circles with radii in the ratio 1:2 touch each other externally. Let the center of the smaller circle be $C$ and the center of the larger circle be $D$. The point of contact is $A$. A straight line $\overline{PAQ}$ is drawn such that $P$ is on the smaller circle and $Q$ is on the larger circle, and $\overline{PAQ}$ does not pass through $C$. Given that $\angle QAD = \alpha$, find the angle between the tangent at $Q$ to the larger circle and the diameter of the smaller circle that passes through $P$ (extended if necessary).

Answer: 90^\circ


Question: Does there exist a Banach space $E$ and a linear functional $\lambda \in E^*$ such that $\lambda$ is an extreme point of the unit ball and yet does not norm any (non-zero) vector of $E$?

Answer: Yes


Question: Determine whether the degree of the field extension \([\mathbb{Q}(\beta): \mathbb{Q}]\) is 3, where \(\beta = 1+\alpha-2\alpha^2\) and \(\alpha\) is a root of the polynomial \(p(x) = x^3 - x + 1\) in \(\mathbb{Q}[x]\).

Answer: 3


Question: Calculate the limit: $$\displaystyle \lim_{\alpha \to -1}\left(2\Gamma(-\alpha-1)+\Gamma\left(\frac{\alpha + 1}{2}\right) \right)$$

Answer: -3\gamma


Question: Evaluate the limit: \[ \lim _{k \rightarrow \infty} \int_{0}^{x} \exp \left(-k t^{2} / 2\right) dt, \] where $x > 0$ is fixed.

Answer: 0


Question: Determine whether the equality \(|z|^n = |z^n|\) holds for complex numbers \(z\) and natural numbers \(n\). Provide a proof or counterexample.

Answer: |z|^n = |z^n|


Question: Compute \( \lim_{n\to\infty} \frac{\sum_{i=1}^{n} X_{i}}{\sum_{i=1}^{n}X_{i}^2 } \) where \(X_i\) are i.i.d. random variables uniformly distributed on \((0,1)\).

Answer: \dfrac{3}{2}


Question: Let $X$ and $Y$ be two independent random variables, each uniformly distributed on the interval $(0,1)$. Compute the conditional expectation $E(X \mid X < Y)$. Provide your answer as a single value.

Answer: \dfrac{1}{3}


Question: Find the last two digits of the number \( N = 2^5 + 2^{5^2} + 2^{5^3} + \cdots + 2^{5^{2015}} \).

Answer: 80


Question: Determine a basis and the dimension of the real vector subspace of complex matrices $A \in \mathbb{C}^{3 \times 3}$ such that the sum of the elements of its diagonal equals zero.

Answer: 16


Question: Consider a polynomial $P(x,y)$ with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically invariant quantity for all surfaces in $\mathbb{R}^{3}$, meaning it is invariant under isometries of surfaces. Is it true that $P(x,y)$ must be of the form $P(x,y)=F(xy)$ for some one-variable polynomial $F$?

Answer: Yes


Question: Evaluate the line integral \( \oint_C z \, dx + x \, dy + y \, dz \) using Stokes' Theorem, where \( C \) is the trace of the cylinder \( x^2 + y^2 = 25 \) on the plane \( y + z = 10 \).

Answer: 50\pi


Question: Evaluate the limit: \[ \lim_{n\to\infty}\int_{[0, \frac{n+1}{n}]} \frac{x \cos(ne^x)}{3n^{\sqrt{2}} + \sin{(e^x + 7n^2)}} \, dx \]

Answer: 0


Question: Let \( f: X \to Y \) be a function. Suppose that for every subset \( A \subseteq X \), the condition \( f(X \setminus A) \subseteq Y \setminus f(A) \) holds. Does this imply that \( f \) is injective?

Answer: Yes


Question: Determine the values of \( x \in \mathbb{R} \) for which the series \( \sum_{n=2}^\infty \frac{1}{n\sqrt{\ln (n+x^2)}} \) converges.

Answer: \emptyset


Question: Find the maximum value of the function \( f(x, y) = x^3 + y^3 \) subject to the constraint \( x^2 + y^2 = 1 \). Determine whether the maximum occurs at the points \((1, 0)\) and \((0, 1)\) or at \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\).

Answer: 1


Question: Evaluate the real integral \( \int_{-\infty}^{\infty}{\frac{\sin x}{x^4-6x^2+10}\,\mathrm dx} \) using the Residue Theorem.

Answer: 0


Question: Find the orthogonal trajectories of the family of curves given by the equation \( y^4 = Cx^5 \).

Answer: 4x^2 + 5y^2 = C


Question: If a ring contains a subring isomorphic to the integers \( \mathbb{Z} \), does it necessarily imply that the ring is an integral domain? Provide a justification for your answer.

Answer: No


Question: Given a matrix $A$ with entries of either 0 or 1, where each row contains at most $r$ ones, and each row of the matrix product $AA^T$ contains at most $z$ non-zero entries, find an upper bound on the number of ones in each column of $A$ as a function of $r$ and $z$. The bound need not be tight but should be expressible as a simple function of $r$ and $z$. Provide your answer as a function of $r$ and $z$.

Answer: z


Question: Evaluate the integral \( \int_\gamma \frac{1}{z^2-4} \, dz \), where \( \gamma(t) = 3e^{it} \) for \( t \in [-\pi, \pi] \).

Answer: 0


Question: Determine whether the function \( f: \mathbb{E}^1 \to X \), defined by \( f(x) = x \), is continuous, where \( X \) is the set of all real numbers with the finite complement topology.

Answer: Yes


Question:
Let $f_n$ be a sequence of differentiable functions on $[0,1]$ converging uniformly to a function $f$ on $[0,1]$. Which of the following statements is true about $f$?

1. $f$ is differentiable and Riemann integrable on $[0,1]$.
2. $f$ is uniformly continuous and Riemann integrable on $[0,1]$.
3. $f$ is continuous, need not be differentiable on $(0,1)$, and need not be Riemann integrable on $[0,1]$.
4. $f$ need not be continuous.

Select the correct statement.

Answer: 2


Question: Given the matrices \(A=\begin{bmatrix} 1&2 \\3&4\end{bmatrix}\) and \(B=\begin{bmatrix} k&2 \\3&9\end{bmatrix}\), find the value of \(k\) such that \((AB)^5 = A^5B^5\).

Answer: 6


Question: Find a measure $\mu$ on $\mathbb{R}$ such that $\mu(\mathbb{R}) = 1$ and a set is a $\mu$-null set if and only if it is a $\lambda$-null set for the Lebesgue measure $\lambda$. Consider how to handle uncountable null sets like the Cantor set.

Answer: d\mu(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \, d\lambda(x)


Question: Find a linear transformation \( T: V \rightarrow W \) where \( V = C^\infty(-\infty, \infty) \), the vector space of real-valued continuous functions with continuous derivatives of all orders on \((-\infty, \infty)\), and \( W = F(-\infty, \infty) \), the vector space of real-valued functions defined on \((-\infty, \infty)\). The kernel of \( T \) should be \( P_3 \), the space of polynomials of degree \( \leq 3 \).

Answer: T(f) = f^{(4)}


Question: Let $M$ and $N$ be smooth manifolds where $M$ has a non-empty boundary. Let $F: M \to N$ be a smooth map, and let $f$ be the restriction of $F$ to $\partial M$. Suppose there exists a homotopy $H: \partial M \times I \to N$ between $f$ and a smooth map $g: \partial M \to N$. Does there necessarily exist a smooth map $G: M \to N$ such that $G$ is homotopic to $F$ and the restriction of $G$ to $\partial M$ is $g$?

Answer: Yes


Question: Find the remainder when \(2^{1990}\) is divided by 1990.

Answer: 1024


Question:
Given a well-founded preorder \((\preceq, P)\), where there is no infinite sequence \(... a_3 \preceq a_2 \preceq a_1\) with all distinct elements, consider the topology on \(P\) generated by upwards closed sets being open. For a set \(X \subseteq P\), determine if there exists a finite \(n\) such that the collection \(S_n\), defined by:

- \(S_0 = \{X\}\)
- \(S_{n+1} = \{\text{int}(Y), Y \cap Z, P \setminus Y \mid Y, Z \in S_n\}\)

is closed under Boolean operations and the interior operation. Is \(S_n\) closed under these operations for some finite \(n\)?

Answer: Yes


Question: Is the module \( \mathbb{Z}/n\mathbb{Z} \) projective as a \( \mathbb{Z}/n\mathbb{Z} \)-module?

Answer: Yes


Question: Given a simple graph $G$ with degree sequence $D_G$, and another sequence of numbers $D$ such that $D_G[i] \ge D[i]$ for all $i$ and $\left( \sum_{i} D_G[i] - \sum_{i} D[i] \right)$ is an even number, determine if there is a polynomial-time algorithm to realize $D$ as the degree sequence of a spanning subgraph of $G$ by removing edges from $G$. Answer 'yes' or 'no'.

Answer: yes


Question: Calculate the area of the surface obtained by rotating the curve given by \(3y^2 = x(1-x)^2\) about the x-axis, for \(0 \leq x \leq 1\).

Answer: \dfrac{\pi}{3}


Question: Is it possible to find a subset $A$ of the real line $\mathbb{R}$ such that the Lebesgue measure of $A$ minus its interior is positive?

Answer: Yes


Question: Given the real symmetric matrix \( A = \begin{bmatrix} a & 2f & 0 \\ 2f & b & 3f \\ 0 & 3f & c \end{bmatrix} \), where \( a, b, c \) are distinct real numbers and \( f \) is a non-zero real number, find the geometric multiplicity of the largest eigenvalue of \( A \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy} \cos(x+y) \]

Answer: 1


Question: Evaluate the limit \( \lim_{(x,y) \to (0,0)} \frac{y^2-x \arctan y}{y} \).

Answer: 0


Question: Determine the values of $z$ for which the generating function $\sum_{n=1}^{\infty} g_{n} z^{n}$, where $g_{n}$ is the number of isomorphism classes of groups of order $n$, converges.

Answer: |z| < 1


Question: Evaluate the limit: \[ \lim_{R\rightarrow\infty} \int_0^R e^{x^2 - R^2}dx \]

Answer: 0


Question: Let $u: \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous subharmonic function with $n>2$, satisfying $0 \leq u \leq 1$ and $\sup_{\mathbb{R}^n}u = 1$. Consider the convolution $u \ast \rho_r(z) := \int_{\mathbb{R}^n} u(y) \rho(\frac{y-z}{r})r^{-n}dy$, where $\rho$ is a smooth mollifier with $\rho|_{B_1(0)} > 0$, $\rho|_{\mathbb{R}^n \backslash B_1(0)} = 0$, and $\int_{\mathbb{R}^n} \rho(y)dy=1$. Determine whether $\lim_{r\rightarrow \infty} u \ast \rho_r(z) = 1$. Provide a justification for your answer.

Answer: 1


Question: Determine whether there are any integer solutions \((x, y)\) to the equation \(x^2 + y^2 = 2015\). If solutions exist, find all such pairs \((x, y)\).

Answer: \emptyset


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

Answer: 78


Question: Suppose $A \in {\mathbb C^{n \times n}}$ is a matrix with $\left\| A \right\| \le \varepsilon$, and $v \in {\mathbb C^n}$ is a vector such that ${v^*}v = 1$. Is it true that $\left\| {{v^*}Av} \right\| \le \varepsilon$?

Answer: Yes


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

Answer: -\dfrac{1}{6}


Question: Determine the region of convergence for the series \( \sum\limits_{n=0}^{\infty} \frac{z^n}{a^n+b^n} \), where \( a \) and \( b \) are real parameters with \( 0<a<b \).

Answer: |z| < b


Question:
Find the determinant of the \((n-1) \times (n-1)\) matrix where the diagonal entries are \(n-1\) and the off-diagonal entries are all \(-1\):
\[
\begin{pmatrix}
n-1 & -1 & \cdots & -1 \\
-1 & n-1 & \cdots & \vdots \\
\vdots & \vdots & \ddots & \vdots \\
-1 & \cdots & \cdots & n-1
\end{pmatrix}
\]

Answer: n^{n-2}


Question: If \((x_n)\) is a Cauchy sequence in \(\ell^1\), does it follow that \((\|x_n\|_1)\) is a Cauchy sequence in \(\mathbb{F}\)? Here, \(\ell^1 = \{x: \mathbb{N} \to \mathbb{F} : \|x\|_1 \text{ is finite}\}\), where \(\mathbb{F}\) is either \(\mathbb{R}\) or \(\mathbb{C}\).

Answer: Yes


Question: Given the inequalities \( \frac{a+b+d}{a'+b'+d'}\le M \) and \( \frac{a+c+d}{a'+c'+d'}\le M \), and the conditions \( \frac{a}{a'}\le \frac{b}{b'}\le\frac{d}{d'} \) and \( \frac{a}{a'}\le \frac{c}{c'}\le\frac{d}{d'} \), where \( a,b,c,d,a',b',c',d' \in (0,1] \), determine if the following inequality holds: \( \frac{a+b+c+d}{a'+b'+c'+d'}\le M \).

Answer: Yes


Question: Solve the equation \( \log_{2x+3}(6x^2+23x+21) = 4 - \log_{3x+7}(4x^2+12x+9) \).

Answer: -\dfrac{1}{4}


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

Answer: e^{1 - a}


Question: Compute the limit of \( \frac{S_p(n)}{n^{p+1}} \) as \( p \to \infty \), where \( S_p(n) = \sum_{k=1}^n k^p \).

Answer: \dfrac{1}{n}


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

Answer: \dfrac{1}{3\pi}


Question: Given the nonlinear recurrence relation \(b_n = \left(\frac{1}{2}b_{n-1} + \frac{1}{2}\right)^2\) with initial condition \(b_0 = \frac{1}{2}\), evaluate \(\lim_{n \to \infty} (b_n)^{2n}\).

Answer: e^{-8}


Question: Does there exist a lacunary sequence with positive upper density?

Answer: No


Question: In any triangle $ABC$, if $\frac{1}{8} \geq \cos A \cdot \cos B \cdot \cos C > y$, find the value of $y$.

Answer: -1


Question: Find the inverse of the function \( h(x) = 1 + x - \sqrt{1+2x} \) for \( x > 0 \).

Answer: x + \sqrt{2x}


Question:
Determine whether the following equality holds:
\[ \int_{B(0,2)} (xy^2+y^2z) \, d\mu(x,y,z) = \int_{B(0,1)} (xy^2+y^2z) \sqrt{x^2+y^2+z^2} \, d\lambda_3(x,y,z) \]
where \( \mu(E) = \int_{E \cap B(0,1)} \sqrt{x^2+y^2+z^2} \, d\lambda_3(x,y,z) \) for any measurable set \( E \subset \mathbb{R}^3 \), and \( \lambda_3 \) is the Lebesgue measure in \( \mathbb{R}^3 \).

Answer: Yes


Question: Calculate the gradient of the function \( f(X) = \text{tr}((AX)^t (AX)) \) with respect to the matrix \( X \).

Answer: 2 A^\top A X


Question: Determine whether there exists a natural number $n$ with arbitrarily many prime factors such that $n$ divides the $n$-th Fibonacci number $F_n$.

Answer: Yes


Question: Determine the values of \(a\) for which the integral \(\int_B ||x||^{-a} \, dx\) exists, where \(B = \{ x \in \mathbb{R}^2 : ||x|| \leq 1 \}\).

Answer: a < 2


Question: Let \( p: (E, e_0) \to (B, b_0) \) be a covering map. Is the induced homomorphism between the first singular homology groups \( H_1(p) : H_1(E) \to H_1(B) \) injective?

Answer: No


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

Answer: 81


Question: What is the largest prime factor of \(\tau(20!)\), where \(\tau(n)\) is the number of divisors of \(n\)?

Answer: 19


Question: Let $X$ and $Y$ be metric spaces, and let $f: X \to Y$ be a continuous and closed map. Is it true that the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$?

Answer: Yes


Question: Calculate the number of 10-letter words that can be formed from a 26-letter alphabet, where no letter appears more than twice consecutively.

Answer: 139557041406250


Question: Find the last two non-zero digits of $70!$. Provide your answer as a two-digit number.

Answer: 68


Question: Evaluate the infinite series \( \sum_{r=1}^{\infty} \frac{(2r+1)^2}{2^r} \).

Answer: 33


Question: Determine whether the function \(f(x) = x^2 \sin\left(\frac{5x+4}{x}\right)\) is differentiable at \(x = 0\).

Answer: Yes


Question: Consider the space $\ell^{\infty}$ and the $\lim$ functional defined on the subspace of convergent sequences. Is the extension of this functional, which assigns zero to all non-convergent sequences, a bounded functional on $\ell^{\infty}$?

Answer: No


Question: Evaluate the limit \( \lim_{a \to \infty} F(a) \), where \( F(a) = \int_0^a (x^2+1)e^{-x/2} \, dx \).

Answer: 18


Question: From a point \(P=(3,4)\), perpendiculars \(PQ\) and \(PR\) are drawn to the line \(3x+4y-7=0\) and a variable line \(y-1=m(x-7)\) respectively. Find the maximum area of \(\triangle PQR\).

Answer: 9


Question: Evaluate \( \int_0^1 x f(x) \, dx \) given that \( \int_0^1 F(x) \, dx = 10 \), \( F'(x) = f(x) \), and \( F(1) = e \).

Answer: e - 10


Question: Calculate the Hausdorff dimension of the set \( \{0, 1, \frac{1}{4}, \frac{1}{9}, \ldots\} \).

Answer: 0


Question: Evaluate the limit \( \lim_{n \to \infty} \left( \frac{1^k + 2^k + \cdots + n^k}{n^k} - \frac{n}{k+1} \right) \) where \( k \in \mathbb{N} \).

Answer: \dfrac{1}{2}


Question: Find the value of \(\alpha\) given by the series \(\sum_{a=2}^\infty\left( \lim_{n \to \infty} \frac{g_n(f_n(a))}{a!} \right)=\alpha\), where for \(n \in \mathbb{N}\), \(g_n(x)= x^{\frac{1}{n}}\) and \(f_n(a)= \int_0^1(1-x^a)^n dx\).

Answer: e - 2


Question: Calculate the value of \( \sin(\pi/2 + i\ln 2) \).

Answer: \dfrac{5}{4}


Question: Determine the rank of the projection map \( \pi : \mathbb{S}^{n} \subseteq \mathbb{R}^{n+1} \rightarrow \mathbb{P}^{n} \mathbb{R} \), where \( \pi \) is obtained by restricting the projection map \( \mathbb{R}^{n+1} \setminus \{0\} \rightarrow \mathbb{P}^{n} \mathbb{R} \). What is the rank of \( \pi \) at any given point?

Answer: n


Question: Evaluate the integral: \[ f(u) = \int_{-\infty}^{\infty} \frac{e^{-itu}}{\sqrt{1+t^2}} \, dt \]

Answer: 2 K_0(|u|)


Question: Evaluate the triple integral \( \iiint_V xyz \, dxdydz \), where \( V \) is the volume in \( \mathbb{R}^3 \) defined by the inequalities \( x^2+y^2+z^2\leq1, \quad x\geq0, \quad y\geq0, \quad z\geq0. \) Use spherical coordinates to perform the integration.

Answer: \dfrac{1}{48}


Question: Let \( \rho : S_n \rightarrow \text{GL}(n, \mathbb{C}) \) be the homomorphism mapping a permutation \( g \) to its permutation matrix. Define \( \chi(g) = \text{Trace}(\rho(g)) \). Compute the value of \( \langle \chi, \chi \rangle = \frac{1}{n!} \sum_{g \in S_n} \chi(g)^2 \). Is this value always 2 for any \( n \)?

Answer: 2


Question: Find the limit: \[ \lim_{n \to \infty} \sum_{k=0}^n \frac{{n\choose k}}{n^k(k+3)} \]

Answer: e - 2


Question: What is the maximum number of positive integers such that any three of them can be the sides of an obtuse triangle? Ensure that for any three integers \(a, b, c\), the conditions \(a^2 + b^2 < c^2\) and \(a + b > c\) are satisfied.

Answer: 3


Question: Compute the integral \( \int_{0}^{\infty}{(2x^2+1)e^{-x^2}}dx \) using techniques of integral calculus, without employing the error function. Verify that the result is \( \sqrt{\pi} \).

Answer: \sqrt{\pi}


Question: Determine whether the product \( p = \prod_{j=2}^\infty \zeta(j)^{-1} \) is greater than zero, assuming the product converges.

Answer: p > 0


Question: Calculate the volume of the region \( T = \{(x,y,z) \in \mathbb{R}^3 : 0 \leq z \leq x^2 + y^2, (x-1)^2 + y^2 \leq 1, y \geq 0\} \).

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


Question: Solve the initial value problem using the Laplace transform: \[ y'' + 3ty' - 6y = 1, \quad y(0) = 0, \; y'(0) = 0 \]

Answer: \dfrac{t^2}{2}


Question: Evaluate \( \lim_{n\to\infty} n^2 \int_0^1 x^n f(x) \, dx \) given that \( f(x) = (x-1) e^{-\frac{1}{x+5}} \) and \( f(1) = 0 \).

Answer: -e^{-\frac{1}{6}}


Question: Evaluate the limit: \[ \lim_{x\to+\infty} \frac{\sqrt{x}(\sin{x}+\sqrt{x}\cos{x})}{x\sqrt{x}-\sin(x\sqrt{x})} \]

Answer: 0


Question: Determine whether there exists an additive function on the real numbers that is unbounded everywhere.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{x\to0}\frac{\sin\left(1-\frac{\sin(x)}{x}\right)}{x^2}. \]

Answer: \dfrac{1}{6}


Question: Given a prime number $q$ and another prime $p = 20q + 1$, determine whether $-1$ has a square root in the multiplicative group of integers modulo $p$, denoted as $\mathbb{Z}_p^*$. Provide a justification for your answer.

Answer: Yes


Question:
Given the system of equations:
\[\begin{cases}
ab+c+d = 13\\
bc+d+a = 27\\
cd+a+b = 30\\
da+b+c = 17
\end{cases}\]
Compute the value of \(a+b+c+d\).

Answer: 16


Question: Evaluate the integral \( \int_{[0,1]} (2 - \omega) \, d\mu \), where \( \mu = 2\lambda_1 + 3\delta_1 \) is a measure on the space \(([0,1], \mathcal{B}([0,1]))\).

Answer: 6


Question: Is there a graph, without loops or parallel edges, that has half the number of vertices as its dual graph? Provide a justification for your answer.

Answer: No


Question:
A group of 20 children, consisting of 10 boys and 10 girls, each receive one hat from a collection of 20 hats: 10 red, 5 blue, and 5 green. After receiving their hats, the children line up randomly. What is the probability that no two children wearing green hats are adjacent in the lineup? Choose the correct answer from the following options:

1) \( \frac{89}{323} \)
2) \( \frac{90}{323} \)
3) \( \frac{91}{323} \)
4) \( \frac{92}{323} \)

Answer: C


Question: Evaluate the integral \( \int\limits_1^3 \frac{\lfloor x^2 \rfloor}{\lfloor (x-4)^2 \rfloor + \lfloor x^2 \rfloor} \, dx \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

Answer: 1


Question: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a locally Lipschitz function. By Rademacher's theorem, $f$ is almost everywhere differentiable. Assume that $\frac{\partial f(x)}{\partial x_i} \geq 0$ for all $i = 1, \ldots, n$ whenever $f$ is differentiable at $x$. Is $f$ nondecreasing with respect to all the variables?

Answer: Yes


Question:
For any given \( \alpha > 0 \), does there exist a sequence \( a_j \) such that \( 0 < a_j < 1 \) and \( 0 < \prod_{j=1}^\infty a_j < 1 \), with the property that the convergence rate of the product satisfies:
\[
\prod_{j=1}^n a_j - \prod_{j=1}^\infty a_j = O(n^{-\alpha})?
\]

Answer: Yes


Question: Determine whether the following statement is true or false: If \( \sum\frac{a_n}{n} \) converges, then \( \frac{a_1+\cdots+a_n}{n} \) converges to 0.

Answer: True


Question: In a grid with coordinates, you start at (0,0) and aim to reach (8,14) by moving only up or right. You cannot pass through any points where both coordinates are odd, such as (1,1), (1,3), (3,1), etc. Calculate the total number of valid paths from (0,0) to (8,14) under these conditions.

Answer: 330


Question: Suppose $a$, $b$, $p$, and $q$ are natural numbers such that $a < 300$ and $\gcd(p, q) = 1$. Is it possible for the expression \( k = \frac{900q^2 + ap^2}{3q^2 + b^2 p^2} \) to be an integer?

Answer: Yes


Question: Evaluate the integral \( \int_{-\infty }^{\infty }\dfrac{dx}{(x^{2}+1)^{3}}. \)

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


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

Answer: 2


Question: Let \( p \) be an odd prime and \( k \in \mathbb{Z}^+ \). If \( pk + 1 \) divides \( \frac{p^2 - 1}{2} \), does it necessarily follow that \( k = 1 \)?

Answer: Yes


Question: Let \( f(x) = 2x^3 + 7x - 1 \), and let \( g(x) \) be the inverse of \( f(x) \). Find \( g'(191/4) \).

Answer: \dfrac{2}{89}


Question: Evaluate the limit: \[ \lim_{n \to \infty} \frac{n(n^{1/n}-1)}{\ln (n+2)! - \ln n!}. \]

Answer: \dfrac{1}{2}


Question:
Compute the value of the double integral \( \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) \, dx \, dy \), where the function \( f(x,y) \) is defined as follows:
\[
f(x,y) =
\begin{cases}
 g(x)/x,&\text{if}\, 0 < y < x\\
 0, &\text{otherwise}
\end{cases}
\]
and \( g(x) \) is a non-negative function defined on \((0, \infty)\) with \( \int_{0}^{\infty} g(x) \, dx = 1 \).

Answer: 1


Question:
Find all functions \( f: \mathbb{Q}_+ \to \mathbb{Q}_+ \) that satisfy the following conditions:

1. \( f(x+1) = f(x) + 1 \) for all \( x \in \mathbb{Q}_+ \).
2. \( f(x^2) = f(x)^2 \) for all \( x \in \mathbb{Q}_+ \).

Answer: f(x) = x


Question: Evaluate the limit: \[ \lim_{x\to \sqrt{2}}\left(\frac{e^{x^2}+e^2(1-x^2)}{[\ln(x^2-3\sqrt{2}x+5)]^2}\right) \]

Answer: 2e^2


Question: Determine the real function $f: \mathbb{R} \to \mathbb{R}$ such that \[ 3f(x) + 2 = 2f(\left \lfloor{x}\right \rfloor) + 2f(\{x\}) + 5x, \] where $\left \lfloor{x}\right \rfloor$ is the floor function and $\{x\} = x - \left \lfloor{x}\right \rfloor$ is the fractional part of $x$. Find $f(x)$.

Answer: 5x + 2


Question: Calculate the limit \( \lim \limits_{n \to \infty} |\sin(\pi \sqrt{n^2+n+1})| \).

Answer: 1


Question: Evaluate the limit: \[ \lim_{x\to0}{\frac{\ln(2-\cos(x))}{\cosh(x)-1}} \]

Answer: 1


Question: Let \( X \) be the number of failures before the first success in a Bernoulli trial with success probability \( p \), i.e., \( X \sim \text{Geo}(p) \). Define \( B \sim \text{Bin}(2X, \frac{1}{2}) \). Prove that \( P(B = X) = \sqrt{p} \).

Answer: \sqrt{p}


Question: Consider the hyperbola \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \). A tangent at any point \( P \) on this hyperbola meets another hyperbola at points \( A \) and \( B \). If \( P \) is the midpoint of \( AB \) for every choice of \( P \), find the floor of the sum of all possible values of the eccentricities of this new hyperbola.

Answer: 2


Question: Let $X_{n1}, X_{n2}, \ldots, X_{nn}$ be independent Poisson random variables with mean $\frac{1}{n}$. Define $X = X_{n1} + X_{n2} + \ldots + X_{nn}$, which is a Poisson random variable with mean 1. Let $M_n = \max\{X_{n1}, X_{n2}, \ldots, X_{nn}\}$. Find \( \lim_{n \to \infty} \mathbb{P}(M_n > \frac{1}{2}) \).

Answer: 1 - \frac{1}{e}


Question: Let $M$ be a closed subset of $\Bbb{R}^n$ consisting of disjoint embedded smooth boundless manifolds, each with codimension at least $1$. Suppose the following two properties are satisfied: (1) Any ray starting from $O$ intersects $M$; (2) If the ray $\overrightarrow{OP}$ intersects $M$ at $P$, then there exists a circle $C$, centered at $O$ and passing through $P$, contained in $M$. Must $M$ contain a sphere $S$ centered at $O$?

Answer: Yes


Question: Let X and Y be independent random variables, each uniformly distributed on the interval [0,1]. Given that the smaller of X and Y is less than 1/4, find the conditional probability that the larger of X and Y is greater than 3/4.

Answer: \dfrac{2}{7}


Question: Given a circle with the equation \(x^2 + y^2 = 4\) and an ellipse with the equation \(2x^2 + 5y^2 = 10\), find the value of \(k^2 + n^2\) for their mutual tangent line \(y = kx + n\).

Answer: 14


Question: Given a $k$-variate normal distribution with mean vector $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$, calculate the variance of a single draw $\mathbf{X} = (X_1, X_2, \ldots, X_k)^T$ from this distribution. Express the variance in terms of $\mathbf{\Sigma}$. Provide your answer as a single expression.

Answer: \mathbf{\Sigma}


Question:
Find the minimum value of the expression:
\[ \sum_{cyc} \sqrt{a^2 + b^2 + ab - 2a - b + 1} \]
where \(0 < a, b, c < 1\).

Answer: \dfrac{3}{2}


Question: Evaluate the improper integral \( \int_a^b \frac{dt}{\sqrt{(b-t)(t-a)}} \) for \( a, b \in \mathbb{R} \) and \( a < b \).

Answer: \pi


Question: Determine the number of solutions to the equation \(\ln(x+2) = e^x - 2\).

Answer: 2


Question: Evaluate the expression \( \frac{\partial}{\partial f}\left( f' + f \right) \) where \( f: \mathbb{R} \to \mathbb{R} \) and \( f' = \frac{df}{dt} \). Specifically, determine the value of \( \frac{\partial f'}{\partial f} \).

Answer: 0


Question:
Determine if the following two double integrals, which describe a triangle with vertices at \((2, 10), (5, 10),\) and \((2, 4)\), yield the same result:
\[ \int_{4}^{10}\int_{2}^{y/2} (5x + 2y) \, dx \, dy \]
\[ \int_{2}^{5}\int_{2x}^{10} (5x + 2y) \, dy \, dx \]

Answer: Yes


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

Answer: \infty


Question: Find the constant \( a \) such that \( a(X_1 - X_2 + 1)^2 \) has a chi-squared distribution, where \( X_1 \sim N(1,1) \) and \( X_2 \sim N(2,2) \) are independent random variables.

Answer: \dfrac{1}{3}


Question: Let $x_n$ be a convergent sequence $x_n \to x$ in a topological space $X$ and $F:X\longrightarrow \mathbb R$ be any function (not necessarily continuous). If there exists a subsequence $x_{n_k}$ such that \(F(x) \leq \liminf_k F(x_{n_k})\), can we conclude that \(F(x) \leq \liminf_n F(x_n)\)? Provide a justification for your answer.

Answer: No


Question: For positive random variables \((X_1, Y_1)\) and \((X_2, Y_2)\), suppose that \((X_1, Y_1)\) and \((X_2, Y_2)\) have the same distribution and are independent. Also, suppose that \(E[Y_1|X_1] = \theta X_1\). Let \(Z=\frac{Y_1 + Y_2}{X_1+X_2}\). Find \(E[Z]\).

Answer: \theta


Question: Let $X$ and $Y$ be independent random variables with the same geometric distribution, where $\mathbb{P}(X=k)=\mathbb{P}(Y=k)=pq^{k-1}$ for $k\geq 1$, and $q=1-p$. Find the conditional probability $\mathbb{P}(X=k \mid X+Y=n+1)$, where $n$ is a positive integer. Identify the distribution of this conditional probability.

Answer: \dfrac{1}{n}


Question: Given that $x, y, z > 0$ and $x + y + z = xyz$, find the minimum value of the expression \[ \frac{x}{y^2} + \frac{y}{z^2} + \frac{z}{x^2}. \]

Answer: \sqrt{3}


Question: Let \( g(n) \) be the smallest prime that divides \( n^4 + 1 \). Determine the remainder when \( g(1) + g(2) + \ldots + g(2015) \) is divided by 8.

Answer: 7


Question: Calculate the limit: \[ \lim_{(x,y)\to(0,0)}\frac{xy^4}{x^2+y^6} \]

Answer: 0


Question: Let $f_{m,n}(x)$ be a sequence of Lebesgue integrable functions on $\mathbb{R}$, dependent on parameters $m$ and $n$. Suppose $f_{m,n}(x) \to 0$ as $m,n \to +\infty$ for almost every $x \in \mathbb{R}$, and $|f_{m,n}(x)| \le g(x)$ for all $m,n \in \mathbb{N}$ and all $x \in \mathbb{R}$, where $g \in L^1(\mathbb{R})$. Can the Lebesgue dominated convergence theorem be applied to conclude that $\int_\mathbb{R} f_{m,n}(x) \, dx \to 0$ as $m,n \to +\infty$?

Answer: Yes


Question: Calculate the integral \( \int_D{f^*\omega } \) where \( f:D \rightarrow \mathbb{R}^3 \) is given by \( f(u,v)=(u,v,u^2+v^2+1) \) with \( D=[0,1]\times [0,1] \), and \( \omega=y\,\mathrm{d}y\wedge \mathrm{d}z+xz\,\mathrm{d}x\wedge \mathrm{d}z \).

Answer: \dfrac{1}{2}


Question:
Let \(f\colon\mathbb{R}^{n}\to\mathbb{R}\) be a linear map such that \(f(0,0,\dots)=0\). Determine the set \(\{f(x_{1},...,x_{n} ):\sum_{j=1}^n x_{j}^2\leq 1\}\). Choose the correct option:

A. \([-a,a]\) for some \(a\in\mathbb{R}\), \(a\geq 0 \).
B. \([0,1]\)
C. \([0,a]\) for some \(a\in\mathbb{R}\), \(a\geq 0 \).
D. \([a,b]\) for some \(a\in\mathbb{R}\), \(0\leq a<b \).

Answer: A


Question: A ball is passed around a circular table with 10 seats, each occupied by a person. In the first run, the ball moves to the $1^{1} = 1$st position. In the second run, it moves an additional $2^{2} = 4$ positions, ending up in the 5th position. In the third run, it moves another $3^{3}$ positions, ending up in the 2nd position. Determine the position of the ball after the 2017th run.

Answer: 1


Question: Find the minimal integer \( k \) such that the number of \( q \)-torsion points of a non-supersingular elliptic curve \( E \) over \( GF(q) \) is equal to \( q \).

Answer: 1


Question: Evaluate the contour integral \( \int_{C}\frac{e^{\pi z}}{\frac{(z-4i)^2}{z-i}}dz \), where \( C \) is the circle \( x^2+y^2=9 \), oriented counterclockwise.

Answer: 0


Question: Let $X$ and $Y$ be random variables on the same probability space such that $Y \ge 0$ almost surely and $X+Y$ and $X$ have the same distribution. Prove or disprove that $\mathbb{P}(Y=0)=1$.

Answer: \mathbb{P}(Y=0)=1


Question: Determine the range of positive values of \( \alpha \) for which the function \((x+y)^{-3}\) is integrable over the region defined by \(0 < x < 1\) and \(0 < y < x^\alpha\).

Answer: (2, \infty)


Question: Evaluate the limit: \[ L = \lim_{x\rightarrow{\pi/2}}{(\cos{x})^{(\pi/2-x)}} \]

Answer: 1


Question: Calculate the limit: $$\lim_{x\to \infty} \left((x+1) \cos \left(2\cos ^{-1}\left(\frac{x}{x + 1}\right) \left(a - \frac{1}{2}\right)\right) - x\right)$$ using cosine expansions.

Answer: 4a(1 - a)


Question: Let $\mu$ be a Radon measure on $[0, 1]$, and $f: [0, 1] \to \mathbb{R}$ a Borel measurable function. Determine whether it is true that for $\mu$-almost every $x \in [0, 1]$, the inequality \( f(x) \leq \mu\text{-esssup}_{[0, x]} \, f \) holds, where the essential supremum is taken with respect to $\mu$. Provide a justification for your answer.

Answer: True


Question: Find the limit \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{1}{2k-1} - a \sum_{k=1}^{n} \frac{1}{k} \right) \) such that the sequence is bounded. Determine the value of \( a \) that ensures the sequence is bounded and find the limit. The correct answer should be \( \ln 2 \).

Answer: \ln 2


Question: Let A be a symmetric 3x3 matrix with eigenvector (0, 0, 1) corresponding to eigenvalue \( \lambda_1 = 0 \) and eigenvector (2, 1, 0) corresponding to eigenvalue \( \lambda_2 = 1 \). The third eigenvalue is negative. Given the vector \( b = (2, 1, 0) \), find one solution for the equation \( Ax = b \).

Answer: (2, 1, 0)


Question: Given the parametric equations \( x(t) = t^2 + 2t \) and \( y(t) = 3t^4 + 4t^3 \) for \( t > 0 \), find the value of the second derivative \( \frac{d^2y}{dx^2} \) at the point \( (8, 80) \).

Answer: 4


Question: Does there exist a sequence \( \{A_n\} \) of subsets in \( B(H) \) that satisfies the DS property, where \( H \) is a separable Hilbert space?

Answer: Yes


Question:
Given that three roots of the equation \(x^4 - px^3 + qx^2 - rx + s = 0\) are \(\tan A\), \(\tan B\), and \(\tan C\), where \(A\), \(B\), and \(C\) are the angles of a triangle, determine the fourth root of the equation.

Choose the correct option:

- \(\frac{p-r}{1-q+s}\)
- \(\frac{p-r}{1+q-s}\)
- \(\frac{p+r}{1-q+s}\)
- \(\frac{p+r}{1+q-s}\)

Answer: A


Question: Given the polynomial \(1-x+x^2-x^3+\ldots-x^{17}\), express it in terms of \(y = x + 1\) as \(a_0 + a_1y + a_2y^2 + \ldots + a_{17}y^{17}\). Determine the value of \(a_{17}\).

Answer: -1


Question: Let $M$ be an abelian group with a torsion subgroup $M_{\text{tors}}$. Suppose there exists a homomorphism $n^{-1}: M \to M$ that serves as an inverse to multiplication by $n$. Is this condition sufficient to ensure that $M_{\text{tors}} = 0$, meaning that $M$ is torsion-free?

Answer: No


Question:
Given the functional equation for any rational numbers $x$ and $y$:
\[ f(x+y) = f(x)f(y) - f(xy) + 1 \]
where $f(x)$ is a real number, and it is known that $f(2017) \neq f(2018)$. Additionally, it is given that:
\[ f\left(\frac{2017}{2018}\right) = \frac{a}{b} \]
where $a$ and $b$ are coprime integers. Find the value of $a-b$.

Answer: 2017


Question: Determine whether the mixed partial derivatives of the function \( f(x,y) = \begin{cases} \frac{x^2y^3}{x^2+y^2} & \text{if} \ (x,y) \neq (0,0) \\ 0 & \text{if} \ (x,y) = (0,0) \end{cases} \) are equal at the point \((0,0)\). Specifically, check if \( \frac{\partial^2 f}{\partial x \partial y}(0,0) = \frac{\partial^2 f}{\partial y \partial x}(0,0) \).

Answer: Yes


Question: Determine whether the following statement is true or false: If $F$ is a coherent sheaf on a noetherian scheme $X$ and $F_x = 0$ for a point $x \in X$, then there exists a neighborhood $U$ of $x$ such that the restriction of $F$ to $U$ is zero.

Answer: True


Question: Maximize the expression \( A = \sqrt{1+x^2} + \sqrt{1+y^2} + \sqrt{1+z^2} + 2(\sqrt{x} + \sqrt{y} + \sqrt{z}) \) given that \( x, y, z \) are positive numbers satisfying \( x+y+z \le 3 \). Find the maximum value of \( A \).

Answer: 6 + 3\sqrt{2}


Question: Consider the set of bijections \( \phi: \mathbb{N} \cup \{0\} \rightarrow \mathbb{Q} \). For any such bijection, define the step size as \( \sup_{n \geq 0} |\phi(n+1) - \phi(n)| \). Determine the smallest possible step size, i.e., find \[ \inf_{\phi \in \Omega} \sup_{n \geq 0} |\phi(n+1) - \phi(n)|. \]

Answer: 0


Question: Determine whether there exists a strictly positive measure on every countable Boolean Algebra.

Answer: Yes


Question: Evaluate the limit: \[ \lim_{n\to\infty}\frac{n!}{n^{n/2}} \]

Answer: \infty


Question: Find the remainder when \( (45^{17})^{17} \) is divided by 204.

Answer: 45


Question: Prove that for any prime number $p > 17$, the expression $p^{32} - 1$ is divisible by 16320.

Answer: 16320


Question: Find the remainder when \( \binom{40}{12} \) is divided by 7.

Answer: 5


Question: There are \( n \) men and \( n \) women. Each man chooses \( k \) women and each woman chooses \( k \) men, where \( k \) is given. A meeting occurs if and only if a man and a woman choose each other. Find the expected number of meetings.

Answer: k^2


Question: Calculate \( \mathbb{E}[Z^2] \) where \( Z = \frac{X - \mu}{\sigma} \) and \( X \sim N(\mu, \sigma^2) \).

Answer: 1


Question: Find the minimum value of \( \frac{a+b}{2} + \frac{2}{ab-b^{2}} \), where \( a, b \in \mathbb{R} \) and \( a > b > 0 \).

Answer: 3


Question: Prove that if \( \|A + C \, \cos\pi z\| \leq e^{\|z\|} \) for all complex numbers \( z \), where \( A \) and \( C \) are constants, then \( C = 0 \).

Answer: C = 0


Question: Find the sum of the infinite series: \[ \sum\limits_{n=0}^{\infty} (2n+1) \left(\frac{1}{2}\right)^n. \]

Answer: 6


Question: Determine the values of \(a\) for which the improper integral \(\int_{-\infty}^\infty \frac{e^{-at}}{1+e^t} \, \mathrm{d}t\) converges.

Answer: (-1, 0)


Question: Find the maximum value of \( k \in \mathbb{R} \) such that the inequality \[ \frac{a}{1+9bc+k(b-c)^2} + \frac{b}{1+9ca+k(c-a)^2} + \frac{c}{1+9ab+k(a-b)^2} \geq \frac{1}{2} \] holds for all non-negative \( a, b, c \) with \( a + b + c = 1 \).

Answer: 4


Question: Given a region in 3-D space with a density function \( f_{x,y,z}(X,Y,Z) = \begin{cases} 1 & \text{if $(x,y,z)\in W$}; \\ 0 & \text{if $(x,y,z)\notin W$}; \end{cases} \), where \( W \) is the set of points inside the pyramid with vertices at \((0,0,0)\), \((2,0,0)\), \((0,3,0)\), and \((0,0,1)\), find the probability \( P\{\frac{1}{3}<Z<\frac{2}{3}\} \).

Answer: \dfrac{7}{27}


Question: Given a single scaled Cauchy distribution \( f_{X|\theta}(s)=\frac{2}{\tau} \cdot \frac{\theta}{\theta^2+s^2} \) and the hypotheses \( H_0: \theta=1.4 \) versus \( H_1: \theta=2.3 \), calculate the Bayes Factor for \( H_0 \) versus \( H_1 \) given that a single data point \( X = 4.1 \) is drawn.

Answer: 0.717


Question: Given that $7^{45} \equiv 32 \pmod{101}$, find the inverse of 32 modulo 101 using the square and multiply algorithm.

Answer: 60


Question: Determine whether the Sylow $p$-subgroups of $S_4$ are also Sylow $p$-subgroups of $S_5$. Justify your answer.

Answer: Yes


Question: Given a closed convex cone $C \subseteq \mathbb{R}^n$, suppose that for every $x$ in the non-negative orthant $\mathbb{R}_{0+}^n$, there exists a $y \in C$ such that $x \cdot y > 0$. Does it follow that the intersection of $C$ with the non-negative orthant contains more than just the origin?

Answer: Yes


Question: Find the exact value of \( \lim_{n\to \infty} \sqrt[n]{(1+1/n)(1+2/n)\cdots(1+n/n)} \).

Answer: \dfrac{4}{e}


Question: If the sequence \( \{t_n\} \) defined by \( t_n = \frac{s_1 + s_2 + \ldots + s_n}{n} \) converges, where \( s_n = a_1 + a_2 + \ldots + a_n \) for a sequence of real numbers \( a_1, a_2, \ldots \), does the sequence \( \{s_n\} \) also converge?

Answer: No


Question:
Solve the radical equation for \( x \):
\[ \frac{x}{\sqrt{x^2+1}} = x^4 - x \]

Answer: 0


Question: Determine the value of \(v\) that minimizes the function \(E(v) = (av^3)\left(\frac{L}{v-u}\right)\), where \(a\), \(L\), and \(u\) are constants.

Answer: \dfrac{3u}{2}


Question: Evaluate the integral: \[ \int_0^{\pi/2} \frac{1}{1+\tan^{2017} x} \, dx \]

Answer: \dfrac{\pi}{4}


Question: Let \( X = (C[0, \pi], \| \cdot \|) \) be the space of continuous functions on \([0, \pi]\) with the supremum norm. For \( f \in C[0, \pi] \), define the operator \( T \) by \( T(f)(x) = \int_0^x \sin(x+y)f(y) \, dy \). Find the operator norm \( \| T \| \).

Answer: 2


Question: Suppose we roll a fair 6-sided die repeatedly. What is the expected number of rolls required to see 3 of the same number in succession?

Answer: 43


Question: Find the value of \( m \geq 0 \) such that the equations \( \sin^4 x + \cos^4 x = m \) and \( \sin^6 x + \cos^6 x = m \) have the same solutions.

Answer: 1


Question: Let $A$ and $B$ be subsets of a set $M$ and let $S_0=\{A,B\}$. For $i\geq 0$, define $S_{i+1}$ inductively to be the collection of subsets $X$ of $M$ that are of the form $C\cup D, C\cap D,$ or $M-C$ (the complement of $C$ in $M$), where $C,D \in S_i$. Let $S=\cup^\infty_{i=0} S_i$. What is the largest possible number of elements of $S$?

Answer: 16


Question: Compute the limit: $$\lim_{n\to\infty} \left[\ln\left(\frac{1}{0!}+\frac{1}{1!}+\cdots+\frac{1}{n!}\right)\right]^n$$

Answer: 1


Question: Find the limit of the sequence sum \( X(n) = \frac{1}{2} + \frac{3}{4} + \frac{5}{8} + \cdots + \frac{2n-1}{2^n} \) as \( n \to \infty \). Determine \( \lim_{n \to \infty} X(n) \).

Answer: 3


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

Answer: \mathbb{F}_8


Question: Find the sum of the residues of the rational function \( h(z) = \frac{5z^{2015} + 7z^{2010} - 38z^5 + z^4 - 2z^3 + 5}{7z^{2016} + 2z^{2013} - 6z^8 + 3z^7 + z^5 - 4z^2 - z + 111} \) at its poles in the complex plane.

Answer: \dfrac{5}{7}


Question: Suppose \( f \) is a harmonic function on \( \mathbb{R}^2 \) and is constant on a neighborhood in \( \mathbb{R}^2 \). Is \( f \) constant on the entire \( \mathbb{R}^2 \)? Provide a justification for your answer.

Answer: Yes


Question: Let $f \in L^p(\mathbb{R}^d)$, with $1 < p < \infty$. Determine whether the following statement is true: \[ \lambda^{\frac{d}{p}}f(\lambda x ) \rightharpoonup 0\quad \text{ weakly in }L^p\text{ as }\lambda\to+\infty. \]

Answer: True


Question:
Determine the number of solutions of the integral equation \( \phi (x) = \cos(7x) +  \lambda \int_{0}^{\pi} \left[ \cos(x)\cos(t) - 2\sin(x)\sin(t) \right]\phi(t) \, dt \) depending on the value of \( \lambda \). Choose the correct option:

1. Solution exists for every value of \( \lambda \).
2. There is some \( \lambda \) for which a solution does not exist.
3. There is some \( \lambda \) for which more than one but finitely many solutions exist.
4. There is \( \lambda \) such that infinitely many solutions exist.

Answer: 4


Question: Determine the value of $n$ for which the polynomial $x^3-x$ is divisible by $2x-1$ in the ring of polynomials $mod \, n$. Provide your answer as a single integer $n$.

Answer: 3


Question:
Calculate the volume of the body bounded by the surface defined by the inequality:
\[ 0 \leq z \leq c \cdot \sin\left(\pi \sqrt{\frac{x^2}{a^2} + \frac{y^2}{b^2}}\right) \]
where \(a\), \(b\), and \(c\) are positive parameters.

Answer: 2abc


Question: Find \(\lim\limits_{x\to +\infty} \frac{x^{2}e^{-x}}{x^{2}+1}\sin(xe^{x^{2}})\).

Answer: 0


Question: Let $a, b, c$ be orthonormal vectors in $\mathbb{R}^n$. Determine whether the sum $\sum_{k=1}^n a_{k}b_{k}c_{k}$ is equal to zero.

Answer: 0


Question: Is the map \( f_3 \) defined by \( f_3(n) = \max_{\phi \in Q_n} (c(\phi)) \) computable, where \( Q_n \) is the set of statements such that there is a proof (in ZFC') of \( ZFC \vdash \phi \) with length \( \leq n \)?

Answer: Yes


Question: Determine whether the following statement is true: For a self-adjoint element $a$ in a $C^*$ algebra, there exist positive elements $a_+$ and $a_-$ such that $a = a_+ - a_-$ and $a_+a_- = a_-a_+ = 0$. Provide a justification for your answer.

Answer: True


Question: Determine the automorphism group of the complete bipartite graph $\mathcal{V}_{m,n}$, where $m, n \in \mathbb{N}$ and $m \ne n$. A complete bipartite graph is a bipartite graph where every vertex of the first set is connected to every vertex of the second set.

Answer: S_m \times S_n


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

Answer: 2\pi i \sin 1


Question:
Suppose $A$ is a finitely generated $\mathbb{Z}$-algebra and $R$ is a finitely generated $A$-algebra. Consider a sequence of finitely generated $R$-modules:
\[ M_1 \rightarrow M_2 \rightarrow M_3 \]
where the composite of the maps in the sequence is zero. If the sequence becomes exact at $M_2 \otimes \mathrm{Frac}(A)$ when tensored with the fraction field of $A$, does it follow that the sequence is exact at $M_2 \otimes A_a$ for some nonzero $a \in A$?

Answer: Yes


Question:
Let \( X_n \) be i.i.d. random variables uniformly distributed on \( (0, 2) \). Find the limit:
\[ \lim_{n \rightarrow \infty} P\left( \frac{X_1^3 + \ldots + X_n^3}{X_1 + \ldots + X_n} < \frac{1}{2} \right) \]

Answer: 0


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

Answer: 1


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

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


Question:
Determine if there exist paths for which the following two line integrals are equal to zero:
\[ \int_C xy^2 \, dx + y \, dy = 0 \]
\[ \int_C \frac{-y \, dx + x \, dy}{x^2 + y^2} = 0 \]

Answer: Yes


Question: Let $X$ be a complete metric space such that the distance between any two points is less than $1$. Is $X$ necessarily compact?

Answer: No


Question: Evaluate the improper integral: \[ \int_0^\infty \frac{1}{1+x^2 \sin^2 x} \, dx. \]

Answer: \infty


Question: Let $f$ be a Riemann integrable function on $[1,b]$ for every $b>1$, and suppose $\int_1^\infty |f| < \infty$. Prove that \[ \lim_{n\to \infty} \int_1^\infty f(x^n)\, dx = 0. \]

Answer: 0


Question: Find the Laurent series of the complex function \( \frac{z}{1-\cos(z)} \) around \( z=0 \). Determine the residue of this function at \( z=0 \).

Answer: 2


Question: Suppose that $f'(z)$ exists on $\mathbb{C}$ and $\Im(f'(z))=6x(2y-1)$, with $f(0)=3-2i$ and $f(1)=5-5i$. Find $f(1+i)$. \(z = x + yi\) is a complex number.

Answer: 5 + 2i


Question: Let \( \psi \) be a 7th primitive root of unity. Determine the number of elements in the set \( \{a = a_1\psi + a_2\psi^2 + a_3\psi^3 + a_4\psi^4 + a_5\psi^5 + a_6\psi^6 : a_i \in (0,1)\} \) such that \( \mathbb{Q}(a) = \mathbb{Q}(\psi) \).

Answer: 54


Question: Find the limit: $$\lim \limits_{n \to \infty} \int \limits^{2006}_{1385}f(nx)\, \mathrm dx$$ where $f\colon [0, +\infty) \to \mathbb{R}$ is a continuous function and $\lim \limits_{x \to +\infty} f(x) = 1$. Determine the value of this limit.

Answer: 621


Question: Given that \( \vec{OA} + \vec{OB} + \vec{OC} = 0 \), \( |\vec{OA}| = \sqrt{6} \), \( |\vec{OB}| = 2 \), and \( |\vec{OC}| = \sqrt{14} \), find the area of triangle ABC.

Answer: 3\sqrt{5}


Question: Decide whether the statement "For a sequence of independent random variables $\{X_n\}$ with $E[X_n] = 0$ and $E[|X_n|] = 1$ for every $n$, it holds that $P(\liminf(X_n) < 0) > 0$" is true or false. Provide a proof or a counterexample.

Answer: False


Question:
Is a discrete uniform distribution fully characterized by its first two moments? Let $U$ be the discrete uniform distribution on $\{0,1,\dots,n\}$. If a discrete random variable $X$ satisfies:
\[
\begin{align}
\mathbb{E}[X] &= \mathbb{E}[U] = n/2 \\
\mathrm{Var}(X) &= \mathrm{Var}(U) = ((n+1)^2-1)/12,
\end{align}
\]
is this sufficient to conclude that $X \equiv U$, given that $X$ has full support over $\{0,1,\dots,n\}$?

Answer: No


Question: Suppose $M$ is a path-connected smooth manifold. Determine whether any two points $p, q \in M$ can be joined by a smooth ($C^{\infty}$) curve on $M$. Provide a justification for your answer.

Answer: Yes


Question: What is the probability that when rolling four four-sided dice, the sum of the results is divisible by 4, and this occurs in each of four sets of rolls?

Answer: \dfrac{1}{256}


Question: Determine whether the following equation holds: \[ \lim_{n\to\infty} \sum_{k=0}^n \binom{n}{k}p^k(1-p)^{n-k}\mathrm{exp}(-2kc) = 0, \] where \(0 < p < 1\) and \(c > 0\) is a constant.

Answer: 0


Question: Kevin needs to compute the product \(9 \times 99 \times 999 \times \cdots \times 999999999\). Kevin takes exactly \(ab\) seconds to multiply an \(a\)-digit integer by a \(b\)-digit integer. What is the minimum number of seconds necessary for Kevin to evaluate the expression by performing eight multiplications?

Answer: 870


Question: In triangle ABC, AC = 6 and BC = 8. The medians drawn from vertices A and B are perpendicular to each other. Find the area of triangle ABC.

Answer: 4\sqrt{11}


Question: Let \( n \) be a positive integer such that \( (n,8)=1 \) and \(-4\) is a square modulo \( n \). Prove that \( n \) can be expressed in the form \( n = x^2 + 4y^2 \) for some integers \( x \) and \( y \).

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


Question: Evaluate the limit: \[ \lim _{n \rightarrow \infty} \left[\frac{1 \cdot 1 !+2 \cdot 2 !+\cdots+n \cdot n !}{(n+1) !}\right]^{(n+1)!} \]

Answer: \dfrac{1}{e}


Question: Determine whether the following statement is true or false: Let \( f \) be a continuous integrable function on \( \mathbb{R} \) such that either \( f(x) > 0 \) or \( f(x) + f(x + 1) > 0 \) for all \( x \in \mathbb{R} \). Then \( \int_{-\infty}^{\infty} f(x) \, dx > 0 \).

Answer: True


Question: Find the value of \( C \in \mathbb{R} \) such that the functions \( f(x) = e^x \) and \( g(x) = \ln(x) + C \) are tangent to each other.

Answer: 2.330


Question: Is every totally disconnected space a discrete topology?

Answer: No


Question: Let $V$ be a real $d$-dimensional vector space, and let $1 < k < d$ be fixed. Consider two bases $v_i$ and $w_i$ for $V$. The basis $v_i$ induces a basis for the $k$-th exterior power $\bigwedge^k V$, given by $v_{i_1} \wedge \dots \wedge v_{i_k}$. Suppose that for each strictly increasing multi-index $I = (i_1, \ldots, i_k)$, there exists a unique multi-index $J = (j_1, \ldots, j_k)$ such that $v^I = \lambda_I w^J$ for some non-zero scalar $\lambda_I$. Is it true that there exists a permutation $\sigma \in S_d$ such that $\text{span}(v_i) = \text{span}(w_{\sigma(i)})$ for all $i$?

Answer: Yes


Question: Determine whether every orthonormal basis is a Riesz basis in a Hilbert space.

Answer: Yes


Question: Find the power series expansion of \( f(z) = \frac{1}{3-z} \) about the point \( 4i \). Determine the radius of convergence of this series.

Answer: 5


Question: Determine the Krull dimension of the local ring \( \mathbb{C}[x,y]_{(0,0)}/(y^2-x^7,y^5-x^3) \).

Answer: 0


Question: Evaluate the limit: \[ \lim_{x\to 0} \frac{1-\cos x \sqrt{\cos 2x}}{x^2} \]

Answer: \dfrac{3}{2}


Question: Evaluate the limit: \( \lim_{n\to\infty} n! \ n^{-\ln(n)} \)

Answer: \infty


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

Answer: e^2


Question: Evaluate the integral \( \int_{-\infty}^{\infty} |1 - |\frac{x}{2}||^2 \, \mathrm{d}x \).

Answer: \infty


Question: Can a finite 2D figure with a nontrivial rotational symmetry have exactly one reflection symmetry?

Answer: No


Question: Find the sum of the infinite series: $$x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + \frac{(-1)^{n+1}x^n}{n} + \cdots$$

Answer: \ln(1 + x)


Question: Find the radius of convergence of the power series \( \sum_{n=1}^{\infty} x^{2^n} \).

Answer: 1


Question: Prove that the inequality \( x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+\frac{3}{4} > 0 \) holds for all real numbers \( x \).

Answer: x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + \frac{3}{4} > 0


Question: Find the value of \(a\) such that the equation \(x^2 = a^x\) has exactly one real root.

Answer: e^{\frac{2}{e}}


Question: Given the sequence defined by $a_1 = a_2 = 1$ and $a_n = (n-1)(a_{n-1} + a_{n-2})$ for $n \geq 3$, find the value of $a_{2020}$.

Answer: 2020! - D_{2020}


Question: Find the sum of all 4-digit perfect squares \(N = (\overline{abcd})\) such that \((\overline{ab}) = 3(\overline{cd}) + 1\). Here, \((\overline{ab})\) and \((\overline{cd})\) represent the two-digit numbers formed by the digits \(a, b\) and \(c, d\) respectively.

Answer: 2809


Question: Determine whether the limit \( \lim_{(x,y)\to (2,-2)} \frac{\sin(x+y)}{x+y} \) exists.

Answer: 1


Question: Assume we have a sequence of i.i.d. random variables $(Y_n)_{n\geq 0}$ with finite expectation. If $\sqrt{n}^{-1}Y_n\rightarrow 0$ almost surely, does it follow that $\sqrt{n}^{-1}Y_{n+m}\rightarrow 0$ almost surely for some $m \in \mathbb{N}$?

Answer: Yes


Question: Evaluate the surface integral \( \iint_{S} \nabla \times \mathbf{F} \cdot \mathbf{n} \, dA \) using Stokes' theorem for the hemisphere \( S: x^{2}+y^{2}+z^{2}=9, \; z\geq 0 \) with its bounding circle \( C: x^{2}+y^{2}=9, \; z=0 \) and the vector field \( \mathbf{F}=y\mathbf{i}-x\mathbf{j} \). Determine the outer unit normal vector to the hemisphere and use it to evaluate the integral.

Answer: -18\pi


Question: Determine the number of zeros of the polynomial \( z^4 + 3z^2 + z + 1 \) that lie in the right half-plane.

Answer: 2


Question: Determine whether the following statement is true or false: If \( \text{Corr}\left(f(X),g(Y)\right)=0 \) for all possible functions \( f(\cdot) \) and \( g(\cdot) \), then the random variables \( X \) and \( Y \) are independent.

Answer: True


Question: A box contains some identical tennis balls. The ratio of the total volume of the tennis balls to the volume of empty space surrounding them in the box is 1:k, where k is an integer greater than one. A prime number of balls is removed from the box, and the new ratio of the total volume of the remaining tennis balls to the volume of empty space is 1:k^2. Find the original number of tennis balls in the box.

Answer: 5


Question: Determine whether there exists an integer polynomial \( f \in \mathbb{Z}[x] \) such that for infinitely many prime numbers \( p \), the condition \( \forall k \in \mathbb{N}: p \not\mid f(k) \) holds.

Answer: Yes


Question: If $A$ and $B$ are two $n \times n$ real matrices such that $A^2 + B^2 = -I_n$, does it necessarily follow that $AB + BA = 0$?

Answer: No


Question: Evaluate the limit \( \lim_{R \to \infty} \int_{|z|=R} \left| \frac{\sin(z)}{z} \right| |dz| \).

Answer: \infty


Question: Consider a one-dimensional Brownian motion \( \{B_1(t):t\geq0\} \). Define \( B^a_1(t):=a^{-1}B_1(a^2t) \) for \( a\in\mathbb{R} \), which is also a Brownian motion. Let \( \{B_2(t):t\geq0\} \) be another independent one-dimensional Brownian motion. Define \( T:=\inf\{t\geq0:B_2(t)=b\} \) and \( T^a:=\inf\{t\geq0:B^a_2(t)=b\} \), where \( B^a_2(t):=a^{-1}B_2(a^2t) \) and \( T \) and \( T^a \) have the same distribution. Determine if \( B_1(T) \) and \( B^a_1(T^a) \) have the same distribution.

Answer: Yes


Question: Given the function \( F(\alpha) = \int\limits_0^{\pi/2} \ln(\alpha^2 - \sin^2 x) \mathrm{d}x \) for \( \alpha > 1 \), and knowing that \( F'(\alpha) = \frac{ \pi }{\sqrt{ \alpha^2 - 1 } } \), find the constant \( C \) such that \( F(\alpha) = \pi \ln \left(\alpha + \sqrt{ \alpha^2 - 1 } \right) + C \).

Answer: -\pi \ln 2


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

Answer: -1


Question: Calculate the integral \( \int_{D} dx \wedge dy \) over the unit disk \( D = \{ v \in \mathbb{R}^2 \mid \Vert v \Vert \leq 1 \} \) using Stokes' theorem.

Answer: \pi


Question:
Find the sum of all values of \(a\) that satisfy the equation:
\[ \int_{0}^{\pi/2} (\sin x + a \cos x)^3 \, dx - \frac{4a}{\pi-2} \int_{0}^{\pi/2} x \cos x \, dx = 2. \]

Answer: -\dfrac{3}{2}


Question: Evaluate the integral: \[ \int_{0}^{2\pi} \frac{1}{\sin^4x + \cos^4x} \, dx \]

Answer: 2\sqrt{2}\pi


Question: Is it true that for any Banach norm \(\|\cdot\|\) on \(\mathbb{R}^n\), the set of points where the function \(f: v \mapsto \|v\|\) is not Fréchet differentiable has Lebesgue measure zero?

Answer: Yes


Question:
Let \( A \) be a 10 by 10 matrix with complex entries such that all its eigenvalues are non-negative real numbers and at least one eigenvalue is positive. Determine which of the following statements is always false:

A. There exists a matrix \( B \) such that \( AB - BA = B \).
B. There exists a matrix \( B \) such that \( AB - BA = A \).
C. There exists a matrix \( B \) such that \( AB + BA = A \).
D. There exists a matrix \( B \) such that \( AB + BA = B \).

Answer: B


Question: Let \( A = \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 7-a^2 \\ 2 & 2+a & 6 \end{pmatrix} \) and \( B = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 9 \end{pmatrix} \), where \( a \in \mathbb{R} \). Find all values of \( a \) for which \( A \) is similar to \( B \).

Answer: 2


Question: Calculate the expected value of the product of the maximum and minimum of two independent uniform random variables X and Y over the interval [0,1]. Specifically, find E[ZW] where Z = max(X,Y) and W = min(X,Y).

Answer: \dfrac{1}{4}


Question: Evaluate the limit: \[ \lim\limits_{x\to 0} \frac{x^2\cos x - 6 \ln(1+x^2) +5x^2}{(e^{\sqrt[4]{1+4x^3+8x^4}}-e)\arcsin(x)} \] without using L'Hospital's rule.

Answer: \dfrac{5}{2e}


Question: Compute the limit: \[ \lim_{n\to\infty}\frac{\frac{n}{1}+\frac{n-1}{2}+\dots+\frac{2}{n-1}+\frac{1}{n}}{\ln(n!)} \]

Answer: 1


Question: Compute the value of \( X = (x \bmod 10) \), where \( x = \gcd(d^{671} + 1, d^{610} - 1) \) and \( d = \gcd(51^{610} + 1, 51^{671} - 1) \). Use the substitution \( y = 51^{61} \) to simplify \( d \) to \( d = (y^{10} + 1, y + 1) \).

Answer: 3


Question: Given that $X$ and $Y$ are standard normal random variables with $\operatorname{Cov}(X,Y) = \frac{1}{\sqrt{2}}$, compute the conditional probability $P(X>0 | Y<0)$. Use the relation $P(X>0 | Y<0) = 2 P(X>0 \text{ and } Y<0)$ to find your answer.

Answer: \dfrac{1}{4}


Question: Consider a double sequence \((a_{m, n})_{m, n}\) of real numbers such that for every \(m\), the sequence \((a_{m, n})_n\) converges to a limit \(x_m\), and for every \(n\), the sequence \((a_{m, n})_m\) converges to a limit \(y_n\). Suppose that for every \(\epsilon > 0\), there exists \(N \in \mathbb{N}\) such that \(|a_{m, n} - x_m| < \epsilon\) and \(|a_{m, n} - y_n| < \epsilon\) for every \(m\), \(n > N\). If both sequences \((x_m)_m\) and \((y_n)_n\) converge, do they converge to the same limit?

Answer: Yes


Question: Is there an extension of number fields $L/K$ such that the relative discriminant ideal $d_{L/K}$ is not principal?

Answer: Yes


Question:
Find the coefficient of \(x^{100}\) in the power series expansion of the function:
\[f(x)=(x+x^2+x^3+ \cdots) \cdot (x^2+x^3+x^4 \cdots) \cdot (x^3+x^4+x^5 \cdots)\]

Answer: 4560


Question: Let \( f: \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R} \) be a symmetric bilinear form such that for all \( g \in M_2(\mathbb{R}) \) and for all \( u, v \in \mathbb{R}^2 \), \( f(gu, gv) = f(u, v) \). Determine the possible values of \( \det(g) \).

Answer: \mathbb{R}


Question: Given a $2n \times 2n$ integer matrix $A$ with no real eigenvalues and eigenvalues $\lambda_1, \overline{\lambda_1}, \dotsc, \lambda_n, \overline{\lambda_n}$, is it possible to construct an $n \times n$ integer matrix $B$ with eigenvalues $\lambda_1 + \overline{\lambda_1}, \dotsc, \lambda_n + \overline{\lambda_n}$? Assume $A$ is diagonalizable.

Answer: Yes


Question: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a non-constant, thrice differentiable function such that \( f(1+1/n)=1 \) for all \( n \in \mathbb{Z} \). Determine the value of \( f''(1) \).

Answer: 0


Question: Find the 6th derivative of the function \( f(x) = \frac{1}{1+\sin(x^2)} \) evaluated at \( x = 0 \).

Answer: -600


Question: Is a $K$-analytic space $X$ countable if every compact subset of $X$ is countable?

Answer: Yes


Question: Find the explicit formula for the sequence \(a_n\) given the initial condition \(a_0 = 0\) and the recurrence relation \(a_{n+1} = (n+1)a_n + 2(n+1)!\) for \(n > 0\).

Answer: 2n \cdot n!


Question: Let $X$ be an inner product space and $z \in X$ be fixed. Define $f(x) = \langle x,z\rangle$. Suppose the mapping $X \to X'$ given by $z \to f$ is surjective, where $X'$ is the dual space of $X$. Does it follow that $X$ is a Hilbert space?

Answer: Yes


Question: Let \( A_1, A_2, \ldots \) be a sequence of measurable sets such that for each \( n \), \( \lambda(A_n) \geq \frac{1}{2} \) and for all \( k \neq s \), \( \lambda(A_s \cap A_k) \leq \frac{1}{4} \). Prove that \( \lambda \left( \bigcup_{k=1}^{\infty} A_k \right) \geq 1 \).

Answer: 1


Question: Let \( B \in \mathbb{R}^{n \times k} \) be a matrix with rank \( k \). Does there exist an invertible matrix \( A \in \mathbb{R}^{n \times n} \) such that \( AB = \begin{bmatrix} I_{k,k} \\ 0_{n-k,k} \end{bmatrix} \), where \( I_{k,k} \) is the \( k \times k \) identity matrix and \( 0_{n-k,k} \) is a zero matrix? Justify your answer.

Answer: Yes


Question: Find a change of variable that will transform the quadratic form \(x_1^2 - x_3^2 - 4x_1x_2 + 4x_2x_3\) into a sum of squares. Express the quadratic form in terms of the new variables.

Answer: y_1^2 - y_2^2


Question: Evaluate the integral \( I = \iint_{D} \frac{(x+y)\ln(1+y/x)}{\sqrt{1-x-y}}\,dx\,dy \), where \( D = \{(x,y) \mid x+y \le 1, x \ge 0, y \ge 0\} \).

Answer: \dfrac{16}{15}


Question: Calculate the limit: $$\lim\limits_{x\to1}{\sqrt{{\pi\over 2}-\arctan{1\over (x-1)^2}}}$$

Answer: 0


Question:
Determine which of the following sequences of functions is uniformly convergent:

A. \( f_n(x) = x^n; \ x \in [0,1] \).

B. \( f_n(x) = 1 - x^n; \ x \in \left[\frac{1}{2}, 1\right] \).

C. \( f_n(x) = \frac{1}{1+nx^2}; \ x \in \left[0, \frac{1}{2}\right] \).

D. \( f_n(x) = \frac{1}{1+nx^2}; \ x \in \left[\frac{1}{2}, 1\right] \).

Answer: D


Question: Find the limit: $$\lim\limits_{t \rightarrow 0}(\cos 2t)^{\frac{1}{t^2}}$$

Answer: e^{-2}