Which of the following are true? $\textbf{(A)}~\exists A\in M_3(\mathbb R)\text{ such that }A^2=-I_3$ $\textbf{(B)}~\exists A,B\in M_3(\mathbb R)\text{ such that }AB-BA=I_3$ $\textbf{(C)}~\forall A\in M_4,\det\left(I_4+A^2\right)\ge0$ $\textbf{(D)}~\text{None of the above}$

Real numbers $x,y,a$ satisfy the equations $$x+y = x^3 +y^3 = x^5 +y^5 = a$$ Find all possible values of $a$.

(a) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $11n$ is twice the sum of the digits of $n$. (b) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $5n + 1$ is six times the sum of the digits of $n$.

Let $\alpha > 1$ be an irrational number and $L$ be a integer such that $L > \frac{\alpha^2}{\alpha - 1}$. A sequence $x_1, x_2, \cdots$ satisfies that $x_1 > L$ and for all positive integers $n$, \[ x_{n+1} = \begin{cases} \left \lfloor \alpha x_n \right \rfloor & \textup{if} \; x_n \leqslant L \\\left \lfloor \frac{x_n}{\alpha} \right \rfloor & \textup{if} \; x_n > L \end{cases}. \] Prove that (i) $\left\{x_n\right\}$ is eventually periodic. (ii) The eventual fundamental period of $\left\{x_n\right\}$ is an odd integer which doesn't depend on the choice of $x_1$.

Let $A$ and $B$ be points in space such that $AB=1.$ Let $\mathcal{R}$ be the region of points $P$ for which $AP \le 1$ and $BP \le 1.$ Compute the largest possible side length of a cube contained in $\mathcal{R}.$

Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.

Assume that a real polynomial $P(x)$ has no real roots. Prove that the polynomial $$Q(x)=P(x)+\frac{P''(x)}{2!}+\frac{P^{(4)}(x)}{4!}+\ldots$$ also has no real roots.

Prove that for all positive integers $n$, $n>1$ the number $\sqrt{ \overline{ 11\ldots 44 \ldots 4 }}$, where 1 appears $n$ times, and 4 appears $2n$ times, is irrational.

A calculating ruler is a ruler for doing algebric calculations. This ruler has three arms, two of them are sationary and one can move freely right and left. Each of arms is gradient. Gradation of each arm depends on the algebric operation ruler does. For eaxample the ruler below is designed for multiplying two numbers. Gradations are logarithmic. [img]http://aycu05.webshots.com/image/5604/2000468517162383885_rs.jpg[/img] For working with ruler, (e.g for calculating $x.y$) we must move the middle arm that the arrow at the beginning of its gradation locate above the $x$ in the lower arm. We find $y$ in the middle arm, and we will read the number on the upper arm. The number written on the ruler is the answer. 1) Design a ruler for calculating $x^{y}$. Grade first arm ($x$) and ($y$) from 1 to 10. 2) Find all rulers that do the multiplication in the interval $[1,10]$. 3) Prove that there is not a ruler for calculating $x^{2}+xy+y^{2}$, that its first and second arm are grade from 0 to 10.

A divisor $d$ of a positive integer $n$ is said to be a [i]close[/i] divisor of $n$ if $\sqrt{n}<d<2\sqrt{n}$. Does there exist a positive integer with exactly $2020$ close divisors?

Three distinct integers are selected at random between $1$ and $2016$, inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd? $\textbf{(A)}\ p<\dfrac{1}{8}\qquad\textbf{(B)}\ p=\dfrac{1}{8}\qquad\textbf{(C)}\ \dfrac{1}{8}<p<\dfrac{1}{3}\qquad\textbf{(D)}\ p=\dfrac{1}{3}\qquad\textbf{(E)}\ p>\dfrac{1}{3}$

Solve the equation: \[3\sqrt{3x-1}=x^2+1\]

Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that \[(f(x))^2 -(f(y))^2 = f(x + y)f(x - y) \text{ for all } x, y \in \mathbb R.\]

Some cities of a country consisting of $n$ cities are connected by round trip flights so that there are at least $k$ flights from any city and any city is reachable from any city. Prove that for any such flight organization these flights can be distributed among $n-k$ air companies so that one can reach any city from any city by using of at most one flight of each air company.

An odd integer $ n \ge 3$ is said to be nice if and only if there is at least one permutation $ a_{1}, \cdots, a_{n}$ of $ 1, \cdots, n$ such that the $ n$ sums $ a_{1} \minus{} a_{2} \plus{} a_{3} \minus{} \cdots \minus{} a_{n \minus{} 1} \plus{} a_{n}$, $ a_{2} \minus{} a_{3} \plus{} a_{3} \minus{} \cdots \minus{} a_{n} \plus{} a_{1}$, $ a_{3} \minus{} a_{4} \plus{} a_{5} \minus{} \cdots \minus{} a_{1} \plus{} a_{2}$, $ \cdots$, $ a_{n} \minus{} a_{1} \plus{} a_{2} \minus{} \cdots \minus{} a_{n \minus{} 2} \plus{} a_{n \minus{} 1}$ are all positive. Determine the set of all `nice' integers.

Let $f$ be a function of the positive reals in the positive reals, such that $$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$ (a) Find $f(1)$. (b) Find $f(x)$.

$$\int_0^{20\pi}|x\sin(x)|\,dx$$ [i]Proposed by Connor Gordon[/i]

A hundred people want to take a photo. They can stand in any number of rows from 1 to 100. Let $N$ be the number of possible photos they can take. What is the largest integer $k$ such that $2^k \mid N$? [i]Lightning 2.4[/i]

Four positive integers $ a,b,c,$ and $ d$ have a product of 8! and satisfy\begin{align*}ab \plus{} a \plus{} b &\equal{} 524\\ bc \plus{} b \plus{} c &\equal{} 146\\ cd \plus{} c \plus{} d &\equal{} 104.\end{align*} What is $ a \minus{} d$? $ \textbf{(A)} \ 4 \qquad \textbf{(B)} \ 6 \qquad \textbf{(C)} \ 8 \qquad \textbf{(D)} \ 10 \qquad \textbf{(E)} \ 12$

Can one draw , on the surface of a Rubik's cube , a closed path which crosses each little square exactly once and does not pass through any vertex of a square? (S . Fomin, Leningrad)

Find all pairs $(x,y)$ of integers which satisfy the equation $(x + y)^2(x^2 + y^2) = 2009^2$

[b]a)[/b] Show that the set of nilpotents of a finite, commutative ring, is closed under each of the operations of the ring. [b]b)[/b] Prove that the number of nilpotents of a finite, commutative ring, divides the number of divisors of zero of the ring.

Find the value of $\frac{2014^3-2013^3-1}{2013\times 2014}$. $ \textbf{(A) }3\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }11 $

In a convex quadrilateral $ABCD$ whose diagonals intersect at point $E$, the equalities$$\dfrac{|AB|}{|CD|}=\dfrac{|BC|}{|AD|}=\sqrt{\dfrac{|BE|}{|ED|}}$$hold. Prove that $ABCD$ is either a paralellogram or a cyclic quadrilateral

One day, before his work time at Jane Street, Sunny decided to have some fun. He saw that there are some real numbers $a_{-1},\ldots,a_{-k}$ on a blackboard, so he decided to do the following process just for fun: if there are real numbers $a_{-k},\ldots,a_{n-1}$ on the blackboard, then he computes the polynomial $$P_n(t)=(1-a_{-k}t)\cdots(1-a_{n-1}t).$$ He then writes a real number $a_n$, where $$a_n=\frac{iP_n(i)-iP_n(-i)}{P_n(i)+P_n(-i)}.$$ If $a_n$ is undefined (that is, $P_n(i)+P_n(-i)=0$), then he would stop and go to work. Show that if Sunny writes some real number on the blackboard twice (or equivalently, there exists $m>n\ge0$ such that $am=an$), then the process never stops. Moreover, show that in this case, all the numbers Sunny writes afterwards will already be written before. (usjl)