$a_{n}$ is a sequence of positive integers such that, for every $n\geq 1$, $0<a_{n+1}-a_{n}<\sqrt{a_{n}}$. Prove that for every $x,y\in{R}$ such that $0<x<y<1$ $x< \frac{a_{k}}{a_{m}}<y$ we can find such $k,m\in{Z^{+}}$.

A convex hexagon $ABCDEF$ is given whose sides $AB$ and $DE$ are parallel. Each of the diagonals $AD, BE, CF$ divides this hexagon into two quadrilaterals of equal perimeters. Show that these three diagonals intersect at one point.

Find the number of integers $n$ between $1$ and $2021$ such that $2^n+2^{n+3}$ is a perfect square.

Consider a (right) square pyramid $ABCDV$ with the apex $V$ and the base (square) $ABCD$. Denote $d=AB/2$ and $\varphi$ the dihedral angle between planes $VAD$ and $ABC$. (1) Consider a line $XY$ connecting the skew lines $VA$ and $BC$, where $X$ lies on line $VA$ and $Y$ lies on line $BC$. Describe a construction of line $XY$ such that the segment $XY$ is of the smallest possible length. Compute the length of segment $XY$ in terms of $d,\varphi$. (2) Compute the distance $v$ between points $V$ and $X$ in terms of $d,\varphi.$

Consider two solid spherical balls, one centered at $(0, 0, \frac{21}{2} )$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac 92$ . How many points $(x, y, z)$ with only integer coordinates (lattice points) are there in the intersection of the balls? $\text{(A)}\ 7 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15$

Let $q(x) = q^1(x) = 2x^2 + 2x - 1$, and let $q^n(x) = q(q^{n-1}(x))$ for $n > 1$. How many negative real roots does $q^{2016}(x)$ have?

1. Find all primes of the form $n^3-1$ .

Suppose $f : \mathbb{N} \longrightarrow \mathbb{N}$ is a function that satisfies $f(1) = 1$ and $f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array}$ $(a)$ Prove that $f(f(n)-n+1)$ is either $n$ or $n+1$. $(b)$ Determine$f$.

Given two real numbers $a, b$ with $a \neq 0$, find all polynomials $P(x)$ which satisfy \[xP(x - a) = (x - b)P(x).\]

If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$. Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?

The plan of a Martian underground is represented by a closed selfintersecting curve, with at most one self-intersection at each point. Prove that a tunnel for such a plan may be constructed in such a way that the train passes consecutively over and under the intersecting parts of the tunnel.

Find all pairs of positive integers $(m,n)$ such that $\frac{m^5+n}{m^2+n^2}$ and $\frac{m+n^5}{m^2+n^2}$ are integers.

Compute the smallest positive integer $n$ such that $\frac{n}{2}$ is a perfect square and $\frac{n}{3}$ is a perfect cube.

Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$.

If $\sqrt{2 + \sqrt{x}} = 3$, then $x =$ $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \sqrt{7} \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 49 \qquad \textbf{(E)}\ 121$

Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$

Let $ABC$ be a scalene triangle, $M$ be the midpoint of $BC,P$ be the common point of $AM$ and the incircle of $ABC$ closest to $A$, and $Q$ be the common point of the ray $AM$ and the excircle farthest from $A$. The tangent to the incircle at $P$ meets $BC$ at point $X$, and the tangent to the excircle at $Q$ meets $BC$ at $Y$. Prove that $MX=MY$.

Which of the following describes the largest subset of values of $y$ within the closed interval $[0,\pi]$ for which $$\sin(x+y)\leq \sin(x)+\sin(y)$$ for every $x$ between $0$ and $\pi$, inclusive? $\textbf{(A) } y=0 \qquad \textbf{(B) } 0\leq y\leq \frac{\pi}{4} \qquad \textbf{(C) } 0\leq y\leq \frac{\pi}{2} \qquad \textbf{(D) } 0\leq y\leq \frac{3\pi}{4} \qquad \textbf{(E) } 0\leq y\leq \pi $

In the acute-angled triangle $ABC$ with circumcircle $\omega$ and orthocenter $H$, points $D$ and $E$ are the feet of the perpendiculars from $A$ onto $BC$ and from $B$ onto $AC$ respecively. Let $P$ be a point on the minor arc $BC$ of $\omega$ . Points $M$ and $N$ are the feet of the perpendiculars from $P$ onto lines $BC$ and $AC$ respectively. Let $PH$ and $MN$ intersect at $R$. Prove that $\angle DMR=\angle MDR$.

Let $ABC$ be a triangle with orthocenter $H$ and $\ell$ a line that passes through $H$, and is parallel to $BC$. Let $m$ and $n$ be the reflections of $\ell$ on the sides of $AB$ and $AC$, respectively, $m$ and $n$ are intersect at $P$. If $HP$ and $BC$ intersect at $Q$, prove that the parallel to $AH$ through $Q$ and $AP$ intersect at the circumcenter of the triangle $ABC$.

Mari and Juri ordered a round pizza. Juri cut the pizza into four pieces by two straight cuts, none of which passed through the centre point of the pizza. Mari can choose two pieces not aside of these four, and Juri gets the rest two pieces. Prove that if Mari chooses the piece that covers the centre point of the pizza, she will get more pizza than Juri.

Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers

A room has $2017$ boxes in a circle. A set of boxes is [i]friendly[/i] if there are at least two boxes in the set, and for each boxes in the set, if we go clockwise starting from the box, we would pass either $0$ or odd number of boxes before encountering a new box in the set. $30$ students enter the room and picks a set of boxes so that the set is friendly, and each students puts a letter inside all of the boxes that he/she chose. If the set of the boxes which have $30$ letters inside is not friendly, show that there exists two students $A, B$ and boxes $a, b$ satisfying the following condition. (i). $A$ chose $a$ but not $b$, and $B$ chose $b$ but not $a$. (ii). Starting from $a$ and going clockwise to $b$, the number of boxes that we pass through, not including $a$ and $b$, is not an odd number, and none of $A$ or $B$ chose such boxes that we passed.

Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 15 $

Each of the digits $1$, $3$, $7$ and $9$ occurs at least once in the decimal representation of some positive integers. Prove that one can permute the digits of this integer such that the resulting integer is divisible by $7$.