Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]

Given a triangular pyramid in which all the plane angles at one of the vertices are right. It is known that there is a point in space located at a distance of $3$ from the indicated vertex and at distances $\sqrt5, \sqrt6, \sqrt7$ from three other vertices. Find the radius of the sphere circumscribed around this pyramid. (The circumscribed sphere for a pyramid is the sphere containing all its vertices.)

Let $\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}$ be the decimal representation of a prime positive integer such that $n>1$ and $a_{n}>1$. Prove that the polynomial $P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}$ cannot be written as a product of two non-constant integer polynomials.

There are $n$ boxes which is numbere from $1$ to $n$. The box with number $1$ is open, and the others are closed. There are $m$ identical balls ($m\geq n$). One of the balls is put into the open box, then we open the box with number $2$. Now, we put another ball to one of two open boxes, then we open the box with number $3$. Go on until the last box will be open. After that the remaining balls will be randomly put into the boxes. In how many ways this arrangement can be done?

The expression $\dfrac{\dfrac{a}{a+y}+\dfrac{y}{a-y}}{\dfrac{y}{a+y}-\dfrac{a}{a-y}}$, a real, $a\neq 0$, has the value $-1$ for: $\textbf{(A)}\ \text{all but two real values of }y \qquad \textbf{(B)}\ \text{only two real values of }y \qquad$ $\textbf{(C)}\ \text{all real values of }y \qquad \textbf{(D)}\ \text{only one real value of }y \qquad \textbf{(E)}\ \text{no real values of }y$

Define a sequence by $a_0=2019$ and $a_n=a_{n-1}^{2019}$ for all positive integers $n$. Compute the remainder when \[a_0+a_1+a_2+\dots+a_{51}\] is divided by $856$. [i]Proposed by Tristan Shin[/i]

Let $a,b,c$ be three positive real numbers such that $a+ b +c =1$. Let $\lambda = min \{ a^3 + a^2bc , b^3 + b^2 ac , c^3 + ab c^2 \}$ Prove that the roots of $x^2 + x + 4 \lambda = 0$ are real.

Given a non-decreasing unbounded sequence $a_n,$ construct a new sequence $b_n$ as follows $$b_n = \frac{a_2 - a_1}{a_2} + \frac{a_3 - a_2}{a_3} + ... + \frac{a_n - a_{n-1}}{a_n}$$ Prove that $b_n$ is also unbounded.

Consider a standard ($8$-by-$8$) chessboard. Bishops are only allowed to attack pieces that are along the same diagonal as them (but cannot attack along a row or column). If a piece can attack another piece, we say that the pieces threaten each other. How many bishops can you place a chessboard without any of them threatening each other?

The prime number $p$ and a positive integer $k$ are given. Assume that $P(x)\in \mathbb Z[X]$ is a polynomial with coefficients in the set $\{0,1,\cdots,p-1\}$ with least degree which satisfies the following property: There exists a permutaion of numbers $1,2,\cdots,p-1$ around a circle such that for any $k$ consecutive numbers $a_1,a_2,\cdots,a_k$ one has $$ p | P(a_1)+P(a_2)+\cdots+ P(a_k). $$ Prove that $P(x)$ is of the form $ax^d+b$. Proposed by [i]Yahya Motevassel[/i]

There are $n\geq 2$ people at a meeting. Show that there exist two people at the meeting who have the same number of friends among the persons at the meeting. (It is assumed that if $A$ is a friend of $B,$ then $B$ is a friend of $A;$ moreover, nobody is his own friend.)

Let $a$, $b$ and $m$ be three positive real numbers and $a>b$. Which of the numbers $A=\sqrt{a+m}-\sqrt{a}$ and $B=\sqrt{b+m}-\sqrt{b}$ is bigger:

The Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions. The Bell Zoo has three more elephants than the Carlton Zoo has lions. The Bell Zoo has the same number of elephants as the Carlton Zoo has rhinoceroses. The Carlton Zoo has two more elephants than rhinoceroses. The Carlton Zoo has twice as many monkeys as it has rhinoceroses, elephants, and lions combined, and it has two more penguins than monkeys. The Bell Zoo has two-thirds as many monkeys as the Carlton Zoo has penguins. The Bell Zoo has two more penguins than monkeys but only half as many lions as penguins. The total of the numbers of rhinoceroses, elephants, lions, monkeys, and penguins in the Bell Zoo is $48$. Find the total of the numbers of rhinoceroses, elephants, lions, monkeys, and penguins in the Carlton Zoo.

Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?

Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$, $BC = CD = 43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. GIven that $OP = 11$, the length $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m + n$? $\textbf{(A)}\: 65\qquad\textbf{(B)}\: 132\qquad\textbf{(C)}\: 157\qquad\textbf{(D)}\: 194\qquad\textbf{(E)}\: 215$

a) Points $A_1, A_2, A_3, A_4, A_5, A_6$ divide a circle of radius $1$ into six equal arcs. Ray $\ell_1$ from $A_1$ connects $A_1$ with $A_2$, ray $\ell_2$ from $A_2$ connects $A_2$ with $A_3$, and so on, ray $\ell_6$ from $A_6$ connects $A_6$ with $A_1$. From a point $B_1$ on $\ell_1$ the perpendicular is drawn on $\ell_6$, from the foot of this perpendicular another perpendicular is drawn on $\ell_5$, and so on. Let the foot of the $6$-th perpendicular coincide with $B_1$. Find the length of segment $A_1B_1$. b) Find points $B_1, B_2,... , B_n$ on the extensions of sides $A_1A_2, A_2A_3,... , A_nA_1$ of a regular $n$-gon $A_1A_2...A_n$ such that $B_1B_2 \perp A_1A_2$, $B_2B_3 \perp A_2A_3$,$ . . . $, $B_nB_1 \perp A_nA_1$.

Let $n \in N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge a_2\ge a_3\ge 2$ have sum $n$ and they satisfy $1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n ) $ and $1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )$. Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .

Suppose that $ ABCD$ is a cyclic quadrilateral and $ CD\equal{}DA$. Points $ E$ and $ F$ belong to the segments $ AB$ and $ BC$ respectively, and $ \angle ADC\equal{}2\angle EDF$. Segments $ DK$ and $ DM$ are height and median of triangle $ DEF$, respectively. $ L$ is the point symmetric to $ K$ with respect to $ M$. Prove that the lines $ DM$ and $ BL$ are parallel.

Prove that there exist infinitely many positive integers which can't be represented as sum of less than $10$ odd positive integers' perfect squares.

For how many permutations $\pi$ of $\{1,2,\ldots,9\}$ does there exist an integer $N$ such that \[N\equiv \pi(i)\pmod{i}\text{ for all integers }1\leq i\leq 9?\]

Allen and Brian are playing a game in which they roll a $6$-sided die until one of them wins. Allen wins if two consecutive rolls are equal and at most 3. Brian wins if two consecutive rolls add up to $7$ and the latter is at most $3$. What is the probability that Allen wins

A positive integer $N$ has the digits $1, 2, 3, 4, 5, 6$ and $7$, so that each digit $i$, $i \in \{1, 2, 3, 4, 5, 6, 7\}$ occurs $4i$ times in the decimal representation of $N$. Prove that $N$ is not a perfect square.

A triangle $ABC$ is given with barycentre $G$ and circumcentre $O$. The perpendicular bisectors of $GA, GB$ meet at $C_1$,of $GB,GC$ meet at $A _1$, and $GC,GA$ meet at $B_1$. Prove that $O$ is the barycenter of the triangle $A_1B_1C_1$.

Prove that for all $a,b>0$ with $a+b=2$, we have $$\left(a^n+1\right)\left(b^n+1\right)\ge4$$ for all $n\in\mathbb N_{\ge2}$.

There are unique integers $ a_2, a_3, a_4, a_5, a_6, a_7$ such that \[ \frac {5}{7} \equal{} \frac {a_2}{2!} \plus{} \frac {a_3}{3!} \plus{} \frac {a_4}{4!} \plus{} \frac {a_5}{5!} \plus{} \frac {a_6}{6!} \plus{} \frac {a_7}{7!},\] where $ 0 \le a_i < i$ for $ i \equal{} 2,3...,7$. Find $ a_2 \plus{} a_3 \plus{} a_4 \plus{} a_5 \plus{} a_6 \plus{} a_7$. $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12$