Find all integers $a,b,c $ such that $a+b+c=abc$

Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$ .

Let $n \geq 1$ be an integer, and $a_0, a_1, \dots, a_{n-1}$ be real numbers such that \[ 1 \geq a_{n-1} \geq a_{n-2} \geq \dots \geq a_1 \geq a_0 \geq 0. \]We assume that $\lambda$ is a real root of the polynomial \[ x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0. \]Prove that $|\lambda| \leq 1$.

We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$ We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$. Prove that for every $n>0$, $y_n$ is the square of an odd integer

Find all the real numbers $x$ such that $\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}$ where $[x]$ denotes the integer part of $x$ and $\{x\}=x-[x]$. For example, $[2.5]=2, \{2.5\} = 0.5$ and $[-1.7]= -2, \{-1.7\} = 0.3$

Find the maximum and minimum of the expression \[ \max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1), \]where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$.

Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$In which cases do we have equality?

Abimbola plays a game with a coin. He tosses the coin a number of times, and records whether each toss was a "heads" or "tails". He stops tossing the coin as soon as he tosses an odd number of heads in a row, followed by a tails. (Note that he stops if the number of heads since the previous time that he tosses tails is odd, and he then tosses another tails. If he has not tossed tails previously, then he stops if the total number of heads is odd, and he then tosses tails.) How many different sequences of coin tosses are there such that he stops after the $n^\text{th}$ coin toss?

On a $50 \times 50$ chessboard, we put, in the lower left corner, a die whose faces are numbered from $1$ to $6$. By convention, the sum of digits on two opposite side of the die equals $7$. Adama wants to move the die to the diagonally opposite corner using the following rule: at each step, Adama can roll the die only on to its right side, or to its top side. We suppose that whenever the die lands on a square, the number on its bottom face is printed on the square. By the end of these operations, Adama wants to find the sum of the $99$ numbers appearing on the chessboard. What are the maximum and minimum possible values of this sum?

The numbers from $1$ to $2017$ are written on a board. Deka and Farid play the following game : each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of $2, 3$ or $5$ loses and the game is over. Is there a winning strategy for Deka ?

We consider a square $ABCD$ and a point $E$ on the segment $CD$. The bisector of $\angle EAB$ cuts the segment $BC$ in $F$. Prove that $BF + DE = AE$.

Let $ABCDE$ be a regular pentagon, and $F$ some point on the arc $AB$ of the circumcircle of $ABCDE$. Show that \[ \frac{FD}{FE + FC} = \frac{FB + FA}{FD} = \frac{-1 + \sqrt{5}}{2}, \]and that $FD + FB + FA = FE + FC$.

Let $ABC$ be a triangle with $H$ its orthocenter. The circle with diameter $[AC]$ cuts the circumcircle of triangle $ABH$ at $K$. Prove that the point of intersection of the lines $CK$ and $BH$ is the midpoint of the segment $[BH]$

Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?

Let $n$ be a positive integer. - Find, in terms of $n$, the number of pairs $(x,y)$ of positive integers that are solutions of the equation : $$x^2-y^2=10^2.30^{2n}$$- Prove further that this number is never a square