OMCPLP 2020

In certain country, the coins have the following values: $2^0, 2^1, 2^2,\dots 2^{10}$ . A cash machine has $1000$ coins of each value and give the money using each coin(of each value) at most once. The customers order all the positive integers: $1,2,3,4,5,\dots$ (in this order) in coins. a) Determine the first integer, such that the cash machine cannot provide. b) In the moment that the first customer can not be attended, by the lack of coins, what are the coins which are not available in the cash machine?

a) Find a pair(s) of integers $(x,y)$ such that: $y^2=x^3+2017$ b) Prove that there isn't integers $x$ and $y$ , with $y$ not divisible by $3$ , such that: $y^2=x^3-2017$

Let $ABC$ be a triangle and on the sides we draw, externally, the squares $BADE, CBFG$ and $ACHI$ . Determine the greatest positive real constant $k$ such that, for any triangle $\triangle ABC$ , the following inequality is true: $[DEFGHI]\geq k\cdot [ABC]$ Note: $[X]$ denotes the area of polygon $X$ .

Let $ABC$ be an acute triangle. Its incircle touches the sides $BC$ , $CA$ and $AB$ at the points $D$ , $E$ and $F$ , respectively. Let $P$ , $Q$ and $R$ be the circumcenters of triangles $AEF$ , $BDF$ and $CDE$ , respectively. Prove that triangles $ABC$ and $PQR$ are similar.

In how many ways can we fill the cells of a $4\times4$ grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is $2020$ ?

Prove that $\lfloor{\sqrt{9n+7}}\rfloor=\lfloor{\sqrt{n}+\sqrt{n+1}+\sqrt{n+2}}\rfloor$ for all postive integer $n$ .