2019 Regional Olympiad of Mexico West

1

We say that a table with three rows or infinite columns is cool if it was filled with natural numbers, and also whenever the same number m appears in two or more different places in the table, the numbers that appear in the cells immediately below said places (when they exist) are equal. For example, the following table is cool: For each of the following two tables, decide whether it is possible to fill in the empty cells before the resulting tables are cool, explaining how to do this, or why it is not possible to do this. In both tables from the fifth column, the number in the third line is two units greater than the number in the first line.

2

Given a square $ABCD$, points $E$ and $F$ are taken inside the segments $BC$ and $CD$ so that $\angle EAF = 45^o$. The lines $AE$ and $AF$ intersect the circle circumscribed to the square at points $G$ and $H$ respectively. Prove that lines $EF$ and $GH$ are parallel.

3

Determine all pairs $(a,b)$ of natural numbers such that the number $$\frac{a^2(b-a)}{b+a}$$is the square of a prime number.

4

Let $ABC$ be a triangle. $M$ the midpoint of $AB$ and $L$ the midpoint of $BC$. We denote by $G$ the intersection of $AL$ with $CM$ and we take $E$ a point such that $G$ is the midpoint of the segment $AE$. Prove that the quadrilateral $MCEB$ is cyclic if and only if $MB = BG$.

5

Prove that for every integer $n > 1$ there exist integers $x$ and $y$ such that $$\frac{1}{n}=\frac{1}{x(x+1)}+\frac{1}{(x+1)(x+2)}+...+\frac{1}{y(y+1)}.$$

6

In Occidentalia there are $20$ different companies, each looking to hire $15$ new employees. A group of $300$ applicants interview each of the companies. Each company qualifies each applicant as suitable or not suitable to work in it, in such a way that each of them finds exactly $p$ suitable applicants, with $p > 15$. and each applicant is found suitable by at least one company. What is the smallest of $p $f or which it is always possible to assign $15$ applicants to each company, given that each company is assigned only applicants that it considers appropriate, and that each of the $300$ applicants is assigned to a company?