Olympiad problems

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Found problems: 3

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2021 Bolivian Cono Sur TST, 1

Tags: Pre TST , TST , Bolivia , Cono Sur TST , number theory , Divisibility

Find the sum of all positive integers $n$ such that $$\frac{n+11}{\sqrt{n-1}}$$ is an integer.

2021 Bolivian Cono Sur TST, 3

Tags: geometry , rectangle , areas , Bolivia , TST , Pre TST , Cono Sur TST

Let $ABCD$ be a rectangle with sides $AB,BC,CD$ and $DA$. Let $K,L$ be the midpoints of the sides $BC,DA$ respectivily. The perpendicular from $B$ to $AK$ hits $CL$ at $M$. Find $$\frac{[ABKM]}{[ABCL]}$$

2021 Bolivian Cono Sur TST, 2

Tags: combinatorics , monovariant , Cono Sur TST , Pre TST , TST , Bolivia

The numbers $1,2,...,100$ are written in a board. We are allowed to choose any two numbers from the board $a,b$ to delete them and replace on the board the number $a+b-1$. What are the possible numbers u can get after $99$ consecutive operations of these?

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