A cute combinatorial identity

Just wanted to record a cute combinatorial argument. Prove that      I read this formula in the article “Pascal functions” in “The American Mathematical Monthly”. They prove it using the new machinery of Pascal functions that they develop in the article. I tried proving it using an algebraic argument, but couldn’t find the right

Some solutions

I solved two math problems today. The solutions to both were uniquely disappointing. The first problem was the first problem from IMO 1986: Let be a number that is not or . Prove that out of the set , we can select two different numbers such that is not a square. A quick check would

Here’s a slightly badly written proof to a competitive math problem. I guess I could expand it slightly if readers find it unreadable. The following is a question from the International Mathematics Competition, 1994. Prove that in any set of different irrational numbers, there exist irrational numbers such that for any such that (1)  and