Today a math contest I organized was held and I think this was the most difficult exercise overall. There are m distinct positive even integers and n distinct positive odd integers that add up to 1987. Find the maximum of 3m + 4n. So we have this sum of integers We can write that Now […]

# Category: Inequalities

## Algebra At Christmas

Why not? I mean, it’s a nice inequality to try out. What if I asked you to find the maximum value of Well, time to apply the AM-GM inequality. First, let’s say that Notice that by the AM-GM inequality applied on the 4 positive real numbers Hence, we get that with equality when

## Shortlist 2006/A5

Let be the sides of a triangle. Prove that Always Try A Substitution We notice that the denominators are all positive, in fact by the triangular inequality. So let . Then . So we get applying in the last step. Similarly, we obtain Thus, now it is sufficient to prove that So let’s assume WLOG […]

## Cauchy-Schwarz And Titu Inequalities

Today I’m going to show two very nice inequalities to know, which really are useful in the mathematical olympiads. What They Say And Their Proof The Cauchy-Schwarz inequality asserts that, for real tuples we have: Here’s the proof: consider the expression It’s obviously true since each term of the sum is non negative. Hence we […]