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: Algebra

## Hidden Telescoping Sum

Here’s an exercise I proposed in an online competition. You have the following sequence: Find the integer part of: Now, notice that Hence By substitution, we have that the sum telescopes and reduces to Since is a very large number (the sequence has a quadratic increasing rate), we have that the integer part of the […]

## 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 […]

## Sum Of Powers Of Roots Of Polynomials

Our olympic course professor gave us some exercises, and among these there is one which resolution is particularly instructive. Find the sum of the fifth powers of the roots of: To solve it, we need to write the recursion which is associated to the polynomial, which is: This recursion will give us the sum of […]

## Using Symmetry

I put this problem in an old team competition I organized. It is not simple, and I asked only for the case where n is even, now I’ll show you the complete version. Let be reals in an interval of lenght 1. Find the maximum value of Now, let’s treat as constants and let be […]

## Algebra Problem From Yesterday’s Simulation

Given two polynomials and , let a, b, c, d be the roots of , find First, notice that We know that plugging a, b, c, or d into gives 0, so we have to find the sum of squares of the roots minus the sum of the roots plus 4. By Vieta’s Theorem, the […]

## Old But Gold Polynomials

This article is about a couple of problems about polynomials which I regard as highly instructive. Seems A Demon But It’s Really Not I first saw this problem in 2018, and I initially thought it was a very hard problem, except it wasn’t. But, at first sight, it could make you feel a bit lost. […]

## 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 […]

## An Old Problem Came To Mind

Two years ago I participated in an online math competition and the problem which was considered as the most difficult was this one: Find Now the trick is finding upper and lower bounds for such quantity. Notice: Similarly If we sum up this relationship for all values we’ll found out it’s telescoping, so we’re left […]