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 […]
Tag: Algebra
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 […]