https://www.youtube.com/watch?v=f8N4biffzhw We prove Chebyshev’s inequality for two number sequences using several applications of the rearrangement inequality. The goal is to make upper and lower bounds on the expression shown in the thumbnail. These inequalities come in handy on math contests and olympiads.
https://www.youtube.com/watch?v=hcK1MoIVbTA In math contest and olympiad circles, Schur’s inequality is known as a result which has special cases that are equivalent to relatively strong results. We prove the ordinary version of Schur’s inequality in this video. Stronger versions have been unearthed in recent years, such as by Valentin Vornicu.
https://www.youtube.com/watch?v=Oe4Qn_T69-I The real number Cauchy-Schwarz inequality is a multivariable inequality that holds for all real inputs. It has a variety of special cases, including the famous Engel’s form, that have found popular use on math contests and olympiads. Outside of contests, it is useful on optimization problems and in linear algebra. Here, we present an …
https://www.youtube.com/watch?v=iNldQpfT_ug The arithmetic mean – geometric mean inequailty is a famous multivariable inequality that allows us to compare the sum of some non-negative numbers with their product, in the presence of some other operations. This amazing multivariable inequality has widespread use across mathematics, including on math contests and olympiads. This proof is even more remarkable, …
https://www.youtube.com/watch?v=v4YVOMJAuGQ The Sophie Germain identity is a remarkable polynomial factorization. It is frequently useful in math contests, competitions, and olympiads. One would not expect this polynomial to factor, but an ingenious trick via the difference of squares factorization allows for the factorization $$x^4+4y^4 = (x^2-2xy+2y^2)(x^2+2xy+y^2).$$
https://www.youtube.com/watch?v=fgBBJnhbMvI There are a couple of related theorems that we interchangeably call the “integer root theorem.” One tells us how to find that integer roots of a polynomial with integer coefficients. The second tells us that all rational roots of a monic polynomial with integer coefficients are integers. We prove both results in this video.