Algebra Proofs: Inequalities

May 29th, 2015
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1. Squares are positive x2≥0, Factoring, Quadraticsa. Prove the Arithmetic Mean- Geometric Mean inequality for two variables: for all a,b≥0. b. Prove by multiplying by 2 and factoring as squares.

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Algebra Proofs:InequalitiesERHS Math Club1. Squares are positive x20, Factoring, Quadraticsa. Prove the Arithmetic Mean- Geometric Mean inequality for two variables: for all a,b0.b. Prove by multiplying by 2 and factoring as squares.c. Prove that by factoring. Conclude the AM-GM inequality for 3 variables.d. Find all values of k such that x2+y2kxy for all real numbers x and y.e. Find the largest value of k such that x4+4x2y2+y4k(x3y+xy3) for all real numbers x and y.f. Prove that x4+24x2y2+16y48x3y+32xy3 for all real x,y.g. Prove that if a>b>c>d and a+d=b+c then bc >ad. (Hint: let s=a+b. Factor bc-ad)h. Prove Aczels Inequality: if then by considering the determinant of the quadratic i. [Romanian Olympiad 1983] Let a1,a5 and x,y be real numbers such that . Prove that j. [IMO 2008/2] Prove that if , and are three real numbers, all different from ,

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