Maths Olympiad Prep

Library / /44 of 520

Algebra Difficulty 5.8 AIME, harder Prove it

Inference 2.1: If a,b,c∈R+a, b, c \in R_{+}, then
ab+c+bc+a+ca+b−32≥(2a−b−c)22(a+b+c)2\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}-\frac{3}{2} \geq \frac{(2 a-b-c)^{2}}{2(a+b+c)^{2}}

Solution

 Prove simply: (ab+c+bc+a+ca+b−32)(a+b+c)≥(2a−b−c)22(a+b+c)⇔a2b+c+b2c+a+c2a+b−a+b+c2≥(2a−b−c)22(a+b+c)\begin{array}{l}\text { Prove simply: } \quad\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}-\frac{3}{2}\right)(a+b+c) \geq \frac{(2 a-b-c)^{2}}{2(a+b+c)} \\ \Leftrightarrow \frac{a^{2}}{b+c}+\frac{b^{2}}{c+a}+\frac{c^{2}}{a+b}-\frac{a+b+c}{2} \geq \frac{(2 a-b-c)^{2}}{2(a+b+c)}\end{array}

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.