MathDB

4

Part of 2012 239 Open Mathematical Olympiad

Problems(2)

Four variable inequality

Source: 239 2012 S4

7/30/2020
For some positive numbers aa, bb, cc and dd, we know that 1a3+1+1b3+1+1c3+1+1d3+1=2. \frac{1}{a^3 + 1}+ \frac{1}{b^3 + 1}+ \frac{1}{c^3 + 1} + \frac{1}{d^3 + 1} = 2. Prove that 1−aa2−a+1+1−bb2−b+1+1−cc2−c+1+1−dd2−d+1≥0. \frac{1 - a}{a^2 - a + 1} + \frac{1-b}{b^2 - b + 1} + \frac{1-c}{c^2 - c + 1} +\frac{1-d}{d^2 - d + 1} \geq 0.
inequalitiesalgebra
Three variable inequality with sum of one

Source: 239 2012 J4

7/30/2020
For positive real numbers aa, bb, and cc with a+b+c=1a+b+c=1, prove that: (a−b)2+(b−c)2+(c−a)2≥1−27abc2. (a-b)^2 + (b-c)^2 + (c-a)^2 \geq \frac{1-27abc}{2}.
inequalities