RMO Problem 2017

Algebra Level 3

Let x , y , z x,y,z be real numbers, each greater than 1. Must the following be true?

x + 1 y + 1 + y + 1 z + 1 + z + 1 x + 1 ≤ x − 1 y − 1 + y − 1 z − 1 + z − 1 x − 1 \frac{x+1}{y+1} +\frac{y+1}{z+1}+\frac{z+1}{x+1}\le \frac{x-1}{y-1} +\frac{y-1}{z-1}+\frac{z-1}{x-1}

No Yes

Aman Thegreat by Aman thegreat , 19, India

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1 solution

Elijah Frank
Oct 29, 2020

X,Y,Z >1, Z>Y>X then Z=5, Y=4, X=3 Operation: 3+1/4+1 + 5+1/5+1 + 6+1/4+1 = 3/5+6/6+7/5 < 3-1/4-1 + 3-1/5-1 + 4-1/2-1 = 2/3+2/4+3/1

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