1980 USAMO Problems/Problem 5

Problem

If $x, y, z$ are reals such that $0\le x, y, z \le 1$ , show that $\frac{x}{y + z + 1} + \frac{y}{z + x + 1} + \frac{z}{x + y + 1} \le 1 - (1 - x)(1 - y)(1 - z)$

Solution

Rewrite the given inequality so that $1$ is isolated on the right side. Set the left side to be $f(x, y, z)$ . Now a routine computation shows

$\frac{\partial^2 f}{\partial x^2} = \frac{2y}{(x + z + 1)^3} + \frac{2z}{(x + y + 1)^3}\geq 0$

which shows that $f$ is convex (concave up) in all three variables. Thus the maxima can only occur at the endpoints, i.e. if and only if $x, y, z \in \{0,1\}$ . Checking all eight cases shows that the value of the expression cannot exceed 1.

1980 USAMO (Problems • Resources)
Preceded by Problem 4	Followed by Last Question
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1980 USAMO Problems/Problem 5

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