67th NMO Selection Tests for JBMO

Question

Let m, n be positive integers and x, y, z [0, 1] be real numbers. Prove that 0 x^m+n + y^m+n + z^m+n - x^m y^n - y^m z^n - z^m x^n 1 and find when equality holds.

Accepted Answer

Without loss of generality, we may assume that

x

is the largest of

x, y, z

; then we have:

x^{m + n} + y^{m + n} + z^{m + n} - x^{m} y^{n} - y^{m} z^{n} - z^{m} x^{n} = (x^{m} - z^{m}) (x^{n} - y^{n}) + (y^{m} - z^{m}) (y^{n} - z^{n}) \geq 0

. The minimum is

0

and is obtained if

x = max {y, z}

and

y = z

, so if

x = y = z

.

The maximum is achieved for

x = 1

, and its value is

1 + y^{m + n} + z^{m + n} - y^{n} - y^{m} z^{n} - z^{m} = 1 - y^{n} (1 - y^{m}) - z^{m} (1 - z^{n}) - y^{m} z^{n} \leq 1,

with equality only if one of

y

or

z

is

0

and the other one is

0

or

1

.

Therefore, there are

6

equality cases, namely

(x, y, z) \in {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1)} .