2018 USAJMO Problems/Problem 2
Let be positive real numbers such that . Prove that
WLOG let . Add to both sides of the inequality and factor to get: By substituting , we get: The last inequality is true by AM-GM. Since all these steps are reversible, the proof is complete.
WLOG let . Note that the equations are homogeneous, so WLOG let . Thus, the inequality now becomes , which simplifies to .
Now we will use the condition. Letting and , we have .
Plugging this into the inequality, we have , which is true since .
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