2018 USAMO Problems/Problem 1
Contents
[hide]Problem 1
Let be positive real numbers such that
. Prove that
Solution
WLOG let . Add
to both sides of the inequality and factor to get:
The last inequality is true by AM-GM. Since all these steps are reversible, the proof is complete.
Solution 2
https://wiki-images.artofproblemsolving.com//6/69/IMG_8946.jpg
-srisainandan6
Solution 3
Similarly to Solution 2, we will prove homogeneity but we will use that to solve the problem differently. Let . Note that
, thus proving homogeneity.
WLOG, we can scale down all variables such that the lowest one is . WLOG, let this be
.
We now have
, and we want to prove
Adding
to both sides and subtracting
gives us
, or
. Let
. Now, we have
By the trivial inequality, this is always true. Since all these steps are reversible, the proof is complete.
~SigmaPiE
2018 USAMO (Problems • Resources) | ||
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All USAMO Problems and Solutions |