2004 IMO Problems/Problem 4
(Hojoo Lee) Let be an integer. Let be positive real numbers such that
Show that , , are side lengths of a triangle for all , , with .
For , suppose (for sake of contradiction) that for ; then (by Cauchy-Schwarz Inequality)
so it is true for . We now claim the result by induction; for , we have
By AM-GM, , so . Then the problem is reduced to proving the statement true for numbers, as desired.
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