1993 USAMO Problems/Problem 3
Consider functions which satisfy
|(i)||for all in ,|
|(iii)||whenever , , and are all in .|
Find, with proof, the smallest constant such that
for every function satisfying (i)-(iii) and every in .
Lemma 1) for
For , (ii)
Assume that it is true for , then
By principle of induction, lemma 1 is proven.
Lemma 2) For any , and , .
(lemma 1 and (iii) )
(because (i) )
, . Thus, works.
Let's look at a function
It clearly have property (i) and (ii). For and WLOG let ,
For , . Thus, property (iii) holds too. Thus is one of the legit function.
but approach to when is extremely close to from the right side.
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