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Problem 6! IMO 2015

The sequence $$a_1,a_2,\dots$$ of integers satisfies the conditions:

(i) $$1\leq a_j\leq 2015$$ for all $$j\geq1$$, (ii) $$k+a_k\not = \ell+ a_\ell$$ for all $$1\leq k< \ell$$.

Prove that there exist two positive integers $$b$$ and $$N$$ for which

$\left \vert \sum_{j=m+1}^n (a_j-b) \right\vert \leq1007^2$

for all integers $$m$$ and $$n$$ such that $$n>m\geq N$$.

This is part of the set IMO 2015

Note by Sualeh Asif
2 years ago