The sequence of integers satisfies the conditions:
(i) for all , (ii) for all .
Prove that there exist two positive integers and for whichfor all integers and such that .
(Australia)
The sequence $a_1,a_2,\dots$ of integers satisfies the conditions:
(i) $1\le a_j\le2015$ for all $j\ge1$,\\
(ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$.
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Prove that there exist two positive integers $b$ and $N$ for which\[\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2\]for all integers $m$ and $n$ such that $n>m\ge N$.
\begin{flushright}\emph{(Australia)}\end{flushright}
Školjka 
of integers satisfies the conditions:
for all 
,
for all 
. 
and 
for which
for all integers 
and 
such that 
.