Problem A. 624. (October 2014)
A. 624. \(\displaystyle a)\) Prove that for every infinite sequence \(\displaystyle x_1,x_2,\ldots\in[0,1]\) there exists some \(\displaystyle C>0\) such that for every positive integer \(\displaystyle r\) there are positive integers \(\displaystyle n\), \(\displaystyle m\) satisfying \(\displaystyle nm\ge r\) and \(\displaystyle x_nx_m<\frac{C}{nm}\).
\(\displaystyle b)\) Show that for every \(\displaystyle C>0\) there exists an infinite sequence \(\displaystyle x_1,x_2,\ldots\in[0,1]\) and a positive integer \(\displaystyle r\) such that \(\displaystyle x_nx_m>\frac{C}{nm}\) holds true for every pair \(\displaystyle n\), \(\displaystyle m\) of positive integers with \(\displaystyle nm\ge r\).
(CIIM6, Costa Rica)
(5 pont)
Deadline expired on 10 November 2014.
Statistics:
5 students sent a solution.  
5 points:  Williams Kada. 
2 points:  2 students. 
0 point:  2 students. 
