MathDB

1

Part of 2003 Silk Road

Problems(1)

International competition SRMC 2003 P-1

Source:

9/8/2010
Let a1,a2,.....,a2003a_1, a_2, ....., a_{2003} be sequence of reals number. Call aka_k leadingleading element, if at least one of expression ak;ak+ak+1;ak+ak+1+ak+2;....;ak+ak+1+ak+2+....+a2003a_k; a_k+a_{k+1}; a_k+a_{k+1}+a_{k+2}; ....; a_k+a{k+1}+a_{k+2}+....+a_{2003} is positive. Prove, that if exist at least one leadingleading element, then sum of all leadingleading's elements is positive.
Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here
algebra proposedalgebracombinatorics proposed