Subcontests
(6)Inequality on product of differences
Let n>2 be an even positive integer and let a1<a2<⋯<an be real numbers such that ak+1−ak≤1 for each 1≤k≤n−1. Let A be the set of ordered pairs (i,j) with 1≤i<j≤n such that j−i is even, and let B the set of ordered pairs (i,j) with 1≤i<j≤n such that j−i is odd. Show that(i,j)∈A∏(aj−ai)>(i,j)∈B∏(aj−ai) Iberoamerican 2017 problem 3
Consider the configurations of integers
a1,1
a_{2,1} a_{2,2}
a_{3,1} a_{3,2} a_{3,3}
\dots \dots \dots
a_{2017,1} a_{2017,2} a_{2017,3} \dots a_{2017,2017}
Where ai,j=ai+1,j+ai+1,j+1 for all i,j such that 1≤j≤i≤2016.
Determine the maximum amount of odd integers that such configuration can contain. Iberoamerican 2017 problem 1
For every positive integer n let S(n) be the sum of its digits. We say n has a property P if all terms in the infinite secuence n,S(n),S(S(n)),... are even numbers, and we say n has a property I if all terms in this secuence are odd. Show that for, 1≤n≤2017 there are more n that have property I than those who have P.