Subcontests
(6)Cycle in a grid
Find the maximum possible value of n, such that the integers 1,2,…n can be filled once each in distinct cells of a 2015×2015 grid and satisfies the following conditions:[*] For all 1≤i≤n−1, the cells with i and i+1 share an edge. Cells with 1 and n also share an edge. In addition, no other pair of numbers share an edge.
[*] If two cells with i<j in them share a vertex, then min{j−i,n+i−j}=2. Inequality with sum of reciprocals
Find the smallest positive real k such that for any positive integer n≥2 and positive reals a0,a1,…,an, a0+a11+a0+a1+a21+…+a0+a1+…+an1<k(a01+a11+…+an1).