Subcontests
(8)Maximum number of rational roots in a family of quadratics
Let a,b,c be three distinct positive integers. Define S(a,b,c) as the set of all rational roots of px2+qx+r=0 for every permutation (p,q,r) of (a,b,c). For example, S(1,2,3)={−1,−2,−1/2} because the equation x2+3x+2 has roots −1 and −2, the equation 2x2+3x+1=0 has roots −1 and −1/2, and for all the other permutations of (1,2,3), the quadratic equations formed don't have any rational roots.Determine the maximum number of elements in S(a,b,c). Number of permutation satisfying a summation relation
Determine the number of permutations a1,a2,…,an of 1,2,…,n such that for every positive integer k with 1≤k≤n, there exists an integer r with 0≤r≤n−k which satisfies
1+2+⋯+k=ar+1+ar+2+⋯+ar+k.