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min of Σx_i^m from i=1 to n, where Σx_i=k, x_i nonnegative real, m,n positive

Source: Vietnamese MO (VMO) 1965

August 22, 2018
algebraInequality

Problem Statement

1) Two nonnegative real numbers x,yx, y have constant sum aa. Find the minimum value of xm+ymx^m + y^m, where m is a given positive integer. 2) Let m,nm, n be positive integers and kk a positive real number. Consider nonnegative real numbers x1,x2,...,xnx_1, x_2, . . . , x_n having constant sum kk. Prove that the minimum value of the quantity x1m+...+xnmx^m_1+ ... + x^m_n occurs when x1=x2=...=xnx_1 = x_2 = ... = x_n.
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