olymmath

VMO 1965 Problem 3

    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 .

    Dataset text AI-MO AoPS. Quoted with source.

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