MathDB
function on a triangle

Source: French MO 1997 P4

April 10, 2021
functiongeometryTriangle

Problem Statement

In a triangle ABCABC, let a,b,ca,b,c be its sides and m,n,pm,n,p be the corresponding medians. For every α>0\alpha>0, let λ(α)\lambda(\alpha) be the real number such that aα+bα+cα=λ(α)α(mα+nα+pα)α.a^\alpha+b^\alpha+c^\alpha=\lambda(\alpha)^\alpha\left(m^\alpha+n^\alpha+p^\alpha\right)^\alpha. (a) Compute λ(2)\lambda(2). (b) Find the limit of λ(α)\lambda(\alpha) as α\alpha approaches 00. (c) For which triangles ABCABC is λ(α)\lambda(\alpha) independent of α\alpha?
Back to ProblemsView on AoPS