12. IMO 2003 Hong Kong Team Selection Test 1

Thusn−1 n−1 n−1 n−12∑kxk = ∑[( k− 1)( k+ 1) + 1] xk = ∑xk + ∑( k− 1)( k+1) xkk= 1 k= 1 k= 1 k=1n−1 n−12≤ n+ ∑( k− 1) nxk= n+ n∑( k− 1) xk= n+ n(2n−2 − n) = n −n.k= 1 k=1n−1 n−12 2 2∑kxk(2 n− k) = 2 n(2n−2) −∑k xk≥2 n(2n−2) − n + n= 3n −3 n.k= 1 k=1The inequality becomes equality if x1 = n− 1, x2 = x3 = = xn2= 0, and xn1= 1.−−(The Paper was set by Andy Chan, K. H.Law, T. W. Leung and Kin Li.)