MathDB

2

Part of 2011 IberoAmerican

Problems(2)

Problem 2, Iberoamerican Olympiad 2011

Source:

10/2/2011
Find all positive integers nn for which exist three nonzero integers x,y,zx, y, z such that x+y+z=0x+y+z=0 and: 1x+1y+1z=1n\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{n}
number theory proposednumber theory
Problem 5, Iberoamerican Olympiad 2011

Source:

10/2/2011
Let x1,…,xnx_1,\ldots ,x_n be positive real numbers. Show that there exist a1,…,an∈{−1,1}a_1,\ldots ,a_n\in\{-1,1\} such that: a1x12+a2x22+…+anxn2≥(a1x1+a2x2+…+anxn)2a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_n)^2
inequalitiesinductionstrong inductionalgebra proposedalgebra