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Part of 2011 India Regional Mathematical Olympiad

Problems(2)

Indian RMO 2011 P3

Source:

12/4/2011
A natural number nn is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting kk from nn and the larger by adding ll to n.n. Prove that n−kln-kl is a perfect square.
LaTeXquadratics
a,b,c in HP; (a^2+b^2) etc in GP [RMO2-2011, India]

Source:

12/31/2011
Let a,b,c>0.a,b,c>0. If 1a,1b,1c\frac 1a,\frac 1b,\frac 1c are in arithmetic progression, and if a2+b2,b2+c2,c2+a2a^2+b^2,b^2+c^2,c^2+a^2 are in geometric progression, show that a=b=c.a=b=c.
inequalitiesarithmetic sequencegeometric sequence