... get n = and hence Prove that a = is a perfect square 2012 2011 Solution Let p = Then 1 02012 = 9p + Hence, www.hexagon.edu.vn 2012 a = p(9p + 1) + 5p + = (3p + 1)2 , which is a perfect ... 20 √ 311 Find the value of (1 + x5 − x7 )2012 √ √ √ √ √ √ Solution Notice that + = ( + 1)2 and − = ( − 1)2 , 20 = then x = That is 311 (1 + x5 − x7 )2012 = √ 1+ √ Arrange the numbers p = 2 , ... (x − y)2 ≥ Now that 16x + = (x2 − y )2 = (x − y)2 (x + y)2 ≥ x2 From this we obtain the inequality, x2 − 16x − < Solving this inequality gives x ∈ {0, 1, · · · , 16} In addition, 16x + is a perfect...