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Prove that min(x1,x2,...,xn) = sqrt n

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September 20, 2010
algebrarecurrence relationSequenceequationfloor functionIMO Shortlist

Problem Statement

Let nn be an integer greater than 11. Define
x1=n,y1=1,xi+1=[xi+yi2],yi+1=[nxi+1],for i=1,2,… ,x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,
where [z][z] denotes the largest integer less than or equal to zz. Prove that min⁡{x1,x2,…,xn}=[n] \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]
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