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sum -1/a_b_i where a_i, b_i are min and max perfect square <= or >= i resp

Source: 2012 Argentina OMA Finals L3 p4

January 15, 2023

number theoryPerfect Square

Problem Statement

For each natural number

n

we denote

a_n

as the greatest perfect square less than or equal to

n

and

b_n

as the least perfect square greater than

n

. For example

a_9=3^2

b_9=4^2

and

a_{20}=4^2

b_{20}=5^2

. Calculate:

\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}