Not a perfect square!

Number Theory Level 4

Suppose a , b a,b are positive integers such that a , b ≤ 1000 a,b \le 1000 and

k = a 2 + b 2 1 + a b . k =\dfrac{a^2 + b^2}{1+ab}.

Given that k k is an integer, the number of pairs ( a , b ) (a,b) such that k k is not a perfect square is:


The answer is 0.

Rajnish Singh by Rajnish Singh , 25, India

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1 solution

Sándor Daróczi
Jun 29, 2017

Problem 6, IMO 1988:

http://artofproblemsolving.com/wiki/index.php?title=1988 IMO Problems/Problem_6

0 Helpful 0 Interesting 0 Brilliant 0 Confused

One of the first instances of Vieta root jumping - very nice and elegant, don't you think?

Zach Abueg - 3 years, 11 months ago

I have been fond of this problem for a long time because of its illusory simplicity but at the same time the complex thinking one needs to be able to solve it. It's impressive that even mathematicians were not able to solve it within 6 hours but there were 11 young students who answered it correctly within 4,5 hours besides two other quite difficult problems...

Sándor Daróczi - 3 years, 11 months ago

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Yes, agreed! Numberphile has an awesome video on it.

Zach Abueg - 3 years, 11 months ago

So actually this is the hardest problem in IMO at 1988?? I have searched wikipedia for this, Wow! I mean.

Kelvin Hong - 3 years, 11 months ago

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