# What paragraph was written by Emanouil Atanassov to solve problem 6 on the International Mathematical Olympiad in 1988?

Emanouil Atanassov, Bulgaria, solved the problem [assumed to be the most difficult one on the 1988 International Mathematics Olympiad] in a paragraph and received a special prize.

The reference links to this page. It does not mention anything about that paragraph. Does anyone have a copy of the actual paragraph written by him to solve the problem?

• Aug 11, 2020 at 13:31
• There's a video by Numberphile starring Zvezdelina Stankova (the other Bulgarian on the Olympiad that year) called "The Notorious Question Six". She basically shows his solution (using induction and quadratic polynomial/Vieta's formulas). It's really quite elegant. Aug 7, 2022 at 15:39

Let $$m$$ be the quotient. We may suppose that $$(a, b)$$ is a minimal positive solution of the equation $$(a^2 + b^2) = m(ab + 1)$$
(i.e. one with the smallest value of $$a + b$$) for this value of $$m$$. Without loss of generality, suppose that $$b ≥ a ≥ 0$$, and set $$b' = ma - b$$. Then $$a^2 = bb' + m$$. If $$b'$$ is positive, then this equation implies $$b' = (a^2 - m)/b < a^2/b < b$$, and $$(a, b')$$ is a smaller positive solution of the equation of which $$(a, b)$$ was supposed to be the minimal solution. If $$b'$$ is negative, then $$m = a^2 - bb' ≥ a^2 + b > b > ma ≥ m$$, a contradiction. Hence $$b' = 0$$ and $$m = a^2$$.
Note: This solution is just the explicit result of applying reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $$x^2 + mxy + y^2$$, which is the unique reduced quadratic form in its equivalence class.