German math Olympiad. Intriguing tricks used to solve an Olympiad math problem

This is really an interesting video showing how simple tricks could be used to solve Olympiad problems on permutation and combination.
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‪@begechorimaths2975‬

[@sarantis40kalaitzis48]

Squring Both Sides. We get a^2+b^2+2ab=36ab so a^2+b^2=34ab. We DIVIDE Both Sides with b^2>0, hence (a^2/b^2)+(b^2/b^2)=34ab/b^2 so (a/b)^2+1=34*(a/b). Setting a/b=x we get x^2+1=34x so x^2-34x+1=0 so x=(34+ -sqrt(34^2-4))/2. Using trick of Difference of squares 34^2-2^2=(34+2)*(34-2)=36*32 so x= (34+ -6*sqrt32)/2= 17+ - 3sqrt32= 17+ - 3*sqrt16*sqrt2=17+ -12*sqrt2.

[@pmac_]

Easier to divide original equation by b giving:
x² + 1 = 6 sqrt(x).
Square both sides gives:
x² - 34x +1 = 0
Simple.

[@begechorimaths2975]

@@pmac_ Thank you!

[@juanalfaro7522]

Let a/b = z. Then z+1 = 6*sqrt (z) --> (z+1) ^2 = 6^2 * z -> z^2 + 2z + 1 = 36z -> z^2 - 34z + 1 = 0 -> z^2 - 34z + 289 = 288 -> (z-17) ^2 = 2*14^2 -> z=a/b = 17 +/- 12*sqrt (2) [I.e., there are 2 solutions]