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German math Olympiad. Intriguing tricks used to solve an Olympiad math problem
Рет қаралды 661
Күн бұрынThis 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 Ай бұрын
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_ Ай бұрын
Easier to divide original equation by b giving: x² + 1 = 6 sqrt(x). Square both sides gives: x² - 34x +1 = 0 Simple.
@begechorimaths2975 Ай бұрын
@@pmac_ Thank you!
@juanalfaro7522 Ай бұрын
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]
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