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Let a,b be positive integers and (a,b)=1. Prove that the quantity cannot be a real number except in the following cases (a,b)=(1,1), (1,3), (3,1).

Let f(x) be a polynomial of degree n with roots α1,,αn. We define the polygon M as the convex hull of the points α1,,αn on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon M.

a) Using geometric considerations, prove that the base and the side of an isosceles triangle with an angle of 36 at the vertex are incommensurable.

b) Invent a geometric proof of the irrationality of 2.