MathDB

3

Part of 2000 Spain Mathematical Olympiad

Problems(2)

There exists a point M depending on C1, C2 [Spain 2000]

Source:

2/3/2011
Two circles C1C_1 and C2C_2 with the respective radii r1r_1 and r2r_2 intersect in AA and B.B. A variable line rr through BB meets C1C_1 and C2C_2 again at PrP_r and QrQ_r respectively. Prove that there exists a point M,M, depending only on C1C_1 and C2,C_2, such that the perpendicular bisector of each segment PrQrP_rQ_r passes through M.M.
geometrygeometric transformationreflectionperpendicular bisectorgeometry unsolved
There is no function f(f(n))=n+1 [Spain 2000] - Maybe old

Source:

2/3/2011
Show that there is no function f:N→Nf : \mathbb N \to \mathbb N satisfying f(f(n))=n+1f(f(n)) = n + 1 for each positive integer n.n.
functionalgebra proposedalgebra