MathDB

3

Part of 1999 Turkey Team Selection Test

Problems(2)

f(x-1-f(x)) = f(x)-x-1

Source: Turkey TST 1999 - P3

7/2/2012
Determine all functions f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} such that the set {f(x)x:x≠0 and x∈R}\left \{ \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \} is finite, and for all x∈Rx \in \mathbb{R} f(x−1−f(x))=f(x)−x−1f(x-1-f(x)) = f(x) - x - 1
functionalgebra proposedalgebra
Covering plane with parabolas

Source: Turkey TST 1999 - P6

7/2/2012
Prove that the plane is not a union of the inner regions of finitely many parabolas. (The outer region of a parabola is the union of the lines not intersecting the parabola. The inner region of a parabola is the set of points of the plane that do not belong to the outer region of the parabola)
conicsparabolacombinatorics proposedcombinatorics