MathDB

3

Part of 2015 International Zhautykov Olympiad

Problems(2)

Hard functional equation

Source: IZHO2015.P3

1/13/2015
Find all functions f ⁣:R→R f\colon \mathbb{R} \to \mathbb{R} such that f(x3+y3+xy)=x2f(x)+y2f(y)+f(xy) f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy) , for all x,y∈R x,y \in \mathbb{R} .
functionalgebrapolynomialalgebra unsolved
pentagon inequality

Source: Izho 2015 problem 6

1/14/2015
The area of a convex pentagon ABCDEABCDE is SS, and the circumradii of the triangles ABCABC, BCDBCD, CDECDE, DEADEA, EABEAB are R1R_1, R2R_2, R3R_3, R4R_4, R5R_5. Prove the inequality R14+R24+R34+R44+R54≥45sin⁡2108∘S2. R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2.
inequalitiesgeometrytrigonometryinequalities unsolved