MathDB

1

Part of 2012 International Zhautykov Olympiad

Problems(2)

The area does not depend on the position of point

Source: International Zhautykov Olympiad 2012 - D1 - P1

1/31/2012
An acute triangle ABCABC is given. Let DD be an arbitrary inner point of the side ABAB. Let MM and NN be the feet of the perpendiculars from DD to BCBC and ACAC, respectively. Let H1H_1 and H2H_2 be the orthocentres of triangles MNCMNC and MNDMND, respectively. Prove that the area of the quadrilateral AH1BH2AH_1BH_2 does not depend on the position of DD on ABAB.
geometrygeometry questionPlane Geometry
Do there exist integers m, n and a function f?

Source: International Zhautykov Olympiad 2012 - D2 - P1

1/31/2012
Do there exist integers m,nm, n and a function f ⁣:R→Rf\colon \mathbb R \to \mathbb R satisfying simultaneously the following two conditions?
∙\bullet i) f(f(x))=2f(x)−x−2f(f(x))=2f(x)-x-2 for any x∈Rx \in \mathbb R; ∙\bullet ii) m≤nm \leq n and f(m)=nf(m)=n.
functioninductionnumber theory proposednumber theory