MathDB

2

Part of 1992 Polish MO Finals

Problems(2)

functional equation in Q+

Source: Problem 2, Polish NO 1992

10/1/2005
Find all functions f:Q+→Q+f : \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}, where Q+\mathbb{Q}^{+} is the set of positive rationals, such that f(x+1)=f(x)+1f(x+1) = f(x) + 1 and f(x3)=f(x)3f(x^3) = f(x)^3 for all xx.
functionalgebra unsolvedalgebra
regular pyramid and 2n-gon as a base

Source: Problem 5, Polish NO 1992

10/1/2005
The base of a regular pyramid is a regular 2n2n-gon A1A2...A2nA_1A_2...A_{2n}. A sphere passing through the top vertex SS of the pyramid cuts the edge SAiSA_i at BiB_i (for i=1,2,...,2ni = 1, 2, ... , 2n). Show that ∑i=1nSB2i−1=∑i=1nSB2i\sum\limits_{i=1}^n SB_{2i-1} = \sum\limits_{i=1}^n SB_{2i}.
geometry3D geometrypyramidspherevectorgeometry unsolved