MathDB

1

Part of 2000 Polish MO Finals

Problems(2)

Find number of solutions in non-negative reals

Source:

11/19/2005
Find number of solutions in non-negative reals to the following equations: \begin{eqnarray*}x_1 + x_n ^2 = 4x_n \\ x_2 + x_1 ^2 = 4x_1 \\ ... \\ x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}
trigonometryfunctionalgebra unsolvedalgebra
Pyramid and inequality

Source:

11/19/2005
PA1A2...AnPA_1A_2...A_n is a pyramid. The base A1A2...AnA_1A_2...A_n is a regular n-gon. The apex PP is placed so that the lines PAiPA_i all make an angle 60⋅60^{\cdot} with the plane of the base. For which nn is it possible to find BiB_i on PAiPA_i for i=2,3,...,ni = 2, 3, ... , n such that A1B2+B2B3+B3B4+...+Bn−1Bn+BnA1<2A1PA_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P?
geometry3D geometrypyramidinequalitiesgeometry unsolved