MathDB

2

Part of 2009 Belarus Team Selection Test

Problems(3)

find n such that x^n+y^n+z^n is constant for all x,y,z x+y+z=0 and xyz=1.

Source: 2009 Belarus TST 1.2

11/8/2020
Find all n∈Nn \in N for which the value of the expression xn+yn+znx^n+y^n+z^n is constant for all x,y,z∈Rx,y,z \in R such that x+y+z=0x+y+z=0 and xyz=1xyz=1.
D. Bazylev
algebraconstantexponential
x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}} cannot an be infitine sequence

Source: 2009 Belarus TST 2.2

11/8/2020
a) Prove that there is not an infinte sequence (xn)(x_n), n=1,2,...n=1,2,... of positive real numbers satisfying the relation xn+2=xn+1−xnx_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}}, ∀n∈N\forall n \in N (*) b) Do there exist sequences satisfying (*) and containing arbitrary many terms?
I.Voronovich
algebraSequence
XA_i=A_{i+2}A_{i+3} inside a convex pentagon

Source: Belarus TST 2009 8.2

6/13/2020
Does there exist a convex pentagon A1A2A3A4A5A_1A_2A_3A_4A_5 and a point XX inside it such that XAi=Ai+2Ai+3XA_i=A_{i+2}A_{i+3} for all i=1,...,5i=1,...,5 (all indices are considered modulo 55) ?
I. Voronovich
pentagonequal segmentsgeometry