MathDB

3

Part of 1995 Polish MO Finals

Problems(2)

Number theory sequence.

Source: Poland

1/10/2005
Let pp be a prime number, and define a sequence by: xi=ix_i=i for i=,0,1,2...,p−1i=,0,1,2...,p-1 and xn=xn−1+xn−px_n=x_{n-1}+x_{n-p} for n≥pn \geq p Find the remainder when xp3x_{p^3} is divided by pp.
inductionmodular arithmeticalgebrapolynomialnumber theory proposednumber theory
3 rays in space

Source: Poland MO 1995

8/24/2005
PA,PB,PCPA, PB, PC are three rays in space. Show that there is just one pair of points B′,CB', C' with B′B' on the ray PBPB and C′C' on the ray PCPC such that PC′+B′C′=PA+AB′PC' + B'C' = PA + AB' and PB′+B′C′=PA+AC′PB' + B'C' = PA + AC'.
geometry unsolvedgeometry