MathDB

1

Part of 1994 Vietnam National Olympiad

Problems(2)

solve system

Source: 32-th Vietnamese Mathematical Olympiad 1994

2/24/2007
Find all real solutions to x3+3x−3+ln⁡(x2−x+1)=y,x^{3}+3x-3+\ln{(x^{2}-x+1)}=y, y3+3y−3+ln⁡(y2−y+1)=z,y^{3}+3y-3+\ln{(y^{2}-y+1)}=z, z3+3z−3+ln⁡(z2−z+1)=x.z^{3}+3z-3+\ln{(z^{2}-z+1)}=x.
logarithmsalgebrasystem of equationsalgebra proposed
$n+1$ containers arranged in a circle

Source: 32-th Vietnamese Mathematical Olympiad 1994

2/24/2007
There are n+1n+1 containers arranged in a circle. One container has nn stones, the others are empty. A move is to choose two containers AA and BB, take a stone from AA and put it in one of the containers adjacent to BB, and to take a stone from BB and put it in one of the containers adjacent to AA. We can take A=BA = B. For which nn is it possible by series of moves to end up with one stone in each container except that which originally held nn stones.
invariantcombinatorics unsolvedcombinatorics