MathDB

10

Part of 2008 JBMO Shortlist

Problems(2)

2008 JBMO Shortlist G10

Source: 2008 JBMO Shortlist G10

10/10/2017
Let Γ\Gamma be a circle of center OO, and δ\delta. be a line in the plane of Γ\Gamma, not intersecting it. Denote by AA the foot of the perpendicular from OO onto δ\delta., and let MM be a (variable) point on Γ\Gamma. Denote by γ\gamma the circle of diameter AMAM , by XX the (other than M ) intersection point of γ\gamma and Γ\Gamma, and by YY the (other than AA) intersection point of γ\gamma and δ\delta. Prove that the line XYXY passes through a fixed point.
geometryJBMO
2^n + 3^n is never a perfect cube

Source: JBMO 2008 Shortlist N10

10/14/2017
Prove that 2n+3n2^n + 3^n is not a perfect cube for any positive integer nn.
JBMOperfect cubenumber theory