MathDB

3

Part of 1987 IberoAmerican

Problems(2)

m must be a perfect square

Source: IberoAmerican 1987 Q3

11/12/2010
Prove that if m,n,rm,n,r are positive integers, and: 1+m+n3=(2+3)2r−11+m+n\sqrt{3}=(2+\sqrt{3})^{2r-1} then mm is a perfect square.
number theory unsolvednumber theory
PQ forms equal angles with AB and CD

Source: IberoAmerican 1987 Q6

11/12/2010
Let ABCDABCD be a convex quadrilateral and let PP and QQ be the points on the sides ADAD and BCBC respectively such that APPD=BQQC=ABCD\frac{AP}{PD}=\frac{BQ}{QC}=\frac{AB}{CD}. Prove that the line PQPQ forms equal angles with the lines ABAB and CDCD.
geometrycircumcirclesymmetryangle bisectorgeometry proposed