MathDB

2

Part of 2008 Bosnia Herzegovina Team Selection Test

Problems(2)

Divisibility

Source: Bosnian TST 2008, 1.st day

5/17/2008
Find all pairs of positive integers m m and n n that satisfy (both) following conditions: (i) m^{2}\minus{}n divides m\plus{}n^{2} (ii) n^{2}\minus{}m divides n\plus{}m^{2}
number theory proposednumber theory
Circumcenter lies on EF

Source: Bosnian TST, 2008, 2.nd day

5/18/2008
Let AD AD be height of triangle △ABC \triangle ABC and R R circumradius. Denote by E E and F F feet of perpendiculars from point D D to sides AB AB and AC AC. If AD\equal{}R\sqrt{2}, prove that circumcenter of triangle △ABC \triangle ABC lies on line EF EF.
geometrycircumcircletrigonometrypower of a pointradical axisgeometry proposed