MathDB

2

Part of 2008 Mexico National Olympiad

Problems(2)

A Bunch of Tangencies

Source: OMM 2008 2

7/19/2014
Consider a circle Γ\Gamma, a point AA on its exterior, and the points of tangency BB and CC from AA to Γ\Gamma. Let PP be a point on the segment ABAB, distinct from AA and BB, and let QQ be the point on ACAC such that PQPQ is tangent to Γ\Gamma. Points RR and SS are on lines ABAB and ACAC, respectively, such that PQ∥RSPQ\parallel RS and RSRS is tangent to Γ\Gamma as well. Prove that [APQ]⋅[ARS][APQ]\cdot[ARS] does not depend on the placement of point PP.
trigonometrygeometryperimeterinradiussimilar trianglesgeometry unsolved
Numbers in the Vertices of a Cube

Source: OMM 2008 5

7/19/2014
We place 88 distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let EE be the sum of the numbers on the edges and VV the sum of the numbers on the vertices.
a) Prove that 23E≤V\frac23E\le V. b) Can E=VE=V?
geometry3D geometrygreatest common divisornumber theory unsolvednumber theory