MathDB

6

Part of 1989 IMO Longlists

Problems(2)

Concyclic if segments are diameters

Source: IMO Longlist 1989, Problem 6

9/18/2008
The circles c1 c_1 and c2 c_2 are tangent at the point A. A. A straight line l l through A A intersects c1 c_1 and c2 c_2 at points C1 C_1 and C2 C_2 respectively. A circle c, c, which contains C1 C_1 and C2, C_2, meets c1 c_1 and c2 c_2 at points B1 B_1 and B2 B_2 respectively. Let ω \omega be the circle circumscribed around triangle AB1B2. AB_1B_2. The circle k k tangent to ω \omega at the point A A meets c1 c_1 and c2 c_2 at the points D1 D_1 and D2 D_2 respectively. Prove that (a) the points C1,C2,D1,D2 C_1,C_2,D_1,D_2 are concyclic or collinear, (b) the points B1,B2,D1,D2 B_1,B_2,D_1,D_2 are concyclic if and only if AC1 AC_1 and AC2 AC_2 are diameters of c1 c_1 and c2. c_2.
geometryparallelogramgeometry unsolved
Function of area of a triangle

Source: IMO Longlist 1989, Problem 105

9/18/2008
Let E E be the set of all triangles whose only points with integer coordinates (in the Cartesian coordinate system in space), in its interior or on its sides, are its three vertices, and let f f be the function of area of a triangle. Determine the set of values f(E) f(E) of f. f.
functiongeometryanalytic geometrygeometry unsolved