MathDB

3

Part of 2003 Federal Competition For Advanced Students, Part 2

Problems(2)

Compute the area of the union of all squares

Source: Austrian Mathematical Olympiad 2003, Part 2, D1, P3

6/18/2011
For every lattice point (x,y)(x, y) with x,yx, y non-negative integers, a square of side 0.92x5y\frac{0.9}{2^x5^y} with center at the point (x,y)(x, y) is constructed. Compute the area of the union of all these squares.
geometrynumber theory unsolvednumber theory
Show that C lies on the line RS.

Source: Austrian Mathematical Olympiad 2003, Part 2, D2, P3

6/18/2011
Let ABCABC be an acute-angled triangle. The circle kk with diameter ABAB intersects ACAC and BCBC again at PP and QQ, respectively. The tangents to kk at AA and QQ meet at RR, and the tangents at BB and PP meet at SS. Show that CC lies on the line RSRS.
projective geometrygeometry unsolvedgeometry