8
Part of 2008 All-Russian Olympiad
Problems(2)
To find fake coin in 3k+1 weighings among 3^2k coins
Source: All-Russian Olympiad 2008 9.8
6/14/2008
We are given apparently identical coins,one of which is fake,being lighter than the others. We also dispose of three apparently identical balances without weights, one of which is broken (and yields outcomes unrelated to the actual situations). How can we find the fake coin in 3k\plus{}1 weighings?
inductioninequalitiesalgorithmlinear algebramatrixcombinatorics proposedcombinatorics
cutting rectangles
Source: ARO 2008
6/12/2008
On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines.
geometryrectangleanalytic geometryinductionvectorfunctiongeometric transformation