MathDB

3

Part of 1994 IMO Shortlist

Problems(2)

The circumcenter

Source: IMO Shortlist 1994, G3

10/22/2005
A circle C C has two parallel tangents L′ L' andL" L". A circle C′ C' touches L′ L' at A A and C C at X X. A circle C" C" touches L" L" at B B, C C at Y Y and C′ C' at Z Z. The lines AY AY and BX BX meet at Q Q. Show that Q Q is the circumcenter of XYZ XYZ
geometrycircumcircleinradiussymmetryincenterIMO Shortlist
A series of transfers

Source: IMO Shortlist 1994, C3

10/22/2005
Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. Prove that Peter can always transfer all his money into two accounts. Can Peter always transfer all his money into one account?
calculusinvariantcombinatoricsgamealgorithmIMO Shortlist