Under French law, the Beaujolais Nouveau
( wine ) is released at 12:01 A.M.
on the third Thursday in November every year.
Last year, prior to the above date 11 wine amateurs deposited 12 small-size barrels at their merchantâ€™s store (A,B,C,D,E,F,G - 7 liters each; S,T,U - 5 liters each and V two barrels: one 7 liters and one 5 liters).
When the wine arrived 70 liters were poured in the above barrels so that each barrel got an integer number of liters. The Customers were billed accordingly.
If every possible distribution of wine among the 12 barrels is equally likely, what is the possibility that V(ictor) had to pay for 11 liters of wine?
The altitude from one of the vertices of an acute-angled triangle ABC meets the opposite side at D. From D, perpendiculars DE and DF are drawn to the other two sides. Prove that the length EF is the same whichever vertex is chosen.
For extra credit: if a=|BC|, b=|CA|, c=|AB|, and d=|EF|, show that Area(ABC)=½√(abcd).
2002 British Mathematical Olympiad, Round 2, Problem 1.