MathDB

4

Part of 2007 Irish Math Olympiad

Problems(2)

flights

Source: Ireland 2007

7/6/2009
Air Michael and Air Patrick operate direct flights connecting Belfast, Cork, Dublin, Galway, Limerick, and Waterord. For each pair of cities exactly one of the airlines operates the route (in both directions) connecting the cities. Prove that there are four cities for which one of the airlines operates a round trip. (Note that a round trip of four cities P,Q,R, P,Q,R, and S S, is a journey that follows the path P→Q→R→S→P P \rightarrow Q \rightarrow R \rightarrow S \rightarrow P.)
combinatorics unsolvedcombinatorics
digits

Source: Ireland 2007

7/6/2009
Find the number of zeros in which the decimal expansion of 2007! 2007! ends. Also find its last non-zero digit.
number theory proposednumber theory