MathDB

3

Part of 2005 Taiwan TST Round 1

Problems(2)

Tournament

Source: Taiwan 1st TST 2005, final exam, first day, problem 3

8/12/2005
nn teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all nn for which this is possible.
linear algebramatrixquadraticscombinatorics proposedcombinatorics
gcd and lcm

Source: Taiwan 1st TST, final exam, second day, problem 6

8/12/2005
Find all positive integer triples (x,y,z)(x,y,z) such that x<y<zx<y<z, gcd⁡(x,y)=6\gcd (x,y)=6, gcd⁡(y,z)=10\gcd (y,z)=10, gcd⁡(x,z)=8\gcd (x,z)=8, and lcm(x,y,z)=2400(x,y,z)=2400. Note that the problems of the TST are not arranged in difficulty (Problem 1 of day 1 was probably the most difficult!)
number theorygreatest common divisorleast common multiplenumber theory proposed