MathDB

The Integration Premier League has

n

teams competing. The tournament follows a round-robin system, that is, where every pair of teams play each other exactly once. So every team plays exactly

n-1

matches. The top

m \leq n

temas at the end of the tournament qualify for the playoffs. Assume there are no tied matches.
Let

A(m,n)

be the minimum number of matches a team has to win to gurantee selection for the playoffs, regardless of what their run rate is. For example,

A(n,n) = 0

(everyone qualifies anyway so no need to win!) and

A(1,n) = n-1

(even if you lose to just one other team, they might defeat everyone and qualify instead of you). Answer the following:

(A)

FInd the value of

A(2,4),A(2,6)

and

A(4,10)

with proof (explain why a smaller value can still lead to the team not qualifying, and show that the respective values themselves are enough).

(B)

Show that

A(n-1,n) = \frac{n}{2}

when

n

even and

= \frac{n+1}{2}

when

n

odd.

(C)

For bonus marks, try to find a general pattern for

A(m,n)

Problem Statement