MathDB

1

Part of 2024 Mathematical Talent Reward Programme

Problems(2)

MTRP OBJECTIVE Q1

Source: MTRP 2024

3/16/2024
Hari the milkman delivers milk to his customers everyday by travelling on his cycle. Each litre of milk costs him Rs. 2020, and he sells it at Rs. 2424. One day while riding his cycle with 2020L, Hari trips and loses 55L of it, and he decides to mix some water with the rest of the milk. His customers can detect if the milk is more than 1010% impure (11L water in 1010L misture). Given that he doesn't wish to make his customers angry, what is his maximum profit for the day? (A)(A) Rs 1212 profit (B)(B) Rs 2424 profit (C)(C) No profit (D)(D) Rs 1212 loss
algebra
MTRP SUBJECTIVE Q1

Source: MTRP 2024

3/16/2024
The Integration Premier League has nn 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−1n-1 matches. The top m≤nm \leq n temas at the end of the tournament qualify for the playoffs. Assume there are no tied matches.
Let A(m,n)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)=0A(n,n) = 0 (everyone qualifies anyway so no need to win!) and A(1,n)=n−1A(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)(A) FInd the value of A(2,4),A(2,6)A(2,4),A(2,6) and A(4,10)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)(B) Show that A(n−1,n)=n2A(n-1,n) = \frac{n}{2} when nn even and =n+12 = \frac{n+1}{2} when nn odd. (C)(C) For bonus marks, try to find a general pattern for A(m,n)A(m,n).
combinatorics