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A2

Part of 2019 Simon Marais Mathematical Competition

Problems(1)

Operation a(b+1)

Source: Simon Marais MC 2019 A2

10/14/2019
Consider the operation ∗\ast that takes pair of integers and returns an integer according to the rule a∗b=a×(b+1).a\ast b=a\times (b+1).
[*]For each positive integer nn, determine all permutations a1,a2,…,ana_1,a_2,\dotsc , a_n of the set {1,2,…,n}\{ 1,2,\dotsc ,n\} that maximise the value of (⋯((a1∗a2)∗a3)∗⋯∗an−1)∗an.(\cdots ((a_1\ast a_2)\ast a_3) \ast \cdots \ast a_{n-1})\ast a_n.[/*] [*]For each positive integer nn, determine all permutations b1,b2,…,bnb_1,b_2,\dotsc , b_n of the set {1,2,…,n}\{ 1,2,\dotsc ,n\} that maximise the value of b1∗(b2∗(b3∗⋯∗(bn−1∗bn)⋯ )).b_1\ast (b_2\ast (b_3\ast \cdots \ast (b_{n-1}\ast b_n)\cdots )).[/*]
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