For n N , consider non-negative integer-valued functions f on 1,2, , n satisfying f(i) f(j) for ij and _ i=1 ^n(i+f(i))=2023 . Choose n such that _ i=1 ^n f(i) is the least. How many such functions exist in that case?

Question

For n N , consider non-negative integer-valued functions f on 1,2, , n satisfying f(i) f(j) for i>j and _ i=1 ^n(i+f(i))=2023 . Choose n such that _ i=1 ^n f(i) is the least. How many such functions exist in that case?

Quantrex Academy · Accepted Answer

Correct answer: 15

aligned & _ i=1 ^n(i+f(i))=2023 & _ i=1 ^n f(i)=2023- n(n+1) 2 aligned For _ i=1 ^n f(i) be least, n=63 _ i=1 ^n f(i) least =7 Total number of possible functions equal to number of possible partitions of 7. i.e. 7=1+1+1+1+1+1+1=1+1+2+3 Total number of partitions =15 Number of possible functions =15

For n N , consider non-negative integer-valued functions f on 1,2, , n satisfying f(i) f(j) for i>j and _ i=1 ^n(i+f(i))=2023 . Choose n such that _ i=1 ^n f(i) is the least. How many such functions exist in that case?

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