My 300 followers problem! (Parth and Pranjal special!)

Algebra Level 5

Two maths lovers Parth and Pranjal competed each other for the number of problems solved correctly in \(300\) days streak at Brilliant.

Let \(a_i\) be the number of problems solved by Parth on the \(i^{th}\) day.

Let \(b_i\) be the number of problems solved by Pranjal on the \(i^{th}\) day.

Note: \(a_i , b_i\) can be rational numbers , since there may be some questions which both may have solved partially.

The two sequences \(a_1 , a_2 , …. a_{300}\) and \(b_1 , b_2 , … b_{300}\) are non-decreasing . The competition was tough , and Pranjal won by \(1\) problem.

If \(\displaystyle \sum_{i=1}^{300} a_i = 1729 ,\displaystyle \sum_{i=1}^{300} b_i = 1730\),

What is the minimum value of \(\displaystyle 30 \sum_{i=1}^{300} a_i b_i \)?

Note: \(1729\) (taxicab number) , \(1730\) are two consecutive sphenic numbers.

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