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 300300 days streak at Brilliant.

Let aia_i be the number of problems solved by Parth on the ithi^{th} day.

Let bib_i be the number of problems solved by Pranjal on the ithi^{th} day.

Note: ai,bia_i , b_i can be rational numbers , since there may be some questions which both may have solved partially.

The two sequences a1,a2,.a300a_1 , a_2 , …. a_{300} and b1,b2,b300b_1 , b_2 , … b_{300} are non-decreasing . The competition was tough , and Pranjal won by 11 problem.

If i=1300ai=1729,i=1300bi=1730\displaystyle \sum_{i=1}^{300} a_i = 1729 ,\displaystyle \sum_{i=1}^{300} b_i = 1730,

What is the minimum value of 30i=1300aibi\displaystyle 30 \sum_{i=1}^{300} a_i b_i ?

Note: 17291729 (taxicab number) , 17301730 are two consecutive sphenic numbers.

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