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

This is a straight forward application of Chebyshev's inequality:

If $a_1 \leq a_2 \leq \dots \leq a_n$ , $b_1 \leq b_2 \leq \dots \leq b_n$ ,

$\dfrac{\displaystyle \sum_{i=1}^n a_ib_i}{n} \geq \left(\dfrac{\displaystyle \sum_{i=1}^n a_i}{n}\right)\left(\dfrac{\displaystyle \sum_{i=1}^n b_i}{n}\right)$

Thus substituting the given information ,

$\dfrac{\displaystyle \sum_{i=1}^n a_ib_i}{300} \geq \left(\dfrac{1729}{300}\right) \left(\dfrac{1730}{300}\right)$

$\displaystyle \sum_{i=1}^n a_ib_i\geq \left(\dfrac{1729 \times1730}{300}\right)$

$\displaystyle \sum_{i=1}^n a_ib_i\geq \left(\dfrac{299117}{30}\right)$

$M = \left(\dfrac{299117}{30}\right)$

$\Rightarrow 30M = 299117$ :)

Since $a_i \in \mathbb{R}$ and $b_i \in \mathbb{R}$ , Least possible value for each person would be when the number of sums done is equally distributed amongst the 300 days. $30\sum_{i=1}^{300}a_ib_i =( \frac{1729}{300}\times\frac{1730}{300} )\times 300 \times 30$ $= 299117$

Chebyshev!