up!up!up! Can someone sovle this plesase

Anyone got solution?

By Jensen's inequality, we want to maximize as many people as possible. At best, $1$ person can have $11$ points. Then the next best person must have at most $10$ points. The third place gets $9$, and so on until the last person gets $0$ points. The sum is $$\left(\frac{11\cdot 12}{2}\right)^2 = \boxed{4356}$$Now suppose two people have the same score, $b$. As they played each other, one of them beat the other. Then it is possible to change their scores to $b-1, b+1$. And note that $$(b-1)^3+(b+1)^3 = 2b^3 + 6b > 2b^3=b^3 + b^3$$and so it is beneficial to do this. Also, we can always keep doing this until we are left with all different scores. (Left to the reader as an exercise.)

Note that $(b-1)^3+(b+1)^3 = 2b^3+6b > b^3+b^3$, so clearly b and b scores are less optimized for max sum than b-1 and b+1. This technique is like smoothing, where we want to push the numbers as far apart from each other with one maximized and others slowly decreasing, until we cannot increase the larger numbers any further. So, 1 person needs 11, then another must have 10 points, etc. Also, clearly we want everyone to play everyone else for max wins. So our sum $1^3+2^3+…+12^3=(1+2+…+12)^2=\boxed{4356}$.