Seven players who represented the university are to be felicitated in a function. They are A, B, C, D, E, F and G. They are to be seated on the dias along one side of a rectangular table. Some of the conditions are:
i. A Gare to be seated at the extreme right, which is closest to the exit because they have to leave early.
ii. B has won the most medals and so has to be seated at the centre.
iii. C and D do not get along well and so have to be seated as far as possible.
iv. E and F are good friends and sit together.
Four teams – Red (R), Blue (B), Green (G), and Yellow (Y) – are competing in the final four rounds of the Inter-School Science Olympiad, labeled Round A, Round B, Round C, and Round D. Each round consists of one match between two teams, and every team plays exactly two matches. No team plays the same opponent more than once.
The final schedule must adhere to the following rules:
(193 words)
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: