| Case 1 | Case 2 | Case 3 | |
| Floor | Name | Name | Name |
| 12 | K | ||
| 11 | |||
| 10 | G | ||
| 9 | |||
| 7 | G | G | |
| 6 | K | ||
| 5 | |||
| 4 | K | ||
| 3 | |||
| 2 | |||
| 1 |
| Case 1 | Case 2 | Case 3 | |
| Floor | Name | Name | Name |
| 12 | K | ||
| 11 | B | ||
| 10 | G | ||
| 9 | |||
| 8 | G | G | |
| 7 | |||
| 6 | J | K | |
| 5 | B | ||
| 4 | K | ||
| 3 | A/C | B | |
| 2 | |||
| 1 | A/C |
| Case 1 | Case 1.A | |
| Floor | Name | Name |
| 12 | K | K |
| 11 | B | B |
| 10 | ||
| 9 | ||
| 8 | G | G |
| 7 | ||
| 6 | J | J |
| 5 | ||
| 4 | K | |
| 3 | A | C |
| 2 | ||
| 1 | C | A |
| Case 1 | Case 1.A | |
| Floor | Name | Name |
| 12 | K | K |
| 11 | B | B |
| 10 | D | D |
| 9 | I | I |
| 8 | G | G |
| 7 | E | |
| 6 | J | J |
| 5 | F | |
| 4 | ||
| 3 | A | C |
| 2 | ||
| 1 | C | A |
| Case 1 | |
| Floor | Name |
| 12 | K |
| 11 | B |
| 10 | D |
| 9 | I |
| 8 | G |
| 7 | E |
| 6 | J |
| 5 | F |
| 4 | L |
| 3 | A |
| 2 | H |
| 1 | C |
| Case 1 | Case 2 | Case 3 | |
| Floor | Name | Name | Name |
| 12 | K | ||
| 11 | |||
| 10 | G | ||
| 9 | |||
| 7 | G | G | |
| 6 | K | ||
| 5 | |||
| 4 | K | ||
| 3 | |||
| 2 | |||
| 1 |
| Case 1 | Case 2 | Case 3 | |
| Floor | Name | Name | Name |
| 12 | K | ||
| 11 | B | ||
| 10 | G | ||
| 9 | |||
| 8 | G | G | |
| 7 | |||
| 6 | J | K | |
| 5 | B | ||
| 4 | K | ||
| 3 | A/C | B | |
| 2 | |||
| 1 | A/C |
| Case 1 | Case 1.A | |
| Floor | Name | Name |
| 12 | K | K |
| 11 | B | B |
| 10 | ||
| 9 | ||
| 8 | G | G |
| 7 | ||
| 6 | J | J |
| 5 | ||
| 4 | K | |
| 3 | A | C |
| 2 | ||
| 1 | C | A |
| Case 1 | Case 1.A | |
| Floor | Name | Name |
| 12 | K | K |
| 11 | B | B |
| 10 | D | D |
| 9 | I | I |
| 8 | G | G |
| 7 | E | |
| 6 | J | J |
| 5 | F | |
| 4 | ||
| 3 | A | C |
| 2 | ||
| 1 | C | A |
| Case 1 | |
| Floor | Name |
| 12 | K |
| 11 | B |
| 10 | D |
| 9 | I |
| 8 | G |
| 7 | E |
| 6 | J |
| 5 | F |
| 4 | L |
| 3 | A |
| 2 | H |
| 1 | C |
| Case 1 | Case 2 | Case 3 | |
| Floor | Name | Name | Name |
| 12 | K | ||
| 11 | |||
| 10 | G | ||
| 9 | |||
| 7 | G | G | |
| 6 | K | ||
| 5 | |||
| 4 | K | ||
| 3 | |||
| 2 | |||
| 1 |
| Case 1 | Case 2 | Case 3 | |
| Floor | Name | Name | Name |
| 12 | K | ||
| 11 | B | ||
| 10 | G | ||
| 9 | |||
| 8 | G | G | |
| 7 | |||
| 6 | J | K | |
| 5 | B | ||
| 4 | K | ||
| 3 | A/C | B | |
| 2 | |||
| 1 | A/C |
| Case 1 | Case 1.A | |
| Floor | Name | Name |
| 12 | K | K |
| 11 | B | B |
| 10 | ||
| 9 | ||
| 8 | G | G |
| 7 | ||
| 6 | J | J |
| 5 | ||
| 4 | K | |
| 3 | A | C |
| 2 | ||
| 1 | C | A |
| Case 1 | Case 1.A | |
| Floor | Name | Name |
| 12 | K | K |
| 11 | B | B |
| 10 | D | D |
| 9 | I | I |
| 8 | G | G |
| 7 | E | |
| 6 | J | J |
| 5 | F | |
| 4 | ||
| 3 | A | C |
| 2 | ||
| 1 | C | A |
| Case 1 | |
| Floor | Name |
| 12 | K |
| 11 | B |
| 10 | D |
| 9 | I |
| 8 | G |
| 7 | E |
| 6 | J |
| 5 | F |
| 4 | L |
| 3 | A |
| 2 | H |
| 1 | C |
| Case 1 | Case 2 | Case 3 | |
| Floor | Name | Name | Name |
| 12 | K | ||
| 11 | |||
| 10 | G | ||
| 9 | |||
| 7 | G | G | |
| 6 | K | ||
| 5 | |||
| 4 | K | ||
| 3 | |||
| 2 | |||
| 1 |
| Case 1 | Case 2 | Case 3 | |
| Floor | Name | Name | Name |
| 12 | K | ||
| 11 | B | ||
| 10 | G | ||
| 9 | |||
| 8 | G | G | |
| 7 | |||
| 6 | J | K | |
| 5 | B | ||
| 4 | K | ||
| 3 | A/C | B | |
| 2 | |||
| 1 | A/C |
| Case 1 | Case 1.A | |
| Floor | Name | Name |
| 12 | K | K |
| 11 | B | B |
| 10 | ||
| 9 | ||
| 8 | G | G |
| 7 | ||
| 6 | J | J |
| 5 | ||
| 4 | K | |
| 3 | A | C |
| 2 | ||
| 1 | C | A |
| Case 1 | Case 1.A | |
| Floor | Name | Name |
| 12 | K | K |
| 11 | B | B |
| 10 | D | D |
| 9 | I | I |
| 8 | G | G |
| 7 | E | |
| 6 | J | J |
| 5 | F | |
| 4 | ||
| 3 | A | C |
| 2 | ||
| 1 | C | A |
| Case 1 | |
| Floor | Name |
| 12 | K |
| 11 | B |
| 10 | D |
| 9 | I |
| 8 | G |
| 7 | E |
| 6 | J |
| 5 | F |
| 4 | L |
| 3 | A |
| 2 | H |
| 1 | C |
| Case 1 | Case 2 | Case 3 | |
| Floor | Name | Name | Name |
| 12 | K | ||
| 11 | |||
| 10 | G | ||
| 9 | |||
| 7 | G | G | |
| 6 | K | ||
| 5 | |||
| 4 | K | ||
| 3 | |||
| 2 | |||
| 1 |
| Case 1 | Case 2 | Case 3 | |
| Floor | Name | Name | Name |
| 12 | K | ||
| 11 | B | ||
| 10 | G | ||
| 9 | |||
| 8 | G | G | |
| 7 | |||
| 6 | J | K | |
| 5 | B | ||
| 4 | K | ||
| 3 | A/C | B | |
| 2 | |||
| 1 | A/C |
| Case 1 | Case 1.A | |
| Floor | Name | Name |
| 12 | K | K |
| 11 | B | B |
| 10 | ||
| 9 | ||
| 8 | G | G |
| 7 | ||
| 6 | J | J |
| 5 | ||
| 4 | K | |
| 3 | A | C |
| 2 | ||
| 1 | C | A |
| Case 1 | Case 1.A | |
| Floor | Name | Name |
| 12 | K | K |
| 11 | B | B |
| 10 | D | D |
| 9 | I | I |
| 8 | G | G |
| 7 | E | |
| 6 | J | J |
| 5 | F | |
| 4 | ||
| 3 | A | C |
| 2 | ||
| 1 | C | A |
| Case 1 | |
| Floor | Name |
| 12 | K |
| 11 | B |
| 10 | D |
| 9 | I |
| 8 | G |
| 7 | E |
| 6 | J |
| 5 | F |
| 4 | L |
| 3 | A |
| 2 | H |
| 1 | C |
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: