| Month & Year | Date | Person |
| January - 2006 | 13th | |
| 14th | ||
| February - 2007 | 13th | |
| 14th | ||
| March - 2008 | 13th | |
| 14th | ||
| April 2009 | 13th | |
| 14th | ||
| May - 2010 | 13th | |
| 14th |
| Month & Year | Date | Person |
| January - 2006 | 13th | |
| 14th | N | |
| February - 2007 | 13th | |
| 14th | ||
| March - 2008 | 13th | |
| 14th | ||
| April 2009 | 13th | |
| 14th | O | |
| May - 2010 | 13th | |
| 14th |
| Case 1 | Case 2 | ||
| Month & Year | Date | Person | Person |
| January - 2006 | 13th | J | |
| 14th | N | N | |
| February - 2007 | 13th | ||
| 14th | |||
| March - 2008 | 13th | R | R |
| 14th | |||
| April 2009 | 13th | ||
| 14th | O | O | |
| May - 2010 | 13th | J | |
| 14th |
| Case 1 | Case 2 | ||
| Month & Year | Date | Person | Person |
| January - 2006 | 13th | J | |
| 14th | N | N | |
| February - 2007 | 13th | ||
| 14th | |||
| March - 2008 | 13th | R | R |
| 14th | P | P | |
| April 2009 | 13th | L | L |
| 14th | O | O | |
| May - 2010 | 13th | J | |
| 14th |
| Case 1 | Case 1a | Case 2 | ||
| Month & Year | Date | Person | Person | Person |
| January - 2006 | 13th | J | J | |
| 14th | N | N | ||
| February - 2007 | 13th | M/Q | ||
| 14th | M/Q | M/Q | ||
| March - 2008 | 13th | R | R | R |
| 14th | P | P | P | |
| April 2009 | 13th | L | L | L |
| 14th | O | O | O | |
| May - 2010 | 13th | M/Q | J | |
| 14th | M/Q | M/Q |
| Case 1 | ||
| Month & Year | Date | Person |
| January - 2006 | 13th | J |
| 14th | N | |
| February - 2007 | 13th | Q |
| 14th | I | |
| March - 2008 | 13th | R |
| 14th | P | |
| April 2009 | 13th | L |
| 14th | O | |
| May - 2010 | 13th | M |
| 14th | K |
| Month & Year | Date | Person |
| January - 2006 | 13th | |
| 14th | ||
| February - 2007 | 13th | |
| 14th | ||
| March - 2008 | 13th | |
| 14th | ||
| April 2009 | 13th | |
| 14th | ||
| May - 2010 | 13th | |
| 14th |
| Month & Year | Date | Person |
| January - 2006 | 13th | |
| 14th | N | |
| February - 2007 | 13th | |
| 14th | ||
| March - 2008 | 13th | |
| 14th | ||
| April 2009 | 13th | |
| 14th | O | |
| May - 2010 | 13th | |
| 14th |
| Case 1 | Case 2 | ||
| Month & Year | Date | Person | Person |
| January - 2006 | 13th | J | |
| 14th | N | N | |
| February - 2007 | 13th | ||
| 14th | |||
| March - 2008 | 13th | R | R |
| 14th | |||
| April 2009 | 13th | ||
| 14th | O | O | |
| May - 2010 | 13th | J | |
| 14th |
| Case 1 | Case 2 | ||
| Month & Year | Date | Person | Person |
| January - 2006 | 13th | J | |
| 14th | N | N | |
| February - 2007 | 13th | ||
| 14th | |||
| March - 2008 | 13th | R | R |
| 14th | P | P | |
| April 2009 | 13th | L | L |
| 14th | O | O | |
| May - 2010 | 13th | J | |
| 14th |
| Case 1 | Case 1a | Case 2 | ||
| Month & Year | Date | Person | Person | Person |
| January - 2006 | 13th | J | J | |
| 14th | N | N | ||
| February - 2007 | 13th | M/Q | ||
| 14th | M/Q | M/Q | ||
| March - 2008 | 13th | R | R | R |
| 14th | P | P | P | |
| April 2009 | 13th | L | L | L |
| 14th | O | O | O | |
| May - 2010 | 13th | M/Q | J | |
| 14th | M/Q | M/Q |
| Case 1 | ||
| Month & Year | Date | Person |
| January - 2006 | 13th | J |
| 14th | N | |
| February - 2007 | 13th | Q |
| 14th | I | |
| March - 2008 | 13th | R |
| 14th | P | |
| April 2009 | 13th | L |
| 14th | O | |
| May - 2010 | 13th | M |
| 14th | K |
| Month & Year | Date | Person |
| January - 2006 | 13th | |
| 14th | ||
| February - 2007 | 13th | |
| 14th | ||
| March - 2008 | 13th | |
| 14th | ||
| April 2009 | 13th | |
| 14th | ||
| May - 2010 | 13th | |
| 14th |
| Month & Year | Date | Person |
| January - 2006 | 13th | |
| 14th | N | |
| February - 2007 | 13th | |
| 14th | ||
| March - 2008 | 13th | |
| 14th | ||
| April 2009 | 13th | |
| 14th | O | |
| May - 2010 | 13th | |
| 14th |
| Case 1 | Case 2 | ||
| Month & Year | Date | Person | Person |
| January - 2006 | 13th | J | |
| 14th | N | N | |
| February - 2007 | 13th | ||
| 14th | |||
| March - 2008 | 13th | R | R |
| 14th | |||
| April 2009 | 13th | ||
| 14th | O | O | |
| May - 2010 | 13th | J | |
| 14th |
| Case 1 | Case 2 | ||
| Month & Year | Date | Person | Person |
| January - 2006 | 13th | J | |
| 14th | N | N | |
| February - 2007 | 13th | ||
| 14th | |||
| March - 2008 | 13th | R | R |
| 14th | P | P | |
| April 2009 | 13th | L | L |
| 14th | O | O | |
| May - 2010 | 13th | J | |
| 14th |
| Case 1 | Case 1a | Case 2 | ||
| Month & Year | Date | Person | Person | Person |
| January - 2006 | 13th | J | J | |
| 14th | N | N | ||
| February - 2007 | 13th | M/Q | ||
| 14th | M/Q | M/Q | ||
| March - 2008 | 13th | R | R | R |
| 14th | P | P | P | |
| April 2009 | 13th | L | L | L |
| 14th | O | O | O | |
| May - 2010 | 13th | M/Q | J | |
| 14th | M/Q | M/Q |
| Case 1 | ||
| Month & Year | Date | Person |
| January - 2006 | 13th | J |
| 14th | N | |
| February - 2007 | 13th | Q |
| 14th | I | |
| March - 2008 | 13th | R |
| 14th | P | |
| April 2009 | 13th | L |
| 14th | O | |
| May - 2010 | 13th | M |
| 14th | K |
| Month & Year | Date | Person |
| January - 2006 | 13th | |
| 14th | ||
| February - 2007 | 13th | |
| 14th | ||
| March - 2008 | 13th | |
| 14th | ||
| April 2009 | 13th | |
| 14th | ||
| May - 2010 | 13th | |
| 14th |
| Month & Year | Date | Person |
| January - 2006 | 13th | |
| 14th | N | |
| February - 2007 | 13th | |
| 14th | ||
| March - 2008 | 13th | |
| 14th | ||
| April 2009 | 13th | |
| 14th | O | |
| May - 2010 | 13th | |
| 14th |
| Case 1 | Case 2 | ||
| Month & Year | Date | Person | Person |
| January - 2006 | 13th | J | |
| 14th | N | N | |
| February - 2007 | 13th | ||
| 14th | |||
| March - 2008 | 13th | R | R |
| 14th | |||
| April 2009 | 13th | ||
| 14th | O | O | |
| May - 2010 | 13th | J | |
| 14th |
| Case 1 | Case 2 | ||
| Month & Year | Date | Person | Person |
| January - 2006 | 13th | J | |
| 14th | N | N | |
| February - 2007 | 13th | ||
| 14th | |||
| March - 2008 | 13th | R | R |
| 14th | P | P | |
| April 2009 | 13th | L | L |
| 14th | O | O | |
| May - 2010 | 13th | J | |
| 14th |
| Case 1 | Case 1a | Case 2 | ||
| Month & Year | Date | Person | Person | Person |
| January - 2006 | 13th | J | J | |
| 14th | N | N | ||
| February - 2007 | 13th | M/Q | ||
| 14th | M/Q | M/Q | ||
| March - 2008 | 13th | R | R | R |
| 14th | P | P | P | |
| April 2009 | 13th | L | L | L |
| 14th | O | O | O | |
| May - 2010 | 13th | M/Q | J | |
| 14th | M/Q | M/Q |
| Case 1 | ||
| Month & Year | Date | Person |
| January - 2006 | 13th | J |
| 14th | N | |
| February - 2007 | 13th | Q |
| 14th | I | |
| March - 2008 | 13th | R |
| 14th | P | |
| April 2009 | 13th | L |
| 14th | O | |
| May - 2010 | 13th | M |
| 14th | K |
| Month & Year | Date | Person |
| January - 2006 | 13th | |
| 14th | ||
| February - 2007 | 13th | |
| 14th | ||
| March - 2008 | 13th | |
| 14th | ||
| April 2009 | 13th | |
| 14th | ||
| May - 2010 | 13th | |
| 14th |
| Month & Year | Date | Person |
| January - 2006 | 13th | |
| 14th | N | |
| February - 2007 | 13th | |
| 14th | ||
| March - 2008 | 13th | |
| 14th | ||
| April 2009 | 13th | |
| 14th | O | |
| May - 2010 | 13th | |
| 14th |
| Case 1 | Case 2 | ||
| Month & Year | Date | Person | Person |
| January - 2006 | 13th | J | |
| 14th | N | N | |
| February - 2007 | 13th | ||
| 14th | |||
| March - 2008 | 13th | R | R |
| 14th | |||
| April 2009 | 13th | ||
| 14th | O | O | |
| May - 2010 | 13th | J | |
| 14th |
| Case 1 | Case 2 | ||
| Month & Year | Date | Person | Person |
| January - 2006 | 13th | J | |
| 14th | N | N | |
| February - 2007 | 13th | ||
| 14th | |||
| March - 2008 | 13th | R | R |
| 14th | P | P | |
| April 2009 | 13th | L | L |
| 14th | O | O | |
| May - 2010 | 13th | J | |
| 14th |
| Case 1 | Case 1a | Case 2 | ||
| Month & Year | Date | Person | Person | Person |
| January - 2006 | 13th | J | J | |
| 14th | N | N | ||
| February - 2007 | 13th | M/Q | ||
| 14th | M/Q | M/Q | ||
| March - 2008 | 13th | R | R | R |
| 14th | P | P | P | |
| April 2009 | 13th | L | L | L |
| 14th | O | O | O | |
| May - 2010 | 13th | M/Q | J | |
| 14th | M/Q | M/Q |
| Case 1 | ||
| Month & Year | Date | Person |
| January - 2006 | 13th | J |
| 14th | N | |
| February - 2007 | 13th | Q |
| 14th | I | |
| March - 2008 | 13th | R |
| 14th | P | |
| April 2009 | 13th | L |
| 14th | O | |
| May - 2010 | 13th | M |
| 14th | K |
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: