1. If we write out the first four rows of the Pascal triangle and add up the columns we get:
1
1 1
1 2 1
1 3 3 1
1 1 4 3 4 1 1
If we write out the first 1024 rows of the triangle and add up the columns, how many of the resulting 2047 totals will be odd?
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| 2. A1B1A2, B1A2B2, A2B2A3, B2A3B3, ... , A13B13A14, B13A14B14, A14B14A1, B14A1B1 are thin triangular plates with all their edges equal length, joined along their common edges. Can the network of plates be folded (along the edges AiBi) so that all 28 plates lie in the same plane? (They are allowed to overlap). | |
| 3. A and B are positive integers. We are given a collection of n integers, not all of which are different. We wish to derive a collection of n distinct integers. The allowed move is to take any two integers in the collection which are the same (m and m) and to replace them by m + A and m - B. Show that we can always derive a collection of n distinct integers by a finite sequence of moves. |
The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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(C) John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 19 Apr 2003