Baltic Way 1992 · Problem 4
Number Theory
Is it possible to draw a hexagon with vertices in the knots of an integer lattice so that the squares of the lengths of the sides are six consecutive positive integers?
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Review
Topics
Divisibility and factorization
Solutions
Solution
Solution:
The sum of any six consecutive positive integers is odd. On the other hand, the sum of the squares of the lengths of the sides of the hexagon is equal to the sum of the squares of their projections onto the two axes. But this number has the same parity as the sum of the projections themselves, the latter being obviously even.
Contest context
Results from Baltic Way 1992
8 teams
- Mean score
- 2.9 / 5
- Scores of 4 or 5
- 4 / 8
- Estonia
- 0 / 5
Score distribution
03
10
20
31
40
54
All team scores
| Team | Score |
|---|---|
| Denmark | 5 / 5 |
| St. Petersburg | 3 / 5 |
| Poland | 5 / 5 |
| Latvia | 0 / 5 |
| Iceland | 5 / 5 |
| Lithuania | 5 / 5 |
| Estonia | 0 / 5 |
| Sweden | 0 / 5 |