2005 Indonesia MO Problems/Problem 1
Problem
Let be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is .
Solution
WLOG, let . The original problem is essentially asking for the number of lattice points that lie within this bound as well as .
By experimenting with smaller graphs, we can split into two cases.
Case 1: is even
Below is the case where .
The line and intersect at . By symmetry, for each of the four line segments from the diagonal, there are lattice points. Since there are a total of lattice points within , by symmetry, each section formed from the diagonals has lattice points. We want the points on lines , and not , so there are points that satisfy the conditions if is even.
Case 1: is odd
Below is the case where .
The line and intersect at , but that value is not an integer. By symmetry, for each of the four line segments from the diagonal, there are lattice points. Since there are a total of lattice points within , by symmetry, each section formed from the diagonals has lattice points. We want the points on lines , and not , so there are points that satisfy the conditions if is odd.
In summary, the number of triangles that satisfy the conditions are and .
See Also
2005 Indonesia MO (Problems) | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 2 |
All Indonesia MO Problems and Solutions |