Solution
Inside the Triangle
Show the problem again
How many points $(x, y)$ with integer coordinates lie strictly inside the triangle with vertices $(0, 0)$, $(20, 0)$ and $(0, 15)$? Points on the boundary do not count. Give an integer.
Worked solution
The answer is 131. Pick's theorem says a lattice polygon has area $A = I + B/2 - 1$, where $I$ and $B$ count interior and boundary lattice points. Here $A = 150$. Boundary points: the legs contribute 20 and 15 segment steps, and the hypotenuse from $(20, 0)$ to $(0, 15)$ contains $\gcd(20, 15) = 5$ steps. So $B = 20 + 15 + 5 = 40$. Then $I = A - B/2 + 1 = 150 - 20 + 1 = 131$.
Source: Classic exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.