Iberoamerican Mathematical Olympiad

Solution:

Let the sides be $a$, $a + d$, $a + 2d$ with $d \geq 0$. Then the altitudes are $k / a$, $k / (a + d)$, $k / (a + 2d)$, where $k$ is twice the area. We claim that $k / a + k / (a + 2d) > 2k / (a + d)$ unless $d = 0$. This is equivalent to $(a + d)(a + 2d) + a(a + d) > 2a(a + 2d)$ or $2d^2 > 0$, which is obviously true. So the altitudes can only form an arithmetic progression if $d = 0$ and hence the triangle is equilateral.