BMO1 A 2020 2

New BMO problems!

So, if the triangle has side lengths a, a, and b, it’s area is \frac{b\sqrt{a^2-\frac{b^2}{4}}}{2}, and thus we have the equality:

Multiplying both sides by 16 we get:
4b^2a^2-b^4 = 32a+16b

Factoring, we have:

Dividing both sides by (2a+b) we get:

From here, since we know that b has to be an integer, we only have to test b=1, 2, 4 and we can get the following pairs of (a, b):
(\frac{17}{2}, 1), (3, 2), (\frac{5}{2}, 4)

1 Like