Lor
2023 AIME I
Problem 15
Find the largest prime number 
for which there exists a complex number 
satisfying
- the real and imaginary part of are both integers;
and
- there exists a triangle whose three side lengths are
the real part of 
and the imaginary part of 
z = a + bi
Prove that Im{z^3}+Re{z^3}>p.sqrt(p)>p
Calculate Im{z^3} and Re{z^3} in terms of a, b
Consider Im{z^3}, Re{z^3}>0
Prove that abs(Im{z^3}-Re{z^3})<p=a^2+b^2
case1) a>0 -> b>0
case2) a<0, b>0 -> b=-a -> p=2
W.L.O.G a,b>0
Prove that a>max{b.sqrt(3), b/sqrt(3)}
Consider a=bk then k>sqrt(3)
Solution