Croatia_2018

If a complex number $z=x+yi$ is Gaussian integer, then $|z{|}^2=x^2+y^2$ is the sum of two squares of integers. The square of an even integer is divisible by 4, while the square of an odd integer gives remainder 1 when divided by 4. Therefore, the sum of squares of two integers can give remainder 0, 1 or 2 when divided by 4. If $n$ is greater than 3, then among any four consecutive positive integers in the sequence $$|z_1{|}^2,|z_2{|}^2,\dots ,|z_n{|}^2$$ there would be a number giving the remainder 3 when divided by 4, which is impossible. Hence, $n$ is less than or equal to 3. Notice that $2+2i$, $3$ and $3+i$ are Gaussian integers whose absolute values squared are $8$, $9$ and $10$, respectively. Therefore, the answer for $n$ is $3$.