Determine all complex numbers $z$ such that $|z|+|\bar{z}|=|z+\bar{z}|$.
We note $z=a+i b$. We observe that $z+\bar{z}=a+i b+a-i b=2 a$ and the equality becomes $\sqrt{a^{2}+b^{2}}+\sqrt{a^{2}+b^{2}}=\sqrt{4 a^{2}}$. However, $2 \sqrt{a^{2}+b^{2}}=\sqrt{4 a^{2}+4 b^{2}} \geq \sqrt{4 a^{2}}$ with equality if and only if $b=0$. The complex solutions are therefore the complexes whose imaginary part is zero, that is, the real numbers.