Peru EGMO TST 2018 Let $a,b,c$ be complex numbers such that $a^2,b^2,c^2$ are respective vertices of triangle $ABC$ lying on unit circle centered at $0$ and midpoints of arcs $AB,BC,CA$ not containing other vertices of the triangle than at their ends are $-ab,-bc,-ca$. Then (here $i$ not to be confused with imaginary unit) $$n=bc,\ t=2a^2-bc,\ i=-(ab+bc+ca),\ m=\frac{b^2+c^2}{2},\ j=ab-bc+ca.$$If two from the points $J,M,I,T$ are equal we are done. If none are equal we have the expression with no division or multiplying by $0$:$$\frac{m-i}{j-i}\cdot\frac{t-j}{t-m}=\frac{\frac12(b+c)(2a+b+c)}{2a(b+c)}\cdot\frac{a(2a-b-c)}{\frac12(2a-b-c)(2a+b+c)}=\frac12\in\mathbb{R}$$QED