1994 IMO Problem #2

Alternate Solution using inversion is here:

[TechToppers]

I swear to god phantom points are one heck of a thing...

[adamham]

Have you make the proof using inversion??

[ChiragBharadwajYT]

2:13 why does the circle with diameter OE necessarily have to go through Q and B? I understand that EBO and OQE are right angles, but what if QE and OQ are super short (short enough so that Q is inside the circle) while still forming a right angle? Or if QE and OQ are super long? From my understanding, not all quadrilaterals can be circumscribed. What is the reasoning behind OQEB being cyclic? Or is it the case that any quadrilateral with opposing angles being 90 degrees is cyclic?