Let ABCD be $a$ unit square. Suppose M and N are points on BC and CD respectively such that the perimeter of triangle MCN is 2. Let O be the circumcentre of triangle MAN and P be the circumcentre of triangle MON. If OP $m$ OA = $n$ for some relatively prime positive integers $m$ and n, find the value of $m$ + n.

Step 1:  OA = circumradius of AMN

Step 2: OP = circumradius of OMN

Step 3: 2 2

Step 4: OP 1

Step 5: So OA = 2cos 

Step 6:

Step 7: Perimeter of MCN = 2 = (1 – x) + (1 – y) + MN

Step 8: => MN = $x$ + y

Step 9: Now rotate ABM about A so that AB overlaps with AD (by 90°)

Step 10: Clearly AMN = AMN

Step 11: So 2 = 90°

Step 12: =>  = 45°

Step 13: OP 1 m

Step 14: Hence =2= n

Step 15: => m+n=3