2025 SSMO Speed Round Problems/Problem 4

Problem

In rectangle let be the circumcircle of , be the line through parallel to and be the intersection of and . Suppose the value of can be expressed as where and are relatively prime positive integers. Find .

Solution

import geometry;
unitsize(3cm);

point B=dir(aSin(0.47)); point C=dir(180-aSin(0.47)); point D=dir(180+aSin(0.47)); point A=dir(-aSin(0.47));
<br /> point X=

draw(A--B--C--D--cycle,p=black+0.3mm);
draw(unitcircle,p=blue+0.3mm);

dot(A,linewidth(4)); dot(B,linewidth(4)); dot(C,linewidth(4)); dot(D,linewidth(4));

label("$A$",A,dir(45));
label("$B$",B,dir(135));
label("$C$",C,dir(215));
label("$D$",D,dir(-45));
label("$\omega$",(0,-0.9));
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