Difference between revisions of "2025 SSMO Speed Round Problems/Problem 4"
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==Solution== | ==Solution== | ||
| + | |||
| + | <asy> | ||
| + | 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)); | ||
| + | </asy> | ||
Revision as of 15:38, 9 September 2025
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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