MathLabs

Problem 6

Let a>b>c>d>0 a>b>c>d>0 be integers satisfying ac+bd=(b+d+a−c)(b+d−a+c) ac+bd=(b+d+a-c)(b+d-a+c). Prove that ab+cd ab+cd is not prime.
Step 2 of 6: Construct the quadrilateral
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WX=a,XY=c,YZ=b,ZW=d,WY2=a2−ac+c2.WX=a,\quad XY=c,\quad YZ=b,\quad ZW=d,\quad WY^2=a^2-ac+c^2.
Detailed analysis

The cosine rule gives angles WXY equal to 60 degrees and WZY equal to 120 degrees, so the quadrilateral is cyclic.