MathLabs

Problem 5

Find the greatest integer nn for which there are n+4n+4 points A,B,C,D,X1,…,XnA,B,C,D,X_1,\dots,X_n in the plane with AB≠CDAB\ne CD such that, for each i=1,2,…,ni=1,2,\dots,n, triangles ABXiABX_i and CDXiCDX_i are congruent.
Step 4 of 6: Conclude the upper bound
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n≤82=4.n\le\frac 82=4.
Detailed analysis

At most half of the eight necessary candidates can survive the pairing argument, so n is at most 4.