Problem 3

Determine all integers

n>3

for which there exist

n

points

A_1,\ldots,A_n

in the plane, no three collinear, and real numbers

r_1,\ldots,r_n

such that for any distinct

i,j,k

, the area of triangle

A_iA_jA_k

r_i+r_j+r_k

Step 3 of 4: Force equal weights in every five-point configuration

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In plain words

The convex-hull cases all give the same linear relation.

r_4=r_5

Detailed analysis

For a convex pentagon, decomposing two quadrilaterals into triangles gives $r_1+r_3=r_2+r_4$ and $r_1+r_3=r_2+r_5$ , hence $r_4=r_5$ . For a quadrilateral hull, choose the interior point so that two analogous quadrilaterals give $r_3=r_5$ . For a triangular hull, partitioning the hull into three triangles using each of the two interior points and comparing total areas gives $r_4=r_5$ . Thus every five-point configuration has two equal weights, contradicting the previous step.