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 1 of 4: Show that four points work

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In plain words

A square makes all four triangle areas equal.

n=4

Detailed analysis

Take the vertices of a unit square and set $r_1=r_2=r_3=r_4=\frac16$ . Every triangle of three vertices has area $\frac12=r_i+r_j+r_k$ , so $n=4$ is attainable.