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 4 of 4: Exclude all larger

n

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In plain words

Any five of the points would inherit the same area rule.

n>5\Longrightarrow\text{a five-point subconfiguration}

Detailed analysis

If $n>5$ , selecting any five points would produce an impossible five-point configuration. Therefore the only integer satisfying the requirement is $n=4$ .