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 2 of 4: Equal weights create three geometric constraints

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In plain words

Equal areas over a fixed base force equal heights.

r_4=r_5\Longrightarrow A_4A_5\parallel A_iA_j\text{ or }\operatorname{mid}(A_4A_5)\in A_iA_j

Detailed analysis

If $r_4=r_5$ , then triangles $A_iA_jA_4$ and $A_iA_jA_5$ have equal areas for each pair among $A_1,A_2,A_3$ . Thus either $A_4A_5\parallel A_iA_j$ (same side of the base) or the midpoint of $A_4A_5$ lies on $A_iA_j$ (opposite sides). Among the three bases, two alternatives of the same type occur, impossible because $A_1,A_2,A_3$ are noncollinear.