MathLabs

Problem 1

Let SS be a set of 9 distinct integers all of whose prime factors are at most 3. Prove that SS contains 3 distinct integers whose product is a perfect cube.
Step 3 of 6: Handle a repeated type
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N(i,j)≥3⟹(i,j)+(i,j)+(i,j)=(0,0)N(i,j)\ge3\Longrightarrow (i,j)+(i,j)+(i,j)=(0,0)
Detailed analysis

Let N(i,j)N(i,j) count elements of type (i,j)(i,j). If some N(i,j)≥3N(i,j)\ge3, three distinct elements have the same type, and their type sum is 3(i,j)=(0,0)3(i,j)=(0,0), so we are done.