Problem 1

Let

S

be a set of 9 distinct integers all of whose prime factors are at most 3. Prove that

S

contains 3 distinct integers whose product is a perfect cube.

Step 2 of 6: Translate the goal into types

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t(n_i)+t(n_j)+t(n_k)=(0,0)\Longrightarrow n_i n_j n_k\text{ is a perfect cube}

Detailed analysis

If three distinct elements have types whose coordinatewise sum is $(0,0)$ in $\mathbb F_3^2$ , then both prime exponents in their product are multiples of 3. Their product is therefore a perfect cube.