XIX Asian Pacific Mathematics Olympiad

Without loss of generality, we may assume that $S$ contains only positive integers. Let $$S=\{2^{a_i}{3}^{b_i}\mid a_i,b_i\in \mathbb{Z},a_i,b_i\ge 0,1\le i\le 9\}.$$ It suffices to show that there are $1\le i_1,i_2,i_3\le 9$ such that $$\begin{array}{c}{a}_{i_1}+a_{i_2}+a_{i_3}\equiv b_{i_1}+b_{i_2}+b_{i_3}\equiv 0(\mathrm{mod}3).\end{array}\text{\hspace{1em}(†)}$$ For $n=2^a{3}^b\in S$, let's call $(a\mathrm{mod}3,b\mathrm{mod}3)$ the type of $n$. Then there are 9 possible types: $$(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2).$$ Let $N(i,j)$ be the number of integers in $S$ of type $(i,j)$. We obtain 3 distinct integers whose product is a perfect cube when (1) $N(i,j)\ge 3$ for some $i,j$, or (2) $N(i,0)N(i,1)N(i,2)\ne 0$ for some $i=0,1,2$, or (3) $N(0,j)N(1,j)N(2,j)\ne 0$ for some $j=0,1,2$, or (4) $N(i_1,j_1)N(i_2,j_2)N(i_3,j_3)\ne 0$, where $\{i_1,i_2,i_3\}=\{j_1,j_2,j_3\}=\{0,1,2\}$. Assume that none of the conditions (1) $\sim$ (3) holds. Since $N(i,j)\le 2$ for all $(i,j)$, there are at least five $N(i,j)$'s that are nonzero. Furthermore, among those nonzero $N(i,j)$'s, no three have the same $i$ nor the same $j$. Using these facts, one may easily conclude that the condition (4) should hold. (For example, if one places each nonzero $N(i,j)$ in the $(i,j)$-th box of a regular $3\times 3$ array of boxes whose rows and columns are indexed by 0, 1 and 2, then one can always find three boxes, occupied by at least one nonzero $N(i,j)$, whose rows and columns are all distinct. This implies (4).)

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Alternative solution.

Up to $(†)$, we do the same as above and get 9 possible types: $$(a\mathrm{mod}3,b\mathrm{mod}3)=(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)$$ for $n=2^a{3}^b\in S$. Note that (i) among any 5 integers, there exist 3 whose sum is $0(\mathrm{mod}3)$, and that (ii) if $i,j,k\in \{0,1,2\}$, then $i+j+k\equiv 0(\mathrm{mod}3)$ if and only if $i=j=k$ or $\{i,j,k\}=\{0,1,2\}$. Let's define $T$: the set of types of the integers in $S$; $N(i)$: the number of integers in $S$ of the type $(i,\cdot )$; $M(i)$: the number of integers $j\in \{0,1,2\}$ such that $(i,j)\in T$. If $N(i)\ge 5$ for some $i$, the result follows from (i). Otherwise, for some permutation $(i,j,k)$ of $(0,1,2)$, $$N(i)\ge 3,N(j)\ge 3,N(k)\ge 1.$$ If $M(i)$ or $M(j)$ is 1 or 3, the result follows from (ii). Otherwise $M(i)=M(j)=2$. Then either $$(i,x),(i,y),(j,x),(j,y)\in T\text{or}(i,x),(i,y),(j,x),(j,z)\in T$$ for some permutation $(x,y,z)$ of $(0,1,2)$. Since $N(k)\ge 1$, at least one of $(k,x),(k,y)$ and $(k,z)$ is contained in $T$. Therefore, in any case, the result follows from (ii). (For example, if $(k,y)\in T$, then take $(i,y),(j,y),(k,y)$ or $(i,x),(j,z),(k,y)$ from $T$.)