10. Suppose 3 distinct numbers are chosen from $1,2, \ldots, 3 n$ with their sum equal to $3 n$. What is the largest possible product of those 3 numbers?
(a) $n^{3}-n$;
(b) $n^{3}$;
(c) $n^{3}+n$;
(d) $n^{3}-7 n+6$;
(e) $n^{3}-7 n-6$.
10 Ans: (a).
From the identity $4 a b=(a+b)^{2}-(a-b)^{2}$, one sees that when $a+b$ is fixed, the product is the largest when $|a-b|$ is the smallest.
Now let $a<b<c$ be 3 distinct positive integers such that $a+b+c=3 n$ and the product $a b c$ is the biggest possible. If $a$ and $b$ differ more than 2, then by increasing $a$ by 1 and decreasing $b$ by 1 , the product $a b c$ will become bigger. Hence $a$ and $b$ differ by at most 2. Similarly $b$ and $c$ differ by at most 2 .
On the other hand, if $c=b+2=a+4$, then one can increase the value of $a b c$ by decreasing $c$ by 1 and increasing $a$ by 1 . So this case is ruled out.
Since $a+b+c$ is divisible by 3 , it is not possible that $c=b+1=a+3$ or $c=b+2=a+3$.
So we must have $c=b+1=a+2$. Hence the three numbers are $n-1, n$ and $n+1$. Thus, their product is $(n-1) n(n+1)=n^{3}-n$.