Prove the inequality $\deg{P(x)}\cdot \deg{Q(x)}\cdot \deg{R(x)}\ge 656$

Let three non-constant polynomials $P(x),Q(x),R(x)\in \mathbb Z[x]$, and suppose that the equation $P(x)Q(x)R(x)=2015$ has $49$ distinct integer roots.

Prove that $$\deg{P(x)}\cdot \deg{Q(x)}\cdot \deg{R(x)}\ge 656$$

Solution 1:

So, composing all comments together:

Let $x_i, i=1..49$ be distinct integer roots.

1. $x_i \in \mathbb Z$ ; $P(x),Q(x),R(x)\in \mathbb Z[x] \Rightarrow P(x_i),Q(x_i),R(x_i)\in \mathbb Z$ for all $i$.

2. Since $2015 = 5 \cdot 13 \cdot 31$ we have for all $i$ that $P(x_i) \in S:=\{ \pm 1,\pm 5,\pm 13, \pm 31, \pm 65, \pm 155, \pm 403,\pm 2015 \}$. Note that $|S|= 16$.

3. Let $k =deg(P)>0$. We know that for each $s$ there are no more than $k$ distinct numbers $x$ such $P(x)=s$. So there are no more than $16k$ distinct numbers $x$ such that $P(x) \in S$. Then we have $16k \geq 49 \Rightarrow deg(P) = k \geq 4$ since $k$ is integer. Same we can say for $deg(Q)$ and $deg(R)$.

4. Additionally we have $deg(P)+deg(Q)+deg(R) \geq 49$ since there are at least 49 roots (some roots may be multiple roots).

5. We end with optimization problem:

   find $min(x \cdot y \cdot z)$ given $ x,y,z \geq 4$ and $x+y+z \geq 49 $ and $x,y,z \in \mathbb Z$.

   Solution of this simple problem tells that $min(x \cdot y \cdot z) = 656 $ (and it is realized for $(x,y,z) = (4,4,41)$ and it permutations).

All this derivations are contained in comments but in implicit form thats why I posted explicit answer. If you want upvote, please do it on comments.