Problem 11.6. (20 points) Quadratic trinomials $P(x)$ and $Q(x)$ with real coefficients are such that together they have 4 distinct real roots, and each of the polynomials $P(Q(x))$ and $Q(P(x))$ has 4 distinct real roots. What is the smallest number of distinct real numbers that can be among the roots of the polynomials $P(x), Q(x), P(Q(x))$, and $Q(P(x))$?
# Answer: 6.

Solution. Note that if among the roots of the polynomial $P(Q(x))$ there is a root of $Q(x)$, say, the number $x_{0}$, then $P\left(Q\left(x_{0}\right)\right)=P(0)=0$, from which 0 is a root of $P(x)$. Similarly, if among the roots of $Q(P(x))$ there is a root of $P(x)$, then 0 is a root of $Q(x)$. However, $P(x)$ and $Q(x)$ cannot both have the root 0, otherwise, in total, they would have fewer than 4 roots.

From this, we can obtain an estimate of the total number of distinct roots. If there are no more than 5, then $P(Q(x))$ and $Q(x)$ have a common root, and $Q(P(x))$ and $P(x)$ have a common root, which cannot be according to the above.

Now let's construct an example where there are exactly 6 distinct roots. Let $P(x)=\frac{1}{2} x(x-3), Q(x)=$ $-\frac{3}{2}(x+1)(x-2)$. Then the roots of $P(x)$ will be the numbers 0 and 3; the roots of $Q(x)$ will be the numbers -1 and 2; the roots of $P(Q(x))$ will be the numbers -1, 0, 1, 2; the roots of $Q(P(x))$ will be the numbers -1, 1, 2, 4. In total, the roots of all polynomials together are the integers from -1 to 4.

Remark. There are other examples where there are exactly 6 distinct roots; in all examples, one of the quadratic polynomials $P(x)$ and $Q(x)$ has the root 0. For example, the pair of quadratic polynomials $P(x)=$ $x(x-3)$ and $Q(x)=-\frac{3}{4}(x+1)(x-4)$ also works.

## Criteria

The points for the estimate and the example are summed.

+6 points. Estimate - it is proven that there cannot be fewer than 6 distinct numbers among the roots of the considered polynomials.

In the absence of this proof, the following progress is evaluated:

+3 points. It is noted that if the polynomials $P(Q(x))$ and $Q(x)$ have a common root, then the polynomial $P(x)$ has the root 0.

+14 points. Example - suitable quadratic polynomials $P(x)$ and $Q(x)$ are provided, and it is shown that among the roots of all four considered polynomials, there are exactly 6 distinct numbers (or it is otherwise proven that such polynomials exist).

In the absence of such an example with justification, the highest of the following criteria is used:

+10 points. Correct example of quadratic polynomials $P(x)$ and $Q(x)$.

+12 points. Correct example of quadratic polynomials $P(x)$ and $Q(x)$, and the work also includes a correct estimate. (Thus, for a solution with a correct estimate and a correct pair of $P(x)$ and $Q(x)$ without proof that they fit, a total of 18 points is given.)

The following progress is not evaluated:

0 points. Only the answer is provided.

0 points. A correct example with 7 or more roots is provided.