Problem 1949

First, observe that no polynomial is $1$-good (because $Q(X)(P(X)+Q(X))$ always has roots modulo $1$) and the polynomial $P(X)=1$ is not $2$-good (because $Q(X)(Q(X)+1)$ is always divisible by $2$).

Now, if $P$ is $d$-good with some $Q$, then $Q \cdot (P+Q)$ has no roots $\bmod d$. Therefore, it certainly has no roots $\bmod n$ for $d \mid n$, so $P$ must be $n$-good. Consequently, it suffices to show that all polynomials are $n$-good whenever $n$ is an odd prime, or $n=4$.

We start by handling the case $n=4$. We will construct a $Q$ such that $Q(X)$ is never divisible by $4$ and $Q(X)+P(X)$ is always odd; this will clearly show that $P$ is $4$-good. Note that any function modulo $2$ must be either constant or linear - in other words, there are $a, b \in \{0,1\}$ such that $P(X)=a X+b \bmod 2$ for all $X$. If $a=0$ then set $Q(X)=4 X^{2}+b+1$, and if $a=1$ then set $Q(X)=X^{2}+b+1$; in all cases, $Q$ will satisfy the required properties.

It remains to prove that any polynomial is $p$-good, where $p$ is an odd prime. We will prove that for any function $f$ defined $\bmod p$, there is a quadratic $Q$ with no roots $\bmod p$ such that $Q(x) \neq f(x) \bmod p$ for all $x$; the statement about $P$ then follows with $f$ replaced by $-P$. For the remainder of the proof, we will consider all equalities modulo $p$.

Suppose that a function $f$ not satisfying the above exists; in other words, $f$ has the property that for any quadratic $Q$ with no roots $\bmod p$, there is some $x$ such that $Q(x)=f(x)$. Without loss of generality, we may assume that $f$ has no roots $\bmod p$. To see why, suppose that $f(u)=0$ for some $u$, and let $g$ be the function such that $g(x)=f(x)$ for $x \neq u$ and $g(u)=1$. For any $Q$ with no roots, we know that there is some $x \neq u$ such that $P(x)=f(x)$, and so $P(x)=g(x)$ for that choice of $x$. In particular, $g$ is also not $p$-good.

Now, suppose first that there is some nonzero $t$ such that $t$ is not in the image of $f$. Then we may take $Q(X)=p X^{2}+t$; this quadratic is never equal to $f$ and is never zero. Thus, $f$ must be surjective onto the nonzero residues $\bmod p$. There are $p$ choices for $X$ and $p-1$ nonzero residues $\bmod p$, so there must be some $x_{1} \neq x_{2} \bmod p$ such that $f\left(x_{1}\right)=f\left(x_{2}\right)$, and $f$ is a bijection from the set of residues $\bmod p$ not equal to $x_{2}$ to the set of nonzero residues $\bmod p$.

Now, note that we may choose any $b$ and $c$ with $b$ nonzero and replace $f(X)$ with $g(X)= f(b X+c)$; if there were some $Q$ with no roots such that $Q(x) \neq g(x)$ for all $x$, then $Q(X / b-c / b)$ would work for $f$. Choose $b$ and $c$ such that $b x_{1}+c=1$ and $b x_{2}+c=-1$; such $b$ and $c$ must exist (we may take $b=2 /(x_{1}-x_{2})$ and $c=(x_{1}+x_{2}) /(x_{2}-x_{1})$ ). Renaming $g$ to $f$, we see that we may assume $f(1)=f(-1)$.

Let $r'$ be a quadratic nonresidue $\bmod p$. Choose $y \neq 0$ such that $f(y)=(1-r') f(0)$, which must exist as the right hand side is nonzero and $1-r'$ is not equal to $1$. Choose $r=y^{2} / r'$, which is a quadratic nonresidue.

Consider $\phi(X)=f(X) /(X^{2}-r)$. By definition, $\phi(1)=\phi(-1)$ and $\phi(0)=\phi(y)$, so there are no more than $p-2$ values in the image of $\phi$. Choose some nonzero $a$ not in the image of $\phi$, so $f(X) /(X^{2}-r)$ is never equal to $a$. The quadratic $Q(X)=a(X^{2}-r)$ is never zero and also never equal to $f(X)$, which completes the proof.

Given $f$ a function $\bmod p$ such that $f$ is surjective onto the nonzero elements of $\mathbb{Z} / p \mathbb{Z}$ and $f(1)=f(-1)$, we provide an alternative approach to construct a nonzero quadratic $Q(X)$ such that $Q(X) \neq f(X)$. Let $r$ be the smallest quadratic nonresidue $\bmod p$ (so $r-1$ is a square) and let $a$ vary over the nonzero elements $\bmod p$; we will show that it is possible to choose $Q_{a}(X)=a(X^{2}-r)$ for some choice of $a$. Note that any quadratic of this form will be nowhere zero.

Suppose that no such $Q_{a}$ works. Then, for each $a$, there exists $x$ such that $a(x^{2}-r)=f(x)$. We may assume that $x \neq -1$, as if the equality holds for $x=-1$ then it also holds for $x=1$. However, $a(x^{2}-r)=f(x)$ implies $a=f(x)/(x^{2}-r)$, so $f(x)/(x^{2}-r)$ must be a surjection from $\{x \neq -1\}$ to the set of nonzero $a$, and so this is a bijection. In particular, for each $a$, there exists a unique $x_{a}$ such that $f(x_{a})=a(x_{a}^{2}-r)$.

We now have
$$\begin{aligned} \prod_{t \neq 0} t & =\prod_{a \neq 0} f(x_{a}) \\ & =\prod_{a \neq 0} a \prod_{a \neq 0}(x_{a}^{2}-r) \\ & =\prod_{a \neq 0} a \prod_{x \neq -1}(x^{2}-r) \end{aligned}$$
where the first equality follows because $f$ is surjective onto the nonzero residues $\bmod p$, and the second equality follows from the definition of $x_{a}$. The two products cancel, which means that $\prod_{x \neq -1}(x^{2}-r)=1$.

However, we also get
$$\prod_{x \neq -1}(x^{2}-r)=(-r)(1-r)\left(\prod_{x=2}^{(p-1)/2}(x^{2}-r)\right)^{2} .$$
However, this is a contradiction as $-r(1-r)=r(r-1)$, which is not a quadratic residue (by our choice of $r$ ).

As in Solution 1, we will reduce to the case of $p$ being an odd prime and $f$ being a function $\bmod p$ with no roots which is surjective onto the set of nonzero residues $\bmod p$, although we make no assumption about the values of $x_{1}$ and $x_{2}$ with $f(x_{1})=f(x_{2})$.

We will again consider quadratics of the form $Q_{a, b, c}(X)=a R(b X+c)$, where $R(X)=X^{2}-r$ for an arbitrary fixed quadratic nonresidue $r$, $a$ and $b$ are nonzero $\bmod p$, and $c$ is any residue $\bmod p$.

For each fixed $b$ and $c$, there must be $n$ pairs $(a, x)$ such that $a R(b x+c)=f(x)$, because there must be exactly one value of $a$ for each $x$. If any $a$ appears in no such pair then we are done, so assume otherwise. In other words, there must be exactly one $a$ such that there are two such $x$, and for all other $a$ there is only one such $x$.

Thus, for each $(b, c)$, there is exactly one unordered pair $\{x_{1}, x_{2}\}$ such that for some $a$ we have $f(x_{i})=a R(b x_{i}+c)$; in other words, there is exactly one unordered pair $\{x_{1}, x_{2}\}$ such that $f(x_{1}) / R(b x_{1}+c)=f(x_{2}) / R(b x_{2}+c)$.

Now, we show that for each unordered pair $\{x_{1}, x_{2}\}$ there must be at least one pair $(b, c)$ such that $f(x_{1}) / R(b x_{1}+c)=f(x_{2}) / R(b x_{2}+c)$. Indeed, let $t=f(x_{1}) / f(x_{2})$. There must be some $x_{1}', x_{2}'$ such that $R(x_{1}') / R(x_{2}')=t$; this is because $R(X)$ and $t R(X)$ both take $\frac{p+1}{2}$ nonzero values $\bmod p$, so the intersection must be nonempty by the pigeonhole principle. Choosing $b$ and $c$ such that $b x_{1}+c=x_{1}'$ and $b x_{2}+c=x_{2}'$ gives the claim.

Note further that if $(b, c)$ and $\{x_{1}, x_{2}\}$ satisfy the relation, then the same is true for $(-b,-c)$ and $\{x_{1}, x_{2}\}$ because $R(b x+c)=R(-b x-c)$. Since $b$ is nonzero, this means that each pair $\{x_{1}, x_{2}\}$ corresponds to at least two pairs ( $b, c$ ). However, since there are $p(p-1)$ pairs ( $b, c$ ) with $b$ nonzero and $p(p-1) / 2$ unordered pairs $\{x_{1}, x_{2}\}$, each $\{x_{1}, x_{2}\}$ must correspond to exactly two pairs $(b, c)$ and $(-b,-c)$ for some $(b, c)$.

Now, since the image of $f$ has only $p-1$ elements, there must be some $x_{1}, x_{2}$ such that $f(x_{1})=f(x_{2})$. Choose any $b, c$ such that $b x_{1}+c=-\left(b x_{2}+c\right)$, so $R(b x_{1}+c)=R(b x_{2}+c)$ and so $f(x_{1}) / R(b x_{1}+c)=f(x_{2}) / R(b x_{2}+c)$. There is such a pair $b, c$ for any nonzero $b$, so there are at least $p-1$ such pairs, and this quantity is greater than $2$ for $p \geqslant 5$.

Finally, for the special case that $p=3$, we observe that there must be at least one allowed value for $Q(x)$ for each $x$, so there must exist such a quadratic $Q$ by Lagrange interpolation.

Comment. We may also handle the case $p=3$ as follows. Recall that we may assume $f$ is nonzero and surjective onto $\{1,2\} \bmod 3$, so the image of $f$ must be $(1,1,2)$ or $(1,2,2)$ in some order. Without loss of generality $f(1)=f(2)$, so we either have $(f(0), f(1), f(2))=(1,2,2)$ or $(2,1,1)$. In the first case, take $Q(X)=2 X^{2}+2$, and in the second case take $Q(X)=X^{2}+1$.

Again, we reduce to the case of $p$ being an odd prime and $f$ being a function $\bmod p$; we will show that there is a quadratic which is nowhere zero such that $Q(x)=f(x)$ has no root. We can handle the case of $p=3$ separately as in Solution 3, so assume that $p \geqslant 5$.

We will prove the following more general statement: let $p \geqslant 5$ be a prime and let $\mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_p$ be subsets of $\mathbb{Z} / p \mathbb{Z}$ with $|\mathcal{A}_i|=2$ for all $i$. Then there exists a polynomial $Q \in \mathbb{Z} / p \mathbb{Z}[X]$ of degree at most $2$ such that $Q(i) \notin \mathcal{A}_i$ for all $i$. Indeed, applying this statement to the sets $\mathcal{A}_i=\{0, f(i)\}$ (and adding $p X^{2}$ if necessary) produces a quadratic $Q$ satisfying the desired property.

Choose the coefficients of $Q$ uniformly at random from $\mathbb{Z} / p \mathbb{Z}$, and let $T$ be the random variable denoting the number of $i$ for which $Q(i) \in \mathcal{A}_i$. Observe that for $k \leqslant 3$, we have
$$\mathbb{E}\left[\binom{T}{k}\right]=2^{k}\binom{p}{k} p^{-k} .$$
To see why, let $k \leqslant 3$. If $\mathcal{S} \subseteq \mathbb{Z} / p \mathbb{Z}$ has size $k$ and $(a_i)_{i \in \mathcal{S}}$ is a $k$-tuple, the probability that $Q(i)=a_i$ on $\mathcal{S}$ is equal to $p^{-k}$; for $k=3$ this follows by Lagrange interpolation, and for $k<3$ it follows from the $k=3$ case by summing. The expectation is therefore equal to the number of $\mathcal{S} \subseteq \mathbb{Z} / p \mathbb{Z}$ of size $k$ times the probability that $Q(i) \in \mathcal{A}_i$ for each $i \in \mathcal{S}$, which is equal to the right hand side as each $\mathcal{A}_i$ has size $2$.

Now, observe that we have the identity $(t-1)(t-3)(t-4)=-12+12\binom{t}{1}-10\binom{t}{2}+6\binom{t}{3}$, so
$$\begin{aligned} \mathbb{E}[(T-1)(T-3)(T-4)] & =-12+12 \mathbb{E}\left[\binom{T}{1}\right]-10 \mathbb{E}\left[\binom{T}{2}\right]+6 \mathbb{E}\left[\binom{T}{3}\right] \\ & =-12+12 \cdot 2-10 \cdot 2\left(1-\frac{1}{p}\right)+6 \cdot \frac{4}{3}\left(1-\frac{1}{p}\right)\left(1-\frac{2}{p}\right) \\ & =-\frac{4}{p}+\frac{16}{p^{2}} \end{aligned}$$
This is negative for $p \geqslant 5$. Because $(t-1)(t-3)(t-4) \geqslant 0$ for all integers $t>0$, it then follows that $T=0$ with positive probability, which implies that there must exist some $Q$ with $Q(i) \notin \mathcal{A}_i$ for all $i$, as desired.