Problem 4. Determine the non-constant polynomial functions $f:[0,1] \rightarrow \mathbb{R}^{*}$, with rational coefficients, which have the following property: for any $x$ in the interval $[0,1]$, there exist two polynomial functions $g_{x}, h_{x}:[0,1] \rightarrow \mathbb{R}$, with rational coefficients, such that $h_{x}(x) \neq 0$ and

$$
\int_{0}^{x} \frac{1}{f(t)} \mathrm{d} t=\frac{g_{x}(x)}{h_{x}(x)}
$$
Solution. Since the closed interval $[0,1]$ is uncountable and the set of polynomial functions with rational coefficients is countable, there exists an uncountable set $S \subseteq [0,1]$ and two coprime polynomial functions $g, h: [0,1] \rightarrow \mathbb{R}$, with rational coefficients, such that $h(x) \neq 0$ and

$$
\int_{0}^{x} \frac{1}{f(t)} \mathrm{d} t=\frac{g(x)}{h(x)}
$$

for any $x$ in $S$.

Since the set of isolated points of $S$ is at most countable, it follows that there are infinitely many points in $S$ that are accumulation points of $S$. At each of these points, the rational functions $1 / f$ and $(g / h)^{\prime}=\left(g^{\prime} h-g h^{\prime}\right) / h^{2}$ are equal, so

$$
h^{2}=f \cdot\left(g^{\prime} h-g h^{\prime}\right)
$$

in $\mathbb{Q}[X]$.

Since $\operatorname{deg} f \geq 1$, it follows that $\operatorname{deg} g \leq \operatorname{deg} h$. If $\operatorname{deg} g=\operatorname{deg} h$, then the remainder of the division of $g$ by $h$ also satisfies $(*)$ and is coprime with $h$, so we can assume that $\operatorname{deg} g<\operatorname{deg} h$ and, therefore, $\operatorname{deg}\left(g^{\prime} h-g h^{\prime}\right)=\operatorname{deg} g+\operatorname{deg} h-1$.

If $\operatorname{deg} h \geq 2$, since $g$ and $h$ are coprime, any root of order $k$ of $g^{\prime} h-g h^{\prime}$ is a root of $h$, of order at least $k+1$. Thus $\operatorname{deg} h \geq \operatorname{deg} g+\operatorname{deg} h-1+d$, where $d$ is the number of distinct roots of $g^{\prime} h-g h^{\prime}$. Therefore, $d=1$, $\operatorname{deg} g=0$, from which, $h=a(X-b)^{m}$, where $a \in \mathbb{Q}^{*}, b \in \mathbb{Q} \backslash[0,1]$, and $m$ is a natural number greater than or equal to 2. It follows that $f=c(X-b)^{m+1}$, where $c$ is a non-zero rational number.

If $\operatorname{deg} h=1$, then $\operatorname{deg} g=0$ and $f=c(X-b)^{2}$, where $b$ and $c$ are as above.

In conclusion, the functions sought are those of the form $f(x)=c(x-b)^{n}, 0 \leq x \leq 1$, where $c \in \mathbb{Q}^{*}$, $b \in \mathbb{Q} \backslash[0,1]$ and $n$ is a natural number greater than or equal to 2, functions which clearly satisfy the condition in the statement.