Determine all positive integers $n$ for which there exist $k_{1}, \ldots, k_{n}$ distinct integers, and a polynomial $P$ of degree at most $n$ such that $P\left(k_{i}\right)=n$ for every integer $i$ satisfying $1 \leqslant i \leqslant n$, and having a root in $\mathbb{Z}$.

## Solutions
Let $Q(X)=P(X)-n$. Since for all indices $i$ between 1 and $n$, $Q\left(k_{i}\right)=P\left(k_{i}\right)-n=n-n=0$, $Q$ has $n$ roots and is of degree $n$. Thus there exists $c$ such that

$$
Q(X)=c\left(X-k_{1}\right)\left(X-k_{2}\right) \cdots\left(X-k_{n}\right)
$$

with $c$ an integer since $P$ has integer coefficients. On the one hand, there exists an integer $z$ such that $P(z)=0$, so $Q(z)=P(z)-n=-n$ and on the other hand $Q(0)=c\left(z-k_{1}\right) \cdot\left(z-k_{2}\right) \cdots\left(z-k_{n}\right)$. Since the $z-k_{i}$ are pairwise distinct and non-zero, at most two of the $z-k_{i}$ have an absolute value equal to 1, so at least $n-2$ of the $k_{i}$ have an absolute value greater than or equal to 2. Thus $\left|c\left(z-k_{1}\right) \cdot\left(z-k_{2}\right) \cdots\left(z-k_{n}\right)\right| \geqslant 1 \cdot 1 \cdot 2 \cdots 2=2^{n-2}$. Thus $n \geqslant 2^{n-2}$.

Let's prove by induction that if $m \geqslant 5$ is an integer, $2^{m-2}>m+2$. If $m=5$, we have $2^{5-2}=8>7=5+2$. Suppose the property holds for $m$, and we want to show that $2^{m+1-2}>m+1$. Now $2^{m-1}=2 \cdot 2^{m-2}>2 m>m+1$ using the induction hypothesis. This completes the induction.

From this discussion, we deduce that $n \leqslant 4$. If $n=1$, the polynomial $X$ satisfies the condition (with $k_{1}=1$). If $n=2$, the polynomial $-(X-1)(X-2)+2$ satisfies the condition (with $k_{1}=1$ and $k_{2}=2$). If $n=3$, the polynomial $-(X-1)(X+1)(X-3)+3$ satisfies the condition (with $k_{1}=1, k_{2}=-1, k_{3}=3$). Finally, if $n=4$, the polynomial $-(X-1)(X+1)(X+2)(X-2)+4$ satisfies the condition (with $k_{1}=1, k_{2}=-1, k_{3}=-2, k_{4}=2$).

The solutions are therefore the integers $1, 2, 3$, and 4.

## 2 Power of a Point and South Pole (Mathieu)

The course covered two very useful tools in geometry: the South Pole theorem and the power of a point with respect to a circle.

## Exercises