Let $n>1$ be an integer. In the space, consider the set  $$ S=\{(x, y, z) \mid x, y, z \in\{0,1, \ldots, n\}, x+y+z>0\} $$  Find the smallest number of planes that jointly contain all $(n+1)^{3}-1$ points of $S$ but none of them passes through the origin.  (Netherlands)  Answer. $3 n$ planes.
It is easy to find $3 n$ such planes. For example, planes $x=i, y=i$ or $z=i$ $(i=1,2, \ldots, n)$ cover the set $S$ but none of them contains the origin. Another such collection consists of all planes $x+y+z=k$ for $k=1,2, \ldots, 3 n$. We show that $3 n$ is the smallest possible number. Lemma 1. Consider a nonzero polynomial $P\left(x_{1}, \ldots, x_{k}\right)$ in $k$ variables. Suppose that $P$ vanishes at all points $\left(x_{1}, \ldots, x_{k}\right)$ such that $x_{1}, \ldots, x_{k} \in\{0,1, \ldots, n\}$ and $x_{1}+\cdots+x_{k}>0$, while $P(0,0, \ldots, 0) \neq 0$. Then $\operatorname{deg} P \geq k n$. Proof. We use induction on $k$. The base case $k=0$ is clear since $P \neq 0$. Denote for clarity $y=x_{k}$. Let $R\left(x_{1}, \ldots, x_{k-1}, y\right)$ be the residue of $P$ modulo $Q(y)=y(y-1) \ldots(y-n)$. Polynomial $Q(y)$ vanishes at each $y=0,1, \ldots, n$, hence $P\left(x_{1}, \ldots, x_{k-1}, y\right)=R\left(x_{1}, \ldots, x_{k-1}, y\right)$ for all $x_{1}, \ldots, x_{k-1}, y \in\{0,1, \ldots, n\}$. Therefore, $R$ also satisfies the condition of the Lemma; moreover, $\operatorname{deg}_{y} R \leq n$. Clearly, $\operatorname{deg} R \leq \operatorname{deg} P$, so it suffices to prove that $\operatorname{deg} R \geq n k$. Now, expand polynomial $R$ in the powers of $y$ : $$ R\left(x_{1}, \ldots, x_{k-1}, y\right)=R_{n}\left(x_{1}, \ldots, x_{k-1}\right) y^{n}+R_{n-1}\left(x_{1}, \ldots, x_{k-1}\right) y^{n-1}+\cdots+R_{0}\left(x_{1}, \ldots, x_{k-1}\right) $$ We show that polynomial $R_{n}\left(x_{1}, \ldots, x_{k-1}\right)$ satisfies the condition of the induction hypothesis. Consider the polynomial $T(y)=R(0, \ldots, 0, y)$ of degree $\leq n$. This polynomial has $n$ roots $y=1, \ldots, n$; on the other hand, $T(y) \not \equiv 0$ since $T(0) \neq 0$. Hence $\operatorname{deg} T=n$, and its leading coefficient is $R_{n}(0,0, \ldots, 0) \neq 0$. In particular, in the case $k=1$ we obtain that coefficient $R_{n}$ is nonzero. Similarly, take any numbers $a_{1}, \ldots, a_{k-1} \in\{0,1, \ldots, n\}$ with $a_{1}+\cdots+a_{k-1}>0$. Substituting $x_{i}=a_{i}$ into $R\left(x_{1}, \ldots, x_{k-1}, y\right)$, we get a polynomial in $y$ which vanishes at all points $y=0, \ldots, n$ and has degree $\leq n$. Therefore, this polynomial is null, hence $R_{i}\left(a_{1}, \ldots, a_{k-1}\right)=0$ for all $i=0,1, \ldots, n$. In particular, $R_{n}\left(a_{1}, \ldots, a_{k-1}\right)=0$. Thus, the polynomial $R_{n}\left(x_{1}, \ldots, x_{k-1}\right)$ satisfies the condition of the induction hypothesis. So, we have $\operatorname{deg} R_{n} \geq(k-1) n$ and $\operatorname{deg} P \geq \operatorname{deg} R \geq \operatorname{deg} R_{n}+n \geq k n$. Now we can finish the solution. Suppose that there are $N$ planes covering all the points of $S$ but not containing the origin. Let their equations be $a_{i} x+b_{i} y+c_{i} z+d_{i}=0$. Consider the polynomial $$ P(x, y, z)=\prod_{i=1}^{N}\left(a_{i} x+b_{i} y+c_{i} z+d_{i}\right) $$ It has total degree $N$. This polynomial has the property that $P\left(x_{0}, y_{0}, z_{0}\right)=0$ for any $\left(x_{0}, y_{0}, z_{0}\right) \in S$, while $P(0,0,0) \neq 0$. Hence by Lemma 1 we get $N=\operatorname{deg} P \geq 3 n$, as desired. Comment 1. There are many other collections of $3 n$ planes covering the set $S$ but not covering the origin.