2019 IMO Shortlist C1

The infinite sequence $a_{0}, a_{1}, a_{2}, \ldots$ of (not necessarily different) integers has the following properties: $0 \leq a_{i} \leq i$ for all integers $i \geq 0$, and $$\binom{k}{a_{0}}+\binom{k}{a_{1}}+\cdots+\binom{k}{a_{k}}=2^{k}$$ for all integers $k \geq 0$. Prove that all integers $N \geq 0$ occur in the sequence (that is, for all $N \geq 0$, there exists $i \geq 0$ with $a_{i}=N$ ). ## (Netherlands)

(不一定不同)整数的无限序列 $a_{0}, a_{1}, a_{2}, \ldots$ 具有以下属性： $0 \leq a_{i} \leq i$ 对于所有整数 $i \geq 0$ 和 $$\binom{k}{a_{0}}+\binom{k}{a_{1}}+\cdots+\binom{k}{a_{k}}=2^{k}$$ 对于所有整数 $k \geq 0$。证明所有整数 $N \geq 0$ 都出现在序列中(即，对于所有 $N \geq 0$，存在 $i \geq 0$ 且 $a_{i}=N$ )。 ＃＃ (荷兰)