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A salesman sells at least $1$ car each day for $100$ consecutive days selling a total of $150$ cars. Prove that for each value of $n$, $1\le n \lt 50$, there is a period of consecutive days during which he sold a total of exactly $n$ cars.
How do you prove this through Pigeonhole? Thanks in advance.
Let $T_k$ be the total number of cars sold from days 1 to $k$.
The goal is to show that $ \{ T_k \} \cap \{ T_k + n \}$ is non-empty.
There are a total of 200 values here. These are our balls.
The range of values is from 1 to $150+n < 200$. These are our holes.
Hence, by PP, there is at one hole with at least 2 balls.
Can the 2 balls both come from $\{T_k\}$? (Explain why the answer is no.)
Can the 2 balls both come from $\{T_k + n\}$? (Explain why the answer is no.)
Hence, these 2 balls give us $T_i = T_j + n$.
So, from days $j+1$ to $i$, he sold $n$ cars.
