Evan is training for baseball season. His coach wants him to pitch or hit 90 times over the course of the 53 days before the season starts. He plans to do at least one pitching or hitting session a day. Prove that there exists a span of consecutive days that Evan will complete exactly 15 sessions.
The way how I tried solving it was by identifying what the pigeons are and what are the pigeon holes. I came up with the conclusion that the pigeons are the remaining sessions that he hasn't done, so "90 sessions required", minus the "53 days before the season starts" which is 37 sessions that need to be done before the season. And the holes are the days before the season, so 53 days because the question said, "He plans to do at least one pitching or hitting session a day." Implying that he can do two or more in a day so don't I just have to distribute the remaining sessions within those 53 days such that they are consecutively added to a total of 15 days to prove that there does exist a span of consecutive days like that? But I got this wrong I think, my friends used inequalities and some arithmetic to prove this question.
Let $x_i$ be the number of times Evan has practiced through day $i$, where $1 \leq i \leq 53$. Since Evan practices at least once each day, $(x_1, x_2, \ldots, x_{53})$ is a strictly increasing sequence, where $1 \leq x_1 < x_2 < \cdots < x_{53} = 90$.
Let $y_i = x_i + 15$ for $1 \leq i \leq 53$. Then $(y_1, y_2, \ldots, y_{53})$ is a strictly increasing sequence, where $16 \leq y_1 < y_2 < \cdots < y_{53} = 105$.
If we take the union of the two sequences, we obtain a sequence with $2 \cdot 53 = 106$ positive integers, the largest of which is $105$. Hence, two elements of the union must be the same. Since $(x_1, x_2, \ldots, x_{53})$ and $(y_1, y_2, \ldots, y_{53})$ are strictly increasing sequences, this can only occur if there exists some $i, j$ such that $x_i = y_j$, where $j < i$. Then during the days $j + 1, j + 2, \ldots, i$, Evan must practice exactly $15$ times.