Bobby has 5 weeks to prepare for his exam. His friend volunteered to tutor for either 15min or 30min periods every day until the test but not for more than 15 hours total.
Show that during some period of consecutive days, Bobby and his tutor will study for exactly 8.75 hours.
Let $t_i$ be the number of 15-min intervals that Bobby is tutored for.
We have $t_i \in \{ 1, 2 \}$ and $\sum_{i=1}^{35} t_i \leq 60 $.
Consider $T_k = \sum_{i=1}^k t_i$.
We have $ 1 \leq T_1 < T_2 < \ldots < T_{35} \leq 60 $.
We want to find $T_k = T_j + 35$.
If any $T_k = 35$, then we are done. Let's assume that $T_k \neq 35$.
Let the pigeons be $T_k$. There are 35 of them.
Let the pigeon holes be $\{1,36\}, \{2, 37\}, \ldots \{25, 60 \},$ and $ \{ 26\}, \{ 27\}, \{ 28\}, \ldots \{ 34 \}$. There are 34 of them.
Hence, 2 pigeons are in the same hole, which gives us $T_k = T_j + 35$.