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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.

Alvks23
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    Welcome to [math.se]. Can you please [edit] your post and write your attempts at solving the problem? If your question is clear and focused on your specific difficulty and you show your effort in solving the problem, it's more likely to get good and helping answers. By the way, take the opportunity to take the [Tour], if you haven't done it already. See also some tips on [ask], on [formatting help](https://math.stackexchange.com/editing-helpand on writing down equations using LaTeX / MathJax. – Ertxiem - reinstate Monica Nov 07 '19 at 17:00
  • What have you tried? – Calvin Lin Nov 07 '19 at 17:00
  • @CalvinLin I've went about it algebraically like solving how many sessions of 30 minute and 15 minutes add up to 15 hours in 35 days. I got 25 * 0.5 hours and 10 * 0.25 hour sessions. – Alvks23 Nov 07 '19 at 17:02

1 Answers1

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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 $T_k = T_j + 35$.

Calvin Lin
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