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A triathlete planned 44 training sessions over the next 30 days, each day with at least 1 training session. Prove that there will be 2 consecutive days that will have the same number of sessions.

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    Have you made any efforts yet? – 2017-01-11
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    @ThomasGrubb sort of, but so far I am only getting numerical proofs instead of general proof. – 2017-01-11
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    What was the context of this? What 'numerical proofs' have you found? – 2017-01-11
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    @DanBarry what do you mean? – 2017-01-11
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    @DanBarry Repeat 1 and 2 15 times, but that would give you 45, so it could only repeat 14 times and then have two ones at the end but that would violate the claim. – 2017-01-11
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    @GerardL. That's the correct proof, and is almost certainly what the problem is looking for (along with the words "by the pigeonhole principle") – 2017-01-11
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    @StellaBiderman but my "proof" seems numerical and just for one case. – 2017-01-11
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    May i ask why this is getting so many downvotes? – 2017-01-11
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    @GerardL. So what if the proof contains numbers? The problem contains numbers! If you want a proof of the General problem that says that with $\lfloor(3n-1)/2\rfloor$ training sessions and $n=2k$ days you need to have two consecutive days, then write that exact same proof using $\lfloor(3n-1)/2\rfloor$ – 2017-01-11
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    @StellaBiderman Do you mean (3n-2)/2 sessions? – 2017-01-11
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    Uh yes, that was a typo – 2017-01-11

2 Answers 2

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Since you have at least one training per day, you only have 14 trainings to distribute on 30 days in order to avoid having two consecutive days with the same number of trainings.

This means you will have at least 16 days with only one training.

Off topic : in my language, Pigeon-hole principle is about drawers and socks...

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One question to get you started: suppose that the claim weren't true. At most how many days can have exactly 1 training session?

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    29, but it would violate the claim – 2017-01-11
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    @GerardL. In order for the claim not to be violated* – 2017-01-11
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    @Faibbus 15 days? – 2017-01-11
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    @GerardL. exactly, now, how many days can you have more than one training ? – 2017-01-11
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    Yes, so what does that imply about the number of sessions that must be scheduled into the remaining days? – 2017-01-11
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    There are 15 more days with 29 more sessions, so one sessions must be another 1. – 2017-01-11