6.A student has 37 days to prepare for an examination. From past experience she knows that she will require no more than 60 hours of study. She also wishes to study at least 1 hour per day. Show that no matter how she schedules her study time (a whole number of hours per day, however), there is a succession of days during which she will have studied exactly 13 hours.
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Answer:
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Answer:
Step-by-step explanation:
Let si , 1<i<37 be the number of hours studied till i-th day. Then, SI <S2 <... <837 < 60 .
Adding 13 to every element in the inequality we have: S1 + 13 < S2 + 13 <... < 537 + 13 < 60 + 13 = 73 .
Note that for all 1<i, j <37, si + S where i72 [since she studies at least one hour everyday].
There exist 73 distinct integer values (pigeonholes) and 74 summands (pigeons).
So, there must exist two summands having same value. i.e., S; = S; + 13 and this implies that S; - = S; = 13 .
At this point we recall that 5 is the hours completed till and including the i-th day.
Thus there exist a period of (i − j) consecutive days (day j + 1, . . . , i) in which she spent 13 hours for studying.