2021 AMC 10A Problems/Problem 22
Contents
Problem
Hiram's algebra notes are pages long and are printed on sheets of paper; the first sheet contains pages and , the second sheet contains pages and , and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly . How many sheets were borrowed?
Solution
Suppose the roommate took pages through , or equivalently, page numbers through . Because there are numbers taken, The first possible solution that comes to mind is if , which indeed works, giving and . The answer is
~Lcz
Solution 2 (Different Variable Choice, Similar Logic)
Suppose the smallest page number removed is and pages are removed. It follows that the largest page number removed is
Remarks:
1. pages are removed means that sheets are removed, from which must be even.
2. must be odd, as the smallest page number removed is on the right side (odd-numbered).
3.
4. The sum of the page numbers removed is
Together, we have The factors of are Since is even, we only have a few cases to consider:
Since only are possible:
If then the notebook will run out if we take pages starting from page
If then the average page number of the remaining pages will be undefined, as there is no page remaining (after taking pages starting from page ).
So, the only possibility is from which pages are taken out, which is sheets.
~MRENTHUSIASM
Video Solution by OmegaLearn (Arithmetic Sequences and System of Equations)
~ pi_is_3.14
See also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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All AMC 10 Problems and Solutions |
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