Difference between revisions of "2021 AMC 10A Problems/Problem 22"
MRENTHUSIASM (talk | contribs) (→Solution 2 (Different Variable Choice, Similar Logic)) |
MRENTHUSIASM (talk | contribs) (→Solution 2 (Different Variable Choice, Similar Logic)) |
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The factors of <math>650</math> are <cmath>1,2,5,10,13,25,26,50,65,130,325,650.</cmath> Since <math>n</math> is even, we only have a few cases to consider: | The factors of <math>650</math> are <cmath>1,2,5,10,13,25,26,50,65,130,325,650.</cmath> Since <math>n</math> is even, we only have a few cases to consider: | ||
− | <cmath>\begin{ | + | <cmath>\begin{array}{ c c c } |
n & 2k+n-39 & k \\ | n & 2k+n-39 & k \\ | ||
\hline | \hline | ||
Line 38: | Line 38: | ||
26 & 25 & 19 \\ | 26 & 25 & 19 \\ | ||
50 & 13 & 1 \\ | 50 & 13 & 1 \\ | ||
− | 130 & 5 & negative \\ | + | 130 & 5 & \text{negative} \\ |
− | 650 & 1 & negative \\ | + | 650 & 1 & \text{negative} \\ |
− | \end{ | + | \end{array}</cmath> |
Since <math>1\leq k \leq 50,</math> only <math>k=47,19,1</math> are possible: | Since <math>1\leq k \leq 50,</math> only <math>k=47,19,1</math> are possible: |
Revision as of 05:16, 14 February 2021
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 10 pages starting from Page 47.
If then the average page number of the remaining pages will be undefined, as there is no page remaining (after taking 50 pages starting from page 1).
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.