Difference between revisions of "2021 AMC 10A Problems/Problem 22"
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==Solution 1== | ==Solution 1== | ||
− | Suppose the roommate took sheets <math>a< | + | Let <math>c</math> be the number of consecutive sheets Hiram’s roommate borrows, and let <math>b</math> be the number of sheets preceding the <math>c</math> borrowed sheets (I.e. if the friend borrows sheets 3, 4 and 5, then <math>c=3</math> and <math>b=2</math>). |
+ | |||
+ | The sum of the page numbers up till <math>b</math> sheets is <math>1+2+3+\cdots + 2b=\frac{2b\cdot(2b+1)}{2} = b(b+1) | ||
+ | |||
+ | ==Solution 2== | ||
+ | Suppose the roommate took sheets </math>a<math> through </math>b<math>, or equivalently, page numbers </math>2a-1<math> through </math>2b<math>. Because there are </math>(2b-2a+2)<math> numbers taken, <cmath>\frac{(2a-1+2b)(2b-2a+2)}{2}+19(50-(2b-2a+2))=\frac{50\cdot51}{2} \implies (2a+2b-39)(b-a+1)=\frac{50\cdot13}{2}=25\cdot13.</cmath> The first possible solution that comes to mind is if </math>2a+2b-39=25, b-a+1=13 \implies a+b=32, b-a=12<math>, which indeed works, giving </math>b=22<math> and </math>a=10<math>. The answer is </math>22-10+1=\boxed{\textbf{(B)} ~13}<math>. | ||
~Lcz | ~Lcz | ||
− | ==Solution | + | ==Solution 3== |
− | Suppose the smallest page number borrowed is <math>k,< | + | Suppose the smallest page number borrowed is </math>k,<math> and </math>n<math> pages are borrowed. It follows that the largest page number borrowed is </math>k+n-1.<math> |
We have the following preconditions: | We have the following preconditions: | ||
<ol style="margin-left: 1.5em;"> | <ol style="margin-left: 1.5em;"> | ||
− | <li><math>n< | + | <li></math>n<math> pages are borrowed means that </math>\frac{n}{2}<math> sheets are borrowed, from which </math>n<math> must be even.</li><p> |
− | <li><math>k< | + | <li></math>k<math> must be odd, as the smallest page number borrowed is on the right side (odd-numbered).</li><p> |
− | <li><math>1+2+3+\cdots+50=\frac{51(50)}{2}=1275.< | + | <li></math>1+2+3+\cdots+50=\frac{51(50)}{2}=1275.<math></li><p> |
− | <li>The sum of the page numbers borrowed is <math>\frac{(2k+n-1)n}{2}.< | + | <li>The sum of the page numbers borrowed is </math>\frac{(2k+n-1)n}{2}.<math></li><p> |
</ol> | </ol> | ||
Together, we have <cmath>\begin{align*} | Together, we have <cmath>\begin{align*} | ||
Line 26: | Line 31: | ||
650&=(2k+n-39)n. | 650&=(2k+n-39)n. | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | The factors of <math>650< | + | 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{array}{c|c|c} | <cmath>\begin{array}{c|c|c} | ||
& & \\ [-2.25ex] | & & \\ [-2.25ex] | ||
Line 39: | Line 44: | ||
650 & 1 & -305 \\ | 650 & 1 & -305 \\ | ||
\end{array}</cmath> | \end{array}</cmath> | ||
− | Since <math>1\leq k \leq 49,< | + | Since </math>1\leq k \leq 49,<math> only </math>k=47,19,1<math> are possible: |
− | * If <math>k=47,< | + | * If </math>k=47,<math> then there will not be sufficient pages when we take </math>10<math> pages out starting from page </math>47.<math> |
− | * If <math>k=1,< | + | * If </math>k=1,<math> then the average page number of all remaining sheets will be undefined, as there will be no sheets remaining after we take </math>50<math> pages (</math>25<math> sheets) out starting from page </math>1.<math> |
− | Therefore, the only possibility is <math>k=19.< | + | Therefore, the only possibility is </math>k=19.<math> We conclude that </math>n=26<math> pages, or </math>\frac n2=\boxed{\textbf{(B)} ~13}<math> sheets, are borrowed. |
~MRENTHUSIASM | ~MRENTHUSIASM | ||
− | ==Solution | + | ==Solution 4== |
− | Let <math>n< | + | Let </math>n<math> be the number of sheets borrowed, with an average page number </math>k+25.5<math>. The remaining </math>25-n<math> sheets have an average page number of </math>19<math> which is less than </math>25.5<math>, the average page number of all </math>50<math> pages, therefore </math>k>0<math>. Since the borrowed sheets start with an odd page number and end with an even page number we have </math>k \in \mathbb N<math>. We notice that </math>n < 25<math> and </math>k \le (49+50)/2-25.5=24<25<math>. |
− | The weighted increase of average page number from <math>25.5< | + | The weighted increase of average page number from </math>25.5<math> to </math>k+25.5<math> should be equal to the weighted decrease of average page number from </math>25.5<math> to </math>19<math>, where the weights are the page number in each group (borrowed vs. remained), therefore |
<cmath>2nk=2(25-n)(25.5-19)=13(25-n) \implies 13 | n \text{ or } 13 | k</cmath> | <cmath>2nk=2(25-n)(25.5-19)=13(25-n) \implies 13 | n \text{ or } 13 | k</cmath> | ||
− | Since <math>n, k < 25< | + | Since </math>n, k < 25<math> we have either </math>n=13<math> or </math>k=13<math>. If </math>n=13<math> then </math>k=6<math>. If </math>k=13<math> then </math>2n=25-n<math> which is impossible. Therefore the answer should be </math>n=\boxed{\textbf{(B)} ~13}<math>. |
~asops | ~asops | ||
− | ==Solution | + | ==Solution 5== |
− | Let <math>(2k-1)-2n< | + | Let </math>(2k-1)-2n<math> be pages be borrowed, the sum of the page numbers on those pages is </math>(2n+2k+1)(n-k)<math> while the sum of the rest pages is </math>1275-(2n+2k+1)(n-k)<math> and we know the average of the rest is </math>\frac{1275-(2n+2k+1)}{50-2n+2k}<math> which equals to </math>19<math>; multiply this out we got </math>950-38(n-k)=1275-(2n+2k+1)(n-k)<math> and we got </math>(2n+2k-37)(n-k)=325<math>. As </math>325=25\cdot13<math>, we can see </math>n-k=13<math> and that is desired </math>\boxed{\textbf{(B)} ~13}$. |
~bluesoul | ~bluesoul |
Revision as of 03:00, 31 October 2022
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 1
Let be the number of consecutive sheets Hiram’s roommate borrows, and let
be the number of sheets preceding the
borrowed sheets (I.e. if the friend borrows sheets 3, 4 and 5, then
and
).
The sum of the page numbers up till sheets is $1+2+3+\cdots + 2b=\frac{2b\cdot(2b+1)}{2} = b(b+1)
==Solution 2==
Suppose the roommate took sheets$ (Error compiling LaTeX. Unknown error_msg)ab
2a-1
2b
(2b-2a+2)
2a+2b-39=25, b-a+1=13 \implies a+b=32, b-a=12
b=22
a=10
22-10+1=\boxed{\textbf{(B)} ~13}$.
~Lcz
==Solution 3==
Suppose the smallest page number borrowed is$ (Error compiling LaTeX. Unknown error_msg)k,n
k+n-1.
n
\frac{n}{2}
n
k
1+2+3+\cdots+50=\frac{51(50)}{2}=1275.
\frac{(2k+n-1)n}{2}.$</li><p>
</ol>
Together, we have <cmath>\begin{align*}
\frac{1275-\frac{(2k+n-1)n}{2}}{50-n}&=19 \\
1275-\frac{(2k+n-1)n}{2}&=19(50-n) \\
2550-(2k+n-1)n&=38(50-n) \\
2550-(2k+n-1)n&=1900-38n \\
650&=(2k+n-39)n.
\end{align*}</cmath>
The factors of$ (Error compiling LaTeX. Unknown error_msg)650
n
1\leq k \leq 49,
k=47,19,1$are possible:
- If$ (Error compiling LaTeX. Unknown error_msg)k=47,
10
47.
k=1,
50
25
1.
k=19.
n=26
\frac n2=\boxed{\textbf{(B)} ~13}$sheets, are borrowed.
~MRENTHUSIASM
==Solution 4==
Let$ (Error compiling LaTeX. Unknown error_msg)nk+25.5
25-n
19
25.5
50
k>0
k \in \mathbb N
n < 25
k \le (49+50)/2-25.5=24<25$.
The weighted increase of average page number from$ (Error compiling LaTeX. Unknown error_msg)25.5k+25.5
25.5
19$, where the weights are the page number in each group (borrowed vs. remained), therefore
<cmath>2nk=2(25-n)(25.5-19)=13(25-n) \implies 13 | n \text{ or } 13 | k</cmath>
Since$ (Error compiling LaTeX. Unknown error_msg)n, k < 25n=13
k=13
n=13
k=6
k=13
2n=25-n
n=\boxed{\textbf{(B)} ~13}$.
~asops
==Solution 5==
Let$ (Error compiling LaTeX. Unknown error_msg)(2k-1)-2n(2n+2k+1)(n-k)
1275-(2n+2k+1)(n-k)
\frac{1275-(2n+2k+1)}{50-2n+2k}
19
950-38(n-k)=1275-(2n+2k+1)(n-k)
(2n+2k-37)(n-k)=325
325=25\cdot13
n-k=13
\boxed{\textbf{(B)} ~13}$.
~bluesoul
Video Solution by OmegaLearn (Arithmetic Sequences and System of Equations)
~ pi_is_3.14
Video Solution by MRENTHUSIASM (English & Chinese)
https://www.youtube.com/watch?v=28te8OUiVxE
~MRENTHUSIASM
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.