2021 AMC 10A Problems/Problem 22

Problem

Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$ , the second sheet contains pages $3$ and $4$ , 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 $19$ . How many sheets were borrowed?

$\textbf{(A)} ~10\qquad\textbf{(B)} ~13\qquad\textbf{(C)} ~15\qquad\textbf{(D)} ~17\qquad\textbf{(E)} ~20$

Solution 1

Let $c$ be the number of consecutive sheets Hiram’s roommate borrows, and let $b$ be the number of sheets preceding the $c$ borrowed sheets (I.e. if the friend borrows sheets 3, 4 and 5, then $c=3$ and $b=2$ ).

The sum of the page numbers up till $b$ 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)a $through$ b $, or equivalently, page numbers$ 2a-1 $through$ 2b $. Because there are$ (2b-2a+2) $numbers taken, <cmath>\frac{(2a-1+2b)(2b-2a+2)}{2}+19(50-(2b-2a+2))=\frac{50\cdot51}{2} \implies (2a+2b-39)(b-a+1)=\frac{50\cdot13}{2}=25\cdot13.</cmath> The first possible solution that comes to mind is if$ 2a+2b-39=25, b-a+1=13 \implies a+b=32, b-a=12 $, which indeed works, giving$ b=22 $and$ a=10 $. The answer is$ 22-10+1=\boxed{\textbf{(B)} ~13}$.

~Lcz

==Solution 3== Suppose the smallest page number borrowed is$ (Error compiling LaTeX. Unknown error_msg)k, $and$ n $pages are borrowed. It follows that the largest page number borrowed is$ k+n-1. $We have the following preconditions: <ol style="margin-left: 1.5em;"> <li>$ n $pages are borrowed means that$ \frac{n}{2} $sheets are borrowed, from which$ n $must be even.</li><p> <li>$ k $must be odd, as the smallest page number borrowed is on the right side (odd-numbered).</li><p> <li>$ 1+2+3+\cdots+50=\frac{51(50)}{2}=1275. $</li><p> <li>The sum of the page numbers borrowed is$ \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 $are <cmath>1,2,5,10,13,25,26,50,65,130,325,650.</cmath> Since$ n $is even, we only have a few cases to consider: <cmath>\begin{array}{c|c|c} & & \\ [-2.25ex] \boldsymbol{n} & \boldsymbol{2k+n-39} & \boldsymbol{k} \\ [0.5ex] \hline & & \\ [-2ex] 2 & 325 & 181 \\ 10 & 65 & 47 \\ 26 & 25 & 19 \\ 50 & 13 & 1 \\ 130 & 5 & -43 \\ 650 & 1 & -305 \\ \end{array}</cmath> Since$ 1\leq k \leq 49, $only$ k=47,19,1$are possible:

If$ (Error compiling LaTeX. Unknown error_msg)k=47, $then there will not be sufficient pages when we take$ 10 $pages out starting from page$ 47. $* If$ k=1, $then the average page number of all remaining sheets will be undefined, as there will be no sheets remaining after we take$ 50 $pages ($ 25 $sheets) out starting from page$ 1. $Therefore, the only possibility is$ k=19. $We conclude that$ n=26 $pages, or$ \frac n2=\boxed{\textbf{(B)} ~13}$sheets, are borrowed.

~MRENTHUSIASM

==Solution 4==

Let$ (Error compiling LaTeX. Unknown error_msg)n $be the number of sheets borrowed, with an average page number$ k+25.5 $. The remaining$ 25-n $sheets have an average page number of$ 19 $which is less than$ 25.5 $, the average page number of all$ 50 $pages, therefore$ k>0 $. Since the borrowed sheets start with an odd page number and end with an even page number we have$ k \in \mathbb N $. We notice that$ n < 25 $and$ k \le (49+50)/2-25.5=24<25$.

The weighted increase of average page number from$ (Error compiling LaTeX. Unknown error_msg)25.5 $to$ k+25.5 $should be equal to the weighted decrease of average page number from$ 25.5 $to$ 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 < 25 $we have either$ n=13 $or$ k=13 $. If$ n=13 $then$ k=6 $. If$ k=13 $then$ 2n=25-n $which is impossible. Therefore the answer should be$ n=\boxed{\textbf{(B)} ~13}$.

~asops

==Solution 5== Let$ (Error compiling LaTeX. Unknown error_msg)(2k-1)-2n $be pages be borrowed, the sum of the page numbers on those pages is$ (2n+2k+1)(n-k) $while the sum of the rest pages is$ 1275-(2n+2k+1)(n-k) $and we know the average of the rest is$ \frac{1275-(2n+2k+1)}{50-2n+2k} $which equals to$ 19 $; multiply this out we got$ 950-38(n-k)=1275-(2n+2k+1)(n-k) $and we got$ (2n+2k-37)(n-k)=325 $. As$ 325=25\cdot13 $, we can see$ n-k=13 $and that is desired$ \boxed{\textbf{(B)} ~13}$.

~bluesoul

Video Solution by OmegaLearn (Arithmetic Sequences and System of Equations)

https://youtu.be/dWOLIdTxwa4

~ pi_is_3.14

Video Solution by MRENTHUSIASM (English & Chinese)

https://www.youtube.com/watch?v=28te8OUiVxE

~MRENTHUSIASM

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
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2021 AMC 10A Problems/Problem 22

Contents

Problem

Solution 1

Video Solution by OmegaLearn (Arithmetic Sequences and System of Equations)

Video Solution by MRENTHUSIASM (English & Chinese)

See also