Difference between revisions of "2021 AIME I Problems/Problem 10"

Revision as of 13:17, 22 March 2021

Problem

Consider the sequence $(a_k)_{k\ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k\ge 1$ , if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , then

$a_{k+1} = \frac{m + 18}{n+19}.$ Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\frac{t}{t+1}$ for some positive integer $t$ .

Solution 1

We know that $a_{1}=\tfrac{t}{t+1}$ when $t=2020$ so $1$ is a possible value of $j$ . Note also that $a_{2}=\tfrac{2038}{2040}=\tfrac{1019}{1020}=\tfrac{t}{t+1}$ for $t=1019$ . Then $a_{2+q}=\tfrac{1019+18q}{1020+19q}$ unless $1019+18q$ and $1020+19q$ are not relatively prime which happens when $q+1$ divides $18q+1019$ or $q+1$ divides $1001$ , so the least value of $q$ is $6$ and $j=2+6=8$ . We know $a_{8}=\tfrac{1019+108}{1020+114}=\tfrac{1127}{1134}=\tfrac{161}{162}$ . Now $a_{8+q}=\tfrac{161+18q}{162+19q}$ unless $18q+161$ and $19q+162$ are not relatively prime which happens the first time $q+1$ divides $18q+161$ or $q+1$ divides $143$ or $q=10$ , and $j=8+10=18$ . We have $a_{18}=\tfrac{161+180}{162+190}=\tfrac{341}{352}=\tfrac{31}{32}$ . Now $a_{18+q}=\tfrac{31+18q}{32+19q}$ unless $18q+31$ and $19q+32$ are not relatively prime. This happens the first time $q+1$ divides $18q+31$ implying $q+1$ divides $13$ , which is prime so $q=12$ and $j=18+12=30$ . We have $a_{30}=\tfrac{31+216}{32+228}=\tfrac{247}{260}=\tfrac{19}{20}$ . We have $a_{30+q}=\tfrac{18q+19}{19q+20}$ , which is always reduced by EA, so the sum of all $j$ is $1+2+8+18+30=\boxed{059}$ .

Solution 2 (Simon's Favorite Factoring Trick and Generalization)

Let $a_{j_1}, a_{j_2}, a_{j_3}, \cdots, a_{j_u}$ be all terms in the form $\frac{t}{t+1},$ where $j_1<j_2<j_3<\cdots<j_u,$ and $t$ is some positive integer. We wish to find $\sum_{i=1}^{u}{j_i}.$ Suppose $a_{j_i}=\frac{m}{m+1}$ for some positive integer $m.$

To find $\boldsymbol{a_{j_{i+1}},}$ we look for the smallest positive integer $\boldsymbol{k'}$ for which $\boldsymbol{a_{j_{i+1}}=a_{j_i+k'}=\frac{m+18k'}{m+1+19k'}}$ is reducible:

If $\frac{m+18k'}{m+1+19k'}$ is reducible, then there exists a common factor $d>1$ for $m+18k'$ and $m+1+19k'.$ By the Euclidean Algorithm, we have $\begin{align*} d|m+18k' \text{ and } d|m+1+19k' &\Longrightarrow d|k'+1 \text{ and } d|m+18k' \\ &\Longrightarrow d|m-18 \text{ and } d|k'+1. \end{align*}$ Since $m-18$ and $k'+1$ are not relatively prime, and $m$ is fixed, the smallest value of $k'$ such that $\frac{m+18k'}{m+1+19k'}$ is reducible occurs when $k'+1$ is the smallest prime factor of $m-18.$

We will prove that for such value of $\boldsymbol{k',}$ the number $\boldsymbol{a_{j_{i+1}}}$ can be written in the form $\boldsymbol{\frac{t}{t+1}:}$ $a_{j_{i+1}}=a_{j_i+k'}=\frac{m+18k'}{m+1+19k'}=\frac{(m-18)+18(k'+1)}{(m-18)+19(k'+1)}=\frac{\frac{m-18}{k'+1}+18}{\frac{m-18}{k'+1}+19}, \ \ (*)$ where $t=\frac{m-18}{k'+1}+18$ is a positive integer.

We start with $m=2020$ and $a_{j_1}=a_1=\frac{2020}{2021},$ then find $a_{j_2}, a_{j_3}, \cdots, a_{j_u}$ by filling out the table below recursively: $\begin{array}{c|c|c|c|c|l} \boldsymbol{i} & \boldsymbol{m} & \boldsymbol{m-18} & \boldsymbol{k'+1} & \boldsymbol{k'} & \ \boldsymbol{a_{j_{i+1}} \ \left(\textbf{by } (*)\right)} \\ [0.5ex] \hline\hline & & & & & \\ [-1.5ex] 1 & 2020 & 2002 & 2 & 1 & \ \ \ \ \ a_2 = \frac{1019}{1020} \\ [1ex] 2 & 1019 & 1001 & 7 & 6 & \ \ \ \ \ a_8 = \frac{161}{162} \\ [1ex] 3 & 161 & 143 & 11 & 10 & \ \ \ \ a_{18} = \frac{31}{32} \\ [1ex] 4 & 31 & 13 & 13 & 12 & \ \ \ \ a_{30} = \frac{19}{20} \\ [1ex] 5 & 19 & 1 & \text{N/A} & \text{N/A} & \ \ \ \ \ \ \ \ \text{N/A} \\ [1ex] \end{array}$ As $\left(j_1,j_2,j_3,j_4,j_5\right)=(1,2,8,18,30),$ the answer is $\sum_{i=1}^{5}{j_i}=\boxed{059}.$

Remark

Alternatively, from $(*)$ we can set $\frac{m+18k'}{m+1+19k'}=\frac{t}{t+1}.$ We cross-multiply, rearrange, and apply Simon's Favorite Factoring Trick to get $\left(k'+1\right)(t-18)=m-18.$ Since $k'+1\geq2,$ to find the smallest $k',$ we need $k'+1$ to be the smallest prime factor of $m-18.$ Now, continue with the last two paragraphs of the solution above.

~MRENTHUSIASM

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=LIjTty3rVso

@@ Line 8: / Line 8: @@
 ==Solution 2 (Simon's Favorite Factoring Trick and Generalization)==
-<b>I am saving progress as I type this solution (to prevent work loss). Will finish up very soon. Please do not edit this solution. A million thanks.</b>
+Let <math>a_{j_1}, a_{j_2}, a_{j_3}, \cdots, a_{j_u}</math> be all terms in the form <math>\frac{t}{t+1},</math> where <math>j_1<j_2<j_3<\cdots<j_u,</math> and <math>t</math> is some positive integer. We wish to find <math>\sum_{i=1}^{u}{j_i}.</math> Suppose <math>a_{j_i}=\frac{m}{m+1}</math> for some positive integer <math>m.</math>
+<i><b>To find <math>\boldsymbol{a_{j_{i+1}},}</math> we look for the smallest positive integer <math>\boldsymbol{k'}</math> for which <cmath>\boldsymbol{a_{j_{i+1}}=a_{j_i+k'}=\frac{m+18k'}{m+1+19k'}}</cmath> is reducible:</b></i>
+If <math>\frac{m+18k'}{m+1+19k'}</math> is reducible, then there exists a common factor <math>d>1</math> for <math>m+18k'</math> and <math>m+1+19k'.</math> By the Euclidean Algorithm, we have
+<cmath>\begin{align*}
+d|m+18k' \text{ and } d|m+1+19k' &\Longrightarrow d|k'+1 \text{ and } d|m+18k' \\
+&\Longrightarrow d|m-18 \text{ and } d|k'+1.
+\end{align*}</cmath>
+Since <math>m-18</math> and <math>k'+1</math> are not relatively prime, and <math>m</math> is fixed, the smallest value of <math>k'</math> such that <math>\frac{m+18k'}{m+1+19k'}</math> is reducible occurs when <math>k'+1</math> is the smallest prime factor of <math>m-18.</math>
+<i><b>We will prove that for such value of <math>\boldsymbol{k',}</math> the number <math>\boldsymbol{a_{j_{i+1}}}</math> can be written in the form <math>\boldsymbol{\frac{t}{t+1}:}</math></b></i> <cmath>a_{j_{i+1}}=a_{j_i+k'}=\frac{m+18k'}{m+1+19k'}=\frac{(m-18)+18(k'+1)}{(m-18)+19(k'+1)}=\frac{\frac{m-18}{k'+1}+18}{\frac{m-18}{k'+1}+19}, \ \ (*)</cmath> where <math>t=\frac{m-18}{k'+1}+18</math> is a positive integer.
+We start with <math>m=2020</math> and <math>a_{j_1}=a_1=\frac{2020}{2021},</math> then find <math>a_{j_2}, a_{j_3}, \cdots, a_{j_u}</math> by filling out the table below recursively:
+<cmath>\begin{array}{c|c|c|c|c|l}
+\boldsymbol{i} & \boldsymbol{m} & \boldsymbol{m-18} & \boldsymbol{k'+1} & \boldsymbol{k'} & \ \boldsymbol{a_{j_{i+1}} \ \left(\textbf{by } (*)\right)} \\ [0.5ex]
+\hline\hline
+& & & & & \\ [-1.5ex]
+& 2020 & 2002 & 2 & 1 & \ \ \ \ \ a_2 = \frac{1019}{1020} \\ [1ex]
+& 1019 & 1001 & 7 & 6 & \ \ \ \ \ a_8 = \frac{161}{162} \\ [1ex]
+& 161 & 143 & 11 & 10 & \ \ \ \ a_{18} = \frac{31}{32} \\ [1ex]
+& 31 & 13 & 13 & 12 & \ \ \ \ a_{30} = \frac{19}{20} \\ [1ex]
+& 19 & 1 & \text{N/A} & \text{N/A} & \ \ \ \ \ \ \ \ \text{N/A} \\ [1ex]
+\end{array}</cmath>
+As <math>\left(j_1,j_2,j_3,j_4,j_5\right)=(1,2,8,18,30),</math> the answer is <math>\sum_{i=1}^{5}{j_i}=\boxed{059}.</math>
+<b><u>Remark</u></b>
+Alternatively, from <math>(*)</math> we can set <cmath>\frac{m+18k'}{m+1+19k'}=\frac{t}{t+1}.</cmath>
+We cross-multiply, rearrange, and apply Simon's Favorite Factoring Trick to get <cmath>\left(k'+1\right)(t-18)=m-18.</cmath>
+Since <math>k'+1\geq2,</math> to find the smallest <math>k',</math> we need <math>k'+1</math> to be the smallest prime factor of <math>m-18.</math> Now, continue with the last two paragraphs of the solution above.
 ~MRENTHUSIASM

2021 AIME I (Problems • Answer Key • Resources)
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