Difference between revisions of "2022 AMC 10A Problems/Problem 1"

Revision as of 01:27, 13 August 2023

The following problem is from both the 2022 AMC 10A #1 and 2022 AMC 12A #1, so both problems redirect to this page.

Problem

What is the value of $3+\frac{1}{3+\frac{1}{3+\frac13}}?$ $\textbf{(A)}\ \frac{31}{10}\qquad\textbf{(B)}\ \frac{49}{15}\qquad\textbf{(C)}\ \frac{33}{10}\qquad\textbf{(D)}\ \frac{109}{33}\qquad\textbf{(E)}\ \frac{15}{4}$

Solution 1

We have $\begin{align*} 3+\frac{1}{3+\frac{1}{3+\frac13}} &= 3+\frac{1}{3+\frac{1}{\left(\frac{10}{3}\right)}} \\ &= 3+\frac{1}{3+\frac{3}{10}} \\ &= 3+\frac{1}{\left(\frac{33}{10}\right)} \\ &= 3+\frac{10}{33} \\ &= \boxed{\textbf{(D)}\ \frac{109}{33}}. \end{align*}$ ~MRENTHUSIASM

Solution 2

Continued fractions with integer parts $q_i$ and numerators all $1$ can be calculated as $\begin{align*} \dfrac{[q_0,q_1,q_2,\ldots,q_n]}{[q_1,q_2,\ldots,q_n]} \end{align*}$ where

\begin{align*}
[]&=1
[q_0,q_1,q_2,\ldots,q_n]&=q_0[q_1,q_2,\ldots,q_n]+[q_2,\ldots,q_n]\\

[3]&=3\\
[3,3]&=3(3)+1=10\\
[3,3,3]&=3(10)+3=33\\
[3,3,3,3]&=3(33)+10=109\\
\dfrac{[q_0,q_1,q_2,\ldots,q_n]}{[q_1,q_2,\ldots,q_n]}&=\dfrac{[3,3,3,3]}{[3,3,3]}\\
&=\boxed{\textbf{(D)}\ \frac{109}{33}}
\end{align*} (Error compiling LaTeX. Unknown error_msg)

~lopkiloinm

Solution 3

It is well known that for continued fractions of form $n+\frac{1}{n+\cdots}$ , the denominator $y$ and numerator $x$ are solutions to the Diophantine equation $(n^2+4)\left(\frac{y}{2}\right)^2-\left(x-\frac{ny}{2}\right)^2=\pm{1}$ . So for this problem, the denominator $y$ and numerator $x$ are solutions to the Diophantine equation $13\left(\frac{y}{2}\right)^2-\left(x-\frac{3y}{2}\right)^2=\pm{1}$ . That leaves two answers. Since the number of $1$ 's in the continued fraction is odd, we further narrow it down to $13\left(\frac{y}{2}\right)^2-\left(x-\frac{3y}{2}\right)^2=-1$ , which only leaves us with $1$ answer and that is $(x,y)=(109,33)$ which means $\boxed{\textbf{(D)}\ \frac{109}{33}}$ .

~lopkiloinm

Video Solution 1 (Quick and Easy)

https://youtu.be/iVvBTapX3Fs

~Education, the Study of Everything

Video Solution 2

https://youtu.be/4zoXEjrBAgk

~Charles3829

2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by First Problem	Followed by Problem 2
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

2022 AMC 12A (Problems • Answer Key • Resources)
Preceded by First Problem	Followed by Problem 2
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 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 17: / Line 17: @@
 == Solution 2 ==
-Continued fractions are expressed as
+Continued fractions with integer parts <math>q_i</math> and numerators all <math>1</math> can be calculated as
 <cmath>\begin{align*}
 \dfrac{[q_0,q_1,q_2,\ldots,q_n]}{[q_1,q_2,\ldots,q_n]}
@@ Line 23: / Line 23: @@
 where
 <cmath>\begin{align*}
+[]&=1
 [q_0,q_1,q_2,\ldots,q_n]&=q_0[q_1,q_2,\ldots,q_n]+[q_2,\ldots,q_n]\\
 [3]&=3\\
 [3,3]&=3(3)+1=10\\

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