Difference between revisions of "2022 AIME I Problems/Problem 2"

(Solution 3)
(Undo revision 185188 by Pooja26678 (talk)the Ivy League file uses English NINE)
(Tag: Undo)
(17 intermediate revisions by 7 users not shown)
Line 19: Line 19:
 
== Solution 2 ==
 
== Solution 2 ==
  
As shown in Solution 1, we get <math>99a = 71b+8c</math>. Note that <math>99</math> and <math>71</math> are big numbers comparatively to <math>8</math>, so we hypothesize that <math>a</math> and <math>b</math> are equal and <math>8c</math> fills the gap between them. The difference between <math>99</math> and <math>71</math> is <math>28</math>, which is a multiple of <math>4</math>. So, if we multiply this by <math>2</math>, it will be a multiple of <math>8</math> and thus the gap can be filled. Therefore, a viable solution is <math>(a,b,c)=(2,2,7)</math>, and the answer is <math>\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}</math>.
+
As shown in Solution 1, we get <math>99a = 71b+8c</math>.
 +
 
 +
Note that <math>99</math> and <math>71</math> are large numbers comparatively to <math>8</math>, so we hypothesize that <math>a</math> and <math>b</math> are equal and <math>8c</math> fills the gap between them. The difference between <math>99</math> and <math>71</math> is <math>28</math>, which is a multiple of <math>4</math>. So, if we multiply this by <math>2</math>, it will be a multiple of <math>8</math> and thus the gap can be filled. Therefore, the only solution is <math>(a,b,c)=(2,2,7)</math>, and the answer is <math>\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}</math>.
  
 
~KingRavi
 
~KingRavi
Line 25: Line 27:
 
== Solution 3 ==
 
== Solution 3 ==
  
As shown in Solution 1, we get <math>99a = 71b+8c.</math> We list a few multiples of <math>99</math> out:  
+
As shown in Solution 1, we get <math>99a = 71b+8c.</math>
\begin{align*}
+
 
99\\198\\297\\396\\
+
We list a few multiples of <math>99</math> out: <cmath>99,198,297,396.</cmath>
\end{align*}
 
 
Of course, <math>99</math> can't be made of just <math>8</math>'s. If we use one <math>71</math>, we get a remainder of <math>28</math>, which can't be made of <math>8</math>'s either. So <math>99</math> doesn't work.  
 
Of course, <math>99</math> can't be made of just <math>8</math>'s. If we use one <math>71</math>, we get a remainder of <math>28</math>, which can't be made of <math>8</math>'s either. So <math>99</math> doesn't work.  
 
<math>198</math> can't be made up of just <math>8</math>'s. If we use one <math>71</math>, we get a remainder of <math>127</math>, which can't be made of <math>8</math>'s. If we use two <math>71</math>'s, we get a remainder of <math>56</math>, which can be made of <math>8</math>'s.  
 
<math>198</math> can't be made up of just <math>8</math>'s. If we use one <math>71</math>, we get a remainder of <math>127</math>, which can't be made of <math>8</math>'s. If we use two <math>71</math>'s, we get a remainder of <math>56</math>, which can be made of <math>8</math>'s.  
Therefore we get <math>99\cdot2=71\cdot2+8\cdot7</math> so <math>a=2,b=2,</math> and <math>c=7</math>. Plugging this back into the original problem shows that this answer is indeed correct. Therefore, <math>\underline{a}\underline{b}\underline{c}=227.</math>
+
Therefore we get <math>99\cdot2=71\cdot2+8\cdot7</math> so <math>a=2,b=2,</math> and <math>c=7</math>. Plugging this back into the original problem shows that this answer is indeed correct. Therefore, <math>\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}.</math>
 +
 
 
~Technodoggo
 
~Technodoggo
 +
 +
== Solution 4 ==
 +
As shown in Solution 1, we get <math>99a = 71b+8c</math>.
 +
 +
We can see that <math>99</math> is <math>28</math> larger than <math>71</math>, and we have an <math>8c</math>. We can clearly see that <math>56</math> is a multiple of <math>8</math>, and any larger than <math>56</math> would result in <math>c</math> being larger than <math>9</math>. Therefore, our only solution is <math>a = 2, b = 2, c = 7</math>. Our answer is <math>\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}</math>.
 +
 +
~Arcticturn
 +
 +
== Solution 5 ==
 +
As shown in Solution 1, we get <math>99a = 71b+8c,</math> which rearranges to <cmath>99(a – b) = 8c – 28 b = 4(2c – 7b) \le 4(2\cdot 9 - 0 ) = 72.</cmath>
 +
So <math>a=b, 2c = 7b \implies c=7, b=2,a=2.</math>
 +
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
 +
 +
== Video Solution by OmegaLearn ==
 +
https://youtu.be/SCGzEOOICr4?t=340
 +
 +
~ pi_is_3.14
  
 
==Video Solution (Mathematical Dexterity)==
 
==Video Solution (Mathematical Dexterity)==
Line 41: Line 61:
  
 
~Steven Chen (www.professorchenedu.com)
 
~Steven Chen (www.professorchenedu.com)
 +
 +
== Video Solution ==
 +
 +
https://youtu.be/MJ_M-xvwHLk?t=392
 +
 +
~ThePuzzlr
 +
 +
==Video Solution by MRENTHUSIASM (English & Chinese)==
 +
https://www.youtube.com/watch?v=v4tHtlcD9ww&t=360s&ab_channel=MRENTHUSIASM
 +
 +
~MRENTHUSIASM
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2022|n=I|num-b=1|num-a=3}}
 
{{AIME box|year=2022|n=I|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 22:06, 29 December 2022

Problem

Find the three-digit positive integer $\underline{a}\,\underline{b}\,\underline{c}$ whose representation in base nine is $\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},$ where $a,$ $b,$ and $c$ are (not necessarily distinct) digits.

Solution 1

We are given that \[100a + 10b + c = 81b + 9c + a,\] which rearranges to \[99a = 71b + 8c.\] Taking both sides modulo $71,$ we have \begin{align*} 28a &\equiv 8c \pmod{71} \\ 7a &\equiv 2c \pmod{71}. \end{align*} The only solution occurs at $(a,c)=(2,7),$ from which $b=2.$

Therefore, the requested three-digit positive integer is $\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}.$

~MRENTHUSIASM

Solution 2

As shown in Solution 1, we get $99a = 71b+8c$.

Note that $99$ and $71$ are large numbers comparatively to $8$, so we hypothesize that $a$ and $b$ are equal and $8c$ fills the gap between them. The difference between $99$ and $71$ is $28$, which is a multiple of $4$. So, if we multiply this by $2$, it will be a multiple of $8$ and thus the gap can be filled. Therefore, the only solution is $(a,b,c)=(2,2,7)$, and the answer is $\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}$.

~KingRavi

Solution 3

As shown in Solution 1, we get $99a = 71b+8c.$

We list a few multiples of $99$ out: \[99,198,297,396.\] Of course, $99$ can't be made of just $8$'s. If we use one $71$, we get a remainder of $28$, which can't be made of $8$'s either. So $99$ doesn't work. $198$ can't be made up of just $8$'s. If we use one $71$, we get a remainder of $127$, which can't be made of $8$'s. If we use two $71$'s, we get a remainder of $56$, which can be made of $8$'s. Therefore we get $99\cdot2=71\cdot2+8\cdot7$ so $a=2,b=2,$ and $c=7$. Plugging this back into the original problem shows that this answer is indeed correct. Therefore, $\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}.$

~Technodoggo

Solution 4

As shown in Solution 1, we get $99a = 71b+8c$.

We can see that $99$ is $28$ larger than $71$, and we have an $8c$. We can clearly see that $56$ is a multiple of $8$, and any larger than $56$ would result in $c$ being larger than $9$. Therefore, our only solution is $a = 2, b = 2, c = 7$. Our answer is $\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}$.

~Arcticturn

Solution 5

As shown in Solution 1, we get $99a = 71b+8c,$ which rearranges to \[99(a – b) = 8c – 28 b = 4(2c – 7b) \le 4(2\cdot 9 - 0 ) = 72.\] So $a=b, 2c = 7b \implies c=7, b=2,a=2.$

vladimir.shelomovskii@gmail.com, vvsss

Video Solution by OmegaLearn

https://youtu.be/SCGzEOOICr4?t=340

~ pi_is_3.14

Video Solution (Mathematical Dexterity)

https://www.youtube.com/watch?v=z5Y4bT5rL-s

Video Solution

https://www.youtube.com/watch?v=CwSkAHR3AcM

~Steven Chen (www.professorchenedu.com)

Video Solution

https://youtu.be/MJ_M-xvwHLk?t=392

~ThePuzzlr

Video Solution by MRENTHUSIASM (English & Chinese)

https://www.youtube.com/watch?v=v4tHtlcD9ww&t=360s&ab_channel=MRENTHUSIASM

~MRENTHUSIASM

See Also

2022 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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