Difference between revisions of "2022 AMC 10A Problems/Problem 17"
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How many three-digit positive integers <math>\underline{a} \ \underline{b} \ \underline{c}</math> are there whose nonzero digits <math>a,b,</math> and <math>c</math> satisfy | How many three-digit positive integers <math>\underline{a} \ \underline{b} \ \underline{c}</math> are there whose nonzero digits <math>a,b,</math> and <math>c</math> satisfy | ||
<cmath>0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?</cmath> | <cmath>0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?</cmath> | ||
− | (The bar indicates repetition, thus <math>0.\overline{\underline{a}~\underline{b}~\underline{c}}</math> | + | (The bar indicates repetition, thus <math>0.\overline{\underline{a}~\underline{b}~\underline{c}}</math> is the infinite repeating decimal <math>0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots</math>) |
<math>\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14</math> | <math>\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14</math> | ||
− | ==Solution | + | ==Solution== |
We rewrite the given equation, then rearrange: | We rewrite the given equation, then rearrange: | ||
Line 32: | Line 32: | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
− | == | + | ==Remark== |
− | + | One way to solve the Diophantine Equation, <math>7a=3b+4c</math> is by taking <math>\pmod{7}</math>, from which the equation becomes <math>0\equiv 3b-3c\pmod{7} \Longrightarrow b\equiv c\pmod{7}</math> so either <math>b=c</math> or WLOG <math>b<c, b+7=c</math>. | |
+ | |||
+ | ==Video Solution 1== | ||
+ | https://youtu.be/p6IauwE8cX8 | ||
+ | |||
+ | ==Video Solution (⚡️Lightning Fast⚡️)== | ||
+ | https://youtu.be/mgcHM0ATUks | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | ==Video Solution 2== | ||
+ | https://www.youtube.com/watch?v=YAazoVATYQA&list=PLmpPPbOoDfgj5BlPtEAGcB7BR_UA5FgFj&index=4 | ||
+ | |||
== See Also == | == See Also == | ||
{{AMC10 box|year=2022|ab=A|num-b=16|num-a=18}} | {{AMC10 box|year=2022|ab=A|num-b=16|num-a=18}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 21:24, 1 May 2024
Contents
Problem
How many three-digit positive integers are there whose nonzero digits
and
satisfy
(The bar indicates repetition, thus
is the infinite repeating decimal
)
Solution
We rewrite the given equation, then rearrange:
Now, this problem is equivalent to counting the ordered triples
that satisfies the equation.
Clearly, the ordered triples
are solutions to this equation.
The expression has the same value when:
increases by
as
decreases by
decreases by
as
increases by
We find more solutions from the
solutions above:
Note that all solutions are symmetric about
Together, we have ordered triples
~MRENTHUSIASM
Remark
One way to solve the Diophantine Equation, is by taking
, from which the equation becomes
so either
or WLOG
.
Video Solution 1
Video Solution (⚡️Lightning Fast⚡️)
~Education, the Study of Everything
Video Solution 2
https://www.youtube.com/watch?v=YAazoVATYQA&list=PLmpPPbOoDfgj5BlPtEAGcB7BR_UA5FgFj&index=4
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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.