Difference between revisions of "2022 AMC 10A Problems/Problem 17"

Revision as of 12:35, 14 November 2022

Problem

How many three-digit positive integers $\underline{a} \ \underline{b} \ \underline{c}$ are there whose nonzero digits $a,b,$ and $c$ satisfy $0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?$ (The bar indicates repetition, thus $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ in the infinite repeating decimal $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$ )

$\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14$

Solution 1

We rewrite the given equation, then rearrange: $\begin{align*} \frac{100a+10b+c}{999} &= \frac13\left(\frac a9 + \frac b9 + \frac c9\right) \\ 100a+10b+c &= 37a + 37b + 37c \\ 63a &= 27b+36c \\ 7a &= 3b+4c. \end{align*}$ Now, this problem is equivalent to counting the ordered triples $(a,b,c)$ that satisfies the equation.

Clearly, the $9$ ordered triples $(a,b,c)=(1,1,1),(2,2,2),\ldots,(9,9,9)$ are solutions to this equation.

The expression $3b+4c$ has the same value when:

$b$ increases by $4$ as $c$ decreases by $3.$

$b$ decreases by $4$ as $c$ increases by $3.$

We find $4$ more solutions from the $9$ solutions above: $(a,b,c)=(4,8,1),(5,1,8),(5,9,2),(6,2,9).$ Note that all solutions are symmetric about $(a,b,c)=(5,5,5).$

Together, we have $9+4=\boxed{\textbf{(D) } 13}$ ordered triples $(a,b,c).$

~MRENTHUSIASM

Solution 2 (fast)

Use the method from Solution $1$ to get $63a = 27b+36c$ . Subtract $27b+36c$ from both sides to get $63a -27b-36c=0$ . From here we look at possible cases. Either one (from $63a$ , $-27b$ , or $-36c$ ) is positive, and the other 2 are negative, or 2 are positive, and one is negative. There are 6 of these cases ( $pnn,npn,nnp,ppn,pnp,npp$ ). For all of these cases, there are 2 distinct pairs of $a,b$ , and $c$ , as you can simply switch the values of the 2 positives or negatives ( $x+y=y+x$ ). This leaves us with $6 \cdot2 =12$ cases. Then, we add the case where $a, b$ , and $c$ are all 0, giving us $12+1=\boxed{\textbf{(D) } 13}$ ordered triples $(a,b,c).$

~iluvme

Note: This solution does not work because a, b, and c are positive integers...

2022 AMC 10A (Problems • Answer Key • Resources)
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Difference between revisions of "2022 AMC 10A Problems/Problem 17"

Revision as of 12:35, 14 November 2022

Contents

Problem

Solution 1

Solution 2 (fast)

See Also

Revision as of 12:35, 14 November 2022 (view source) Andy lee (talk \| contribs) (→‎Solution 2 (fast)) ← Older edit		Revision as of 12:35, 14 November 2022 (view source) Andy lee (talk \| contribs) (→‎Solution 2 (fast)) Newer edit →
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	~iluvme		~iluvme
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	Note: This solution does not work because a, b, and c are positive integers...		Note: This solution does not work because a, b, and c are positive integers...