Difference between revisions of "2018 AMC 10A Problems/Problem 25"
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<math>\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}</math> | <math>\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}</math> | ||
− | == Solution == | + | == Solution 1== |
+ | |||
+ | Observe <math>A_n = a(1 + 10 + \dots + 10^{n - 1}) = a \cdot \tfrac{10^n - 1}{9}</math>; similarly <math>B_n = b \cdot \tfrac{10^n - 1}{9}</math> and <math>C_n = c \cdot \tfrac{10^{2n} - 1}{9}</math>. The relation <math>C_n - B_n = A_n^2</math> rewrites as | ||
+ | <cmath>c \cdot \frac{10^{2n} - 1}{9} - b \cdot \frac{10^n - 1}{9} = a^2 \cdot \left(\frac{10^n - 1}{9}\right)^2.</cmath>Since <math>n > 0</math>, <math>10^n > 1</math> and we may cancel out a factor of <math>\tfrac{10^n - 1}{9}</math> to obtain | ||
+ | <cmath>c \cdot (10^n + 1) - b = a^2 \cdot \frac{10^n - 1}{9}.</cmath>This is a linear equation in <math>10^n</math>. Thus, if two distinct values of <math>n</math> satisfy it, then all values of <math>n</math> will. Matching coefficients, we need | ||
+ | <cmath>c = \frac{a^2}{9} \quad \text{and} \quad c - b = -\frac{a^2}{9} \implies b = \frac{2a^2}{9}.</cmath>To maximize <math>a + b + c = a + \tfrac{a^2}{3}</math>, we need to maximize <math>a</math>. Since <math>b</math> and <math>c</math> must be integers, <math>a</math> must be a multiple of 3. If <math>a = 9</math> then <math>b</math> exceeds 9. However, if <math>a = 6</math> then <math>b = 8</math> and <math>c = 4</math> for an answer of <math>\boxed{\textbf{(D)} \text{ 18}}</math>. (CantonMathGuy) | ||
+ | |||
+ | == Solution 2 (quicker?) == | ||
+ | |||
+ | Immediately start trying <math>n = 1</math> and <math>n = 2</math>. These give the system of equations <math>11c - b = a^2</math> and <math>1111c - 11b = (11a)^2</math> (which simplifies to <math>101c - b = 11a^2</math>). These imply that <math>a^2 = 9c</math>, so the possible <math>(a, c)</math> pairs are <math>(9, 9)</math>, <math>(6, 4)</math>, and <math>(3, 1)</math>. The first puts <math>b</math> out of range but the second makes <math>b = 8</math>. We now know the answer is at least <math>6 + 8 + 4 = 18</math>. | ||
+ | |||
+ | We now only need to know whether <math>a + b + c = 20</math> might work for any larger <math>n</math>. We will always get equations like <math>100001c - b = 11111a^2</math> where the <math>c</math> coefficient is very close to being nine times the <math>a</math> coefficient. Since the <math>b</math> term will be quite insignificant, we know that once again <math>a^2</math> must equal <math>9c</math>, and thus <math>a = 9, c = 9</math> is our only hope to reach <math>20</math>. Substituting and dividing through by <math>9</math>, we will have something like <math>100001 - b/9 = 99999</math>. No matter what <math>n</math> really was, <math>b</math> is out of range (and certainly isn't <math>2</math> as we would have needed). | ||
+ | |||
+ | The answer then is <math>\boxed{\textbf{(D)} \text{ 18}}</math>. | ||
+ | |||
+ | == Solution 3 == | ||
Notice that <math>(0.\overline{3})^2 = 0.\overline{1}</math> and <math>(0.\overline{6})^2 = 0.\overline{4}</math>. Setting <math>a = 3</math> and <math>c = 1</math>, we see <math>b = 2</math> works for all possible values of <math>n</math>. Similarly, if <math>a = 6</math> and <math>c = 4</math>, then <math>b = 8</math> works for all possible values of <math>n</math>. The second solution yields a greater sum of <math>\boxed{\textbf{(D)} \text{ 18}}</math>. | Notice that <math>(0.\overline{3})^2 = 0.\overline{1}</math> and <math>(0.\overline{6})^2 = 0.\overline{4}</math>. Setting <math>a = 3</math> and <math>c = 1</math>, we see <math>b = 2</math> works for all possible values of <math>n</math>. Similarly, if <math>a = 6</math> and <math>c = 4</math>, then <math>b = 8</math> works for all possible values of <math>n</math>. The second solution yields a greater sum of <math>\boxed{\textbf{(D)} \text{ 18}}</math>. | ||
Revision as of 13:04, 19 February 2018
Problem
For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?
Solution 1
Observe ; similarly and . The relation rewrites as Since , and we may cancel out a factor of to obtain This is a linear equation in . Thus, if two distinct values of satisfy it, then all values of will. Matching coefficients, we need To maximize , we need to maximize . Since and must be integers, must be a multiple of 3. If then exceeds 9. However, if then and for an answer of . (CantonMathGuy)
Solution 2 (quicker?)
Immediately start trying and . These give the system of equations and (which simplifies to ). These imply that , so the possible pairs are , , and . The first puts out of range but the second makes . We now know the answer is at least .
We now only need to know whether might work for any larger . We will always get equations like where the coefficient is very close to being nine times the coefficient. Since the term will be quite insignificant, we know that once again must equal , and thus is our only hope to reach . Substituting and dividing through by , we will have something like . No matter what really was, is out of range (and certainly isn't as we would have needed).
The answer then is .
Solution 3
Notice that and . Setting and , we see works for all possible values of . Similarly, if and , then works for all possible values of . The second solution yields a greater sum of .
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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 |
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Problem |
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.