Difference between revisions of "2019 AMC 12A Problems/Problem 15"

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Since all four terms on the left are positive integers, from <math>\sqrt{\log{a}}</math>, we know that both <math>\log{a}</math> has to be perfect square and <math>a</math> has to be a power of ten. The same applies to <math>b</math> for the same reason. Setting <math>a</math> and <math>b</math> to <math>10^x</math> and <math>10^y</math>, where <math>x</math> and <math>y</math> are the perfect squares, <math>ab = 10^{x+y}</math>. By listing all the [https://artofproblemsolving.com/wiki/index.php/Perfect_square perfect squares] up to <math>14^2</math> (as <math>15^2</math> is larger than the largest possible sum of <math>x</math> and <math>y</math> of <math>200</math> from answer choice <math>E</math>), two of those perfect squares must add up to one of the possible sums of <math>x</math> and <math>y</math> given from the answer choices (<math>52</math>, <math>100</math>, <math>144</math>, <math>164</math>, or <math>200</math>).  
 
Since all four terms on the left are positive integers, from <math>\sqrt{\log{a}}</math>, we know that both <math>\log{a}</math> has to be perfect square and <math>a</math> has to be a power of ten. The same applies to <math>b</math> for the same reason. Setting <math>a</math> and <math>b</math> to <math>10^x</math> and <math>10^y</math>, where <math>x</math> and <math>y</math> are the perfect squares, <math>ab = 10^{x+y}</math>. By listing all the [https://artofproblemsolving.com/wiki/index.php/Perfect_square perfect squares] up to <math>14^2</math> (as <math>15^2</math> is larger than the largest possible sum of <math>x</math> and <math>y</math> of <math>200</math> from answer choice <math>E</math>), two of those perfect squares must add up to one of the possible sums of <math>x</math> and <math>y</math> given from the answer choices (<math>52</math>, <math>100</math>, <math>144</math>, <math>164</math>, or <math>200</math>).  
  
Only a couple possible sums are seen: <math>16+36=52</math>, <math>36+64=100</math>, <math>64+100=164</math>, <math>100+100=200</math>, and <math>4+196=200</math>. By testing each of these (by seeing whether <math>\sqrt{x} + \sqrt{b} + \frac{x}{2} + \frac{y}{2} = 100</math>), only the pair <math>x = 64</math> and <math>y=100</math> work. Therefore, <math>a</math> and <math>b</math> are <math>10^{64}</math> and <math>10^{100}</math>, and our answer is <math>\boxed{\textbf{(D) } 100^{164}}</math>.
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Only a couple possible sums are seen: <math>16+36=52</math>, <math>36+64=100</math>, <math>64+100=164</math>, <math>100+100=200</math>, and <math>4+196=200</math>. By testing each of these (by seeing whether <math>\sqrt{x} + \sqrt{b} + \frac{x}{2} + \frac{y}{2} = 100</math>), only the pair <math>x = 64</math> and <math>y=100</math> work. Therefore, <math>a</math> and <math>b</math> are <math>10^{64}</math> and <math>10^{100}</math>, and our answer is <math>\boxed{\textbf{(D) } 10^{164}}</math>.
  
 
==Solution 2==
 
==Solution 2==

Revision as of 11:00, 10 February 2019

Problem

Positive real numbers $a$ and $b$ have the property that \[\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100\] and all four terms on the left are positive integers, where log denotes the base 10 logarithm. What is $ab$?

$\textbf{(A) }   10^{52}   \qquad        \textbf{(B) }   10^{100}   \qquad    \textbf{(C) }   10^{144}   \qquad   \textbf{(D) }  10^{164} \qquad  \textbf{(E) }   10^{200}$

Solution 1

Since all four terms on the left are positive integers, from $\sqrt{\log{a}}$, we know that both $\log{a}$ has to be perfect square and $a$ has to be a power of ten. The same applies to $b$ for the same reason. Setting $a$ and $b$ to $10^x$ and $10^y$, where $x$ and $y$ are the perfect squares, $ab = 10^{x+y}$. By listing all the perfect squares up to $14^2$ (as $15^2$ is larger than the largest possible sum of $x$ and $y$ of $200$ from answer choice $E$), two of those perfect squares must add up to one of the possible sums of $x$ and $y$ given from the answer choices ($52$, $100$, $144$, $164$, or $200$).

Only a couple possible sums are seen: $16+36=52$, $36+64=100$, $64+100=164$, $100+100=200$, and $4+196=200$. By testing each of these (by seeing whether $\sqrt{x} + \sqrt{b} + \frac{x}{2} + \frac{y}{2} = 100$), only the pair $x = 64$ and $y=100$ work. Therefore, $a$ and $b$ are $10^{64}$ and $10^{100}$, and our answer is $\boxed{\textbf{(D) } 10^{164}}$.

Solution 2

Given that $\sqrt{\log{a}}$ and $\sqrt{\log{b}}$ are both integers, $a$ and $b$ must be in the form $10^{m^2}$ and $10^{n^2}$, respectively for some positive integers $m$ and $n$. Note that $\log \sqrt{a} = \frac{m^2}{2}$. By substituting for a and b, the equation becomes $m + n + \frac{m^2}{2} + \frac{n^2}{2} = 100$. After multiplying the equation by 2 and completing the square with respect to $m$ and $n$, the equation becomes $(m + 1)^2 + (n + 1)^2 = 202$. Testing squares of positive integers that add to $202$, $11^2 + 9^2$ is the only option. WLOG, let $m = 10$ and $n = 8$. Plugging $m$ and $n$ to solve for $a$ and $b$ gives us $a = 10^{100}$ and $b = 10^{64}$. Therefore, $ab = \boxed{\textbf{(D) } 100^{164}}$.

See Also

2019 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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All AMC 12 Problems and Solutions

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