Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Mock AIME I 2015 Problems/Problem 9

Revision as of 20:26, 1 March 2020 by Piis3141592653 (talk | contribs) (original solution was incorrect, updated with correct answers)

Since $a$ is a multiple of $b$, let $a=kb$.

We can rewrite the first and second conditions as:

(a) $(bk)bc$ is a perfect square, or $ck$ is a perfect square.

(b) $b(k+7)c$ is a power of $2$, so it follows that $b$, $c$, and $k+7$ are all powers of $2$.

Now we use casework on $k$. Since $k+7$ is a power of $2$, $k$ is $1, 9, 25, 57, 121,$ or $249$ or $k>500$.

If $k>500$, then no value of $b$ makes $1\leq a, b\leq 500$.

If $k=57$ or $k=249$, then no value of $c$ that is a power of $2$ makes $ck$ a perfect square.

If $k=1$, then $c=1, 4, 16, 64, 256$ and $b=1, 2, 4, 8, 16, 32, 64, 128, 256$ for $5\cdot 9=45$ solutions.

If $k=9$, then $c=1, 4, 16, 64, 256$ and $b=1, 2, 4, 8, 16, 32$ for $5\dot 6=30$ solutions.

If $k=25$, then $c=1, 4, 16, 64, 256$ and $b=1, 2, 4, 8, 16$ for $5\dot 5=25$ solutions.

If $k=121$, then $c=1, 4, 16, 64, 256$ and $b=1, 2, 4$ for $5\dot 3=15$ solutions.

This is a total of $\fbox{115}$ solutions.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Mock_AIME_I_2015_Problems/Problem_9&oldid=118618"