2024 SSMO Team Round Problems/Problem 1
Problem
How many ordered triples of positive integers 
satisfy the equation 
?
Solution
We have
\begin{align*}
2(a^b)^c+1 &= 513\\\implies
a^{bc}&=256\implies a = 2^{a_1} \mid a_1 \in \mathbb{Z}_{\ge 0}\\\implies
2^{a_1bc} &= 2^{8}\\\implies
a_1bc &= 2^3\implies a_1 = 2^{a_2},b=2^{b_1}, c = 2^{c_1} \mid a_2,b_1,c_1 \in \mathbb{Z}_{\ge 0}\\\implies
2^{a_2+b_1+c_1} &=2^3\\\implies
a_2+b_1+c_1&=3.
\end{align*}
From the Hockey Stick Identity, it follows that this equation has 
solutions.
~SMO_Team
