HMMT-Feb 2022 guts, Problem 17

Compute the number of positive real numbers $x$ that satisfy \[ \left(3 \cdot 2^{\left\lfloor\log _{2} x\right\rfloor}-x\right)^{16}=2022 x^{13} \]

Solution: Solution: Let $f(x)=3 \cdot 2^{\left\lfloor\log _{2} x\right\rfloor}-x$. Note that for each integer $i$, if $x \in\left[2^{i}, 2^{i+1}\right)$, then $f(x)=3 \cdot 2^{i}-x$. This is a line segment from $\left(2^{i}, 2^{i+1}\right)$ to $\left(2^{i+1}, 2^{i}\right)$, including the first endpoint but not the second. Now consider the function $f(x)^{16} / x^{13}$. This consists of segments of decreasing functions connecting $\left(2^{i}, 2^{3 i+16}\right)$ and $\left(2^{i+1}, 2^{3 i-13}\right)$. Note that for each $-1 \leq i \leq 7$, we have that $2^{3 i-13} \leq 2^{10}<2022<$ $2^{11} \leq 2^{3 i+16}$. This gives us 9 solutions in total. $\fbox{9}$.

Answer: 9