HMMT 2023 U33

Given a function $f$, let $\pi(f)=f \circ f \circ f \circ f \circ f$. The attached sheet has the graphs of ten smooth functions from the interval $(0,1)$ to itself. The left-hand side consists of five functions:

• $F_{1}(x)=0.005+\frac{1}{2} \sin 2 x+\frac{1}{4} \sin 4 x+\frac{1}{8} \sin 8 x+\frac{1}{16} \sin 16 x+\frac{1}{32} \sin 32 x ;$
• $F_{2}(x)=F_{1}\left(F_{1}(x+0.25)\right)$;
• $F_{3}(x)=F_{1}\left((1-x) F_{1}\left((1-x)^{2}\right)\right)$;
• $F_{4}(x)=F_{1}(x)+0.05 \sin (2 \pi x)$;
• $F_{5}(x)=F_{1}(x+1.45)+0.65$.

The right-hand side consists of the five functions $A, B, C, D$, and $E$, which are $\pi\left(F_{1}\right), \ldots, \pi\left(F_{5}\right)$ in some order. Compute which of the functions $\{A, B, C, D, E\}$ correspond to $\pi\left(F_{k}\right)$ for $k=1,2,3,4,5$. Your answer should be a five-character string containing $A, B, C, D, E$, or $X$ for blank. For instance, if you think $\pi\left(F_{1}\right)=A$ and $\pi\left(F_{5}\right)=E$, then you would answer $A X X X E$. If you attempt to identify $n$ functions and get them all correct, then you will receive $n^{2}$ points. Otherwise, you will receive 0 points.