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Question Number 218399 by Nicholas666 last updated on 09/Apr/25

Answered by MrGaster last updated on 10/Apr/25

$$\overline{)\begin{array}{c}f\left(n\right)=\frac{n^2}{2}\end{array}}\mathrm{when}n\mathrm{is}\mathrm{even}$$$$\cancel{f\left(n\right)=\frac{n^2}{2}}$$$$f\left(n\right)=\frac{n^2-\tau \left(n\right)}{2}$$$$f\left(n\right)+f\left(f\left(n\right)\right)=\frac{n^2-\tau \left(n\right)}{2}+\frac{{\left(\frac{n^2-\tau \left(n\right)}{2}\right)}^2-\tau \left(\frac{n^2-\tau \left(n\right)}{2}\right)}{2}$$$$=\frac{n^2-\tau \left(n\right)}{2}+\frac{n^4-2n^2\tau \left(n\right)+\tau {\left(n\right)}^2-4\tau \left(\frac{n^2-\tau \left(n\right)}{2}\right)}{8}$$$$\cancel{\mathrm{does}\mathrm{not}\mathrm{hold}}$$$$f\left(n\right)+f\left(f\left(n\right)\right)=\tau \left(n\right)n+\tau \left(\tau \left(n\right)n\right)\cdot \tau \left(n\right)n=n^2$$$$\Rightarrow \tau \left(n\right)(n+\tau \left(\tau \left(n\right)n\right)\cdot \tau \left(n\right)n)=n^2$$$$\cancel{\mathrm{no}\mathrm{solution}}$$$$\overline{)\begin{array}{c}f\left(n\right)=n^2-\tau {\left(n\right)}^2\end{array}}$$$$f\left(n\right)+f\left(f\left(n\right)\right)=n^2-\tau {\left(n\right)}^2+{(n^2-\tau {\left(n\right)}^2)}^2-\tau {(n^2-\tau {\left(n\right)}^2)}^2=n^2$$$$\cancel{\mathrm{does}\mathrm{not}\mathrm{hold}}$$$$\overline{)\begin{array}{c}f\left(n\right)=\frac{n^2}{1+\tau \left(n\right)}\end{array}}$$$$\Rightarrow \frac{n^2}{1+\tau \left(n\right)}+\frac{{\left(\frac{n^2}{1+\tau \left(n\right)}\right)}^2}{1+\tau \left(\frac{n^2}{1+\tau \left(n\right)}\right)}=n^2$$$$\cancel{\mathrm{does}\mathrm{not}\mathrm{hold}}$$$$\overline{)\begin{array}{c}f\left(n\right)=\frac{n^2}{2}\mathrm{when}\tau \left(n\right)\mathrm{is}\mathrm{even},f\left(n\right)=\frac{n^2-1}{2}\mathrm{when}\tau \left(n\right)\mathrm{is}\mathrm{odd}\end{array}}$$$$\mathrm{verify}n=2(\tau =2\mathrm{even}):f\left(2\right)=2,f\left(2\right)+f\left(2\right)=4=2^2$$$$n=4(\tau =3\mathrm{odd}):f\left(4\right)=\frac{16-1}{2}=7.5\cancel{\mathrm{not}\mathrm{an}\mathrm{integer}}$$$$\overline{)\begin{array}{c}f\left(n\right)=\{\begin{array}{l}\frac{n^2}{2}n\mathrm{evem}\\ \frac{n^2-1}{2}nodd\end{array}\end{array}}$$$$n=3:f\left(3\right)=4,f\left(4\right)=8\Rightarrow 4+8=12\ne 9\cancel{\mathrm{error}}$$$$\overline{)\begin{array}{c}f\left(n\right)=\frac{n^2}{2}\mathrm{\forall }n\in {\mathbb{Z}}^{+}\end{array}}\mathrm{but}\mathrm{only}\mathrm{when}n=2\mathrm{holds}$$$$\mathrm{final}\mathrm{solution}\mathrm{must}\mathrm{satisfy}f\left(n\right)+f\left(f\left(n\right)\right)=n^2\mathrm{and}f\left(n\right)\mathrm{is}\mathrm{related}\mathrm{to}\tau \left(n\right)$$$$\overline{)\begin{array}{c}f\left(n\right)=n^2-\tau \left(n\right)\end{array}}\mathrm{partially}\mathrm{holds}\mathrm{e}.\mathrm{g}.,n=2,f\left(2\right)=2,f\left(f\left(2\right)\right)=2\Rightarrow 2+2=4=2^2$$

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