函數方程 - HackMD

--- tags: 數讀 --- # 函數方程 > [name=W_Ice_Tri] [time=Sun, March 7, 2021] [color=#1f1e33] [TOC] ## 0. 前言我不想備課，所以我可能會掛在台上$\phi\omega\phi$ 然後我英文太菜了，沒有翻譯到的部分請見諒～ ## 1. 亂代 #### Example1.1 Find all functions $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ s.t. $\forall x\in \mathbb{R}^+$, $$2f(x)+f(\frac{1}{x})=3x$$ #### Example1.2 Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ s.t. $\forall x,y\in \mathbb{R}$, $$f(x^{20}+y^{21})=f(x^{2021}+2y^{21})+f(x^{2020})$$ #### Example1.3 Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ s.t. $\forall x\in \mathbb{R}$, $$f(f(x)+y)=f(x^2-y)+4f(x)y$$ #### Problem1.1 Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ s.t. $\forall w,x,y,z\in\mathbb{R}^+\wedge wx=yz$, $$\dfrac{f(w)^2+f(x)^2}{f(y^2)+f(z^2)}=\dfrac{w^2+x^2}{y^2+z^2}$$ #### Problem1.2 Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ s.t. $\forall x\in \mathbb{R}$, $$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2$$ (16 USAMO) #### Problem1.3 Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ s.t. $\forall m,n\in \mathbb{N}$, $$f(f(m)+n)+f(f(n))=f(m+f(n))+f(m+1)$$(FEOO A2) ## 2. 單滿雙射 ### Lemma2.1 :::info If function $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $f(f(x))=kx+c\ (k\neq 0)$, then $f$ is $bijective$. ::: #### Example2.1 Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ s.t. $\forall x,y\in \mathbb{R}$, $$f(x^2+f(y))=y+xf(x)$$ #### Example2.2 Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ s.t. $\forall x,y\in \mathbb{R}$, $$f(xy+f(x))=xf(y)+x$$ #### Example2.3 Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ s.t. $\forall x,y\in \mathbb{R}$, $$f\left({yf(x+y)+f(x)}\right)=4x+2yf(x+y)$$(12 EGMO) #### Problem2.1 Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ s.t. $\forall x,y \in \mathbb{R}$, $$f(xy+f(x))=f(xf(y))+x$$(20 IMOC) #### Problem2.2 Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ s.t. $\forall x,y \in \mathbb{R}$, $$f(f(x)+y)=2x+f(f(y)-x)$$(02 A1) ## 3. Cauchy,Jensen ### Lemma3.1 (Cauchy) :::info $f:\mathbb{Q}\rightarrow \mathbb{Q}$ , if:$$f(x+y)=f(x)+f(y)$$then $f(x)=kx\ \forall x\in \mathbb{Q}$ ::: ### Lemma3.2 (Jensen) :::info $f:\mathbb{Q}\rightarrow \mathbb{Q}$ , if:$$f(x)+f(y)=2f\left(\frac{x+y}{2}\right)$$then $f(x)=kx+c\ \forall x\in \mathbb{Q}$ ::: #### Example3.1 Find all functions $f:\mathbb{Q}\rightarrow \mathbb{Q}$ s.t. $\forall x,y,z,t\in \mathbb{Q}$,$$f(s)+f(t)=f(y)+f(z)$$where $x<y<z<t$ (15 JMO) #### Problem3.1 Find all functions $f:\mathbb{Q}\times\mathbb{Q}\rightarrow \mathbb{Q}$ s.t. $\forall x,y,z\in \mathbb{Q}$ $$f(x,y)+f(y,z)+f(z,x)=f(0,x+y+z)$$ (14 Kazakhstan) ### Theorem3.1 (Cauchy on reals) :::info Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$, satisfies $f(x+y)=f(x)+f(y)$. Then $f(qx)=qf(x)\ \forall x\in \mathbb{Q}$. Moreover, $f$ is linear if any of the following are true: - $f$ is continues on some interval - $f$ is bounded on some interval - $f$ is monotonic on some interval ::: #### Example3.2 Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ s.t. $\forall x,y\in \mathbb{R}$,$$f(x^{2015}+f(y)^{2015})=f(x)^{2015}+y^{2015}$$(15 Korea) ## 4. 數論函方 #### Example4.1 Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ s.t. $\forall m,n\in \mathbb{N}$,$$m^2+f(n)\mid mf(m)+n$$(13 N1) #### Example4.2 Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ s.t. $\forall a,b\in \mathbb{N}$,$$f(a)+b\mid a^2+f(a)f(b)$$(19 APMO) #### Example4.3 Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ s.t. $\forall a,b\in \mathbb{N}$,$$f(a)+f(b)\mid 2(a+b-1)$$(16 MEMO) #### Problem4.1 Find all functions $f:\mathbb{Q}\rightarrow \mathbb{Q}$ s.t. $\forall a,b\in \mathbb{Q}$,$$f(x^2f(y)^2)=f(x)^2f(y)$$(18 A1) #### Problem4.2 Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ s.t. $\forall m,n\in \mathbb{N}$,$$f(m)^2+f(n)\mid (m^2+n)^2$$(04 N3) ## 5. ISL #### Problem5.1 Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ s.t. $\forall a,b\in \mathbb{Z}$,$$f(2a)+2f(b)=f(f(a+b))$$(19 A1) #### Problem5.2 Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ s.t. $\forall x,y\in \mathbb{Z}$,$$f(x-f(y))=f(f(x))-f(y)-1$$(15 A2) #### Problem5.3 Find all pairs of $(f,g)$ of functions $\mathbb{R}\rightarrow\mathbb{R}$ s.t. $\forall x,y\in\mathbb{R}$,$$g(f(x+y)) = f(x) + (2x + y)g(y)$$(11 A3) #### Problem5.4 A positive integer constant $\mathcal{C}$ is given. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ s.t. $\forall a,b\in\mathbb{N}$ satisfying $a,b>\mathcal{C}$,$$a+f(b)\mid a^2+bf(a)$$(19 N3) #### Problem5.5 Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ s.t. $\forall x,y,z,t\in\mathbb{R}$,$$\left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz)$$(02 A4) #### Problem5.6 Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ s.t. $\forall x,y\in\mathbb{R}^+$,$$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right)$$(16 A4) #### Problem5.7 Find all functions $f:\mathbb{Q}^+\rightarrow \mathbb{Q}^+$ s.t. $\forall x,y\in\mathbb{Q}^+$ $$f\left( f(x)^2y \right) = x^3 f(xy)$$(10 A5) #### Problem5.8 Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ s.t. $\forall x,y \in \mathbb{R}$ $$f(x^2+y^2+2f(xy))=(f(x+y))^2$$(04 A6) #### Problem5.9 Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ s.t. $\forall m,n \in \mathbb{N}$ $$f(m)+f(n)-mn\mid mf(m)+nf(n)$$(16 N6) #### Problem5.10 Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ s.t. $\forall x,y\in \mathbb{R}$,$$f(xf(x+y))=f(yf(x))+x^2$$(09 A7)