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This is another of $Cauchy's functional equations$.} + +\item{Find all functions $f: \mathbb{Z^+} \to \mathbb{Z^+}$, sucht that +\[m^2+f(n) | mf(m)+n\] +for all $m,n \in \mathbb{Z^+}$.} + +\item{The strictly increasing function $f(x)$ is defined for all $x>0$ and it assumes positive integral values for all $n \geq 1$.In addition, it satisfies the condition $f[f(n)]=3*n$. Find $f(1994)$.} + + +\item{Find all functions $f,g: \mathbb{R} \to \mathbb{R}$ such that +\[g(f(x+y))=f(x)+(2x+y)g(y)\] +for all $x,y \in \mathbb{R}$.} + +\item{Let $f: \mathbb{R} \to \mathbb{N}$ be a function that satisfies +\[f\left(x+\frac{1}{f(y)}\right)=f\left(y+\frac{1}{f(x)}\right),\] +for all $x,y\in \mathbb{R}$. Prove that there is a positive integer that is not a value of $f$.} + + +\end{enumerate} + +\end{document} + diff --git a/1617sem1/5_Colouring_Proofs_1/Colouring_Proofs_1.pdf b/1617sem1/5_Colouring_Proofs_1/Colouring_Proofs_1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6ba4985ed1c8207235cc2c69ca6a60f765a39b39 GIT binary patch literal 110781 zcmcG#1z6QzvnWhTcOxa;v0-l-q@}yNySt=AT2euzL8T={xA2OSywVj)}3k92`ow1v_xVfpLnK`nM5VEVAi@C8qve!(QrfkwI2(|gJ zTuJe@rAZjZE8N*_w{(RW>Z00XISgrZooEAV4Qx{BxT@Qq@QD41o;8DAw9hf&f-o?e z7UmnTGIt|)#gCX{Jt87yGwHF+5qX1Fz5*ons!50}xuq|X1jeY* zx#vK&1+e6LrzruwwAHCxE@=GDp${%YbztPQpp0~j``v^IS4yLId zRUT~v+0=y9z`@JzmmuTrQo2EM93NH)g8O+TH`loE-3Kw7%NLLjt%EwD*@&-?nz zQqpbsaSA`Lm8y$(k=9r;txms`r^UPXs_oZ1iZQZBJn`4)FItG5ecPeJe;U64={ot| zOnCUpM(n=6L$I90vFZe=~lvZI`lsgzhpcZV>1s?*rR*$2eK@7>Fo`7qK<*@w$V zj1KjrP(1|_Z6kcn0N!3FsKI3TBDNgccDu&eHVB=Hx}w-+i}-DtJ`JC<>1C7FR%XaO ztwNHM49$=WCps`(-)0m8Q<*-r>VT9Qgd$Sb~rCEGcc7~U^-w$Axet`e?);JhPClw3F5Pynr^b?!tnHsfRH 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