Write a full solution.

1.) Find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) that satisfy the inequality for all \(n \in \mathbb{N}\)

\[f(n+1) > f(f(n))\]

2.) Let \(f: \mathbb{N} \rightarrow \mathbb{N}\) be a function that satisfy the equation for all \(m,n \in \mathbb{N}\)

\[f(m+f(n)) = n+f(m+58)\]

Prove that there exists only 1 solution of functional equation above, and find the value of \(f(1)+f(2)+\dots + f(15)\).

3.) Find all functions \(f: \mathbb{R_{0}^{+}} \rightarrow \mathbb{R_{0}^{+}}\) such that for all \(x,y \in \mathbb{R_{0}^{+}}\)

\[f(x+f(y^{2})) = f(x) + y^{2}\]

Where \(\mathbb{R_{0}^{+}}\) is set of non-negative real numbers.

4.) Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for all \(x,y,z \in \mathbb{R}\)

\[f(x^{2}(z^{2}+1)+f(y)(z+1)) = 1 - f(z)(x^{2}+f(y)) - z((1+z)x^{2}+2f(y))\]

5.) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function that satisfy the equation for all \(x,y \in \mathbb{R}\)

\[f^{3}(x+y)+f^{3}(x-y) = (f(x)+f(y))^{3}+(f(x)-f(y))^{3}\]

Prove that \(f(x+y) = f(x)+f(y)\) for all \(x,y \in \mathbb{R}\).

This note is a part of Thailand Math POSN 2nd round 2015

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## Comments

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TopNewestSamuraiwarm Tsunayoshi, I wanted to ask that for Q-5, can we use a particular \(f(x)\) that satisfies the given condition to prove the required statement?

If so, then one can note that the class of functions \(f:\Bbb{R}\mapsto\Bbb{R}\) defined by \(f(x)=\lambda x~,~\lambda\) is some constant satisfies the given condition. Then, the proof becomes trivial since,

\[f(x+y)=\lambda (x+y)=\lambda x + \lambda y = f(x)+f(y)\]

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Nope, you have to prove that no other functions exist.

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