[Algebra]
Problem: Determine all functions $f$ from the set of positive integers into the set of positive integers such that for all $x$ and $y$ there exists a non degenerated triangle with sides of lengths

$$
x, \quad f(y) \text { and } f(y+f(x)-1) .
$$

Solution: The identity function $f(x)=x$ is the only solution of the problem.

If $f(x)=x$ for all positive integers $x$, the given three lengths are $x, y=f(y)$ and $z=$ $f(y+f(x)-1)=x+y-1$. Because of $x \geq 1, y \geq 1$ we have $z \geq \max \{x, y\}>|x-y|$ and $z<x+y$. From this it follows that a triangle with these side lengths exists and does not degenerate. We prove in several steps that there is no other solution.

Step 1. We show $f(1)=1$.

If we had $f(1)=1+m>1$ we would conclude $f(y)=f(y+m)$ for all $y$ considering the triangle with the side lengths $1, f(y)$ and $f(y+m)$. Thus, $f$ would be $m$-periodic and, consequently, bounded. Let $B$ be a bound, $f(x) \leq B$. If we choose $x>2 B$ we obtain the contradiction $x>2 B \geq f(y)+f(y+f(x)-1)$.

Step 2. For all positive integers $z$, we have $f(f(z))=z$.

Setting $x=z$ and $y=1$ this follows immediately from Step 1 .

Step 3. For all integers $z \geq 1$, we have $f(z) \leq z$.

Let us show, that the contrary leads to a contradiction. Assume $w+1=f(z)>z$ for some $z$. From Step 1 we know that $w \geq z \geq 2$. Let $M=\max \{f(1), f(2), \ldots, f(w)\}$ be the largest value of $f$ for the first $w$ integers. First we show, that no positive integer $t$ exists with

$$
f(t)>\frac{z-1}{w} \cdot t+M,
\tag{1}
$$

otherwise we decompose the smallest value $t$ as $t=w r+s$ where $r$ is an integer and $1 \leq s \leq w$. Because of the definition of $M$, we have $t>w$. Setting $x=z$ and $y=t-w$ we get from the triangle inequality

$$
z+f(t-w)>f((t-w)+f(z)-1)=f(t-w+w)=f(t)
$$

Hence,

$$
f(t-w) \geq f(t)-(z-1)>\frac{z-1}{w}(t-w)+M
$$

a contradiction to the minimality of $t$.

Therefore the inequality (1) fails for all $t \geq 1$, we have proven

$$
f(t) \leq \frac{z-1}{w} \cdot t+M
\tag{2}
$$

instead.



Now, using (2), we finish the proof of Step 3. Because of $z \leq w$ we have $\frac{z-1}{w}<1$ and we can choose an integer $t$ sufficiently large to fulfill the condition

$$
\left(\frac{z-1}{w}\right)^{2} t+\left(\frac{z-1}{w}+1\right) M<t .
$$

Applying (2) twice we get

$$
f(f(t)) \leq \frac{z-1}{w} f(t)+M \leq \frac{z-1}{w}\left(\frac{z-1}{w} t+M\right)+M<t
$$

in contradiction to Step 2, which proves Step 3.

Final step. Thus, following Step 2 and Step 3, we obtain

$$
z=f(f(z)) \leq f(z) \leq z
$$

and $f(z)=z$ for all positive integers $z$ is proven.

Final answer: $f(z)=z$