Ответ. 4049.

For a natural number $n$, define $g_n(x) = f(x+n) - f(x)$. Then the number of chords of length $n$ equals the number of zeros of the function $g_n(x)$.

As an example, consider the following piecewise linear function $f$: $f(x) = x$ for $x \le 2024\frac{9}{10}$ and $f(x) = 20249 \cdot |2025 - x|$ for $x \ge 2024\frac{9}{10}$. Note that for $a \notin [0; 2025\frac{1}{10}]$, the function $f(x)$ takes the value $f(a)$ only at point $a$. Therefore, if $g_n(x) = 0$, then both points $x$ and $x+n$ must lie in the interval $[0, 2025 \frac{1}{10}]$. In particular, $n \le 2025$, and the zeros of $g_n(x)$ lie in $[0; 2025 \frac{1}{10} - n]$.

For $n = 2025$ and $x \in [0, \frac{1}{10}]$, we have $g_{2025}(x) = 20248x$. Thus, $g_{2025}(x)$ has a unique zero at $x = 0$, meaning the function $f$ has exactly one chord of length $2025$. For natural $n \le 2024$, the function $g_n(x)$ is monotonically decreasing on $[0, 2025 - n]$ and monotonically increasing on $[2025 - n, 2025 \frac{1}{10} - n]$, with $g_n(0) > 0$, $g_n(2025 - n) < 0$, and $g_n(2025 \frac{1}{10} - n) > 0$. Therefore, this function has exactly two zeros, meaning $f$ has two chords of each length $n = 1, 2, \dots, 2024$. In total, it has $4049$ distinct chords.

$g_k(0) = 0$. The differences $d_j = f(r_{j+1}) - f(r_j)$ satisfy: if $r_{j+1} > r_j$ then $d_j = g_k(r_j) \ge 0$; if $r_{j+1} < r_j$ then $d_j = -g_m(r_{j+1})$.
The sum $\sum d_j = 0$ must contain both positive and negative terms, implying $g_m(t) > 0$ for some $t \in (0, k)$. Since $g_k(0) + g_m(k) = g_m(0) + g_k(m) = 0$ and $g_k \ge 0$, we get $g_m(0) \le 0$ and $g_m(k) \le 0$. This leads to $g_m$ having at least $3$ zeros (at $0$, $k$, and between them), plus $g_k$ having at least $1$ zero, totaling $4$ zeros as required. The case when $g_k(m) = 0$ is similar.