Question 67
Given a $\triangle ABC$ and a function $f : \mathbb{R}^+ \to \mathbb{R}$ with the following property: for any segment $DE$ of the interior of the triangle and its midpoint $M$ one has that
$$
f(d(D)) + f(d(E)) \le 2f(d(M)).
$$
where $d(X)$ denotes the distance from $X$ to the boundary of $\triangle ABC$. Prove that for any segment $PQ$ of the interior of $\triangle ABC$ and any point $N$ on this segment we have
$$
|QN| \cdot f(d(P)) + |PN| \cdot f(d(Q)) \le |PQ| \cdot f(d(N)).
$$
$$
f(d(D)) + f(d(E)) \le 2f(d(M)).
$$
where $d(X)$ denotes the distance from $X$ to the boundary of $\triangle ABC$. Prove that for any segment $PQ$ of the interior of $\triangle ABC$ and any point $N$ on this segment we have
$$
|QN| \cdot f(d(P)) + |PN| \cdot f(d(Q)) \le |PQ| \cdot f(d(N)).
$$