IMC 1994 Problem 6

Let $f\in C^2[0,N]$ and $|f'(x)|<1$ , $f''(x)>0$ for every $x\in [0, N]$ . Let $0\leq m_0\ <m_1 < \cdots < m_k\leq N$ be integers such that $n_i=f(m_i)$ are also integers for $i=0,1,\ldots, k$ . Denote $b_i=n_i-n_{i-1}$ and $a_i=m_i-m_{i-1}$ for $i=1,2,\ldots, k$ .

a) Prove that $$-1<\frac{b_1}{a_1}<\frac{b_2}{a_2}<\cdots < \frac{b_k}{a_k}<1$$ b) Prove that for every choice of $A>1$ there are no more than $N / A$ indices $j$ such that $a_j>A$ .

c) Prove that $k\leq 3N^{2/3}$ (i.e. there are no more than $3N^{2/3}$ integer points on the curve $y=f(x)$ , $x\in [0,N]$ ).