# Polynomial $f(x)$ such that $\dfrac{f(k)-f(m)}{k-m}\in\mathbb{Z}$

I'm dealing with the test of the International Mathematics Competition for University Students, 2011, and I've had a lot of difficulties, so I hope someone could help me to discuss the questions.

I've posted before the questions 2, 3 and 5, these last still open.

The question 4 says:

Let be $$f(x)$$ a polynomial with real coeficients and degree $$n$$. Supose that $$\dfrac{f(k)-f(m)}{k-m}$$ is integer for all the integers $$0\leq k\lt m\leq n$$. Prove that $$a-b$$ divides $$f(a)-f(b)$$ for any couple of distinct integers $$a$$ and $$b$$.

The only substantial thing that I've got is that:

Given any $$k\in\{1,2,3,\dots,n-1\}$$, we have

$$\dfrac{f(k)-f(0)}{k}\in\mathbb{Z}$$.

So,

$$f(k)-f(0)$$ is integer by any $$k\in\{1,2,3,\dots,n-1\}$$.

I also have thinked that derivatives can be some relation, because of the type of fraction...

I thank very much.

Important Edit (October, 04)

I've found a document with these solutions and I'm studying them. These are the documents: http://www.imc-math.org.uk/imc2011/imc2011-day2-solutions.pdf and http://www.imc-math.org.uk/imc2011/imc2011-day1-solutions.pdf.

• Seems like maybe $f(x)$ must have only integer coefficients (except possibly for the constant). – Alex R. Oct 4 '18 at 21:25

Let $$g(x)=\frac{f(x)-f(0)}{x}$$. Then $$g(1),g(2),\dots,g(n-1)\in\mathbb{Z}$$. Moreover, since $$g$$ is a polynomial of degree $$n-1$$, we have $$0=\Delta^n[g](x)=\sum_{k=0}^n\binom{n}{k}(-1)^{n-k}g(x+k)$$ giving $$g(\mathbb{Z})\subseteq\mathbb{Z}$$ inductively. Thus $$g(x)=\sum_{k=0}^{n-1} c_k\binom{x}{k}$$ with $$c_k\in\mathbb{Z}$$, i.e., $$f(x)=f(0)+x\sum_{k=0}^{n-1} c_k\binom{x}{k}=f(0)+\sum_{k=1}^n c'_k\binom{x}{k}.$$ with $$c'_k\in\mathbb{Z}$$.
Moreover, by examining deeper into the conditon $$\dfrac{f(k)-f(m)}{k-m}\in\mathbb{Z}$$ for $$0\leq k and induction on $$k$$ noting \begin{align*} \binom{a}{k}-\binom{b}{k}&=\sum_{j=1}^k\binom{a-b}{j}\binom{b}{k-j}\\ &=(a-b)\sum_{j=1}^k\frac1j\binom{a-b-1}{j-1}\binom{b}{k-j} \end{align*} one can prove $$c'_k$$ is an integer multiple of $$L_k:=\operatorname{lcm}(1,2,\dots,k)$$ for each $$k$$. Hence the result follows.
• Recall one way to define the binoimial coefficient $\binom{n}{k}$ is as the number of $k$-subsets of an $n$-set. The identity is easy obtained by splitting $[a]$ into a $b$-subset and a $a-b$-subset and consider how to choose a $k$-subset. The $\Delta$ is the forward difference operator $\Delta[f](n):=f(n+1)-f(n)$. – user10354138 Oct 12 '18 at 21:14