# 2015 USAJMO Problems/Problem 4

### Problem

Find all functions such thatfor all rational numbers that form an arithmetic progression. ( is the set of all rational numbers.)

### Solution 1

According to the given, , where x and a are rational. Likewise . Hence , namely . Let , then consider , where .

, . Easily, by induction, for all integers . Therefore, for nonzero integer m, , namely Hence . Let , we obtain , where is the slope of the linear functions, and .

### Solution 2

We have and Subtracting these two and rearranging gives and since we get from which we get Then we have . Setting , we let to get . This is Cauchy's functional equation, so it has solutions at , so the answer is .