International Mathematics Competition
for University Students
2023

IMC2023: Day 1, Problem 1

Problem 1. Find all functions \(\displaystyle f:\) \(\displaystyle \mathbb{R}\to \mathbb{R}\) that have a continuous second derivative and for which the equality \(\displaystyle f(7x+1) = 49f(x)\) holds for all \(\displaystyle x\in\mathbb{R}\).

Alex Avdiushenko, Neapolis University Paphos, Cyprus

Solution. Differentiating the equation twice, we get

\(\displaystyle f''(7x+1) = f''(x) \quad\text{or}\quad f''(x) = f''\left(\frac{x - 1}{7}\right). \)

\(\displaystyle (1) \)

Take an arbitrary \(\displaystyle x\in\mathbb{R}\), and construct a sequence by the recurrence

\(\displaystyle x_0 = x, \quad x_{k+1} = \frac{x_k - 1}{7}. \)

By \(\displaystyle (1)\), the values of \(\displaystyle f''\) at all points of this sequence are equal. The limit of this sequence is \(\displaystyle -\frac{1}{6}\), since \(\displaystyle \left|x_{k+1} + \frac{1}{6}\right| = \frac{1}{7}\left|x_k + \frac{1}{6}\right|\).

Due to the continuity of \(\displaystyle f''\), the values of \(\displaystyle f''\) at all points of this sequence are equal to \(\displaystyle f''\left(-\frac{1}{6}\right)\), which means that \(\displaystyle f''(x)\) is a constant.

Then \(\displaystyle f\) is an at most quadratic polynomial, \(\displaystyle f(x) = ax^2 + bx + c\). Substituting this expression into the original equation, we get a system of equations, from which we find \(\displaystyle a = 36c\), \(\displaystyle b = 12c\), and hence

\(\displaystyle f(x) = c(6x + 1)^2. \)