International Mathematics Competition
for University Students
2025

IMC2025: Day 1, Problem 4

Problem 4. Let \(\displaystyle a\) be an even positive integer. Find all real numbers \(\displaystyle x\) such that

\(\displaystyle \left\lfloor \sqrt[a]{b^{a} + x} \cdot b^{a - 1} \right\rfloor = b^{a} + \left\lfloor x/a\right\rfloor \tag{1} \)

holds for every positive integer \(\displaystyle b\).

(Here \(\displaystyle \lfloor x\rfloor\) denotes the largest integer that is no greater than \(\displaystyle x\).)

Yagub Aliyev, ADA University, Baku, Azerbaijan

  Solution