International Mathematics Competition
for University Students
2026

IMC2026: Problems on Day 1

Problem 1. Show that the equation \(\displaystyle \cos(\cos x) = \sin(\sin x)\) has no real solutions.

Alexander Slávik, Charles University, Prague

  Solution  

Problem 2. Let \(\displaystyle n\) be a positive integer. Suppose that \(\displaystyle A\) and \(\displaystyle B\) are \(\displaystyle n\times n\) matrices with real entries such that

\(\displaystyle A^\top A + BB^\top = AB + BA,\)

where \(\displaystyle X^\top\) denotes the transpose of matrix \(\displaystyle X\).

Does this imply that \(\displaystyle AB = BA\)?

Nikolaos Kolliopoulos, University of Cyprus

  Solution  

Problem 3. Consider a deck of \(\displaystyle n\geq 2\) cards labeled \(\displaystyle 1,2,\ldots,n\). An alternating shuffle of the deck is performed as follows. We split the deck into two non-empty stacks. We then sort the first stack in increasing order, and the second stack in decreasing order. Finally, we alternately take cards from the first and second stacks (starting with the first). If one of the stacks runs out, the remaining cards from the other stack are placed at the end. How many different final orders of the deck can be obtained in this way?
[0.5cm] Example: Suppose \(\displaystyle n=6\), the first stack is \(\displaystyle A=(1,\,3)\), and the second stack is \(\displaystyle B=(6,\,5,\,4,\,2)\). Then the order resulting from the alternating shuffle is \(\displaystyle (1,\,6,\,3,\,5,\,4,\,2)\).

Daniel Volostnov, Neapolis University Paphos, Cyprus

  Solution  

Problem 4. Let \(\displaystyle x_{1} > 0\). Define the sequence \(\displaystyle \{x_{n}\}\) by the recurrence

\(\displaystyle x_{n+1}=\arctan\left(\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right) \text{ for all }n\geq 1. \)

Find \(\displaystyle \lim\limits_{n \to \infty} x_{n} \sqrt{\ln n}\), where \(\displaystyle \ln x\) denotes the natural logarithm of \(\displaystyle x\).

Wanlong Han, Henan, China

  Solution  

Problem 5. Prove that there exists a constant \(\displaystyle C>0\) such that for every pair \(\displaystyle A,B\) of positive integers, there is a real polynomial \(\displaystyle p(x)\) with

\(\displaystyle p(0)^2 > \sum_{i=1}^A p(-i)^2 + \sum_{i=1}^B p(i)^2 \quad\text{and}\quad \deg p < C\sqrt{AB}. \)

Géza Kós, Loránd Eötvös University, Budapest

  Solution