International Mathematics Competition
for University Students
2026

IMC2026: Problems on Day 2

Problem 6.

(a) Is there a differentiable function \(\displaystyle f\colon \mathbb{R} \to \mathbb{R}\) such that f'(f(x)) = x for every \(\displaystyle x \in \mathbb{R}?\)

(b) Is there a differentiable function \(\displaystyle f\colon \mathbb{R} \to \mathbb{R}\) such that f'(f(x)) = |x| for every \(\displaystyle x \in \mathbb{R}?\)

Nikolaos Kolliopoulos, University of Cyprus

  Solution  

Problem 7. For a continuous function \(\displaystyle f\colon[0,1]\to\RR\), let \(\displaystyle S(f)\) be the union of all straight segments in the plane joining points \(\displaystyle (x,0)\) and \(\displaystyle (f(x),1)\), where \(\displaystyle x\in [0,1]\). Let \(\displaystyle A(f)\) be the area of \(\displaystyle S(f)\). Find the infimum of \(\displaystyle A(f)\) over all continuous \(\displaystyle f\).

David Preiss, University of Warwick, UK

  Solution  

Problem 8. Let \(\displaystyle n\geq 5\), and suppose that \(\displaystyle A = (a_{ij})\) is a real symmetric \(\displaystyle n\times n\) matrix such that

\(\displaystyle a_{ii} = 0 \quad\text{and}\quad a_{ij}\in\{-1,1\} \text{ for }i\neq j . \)

Assume that the scalar products of any two distinct rows of \(\displaystyle A\) have the same value. Let \(\displaystyle \lambda_1,\ldots,\lambda_n\) be the eigenvalues of \(\displaystyle A\). Prove that

\(\displaystyle \sum_{i=1}^n\lvert\lambda_i\rvert \geq 2n-2 \)

and determine all matrices for which equality holds.

Slobodan Filipovski, University of Primorska, Koper

  Solution  

Problem 9. Let \(\displaystyle a_1,a_2,\ldots\) be an infinite sequence of positive real numbers satisfying

\(\displaystyle a_1 + a_2 + \cdots + a_{2n-1} = a_n^2 \)

for all positive integers \(\displaystyle n\). Prove that \(\displaystyle a_n \geq 2n-1\) for all positive integers \(\displaystyle n\).

Ilya I. Bogdanov, MIPT, Moscow and Aleksandr Kuznetsov, SPbU, Saint Petersburg

  Solution  

Problem 10. An infinite chessboard of size \(\displaystyle d>0\) is obtained by colouring the interiors of the squares of an infinite square grid of side length \(\displaystyle d\) alternately white and black following the usual chessboard pattern. The points belonging to the grid lines have neither colour and the grid may be translated and rotated arbitrarily in the plane.

Is it true that for any finite set of points \(\displaystyle p_1,\ldots,p_n\) in the plane, there exist \(\displaystyle d\in(0,1)\) and an infinite chessboard of size \(\displaystyle d\) such that all the points \(\displaystyle p_i\) lie in white squares?

David Hruška, Czech Academy of Sciences, Prague

  Solution