International Mathematics Competition
for University Students
2025

IMC2025: Problems on Day 2

Problem 6. Let \(\displaystyle f\colon (0,\infty) \to \mathbb{R}\) be a continuously differentiable function, and let \(\displaystyle b>a>0\) be real numbers such that \(\displaystyle f(a)=f(b) = k\). Prove that there exists a point \(\displaystyle \xi\in(a,b)\) such that

\(\displaystyle f(\xi)-\xi f'(\xi)=k . \)

Alberto Cagnetta, Università degli Studi di Udine

  Solution  

Problem 7. Let \(\displaystyle \mathbb{Z}_{>0}\) be the set of positive integers. Find all nonempty subsets \(\displaystyle M \subseteq \mathbb{Z}_{>0}\) satisfying both of the following properties:

(a) if \(\displaystyle x \in M\), then \(\displaystyle 2x\in M\),

(b) if \(\displaystyle x, y\in M\) and \(\displaystyle x+y\) is even, then \(\displaystyle \displaystyle\frac{x+y}{2}\in M\).

Alexandr Bolbot, Novosibirsk State University

  Solution  

Problem 8. For an \(\displaystyle n\times n\) real matrix \(\displaystyle A\in M_n(\RR)\), denote by \(\displaystyle A^{\mathsf{R}}\) its counter-clockwise \(\displaystyle 90^\circ\) rotation. For example,

\(\displaystyle \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}^{\mathsf{R}} = \begin{bmatrix} 3 & 6 & 9 \\ 2 & 5 & 8 \\ 1 & 4 & 7 \end{bmatrix}. \)

Prove that if \(\displaystyle A = A^{\mathsf{R}}\) then for any eigenvalue \(\displaystyle \lambda\) of \(\displaystyle A\), we have \(\displaystyle \operatorname{Re}\lambda = 0\) or \(\displaystyle \operatorname{Im}\lambda = 0\).

Jan Kuś, University of Warwick

  Solution  

Problem 9. Let \(\displaystyle n\) be a positive integer. Consider the following random process which produces a sequence of \(\displaystyle n\) distinct positive integers \(\displaystyle X_1,X_2,\ldots,X_n\).

First, \(\displaystyle X_1\) is chosen randomly with \(\displaystyle \mathbb{P}(X_1=i)=2^{-i}\) for every positive integer \(\displaystyle i\). For \(\displaystyle 1\leq j\leq n-1\), having chosen \(\displaystyle X_1,\ldots,X_j\), arrange the remaining positive integers in increasing order as \(\displaystyle n_1<n_2<\cdots\), and choose \(\displaystyle X_{j+1}\) randomly with \(\displaystyle \mathbb{P}(X_{j+1}=n_i)=2^{-i}\) for every positive integer \(\displaystyle i\).

Let \(\displaystyle Y_n=\max\{X_1,\ldots,X_n\}\). Show that

\(\displaystyle \mathbb{E}[Y_n]=\sum_{i=1}^{n}\frac{2^i}{2^i-1} \)

where \(\displaystyle \mathbb{E}[Y_n]\) is the expected value of \(\displaystyle Y_n\).

Jan Kuś and Jun Yan, University of Warwick

  Solution  

Problem 10. For any positive integer \(\displaystyle N\), let \(\displaystyle S_N\) be the number of pairs of integers \(\displaystyle 1\leq a, b\leq N\) such that the number \(\displaystyle (a^2+a)(b^2+b)\) is a perfect square. Prove that the limit

\(\displaystyle \lim_{N\to\infty} \frac{S_N}{N} \)

exists and find its value.

Besfort Shala, University of Bristol

  Solution