Editor
We announce contests in mathematics, physics and informatics, altogether in 20 categories with different difficulties. Each contest lasts for 9 months, from September 2025 until the beginning of June 2026; new problems are posted in every month from September to May, and solutions can be submitted before early next month.
1. Fix a positive integer \(\displaystyle n\) and write the numbers \(\displaystyle 0, 1, \ldots, n-1\) on a whiteboard in some order. Two numbers form an inversion if the greater precedes the smaller. A number \(\displaystyle k\) is called special if \(\displaystyle k\) forms inversions with exactly \(\displaystyle k\) other numbers. At most how many special numbers can we have on the board?
