MathDB

1

Part of 2021 Saint Petersburg Mathematical Olympiad

Problems(3)

Splitting [p] into blocks with equal sum

Source: St Petersburg 2021 11.1

12/23/2021
Let pp be a prime number. All natural numbers from 11 to pp are written in a row in ascending order. Find all pp such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum.
A. Khrabov
number theoryprime numbers
Solve trigonometric system of equations

Source: St Petersburg 2021 10.1

12/23/2021
Solve the following system of equations sin⁡2x+cos⁡2y=y4.\sin^2{x} + \cos^2{y} = y^4. sin⁡2y+cos⁡2x=x2.\sin^2{y} + \cos^2{x} = x^2.
A. Khrabov
algebrasystem of equations
Kostya marks points on a circle

Source: St Petersburg 2021 9.1

12/24/2021
There are 20212021 points on a circle. Kostya marks a point, then marks the adjacent point to the right, then he marks the point two to its right, then three to the next point's right, and so on. Which move will be the first time a point is marked twice?
K. Kokhas
combinatorics