MathDB

1

Part of 2024 Dutch IMO TST

Problems(3)

Arithmetic mean of divisors and of coprime numbers

Source: Dutch TST 2024, 1.1

6/28/2024
For a positive integer nn, let α(n)\alpha(n) be the arithmetic mean of the divisors of nn, and let β(n)\beta(n) be the arithmetic mean of the numbers k≤nk \le n with gcd(k,n)=1\text{gcd}(k,n)=1. Determine all positive integers nn with α(n)=β(n)\alpha(n)=\beta(n).
number theorynumber theory proposedDivisorscoprime
A well-known geo configuration revisited

Source: Dutch TST 2024, 2.1

6/28/2024
Let ABCABC be a triangle with orthocenter HH and circumcircle Γ\Gamma. Let DD be the reflection of AA in BB and let EE the reflection of AA in CC. Let MM be the midpoint of segment DEDE. Show that the tangent to Γ\Gamma in AA is perpendicular to HMHM.
geometrygeometry proposedcircumcircletangent
Beetles moving on a 2023x2023 board, will their ways cross?

Source: Dutch TST 2024, 3.1

6/28/2024
On a 2023×20232023 \times 2023 board, there are beetles on some of the cells, with at most one beetle per cell. After one minute, each beetle moves a cell to the right or to the left or to the top or to the bottom. After each further minute, the beetles continue to move to adjacent fields, but they always make a 90∘90^\circ turn, i.e. when a beetle just moved to the right or to the left, it now moves to the top or to the bottom, and vice versa. What is the minimal number of beetles on the board such that no matter where they start and how they move (according to the rules), at some point two beetles will end up in the cell?
combinatoricscombinatorics proposed