MathDB

3

Part of 2020 Dutch IMO TST

Problems(3)

1x1,1x2, ..., 1xn tiles in a nxn board, red n (n + 1)/2 cells

Source: 2020 Dutch IMO TST 1.3

11/21/2020
For a positive integer nn, we consider an n×nn \times n board and tiles with dimensions 1×1,1×2,...,1×n1 \times 1, 1 \times 2, ..., 1 \times n. In how many ways exactly can 12n(n+1)\frac12 n (n + 1) cells of the board are colored red, so that the red squares can all be covered by placing the nn tiles all horizontally, but also by placing all nn tiles vertically?
Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.
combinatoricsTilingtiles
a + b = \phi (a) + \phi (b) + gcd (a, b)

Source: 2020 Dutch IMO TST 2.3

11/22/2020
Find all pairs (a,b)(a, b) of positive integers for which a+b=ϕ(a)+ϕ(b)+gcd(a,b)a + b = \phi (a) + \phi (b) + gcd (a, b).
Here ϕ(n) \phi (n) is the number of numbers kk from {1,2,...,n}\{1, 2,. . . , n\} with gcd(n,k)=1gcd (n, k) = 1.
number theoryrelatively prime
f( - f (x) - f (y))= 1 -x - y , in Zxz

Source: 2020 Dutch IMO TST 3.3

11/22/2020
Find all functions f:Z→Zf: Z \to Z that satisfy f(−f(x)−f(y))=1−x−yf(-f (x) - f (y))= 1 -x - y for all x,y∈Zx, y \in Z
functional equationfunctional equation in Zfunctionalalgebra