MathDB

2

Part of 2023 Iran MO (3rd Round)

Problems(5)

Two random bijections

Source: Iran MO 3rd round 2023 , D1P2

8/16/2023
Does there exist bijections f,gf,g from positive integers to themselves st: g(n)=f(1)+f(2)+⋅⋅⋅+f(n)ng(n)=\frac{f(1)+f(2)+ \cdot \cdot \cdot +f(n)}{n} holds for any nn?
algebra
The main dish of IranNT2023

Source: Iran MO 3rd round 2023 NT exam , P2

8/17/2023
Let NN be the number of ordered pairs (x,y)(x,y) st 1≤x,y≤p(p−1)1 \leq x,y \leq p(p-1) and : xy≡yx≡1(modp)x^{y} \equiv y^{x} \equiv 1 \pmod{p} where pp is a fixed prime number. Show that : (ϕ(p−1)d(p−1))2≤N≤((p−1)d(p−1))2(\phi {(p-1)}d(p-1))^2 \leq N \leq ((p-1)d(p-1))^2 where d(n)d(n) is the number of divisors of nn
number theoryprime numbers
Permutation x divisibility

Source: Iran MO 2023 3rd round , Combinatorics exam P2

8/19/2023
Find the number of permutations of {1,2,...,n}\{1,2,...,n\} like {a1,...,an}\{a_1,...,a_n\} st for each 1≤i≤n1 \leq i \leq n: ai∣2ia_i | 2i
combinatorics
Complex FE

Source: Iran MO 2023 3rd round , Algebra exam P2

8/21/2023
find all f:C→Cf : \mathbb{C} \to \mathbb{C} st: f(f(x)+yf(y))=x+∣y∣2f(f(x)+yf(y))=x+|y|^2 for all x,y∈Cx,y \in \mathbb{C}
functionalgebra
Maybe a P1

Source: Iran MO 2023 3rd round , geometry exam P2

8/23/2023
In triangle △ABC\triangle ABC , MM is the midpoint of arc (BAC)(BAC) and NN is the antipode of AA in (ABC)(ABC). The line through BB perpendicular to AMAM , intersects AM,(ABC)AM , (ABC) at D,PD,P respectively and a line through DD perpendicular to ACAC , intersects BC,ACBC,AC at F,EF,E respectively. Prove that PE,MF,NDPE,MF,ND are concurrent.
geometry