MathDB

3

Part of 2020 Iran MO (3rd Round)

Problems(4)

nice problem for Iran

Source: Iranian Third Round 2020 Algebra exam Problem3

11/20/2020
find all kk distinct integers a1,a2,...,aka_1,a_2,...,a_k such that there exists an injective function ff from reals to themselves such that for each positive integer nn we have {fn(x)−x∣x∈R}={a1+n,a2+n,...,ak+n}\{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\}.
functional equationSubsetsalgebra
Where is the center? on M I_a

Source: Iranian Third Round 2020 Geometry exam Problem3

11/18/2020
The circle Ω\Omega with center IAI_A, is the AA-excircle of triangle ABCABC. Which is tangent to AB,ACAB,AC at F,EF,E respectivly. Point DD is the reflection of AA through IABI_AB. Lines DIADI_A and EFEF meet at KK. Prove that ,circumcenter of DKEDKE , midpoint of BCBC and IAI_A are collinear.
geometryexcircle
latin squares turn to each other

Source: Iranian Third Round 2020 Combinatorics exam Problem3

11/18/2020
Consider a latin square of size nn. We are allowed to choose a 1×11 \times 1 square in the table, and add 11 to any number on the same row and column as the chosen square (the original square will be counted aswell) , or we can add −1-1 to all of them instead. Can we with doing finitly many operation , reach any latin square of size n?n?
latin squarescombinatorics
Wierd but nice functional.

Source: Iranian Third Round 2020 Number Theory exam Problem3

11/21/2020
Find all functions ff from positive integers to themselves, such that the followings hold. 1)1).for each positive integer nn we have f(n)<f(n+1)<f(n)+2020f(n)<f(n+1)<f(n)+2020. 2)2).for each positive integer nn we have S(f(n))=f(S(n))S(f(n))=f(S(n)) where S(n)S(n) is the sum of digits of nn in base 1010 representation.
number theoryfunctionsum of digits