MathDB

1

Part of 2020 Iran MO (3rd Round)

Problems(4)

25% disscount

Source: Iranian Third Round 2020 Algebra exam Problem1

11/20/2020
find all functions from the reals to themselves. such that for every real x,yx,y. f(y−f(x))=f(x)−2x+f(f(y))f(y-f(x))=f(x)-2x+f(f(y))
rhombus Problem for the start

Source: Iranian Third Round 2020 Geometry exam Problem1

11/18/2020
Let ABCDABCD be a Rhombus and let ww be it's incircle. Let MM be the midpoint of ABAB the point KK is on ww and inside ABCDABCD such that MKMK is tangent to ww. Prove that CDKMCDKM is cyclic.
geometryrhombus
Theorem which became a problem.

Source: Iranian Third Round 2020 Combinatorics exam Problem1

11/18/2020
1)1). Prove a graph with 2n2n vertices and n+2n+2 edges has an independent set of size nn (there are nn vertices such that no two of them are adjacent ). 2)2).Find the number of graphs with 2n2n vertices and n+3n+3 edges , such that among any nn vertices there is an edge connecting two of them
graph theorycombinatoricsindependent sets
A rough start

Source: Iranian Third Round 2020 Number Theory exam Problem1

11/21/2020
Find all positive integers nn such that the following holds. τ(n)∣2σ(n)−1\tau(n)|2^{\sigma(n)}-1
number theoryDivisibilitytotient function