MathDB

2

Part of 2007 Iran MO (3rd Round)

Problems(4)

a,b,c different positive real numbers

Source: Iranian National Olympiad (3rd Round) 2007

8/27/2007
a,b,c a,b,c are three different positive real numbers. Prove that: \left|\frac{a\plus{}b}{a\minus{}b}\plus{}\frac{b\plus{}c}{b\minus{}c}\plus{}\frac{c\plus{}a}{c\minus{}a}\right|>1
inequalitiesLaTeXinequalities proposed
Isosceles Triangle

Source: Iranian National Olympiad (3rd Round) 2007

8/27/2007
a) Let ABC ABC be a triangle, and O O be its circumcenter. BO BO and CO CO intersect with AC,AB AC,AB at B′,C′ B',C'. B′C′ B'C' intersects the circumcircle at two points P,Q P,Q. Prove that AP\equal{}AQ if and only if ABC ABC is isosceles. b) Prove the same statement if O O is replaced by I I, the incenter.
geometrycircumcirclegeometry proposed
Reduced residue system

Source: Iranian National Olympiad (3rd Round) 2007

8/29/2007
Let m,n m,n be two integers such that \varphi(m) \equal{}\varphi(n) \equal{} c. Prove that there exist natural numbers b1,b2,…,bc b_{1},b_{2},\dots,b_{c} such that {b1,b2,…,bc} \{b_{1},b_{2},\dots,b_{c}\} is a reduced residue system with both m m and n n.
graph theorynumber theory proposednumber theory
Degree mapping

Source: Iranian National Olympiad (3rd Round) 2007

9/10/2007
We call the mapping Δ:Z\{0}⟶N \Delta:\mathbb Z\backslash\{0\}\longrightarrow\mathbb N, a degree mapping if and only if for each a,b∈Z a,b\in\mathbb Z such that b≠0 b\neq0 and b∤a b\not|a there exist integers r,s r,s such that a \equal{} br\plus{}s, and Δ(s)<Δ(b) \Delta(s) <\Delta(b). a) Prove that the following mapping is a degree mapping: \delta(n)\equal{}\mbox{Number of digits in the binary representation of }n b) Prove that there exist a degree mapping Δ0 \Delta_{0} such that for each degree mapping Δ \Delta and for each n≠0 n\neq0, Δ0(n)≤Δ(n) \Delta_{0}(n)\leq\Delta(n). c) Prove that \delta \equal{}\Delta_{0} http://i16.tinypic.com/4qntmd0.png
floor functionnumber theory proposednumber theory