MathDB

1

Part of 2006 Iran MO (3rd Round)

Problems(6)

Order

Source: Iranian National Olympiad (3rd Round) 2006

8/26/2006
nn is a natural number. dd is the least natural number that for each aa that gcd(a,n)=1gcd(a,n)=1 we know ad≡1(modn)a^{d}\equiv1\pmod{n}. Prove that there exist a natural number that \mbox{ord}_{n}b=d
modular arithmeticabstract algebranumber theoryleast common multiplefunctiongroup theorynumber theory proposed
x_1,...,x_n

Source: Iranian National Olympiad (3rd Round) 2006

9/19/2006
For positive numbers x1,x2,…,xsx_{1},x_{2},\dots,x_{s}, we know that ∏i=1sxk=1\prod_{i=1}^{s}x_{k}=1. Prove that for each m≥nm\geq n ∑k=1sxkm≥∑k=1sxkn\sum_{k=1}^{s}x_{k}^{m}\geq\sum_{k=1}^{s}x_{k}^{n}
inequalitiesinequalities proposed
Rank

Source: Iranian National Olympiad (3rd Round) 2006

9/21/2006
Suppose that A∈Mn(R)A\in\mathcal M_{n}(\mathbb R) with Rank(A)=k\text{Rank}(A)=k. Prove that AA is sum of kk matrices X1,…,XkX_{1},\dots,X_{k} with Rank(Xi)=1\text{Rank}(X_{i})=1.
linear algebramatrixlinear algebra unsolved
Excirlces

Source: Iranian National Olympiad (3rd Round) 2006

9/21/2006
Prove that in triangle ABCABC, radical center of its excircles lies on line GIGI, which GG is Centroid of triangle ABCABC, and II is the incenter.
geometryincentergeometric transformationhomothetyratioradical axisangle bisector
Choombam

Source: Iranian National Math Olympiad (Final exam) 2006

9/19/2006
A regular polyhedron is a polyhedron that is convex and all of its faces are regular polygons. We call a regular polhedron a "Choombam" iff none of its faces are triangles. a) prove that each choombam can be inscribed in a sphere. b) Prove that faces of each choombam are polygons of at most 3 kinds. (i.e. there is a set {m,n,q}\{m,n,q\} that each face of a choombam is nn-gon or mm-gon or qq-gon.) c) Prove that there is only one choombam that its faces are pentagon and hexagon. (Soccer ball) http://aycu08.webshots.com/image/5367/2001362702285797426_rs.jpg d) For n>3n>3, a prism that its faces are 2 regular nn-gons and nn squares, is a choombam. Prove that except these choombams there are finitely many choombams.
geometry3D geometryprismspheretetrahedronsymmetryperpendicular bisector
Equality in Sperner

Source: Iranian National Olympiad (3rd Round) 2006

9/11/2006
Let AA be a family of subsets of {1,2,…,n}\{1,2,\ldots,n\} such that no member of AA is contained in another. Sperner’s Theorem states that ∣A∣≤(n⌊n2⌋)|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}. Find all the families for which the equality holds.
floor functioninequalitiesceiling functioncombinatorics proposedcombinatorics