MathDB

4

Part of 2006 Iran MO (3rd Round)

Problems(6)

a,b,c,t

Source: Iranian National Olympiad (3rd Round) 2006

8/26/2006
a,b,c,ta,b,c,t are antural numbers and k=ctk=c^{t} and n=ak−bkn=a^{k}-b^{k}. a) Prove that if kk has at least qq different prime divisors, then nn has at least qtqt different prime divisors. b)Prove that φ(n)\varphi(n) id divisible by 2t22^{\frac{t}{2}}
number theory proposednumber theory
p(x)\geq0 [Similiar to Shortlist]

Source: Iranian National Olympiad (3rd Round) 2006

9/19/2006
p(x)p(x) is a real polynomial that for each x≥0x\geq 0, p(x)≥0p(x)\geq 0. Prove that there are real polynomials A(x),B(x)A(x),B(x) that p(x)=A(x)2+xB(x)2p(x)=A(x)^{2}+xB(x)^{2}
algebrapolynomialalgebra proposed
Bijection

Source: Iranian National Olympiad (3rd Round) 2006

9/21/2006
f:Rn⟶Rnf: \mathbb R^{n}\longrightarrow\mathbb R^{n} is a bijective map, that Image of every n−1n-1-dimensional affine space is a n−1n-1-dimensional affine space. 1) Prove that Image of every line is a line. 2) Prove that ff is an affine map. (i.e. f=gohf=goh that gg is a translation and hh is a linear map.)
geometrygeometric transformationvectorparallelograminductionlinear algebralinear algebra unsolved
Erdos-Ko-Rado generalization

Source: Iranian National Olympiad (3rd Round) 2006

9/11/2006
Let DD be a family of ss-element subsets of {1.…,n}\{1.\ldots,n\} such that every kk members of DD have non-empty intersection. Denote by D(n,s,k)D(n,s,k) the maximum cardinality of such a family. a) Find D(n,s,4)D(n,s,4). b) Find D(n,s,3)D(n,s,3).
combinatorics proposedcombinatorics
Tiling of space

Source: Iranian National Math Olympiad (Final exam) 2006

9/14/2006
The image shown below is a cross with length 2. If length of a cross of length kk it is called a kk-cross. (Each kk-cross ahs 6k+16k+1 squares.) http://aycu08.webshots.com/image/4127/2003057947601864020_th.jpg a) Prove that space can be tiled with 11-crosses. b) Prove that space can be tiled with 22-crosses. c) Prove that for k≥5k\geq5 space can not be tiled with kk-crosses.
analytic geometryalgorithmpigeonhole principlegeometrycombinatorics proposedcombinatorics
Locus

Source: Iranian National Olympiad (3rd Round) 2006

9/21/2006
Circle Ω(O,R)\Omega(O,R) and its chord ABAB is given. Suppose CC is midpoint of arc ABAB. XX is an arbitrary point on the cirlce. Perpendicular from BB to CXCX intersects circle again in DD. Perpendicular from CC to DXDX intersects circle again in EE. We draw three lines ℓ1,ℓ2,ℓ3\ell_{1},\ell_{2},\ell_{3} from A,B,EA,B,E parralell to OX,OD,OCOX,OD,OC. Prove that these lines are concurrent and find locus of concurrncy point.
geometrygeometry proposed