MathDB

4

Part of 2007 Iran MO (3rd Round)

Problems(4)

Convergent sequence

Source: Iranian National Olympiad (3rd Round) 2007

8/27/2007
a) Let n1,n2,… n_{1},n_{2},\dots be a sequence of natural number such that ni≥2 n_{i}\geq2 and ϵ1,ϵ2,… \epsilon_{1},\epsilon_{2},\dots be a sequence such that ϵi∈{1,2} \epsilon_{i}\in\{1,2\}. Prove that the sequence: \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots\plus{}\sqrt[n_{k}]{\epsilon_{k}}}}is convergent and its limit is in (1,2] (1,2]. Define \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}} to be this limit. b) Prove that for each x∈(1,2] x\in(1,2] there exist sequences n1,n2,⋯∈N n_{1},n_{2},\dots\in\mathbb N and ni≥2 n_{i}\geq2 and ϵ1,ϵ2,… \epsilon_{1},\epsilon_{2},\dots, such that ni≥2 n_{i}\geq2 and ϵi∈{1,2} \epsilon_{i}\in\{1,2\}, and x\equal{}\sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}
inductionlimitreal analysisreal analysis unsolved
Three pairwise tangent cicrcles

Source: Iranian National Olympiad (3rd Round) 2007

8/27/2007
Let ABC ABC be a triangle, and D D be a point where incircle touches side BC BC. M M is midpoint of BC BC, and K K is a point on BC BC such that AK⊥BC AK\perp BC. Let D′ D' be a point on BC BC such that D′MD′K=DMDK \frac{D'M}{D'K}=\frac{DM}{DK}. Define ωa \omega_{a} to be circle with diameter DD′ DD'. We define ωB,ωC \omega_{B},\omega_{C} similarly. Prove that every two of these circles are tangent.
geometrygeometric transformationrotationradical axisgeometry proposed
Find Solutions

Source: Iranian National Olympiad (3rd Round) 2007

8/29/2007
Find all integer solutions of x^{4}\plus{}y^{2}\equal{}z^{4}
number theory proposednumber theory
Triangular Lattice

Source: Iranian National Olympiad (3rd Round) 2007

9/10/2007
In the following triangular lattice distance of two vertices is length of the shortest path between them. Let A1,A2,…,An A_{1},A_{2},\dots,A_{n} be constant vertices of the lattice. We want to find a vertex in the lattice whose sum of distances from vertices is minimum. We start from an arbitrary vertex. At each step we check all six neighbors and if sum of distances from vertices of one of the neighbors is less than sum of distances from vertices at the moment we go to that neighbor. If we have more than one choice we choose arbitrarily, as seen in the attached picture. Obviusly the algorithm finishes a) Prove that when we can not make any move we have reached to the problem's answer. b) Does this algorithm reach to answer for each connected graph?
algorithmgeometry proposedgeometry