MathDB

2

Part of 2008 Iran MO (3rd Round)

Problems(6)

Find the largest K

Source: Iranian National Olympiad (3rd Round) 2008

8/30/2008
Find the smallest real K K such that for each x,y,z\in\mathbb R^{ \plus{} }: x\sqrt y \plus{} y\sqrt z \plus{} z\sqrt x\leq K\sqrt {(x \plus{} y)(y \plus{} z)(z \plus{} x)}
functioninequalitiesinequalities proposed
13|p^3+1

Source: Iranian National Olympiad (3rd Round) 2008

8/30/2008
Prove that there exists infinitely many primes p p such that: 13|p^3\plus{}1
arithmetic sequencenumber theory proposednumber theory
Number of partitions

Source: Iranian National Olympiad (3rd Round) 2008

8/31/2008
Prove that the number permutations α \alpha of {1,2,…,n} \{1,2,\dots,n\} s.t. there does not exist i<j<n i<j<n s.t. \alpha(i)<\alpha(j\plus{}1)<\alpha(j) is equal to the number of partitions of that set.
combinatorics proposedcombinatorics
Two continuous functions

Source: Iranian National Olympiad (3rd Round) 2008

9/12/2008
Let g,f:C⟶C g,f: \mathbb C\longrightarrow\mathbb C be two continuous functions such that for each z≠0 z\neq 0, g(z)\equal{}f(\frac1z). Prove that there is a z∈C z\in\mathbb C such that f(\frac1z)\equal{}f(\minus{}\bar z)
functioncomplex analysiscomplex analysis unsolved
Parallel to Euler line

Source: Iranian National Olympiad (3rd Round) 2008

9/12/2008
Let la,lb,lc l_a,l_b,l_c be three parallel lines passing through A,B,C A,B,C respectively. Let la′ l_a' be reflection of la l_a into BC BC. lb′ l_b' and lc′ l_c' are defined similarly. Prove that la′,lb′,lc′ l_a',l_b',l_c' are concurrent if and only if la l_a is parallel to Euler line of triangle ABC ABC.
Eulergeometrygeometric transformationreflectiongeometry proposed
Triangles can not be separated

Source: Iranian National Olympiad (3rd Round) 2008

9/20/2008
Consider six arbitrary points in space. Every two points are joined by a segment. Prove that there are two triangles that can not be separated. http://i38.tinypic.com/35n615y.png
geometrygeometry proposed