MathDB

2

Part of 2005 Romania Team Selection Test

Problems(5)

dilworth particularization

Source: Romanian IMO TST 2005 - day 1, problem 2

3/31/2005
Let n≥1n\geq 1 be an integer and let XX be a set of n2+1n^2+1 positive integers such that in any subset of XX with n+1n+1 elements there exist two elements x≠yx\neq y such that x∣yx\mid y. Prove that there exists a subset {x1,x2,…,xn+1}∈X\{x_1,x_2,\ldots, x_{n+1} \} \in X such that xi∣xi+1x_i \mid x_{i+1} for all i=1,2,…,ni=1,2,\ldots, n.
pigeonhole principleceiling functionnumber theory proposednumber theory
sum with residues, m is even and n is odd

Source: Romanian IMO TST 2005 - day 2, problem 2

4/1/2005
Let m,nm,n be co-prime integers, such that mm is even and nn is odd. Prove that the following expression does not depend on the values of mm and nn: 12n+∑k=1n−1(−1)[mkn]{mkn}. \frac 1{2n} + \sum^{n-1}_{k=1} (-1)^{\left[ \frac{mk}n \right]} \left\{ \frac {mk}n \right\} . Bogdan Enescu
calculusintegrationnumber theory proposednumber theory
classical computational - triangle inscribed in triangle

Source: Romanian IMO TST 2005 - day 3, problem 2

4/19/2005
Let ABCABC be a triangle, and let DD, EE, FF be 3 points on the sides BCBC, CACA and ABAB respectively, such that the inradii of the triangles AEFAEF, BDFBDF and CDECDE are equal with half of the inradius of the triangle ABCABC. Prove that DD, EE, FF are the midpoints of the sides of the triangle ABCABC.
geometryinradiusconicsinequalitiesperimeterincentergeometry proposed
oriented graph hidden in convex polyhedra

Source: Romanian IMO TST 2005 - day 4, problem 2

4/23/2005
On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out. Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit. Dan Schwartz
Eulercombinatorics proposedcombinatorics
muirhead strikes again

Source: Romanian IMO TST 2005 - day 5, problem 2

4/24/2005
Let n≥2n\geq 2 be an integer. Find the smallest real value ρ(n)\rho (n) such that for any xi>0x_i>0, i=1,2,…,ni=1,2,\ldots,n with x1x2⋯xn=1x_1 x_2 \cdots x_n = 1, the inequality ∑i=1n1xi≤∑i=1nxir \sum_{i=1}^n \frac 1{x_i} \leq \sum_{i=1}^n x_i^r is true for all r≥ρ(n)r\geq \rho (n).
inequalitiesinequalities proposed