MathDB

3

Part of 2006 Vietnam Team Selection Test

Problems(2)

Problem 3 vietnamese tst 2006

Source: Vietnamese TST 2006

4/17/2006
In the space are given 20062006 distinct points, such that no 44 of them are coplanar. One draws a segment between each pair of points. A natural number mm is called good if one can put on each of these segments a positive integer not larger than mm, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on. Find the minimum value of a good number mm.
inductioncombinatorics proposedcombinatorics
Problem 6 vietnamese tst 2006

Source: Vietnamese TST 2006

4/18/2006
The real sequence {an∣n=0,1,2,3,...}\{a_n|n=0,1,2,3,...\} defined a0=1a_0=1 and an+1=12(an+13⋅an). a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ). Denote An=33⋅an2−1. A_n=\frac{3}{3 \cdot a_n^2-1}. Prove that AnA_n is a perfect square and it has at least nn distinct prime divisors.
inductionnumber theory proposednumber theory