MathDB

3

Part of 1995 Vietnam National Olympiad

Problems(2)

similar triangles

Source: Vietnam NMO 1995, Problem 3

9/7/2008
Let a non-equilateral triangle ABC ABC and AD,BE,CF AD,BE,CF are its altitudes. On the rays AD,BE,CF, AD,BE,CF, respectively, let A′,B′,C′ A',B',C' such that \frac {AA'}{AD} \equal{} \frac {BB'}{BE} \equal{} \frac {CC'}{CF} \equal{} k. Find all values of k k such that △A′B′C′∼△ABC \triangle A'B'C'\sim\triangle ABC for any non-triangle ABC. ABC.
geometry proposedgeometry
Coloring regular 2n-polygon

Source: Vietnam MO 1995

2/20/2004
Given an integer n≥2 n\ge 2 and a reular 2n-gon. Color all verices of the 2n-gon with n colors such that: (i) Each vertice is colored by exactly one color. (ii) Two vertices don't have the same color. Two ways of coloring, satisfying the conditions above, are called equilavent if one obtained from the other by a rotation whose center is the center of polygon. Find the total number of mutually non-equivalent ways of coloring. Alternative statement: In how many ways we can color vertices of an regular 2n-polygon using n different colors such that two adjent vertices are colored by different colors. Two colorings which can be received from each other by rotation are considered as the same.
geometrygeometric transformationrotationinvariantcombinatorics unsolvedcombinatorics