MathDB

2

Part of 2011 Iran MO (2nd Round)

Problems(2)

angle less than 30 degree

Source: Iran second round 2011

4/28/2011
In triangle ABCABC, we have ∠ABC=60\angle ABC=60. The line through BB perpendicular to side ABAB intersects angle bisector of ∠BAC\angle BAC in DD and the line through CC perpendicular BCBC intersects angle bisector of ∠ABC\angle ABC in EE. prove that ∠BED≤30\angle BED\le 30.
linear algebramatrixtrigonometrygeometryangle bisectorgeometry proposed
maximum days rainbow can live

Source: Iran second round 2011

4/29/2011
rainbow is the name of a bird. this bird has nn colors and it's colors in two consecutive days are not equal. there doesn't exist 44 days in this bird's life like i,j,k,li,j,k,l such that i<j<k<li<j<k<l and the bird has the same color in days ii and kk and the same color in days jj and ll different from the colors it has in days ii and kk. what is the maximum number of days rainbow can live in terms of nn?
inductioncombinatorics proposedcombinatorics