1999 Mathematical Olympiad of Iran

First Day
May 5th, 1999
Time : 4 hours

1. Does there exist a positive integer which is a power of 2, such that we can obtain another power of 2 by rearranging it's digits?

2. Consider the triangle ABC with the angles B and C each larger than 45 degrees. We build the right-angled and isosceles triangles CAM and BAN outside the triangle ABC such that the right angles are <CAM and <BAN and BPC inside ABC such that it's right angle is <BPC . Prove that the triangle MNP is also right-angled and isosceles.

3. We have a 100 x 100 lattice with a tree on each of the 10000 points. (The points are equally spaced.) Find the maximum number of trees we can cut such that if we stand on any cut tree, we see no trees which have been cut. (In other words, on the line connecting any two trees that have been cut, there should be at least one tree which hasn't been cut.)

 Second Day
May 6th, 1999
Time : 4 hours

4. Find all natural numbers m such that :

m = 1/a1 +2/a2 + 3/a3 + ... + 1378/a1378
where a1 , ... , a1378 are natural numbers.

5. Consider the triangle ABC . P , Q and R are points on the sides AB , AC and BC respectively. A' , B' and C' are points on the lines PQ , PR and QR respectively such that ABllA'B' , ACllA'C' and BCllB'C' . Prove that  AB/A'B' = SABC/SPQR     (SABC means the area of the figure ABC)

6. A1 , A2 , ... , An are n distinct points on the plane. We color the middle of each line AiAj (i & j are not equal.) red. Find the minimum number of red points.