**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/a*_{1} +2/a_{2} + 3/a_{3} + ... + 1378/a_{1378}where *a*_{1} , ... , *a*_{1378
}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
*AB*ll*A'B' *, *AC*ll*A'C'* and
*BC*ll*B'C'* . Prove that *AB/A'B' = S*_{ABC}/S_{PQR}
(*S*_{ABC} means the area of
the figure *ABC*)

6. *A*_{1} , *A*_{2} , ... ,
*A*_{n} are* n *distinct
points on the plane. We color the middle of each line *A*_{i}A_{j }(*i
*& *j* are not equal.) red. Find the minimum number of
red points.