1998 Mathematical Olympiad of Iran

First Day
April 29th, 1998
Time : 4 hours

1. a1 ,a2 , ... ,an are n real numbers such that a1 < a2 < ... < an . Prove that :

a1a24 + a2a34+ ... + an-1an4 + ana14>= a2a14 + a3a24+ ... + anan-14 + a1an4

2. Consider the triangle ABC and I, it's incenter . D is the other point of intersection of the line AI with the circumcircle of ABC. E and F are the feet of the altitudes drawn from I on BD and CD respectively. If IE + IF = AD/2, find the angle BAC.

3. Consider a natural number n. We say the n-tuple (a1 , a2 , ... , an) of natural numbers is "good" when:
i) a1 + a2 + ... + an = 2n
ii) The sum of no k ai's equals n . (0 < k < n)
For example ( 1 , 1 , 4 ) is "good", but ( 1 , 2 , 1 , 2 , 4 ) isn't "good".
Find all "good" n-tuples .

Second Day
April 30th, 1998
Time : 4 hours

4. We know that the natural number n, has at least 4 distinct positive divisors, and 0 < d1 < d2 < d3< d4 are it's smallest four positive divisors . Find all n's such that :    n = d12 + d22+ d32 + d42

5. In the triangle ABC, we know that BC > CA > AB . D is a point on BC, and E is a point on the extension of AB (near A) such that BD = BE = AC . The circumcircle of BED intersects AC  at P . BP intersects the circumcircle of ABC at Q . Prove that :                                                                   AQ + CQ = BP

6. A = (a1 , a2 , ... , an) and B = (b1, b2 , ... , bn) are two n-tuples of 0 and 1. The distance between A and B is the number of i's such that ai is not equal to bi (0 < i < n+1). We know that A, B and C are three n-tuples of 0 and 1 such that the distance of any two of them is equal to d.
a) Prove that d is an even number.
b) Prove that there exists an n-tuple of 0 and 1 like D, such that it's distance from A, B and C is equal to d/2.