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

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 , ... , a_n) of natural numbers is "good" when:
i) a₁ + a₂ + ... + a_n = 2n
ii) The sum of no k a_i'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 .

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 , ... , a_n) and B = (b₁, b₂ , ... , b_n) are two n-tuples of 0 and 1. The distance between A and B is the number of i's such that a_i is not equal to b_i (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.