**1998
Mathematical Olympiad of Iran**
**First Day**

**April 29th,
1998**

**Time : 4 hours**

1. *a*_{1} ,a_{2} , ... ,a_{n} are *n* real numbers such that
*a*_{1} < a_{2}
< ... < a_{n} . Prove that :

*a*_{1}a_{2}^{4} +
a_{2}a_{3}^{4}+ ... + a_{n-1}a_{n}^{4} +
a_{n}a_{1}^{4}>= a_{2}a_{1}^{4} +
a_{3}a_{2}^{4}+ ... + a_{n}a_{n-1}^{4} +
a_{1}a_{n}^{4}
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 *(a*_{1
}, a_{2} , ... , a_{n}) of natural numbers is "*good*"*
*when:

i) *a*_{1} + a_{2} + ... + 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 .

**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 <
d*_{1} < d_{2} <
d_{3}< d_{4} are it's smallest four positive divisors . Find
all *n*'s such that : *n = d*_{1}^{2} + d_{2}^{2}+ d_{3}^{2} + d_{4}^{2}

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 = (a*_{1} , a_{2} , ... , a_{n}) and *B = (b*_{1}, b_{2} , ... , 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*.