Σάββατο, 9 Ιουνίου 2012

▪ Spain Mathematical Olympiad 2012

Ημέρα
1. Determine if the number
   
is irrational for all non-negative integers
2. Find all functions such that
for all
3. Let and be integers such that . We have separate boxes and identical balls. Define as the number of ways that the balls can be distributed into the boxes. Let be a prime number. Find the integers greater than such that the prime number is a divisor of for all
Ημέρα
1. Find all positive integers and such that
 
2. A sequence of integers is defined by the recurrence
Prove that all terms of the sequence are integers and find an explicit formula for
3. Let be an acute-angled triangle. Let be the inscribed circle with centre , be the circumscribed circle with centre and be the midpoint of the altitude where lies on . The circle be tangent to the side at the point . The line cuts at a second point and the perpendicular from to cuts at . The lines and are tangent to the circle at and respectively. Prove that the points and lie on the same circle.
Η λύση του 3ου προβλήματος (2η ημέρα): Φραγκάκης Νίκος

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