03 Dec Regional Mathematical Olympiad – 2013
Time: 3 hours
December 01, 2013
· Calculators (in any form) and protractors are not allowed.
· Rulers and compasses are allowed.
· Answer all the questions.
· All questions carry equal marks. Maximum marks: 102.
· Answer to each question should start on a new page. Clearly indicate the question number.
- Let be an acute-angled triangle. The circle with as diameter intersects and again at and respectively. Determine given that the orthocenter of triangle lies on
- Let and , where are integers with Suppose that the following conditions holds:
(b) the roots of are the square of the roots of
Find the value of
- Find all primes and such that divides and divides
- Find the number of 10-tuples of integers such that and
- Let be a triangle with and Let and be points on the segment such that Prove that
- Suppose that and are integers such that both the quadratic equations and have integer roots. Prove that is divisible by 6.