## 03 Dec Regional Mathematical Olympiad – 2013

Time: 3 hours

December 01, 2013

Instructions:

· 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:

(a)

(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.