## 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.

1. 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

2. 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

3. Find all primes and such that divides and divides

4. Find the number of 10-tuples of integers such that and

5. Let be a triangle with and Let and be points on the segment such that Prove that

6. Suppose that and are integers such that both the quadratic equations and have integer roots. Prove that is divisible by 6.

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