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 ABC be an acute-angled triangle. The circle Gamma with BC as diameter intersects AB and AC again at P and Q, respectively. Determine angle BAC given that the orthocenter of triangle APQ lies on Gamma.

 

2. Let f(x)=x^3+ax^2+bx+c and g(x)=x^3+bx^2+cx+a, where a,b,c are integers with cneq 0. Suppose that the following conditions holds:

(a) f(1)=0;

(b) the roots of g(x)=0 are the square of the roots of f(x)=0.

Find the value of a^{2013}+b^{2013}+c^{2013}.

 

3. Find all primes p and q such that p divides q^2-4 and q divides p^2-1.

 

4. Find the number of 10-tuples (a_1,a_2,dots ,a_{10}) of integers such that |a|leq 1 and

a_1^2+a_2^2+dots +a_{10}^2-a_1a_2-a_2a_3-a_3a_4-dots -a_9a_{10}-a_{10}a_1=2.

 

5. Let ABC be a triangle with angle A=90^{circ} and AB=AC. Let D and E be points on the segment BC such that BD:DE:EC=3:5:4. Prove that angle DAE=45^{circ}.

 

6. Suppose that m and n are integers such that both the quadratic equations x^2+mx-n=0 and x^2-mx+n=0 have integer roots. Prove that n is divisible by 6.

 

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