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.

· 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.