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.
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