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Posted By: arunkumar Member Level: Gold Posted Date: 11 Jun 2008
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2007 Regional Mathematical Olympiad- 2006 Question paper
Regional Mathematical Olympiad- 2006
Maharashtra and Goa Region
17th December 2006
Max. Marks: 100 Time : 4 hours
N.B.(i) There are 8 questions. All questions are compulsory.
(ii) Mathematical reasoning will be taken into
consideration while assessing the answer.
(iii) Figures to the right indicate full marks for the question.
1. Let a, b, c be positive real numbers. Prove that
b2 c2
a2 3
+ + [10]
= .
(a + b)(a + c) (b + c)(b + a) (c + a)(c + b) 4
2. Find all positive integers n such that the number n(2n-1 ) + 1
is a perfect square. [10]
3. In how many ways can 7 X’s be written so that each unit
square contains at most one X and no row is empty in the
following figure? [10]
4. Let P A be a common chord of circles C1 and C2 . Extend P A
to Q such that A is midpoint of P Q. Let the tangent to the
circle C1 drawn at P intersect C2 at R and the tangent to the
circle C2 drawn at P intersect C1 at S. Show that P, Q, R, S
are concyclic. [12]
5. Let ABC be a triangle with ?B as an obtuse angle and
?A < 60? . Let P be a point on the side AB such that
?CP B = 60? . Let D be the point on CP which also lies
on the internal angle bisector of ?A. If ?CBD = 30? , prove
that CP trisects ?ACB. [13]
1
6. A person starts from the origin O(0, 0) in the X-Y plane. He
takes steps of one unit along the X-axis (positive as well as
negative direction)or the Y -axis (positive as well as negative
direction). Traveling in this manner, find the total number of
ways he can reach A(4, 3) by using exactly 11 steps? [15]
7. Find all the real numbers x, y, z such that
1 1
1 x y z
= + 1, = + 1, = + 1. [15]
xy z yz x zx y
8. Find all the positive integers (x, y, z) such that
xyz = 5(x + y + z).
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