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Posted By: arunkumar Member Level: Gold Posted Date: 11 Jun 2008
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2006 Regional Mathematical Olympiad- 2006 Question paper
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Regional Mathematical Olympiad- 2006
Maharashtra and Goa Region
3rd 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 x, y, z be positive real numbers such that xyz = 1. If
1 1 1
+ + = x + y + z, prove that
x y z
1 1
1
+ k + k = xk + y k + z k ,
k
x y z
for every positive integer k. [10]
2. Find all integers m such that m + 3 and m2 + 3m + 3 are
perfect cubes. [10]
3. Each year 8 subjects are taught by 4 teachers in a school.
Every teacher teaches two subjects. At the end of this year
they will meet to decide the course allotment for the next year.
Find the number of ways in which the course distribution can
be done so that each teacher teaches two courses and each
teacher teaches at least one subject different from the subjects
which he taught this year. [10]
4. Let C be a point on the circle with centre O and radius r.
Chord AB of length r is parallel to radius OC. Let the line
AO cut the circle in E and the tangent to the circle at C in
F. If the chord BE cuts OC in L and if AL cuts CF in M,
CF
[10]
find the ratio .
CM
1
5. In the set of complex numbers solve the system of equations
x(x - y)(x - z) = 3,
y(y - x)(y - z) = 3,
z(z - x)(z - y) = 3. [15]
6. An 8 × 8 board is divided into unit squares. Each unit square
is painted red or blue. Find the number of ways of doing this
so that each row and each column has odd number of blue
[15]
squares.
7. Find all natural numbers x, y, z such that
2006 2006 2006
+ +
x+y y+z z+x
is a natural number. [15]
8. Consider circle with centre O and radius OA. Let C be a point
on radius OA. Let P be a variable point on the circle. Join
P and C. Q is a point on the circle such that P and Q are
on the same side of line OA and ?P CO = ?QCA. Find the
locus of the point of intersection of the line P Q and the line
OA. [15]
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