8/18/2019 Regional Mathematical Olympiad - 2012(Question Paper).

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L I M I T E D

( F O R U M F O R I I T - J E E )  

1. Let ABCD be a unit square. Draw a quadrant of circle with  A as centre and B, D as end

points of thearc similarly, draw a quadrant of a circle with B as centre A, C as end points of 

the arc. Inscribe a circle touching the arcs AC and BD both externally and also touching the

side CD. Find the radius of the circle

2. Let a, b, c be positive integers such that a divides b5, b divides c

5and c divides a

5

Prove that abc divides (a + b + c)31

3. Let a and b be positive real numbers such that a + b = 1. Prove that aabb + abba

≤

 1

4. Let X = {1, 2, 3,....., 10}. Find the number of pairs {A, B} such A

⊆

 X. B

⊆

 X. A

≠

 B and

 A 

∩

B = {5, 7, 8}.

5. Let ABC be a triangle. Let D, E be points on the segment BC such that BD = DE = EC. Let F

be the mid- point of AC. Let BF intersect AD in P and AE in Q respectively. Determine the

ratio of the area of the triangle APQ to that of the quadrilateral PDEQ.

6. Find all positive integers n such that 32n + 3n2 + 7 is a perfect square.

Regional Mathematical Olympiad - 2012(Question Paper)

Held on Date: 02-12-2012