RMO–2006

1. Let ABC be an acute-angled triangle and let D, E, F be the feet

of perpendiculars from A,B,C respectively to BC,CA,AB. Let the

perpendiculars from F to CB,CA,AD,BE meet them in P, Q,M,N

respectively. Prove that P, Q,M,N are collinear.

2. Find the least possible value of a + b, where a, b are positive integers

such that 11 divides a + 13b and 13 divides a + 11b.

3. If a, b, c are three positive real numbers, prove that

(a² + 1)/(b + c)+(b² + 1)/(c + a)+(c² + 1)/(a + b)>= 3.

4. A 6×6 square is dissected into 9 rectangles by lines parallel to its sides

such that all these rectangles have only integer sides. Prove that there

are always two congruent rectangles.

5. Let ABCD be a quadrilateral in which AB is parallel to CD and

perpendicular to AD; AB = 3CD; and the area of the quadrilateral is

4. If a circle can be drawn touching all the sides of the quadrilateral,

find its radius.

6. Prove that there are infinitely many positive integers n such that n(n+

1) can be expressed as a sum of two positive squares in at least two

different ways. (Here a² + b² and b² + a² are considered as the same

representation.)

7. Let X be the set of all positive integers greater than or equal to 8 and

let f : X-->X be a function such that f(x + y) = f(xy) for all x >= 4,

y>= 4. If f(8) = 9, determine f(9).