M CUBE: Math-e-Matics by Maheshwari PRMO

Let S be a circle with centre be O. A chord AB, not the diameter, divides S into two regions R1 and R2 such that O belongs to R2. Let S1 be a circle with centre in R1, touching AB at X and S internally. Let S2 be a circle with centre in R2, touching AB at Y, the circle S internally, and passing through the centre of S. The point X lies on the diameter passing through the centre of S2 and angle YXO = 30 degrees. If the radius of S2 is 100 units, find the length of the radius of S1?

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M CUBE: Math-e-Matics by Maheshwari Mathematics Olympiads Info

RMO

RMO or Regional Math Olympiad is the first round of mathematics contest (in India) leading to the prestigious International Mathematics Olympiad. It is held in December (the first Sunday of December). The test is conducted in each of the 19 regions of India. From each region, about 30 students are selected for the next level … Continue reading RMO

M CUBE: Math-e-Matics by Maheshwari RMO

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 sides of the quadrilateral, then find its radius.

RMO 2006 Solution: Let P, Q, R, S be the points of contact of in-circle with the sides AB, BC, CA, DA respectively. Since AD is perpendicular to AB and AB || DC we see that, AP=AS=SD=DR=r, radius of the in-circle. Let BP=BQ=y and CQ=CR=x . Using AB=3CD, we get (r+y)=3(r+x)