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 INMO

Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC is equilateral.

INMO 2011 Solution 1: Solution 2: Here is a pure geometric solution. Consider the triangle BDF, CED and AFE with BD, CE and AF as bases. The sides DF, ED and FE make equal angles θ with the bases of respective triangles. If B ≥ C ≥ A, then it is easy to see that … Continue reading Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC is equilateral.

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)