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Pre - rmo Mumbai region

Hello people! Today I gave my pre rmo in mumbai. The paper was pretty easy and I am confident of 14-15 right answers. But I have a few doubts and would love if anyone could solve them:

1) In rectangle ABCD, AB=8 and BC=20. Let P be a point on AD such that angle BPC=90. If r1, r2, r3 are the radii of incircles of triangles APB, BPC and CPD, what is the value of r1 + r2 + r3?

2) Let a,b and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b - c = 7. What is the value of a^2 - b^2 + c^2?

3) The circle C1 touches the circle C2 internally at P. The centre O of C2 is outside C1. Let XY be a diameter of C2 which is also tangent to C1. Assume PY>PX. Let PY intersect C1 at Z. If YZ = 2PZ, what is the magnitude of angle PYX in degrees?

Thank You!

Note by Aaryaman Gupta
1 year ago

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