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!

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Comments

formula of inradius of a right triangle is= 1/2(p+b-h)

in △APB, r1=(AP+AB−PB)/2

Similarly, r2=(PB+PC−BC)/2

And, r3=(PD+CD−PC)/2

∴r1+r2+r3=(AP+AB−PB)+(PB+PC−BC)+(PD+CD−PC)/2

∴r1+r2+r3(=AD+AB+CD−BC/)2

Since AD=BC and AB=CD, r1+r2+r3=AB=8

Pranay Kr - 3 years, 10 months ago

This is an AMC 10 problem. I'll quickly go through the solution here.

Rearrange the equations to become a+8c=4+7b and 8a−c=7−4b

. Now square both equations to get:

a2+16ac+64c2=16+56b+49b2,

and 64a2−16ac+c2=16b2−56b+49.

Adding the two equations, we get 65a2+65c2=65b2+65 . So 65a2−65b2+65c2=65⇒a2−b2+c2=1

This is a pretty decent manipulation problem.

Pranay Kr - 3 years, 10 months ago

good and easy questions

Dibyam Parida - 3 years, 10 months ago

but tricky

Dibyam Parida - 3 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$