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Math Olympiad

Circle in Circle – PRMO 2017 | Problem 27

PRMO 2017 (pre regional math olympiad) has this beautiful geometry problem of circle in a circle. Try with sequential hints.

Understand the problem

Let \( \Omega_1 \) be a circle with center O and let AB be a diameter of \( \Omega_1 \). Let P be a point on the segment OB different from O. Suppose another circle \( \Omega_2 \) with center P lies in the interior of \( \Omega_1 \). Tangents are drawn from A and B to the circle \( \Omega_2 \) intersecting \( \Omega_1 \) again at \(A_1\) and \(B_1\) respectively such that \(A_1 \) and \(B_1\) are on the opposite sides of AB. Given that \(A_1B = 5, AB_1 = 15 \) and \( OP = 10\), find the radius of \( \Omega_1 \).

Source of the problem

Pre Regional Math Olympiad India 2017, Problem 27

Topic

Geometry

Difficulty Level

Medium

Suggested Book

Challenges and Thrills of Pre- College Mathematics.

Start with hints

Do you really need a hint? Try it first!

Draw a diagram carefully.

PRMO 2017 Problem 27

Suppose the point of tangencies are at C and D. Join PC and PD.

Can you find two pairs of similar triangles?

PRMO 2017 Problem 27 Hint 2

\( \Delta APC \sim \Delta AA_1B \)

Why? 

Notice that AC is perpendicular to \( AA_1 \) as the radius is perpendicular to the tangent.

Also \( \angle A \) is common to both triangles. Hence the two triangles are similar (equiangular implies similar). 

Similarly \( \Delta BPD \sim \Delta BAB_1 \). 

Use the ratio of sides to find OA.

 

Suppose OA = R (radius of the big circle).

OC  = r (radius of the small circle).

We already know OP = 10, \( A_1 B = 5, AB_1 = 15\)

PRMO 2017 Problem 27 Hint 2

Since \( \Delta AA_1B \) and \( ACP \) are similar we have \( \frac{AP}{AB} = \frac{PC}{A_1B}\). This implies  \( \frac{R+10}{2R} = \frac{r}{5}\) (1)

Similarlly since \( \Delta BPD \) and \( BAB_1 \) are similar we have \( \frac{BP}{BA} = \frac{PD}{AB_1}\). This implies  \( \frac{R-10}{2R} = \frac{r}{15}\) (2)

Multiply the reciprocal of (2) with (1) to get R = 20.

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By Dr. Ashani Dasgupta

Ph.D. in Mathematics, University of Wisconsin, Milwaukee, United States.

Research Interest: Geometric Group Theory, Relatively Hyperbolic Groups.

Founder, Cheenta

7 replies on “Circle in Circle – PRMO 2017 | Problem 27”

radius of omega1 is 30 units. The problem is good but send me some elementary and difficult number theory, algebra,combinatorics problems and some good study material lecture vidoes or books for PRE- RMO, RMO, INMO and also for ISI and CMI PREPARATION. Send me as early as possible.

Our computations show it is 20. We double checked with HBCSE. Can you check your calculations? If after reviewing if you think your calculations are correct, then do share it with us.

Nicely Illustrated & Hinted in stages.I like it,It is valuable for all concerned with it.I wish a grand success for thiss endeavour.

Triangle APC and Triangle ABA1 are similar. If r is the radius of smaller circle and R is the radius of bigger circle then,we obtain the equation:
(r/5)=(R+10)/2R.
Similarly, Triangle BPD and Triangle BAB1 are also similar. Hence we get another equation as:
(r/15)=(R-10)/2R.
Hence on solving for R we get 2R=10+30, or, R=20… That’s the answer. It’s easy but a good problem. THANKYOU. I would like to request you to provide few problems for CMI and IOMA preparation as ISI is knocking the door!

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