2025 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 22 Easy

A circle of radius r is surrounded by three circles, whose radi are 1, 2, and 3, all externally tangent to the inner circle and externally tangent to each other, as shown in the diagram below.

What is r?

  • A.

    \frac { 1 } { 4 }

  • B.

    \frac { 6 } { 2 3 }

  • C.

    \frac { 3 } { 1 1 }

  • D.

    \frac { 5 } { 1 7 }

  • E.

    \frac {3 } { 10 }

Answer:B

 

Let A, B, C denote the centers of the three circles, with O as the small circle's center. AC = 1 + 3 = 4, \quad AB = 1 + 2 = 3, \quad BC = 2 + 3 = 5

Since AC^2 + AB^2 = BC^2, we have \angle BAC = 90°.

Establish coordinates: A(0,0), B(3,0), C(0,4).

Let O = (x, y). Since AO = 1 + r, BO = 2 + r, CO = 3 + r: - Equation (1): x^2 + y^2 = (1+r)^2 - Equation (2): (x-3)^2 + y^2 = (2+r)^2 - Equation (3): x^2 + (y-4)^2 = (3+r)^2

From equations (1) and (2): x = 1 - \frac{r}{3}.

From equations (1) and (3): y = 1 - \frac{r}{2}.

Substituting into (1): \left(1 - \frac{r}{3}\right)^2 + \left(1 - \frac{r}{2}\right)^2 = (1+r)^2

This simplifies to 23r^2 + 132r - 36 = 0, giving r = \frac{6}{23} or r = -6.

Since r > 0: r = \frac{6}{23}.

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