Circles and Rectangles.

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In the above rectangle, B C ‾ \overline{BC} is tangent to the red and green circles at E E and F F , A D ‾ \overline{AD} is tangent to the blue and green circles at G G and H H , A B ‾ \overline{AB} is tangent to the red circle at I I , C D CD is tangent to the green circle at J J and the blue circle is tangent to both the red and the green circles and the red, blue and green circles have diameters a a , a + 1 a + 1 and a + 3 a + 3 as shown above.

If the value of a a for which E F ‾ = a + 3 \overline{EF} = a + 3 can be expressed as a = β α − β γ a = \dfrac{\beta\sqrt{\alpha} - \beta}{\gamma} , where α , β \alpha, \beta and γ \gamma are coprime positive integers, find α + β + γ \alpha + \beta +\gamma .


The answer is 24.

Rocco Dalto by Rocco Dalto , 67, USA

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1 solution

Rocco Dalto
Dec 17, 2020

m 2 = ( a + 2 ) 2 − 1 = a 2 + 4 a + 3 = ( a + 3 ) ( a + 1 ) ⟹ m = ( a + 3 ) ( a + 1 ) m^2 = (a + 2)^2 - 1 = a^2 + 4a + 3 = (a + 3)(a + 1) \implies m = \sqrt{(a + 3)(a + 1)}

n 2 = ( 2 a + 1 ) 2 4 − 25 4 = 4 a 2 + 4 a − 24 4 = a 2 + a − 6 = ( a + 3 ) ( a − 2 ) n^2 = \dfrac{(2a + 1)^2}{4} - \dfrac{25}{4} = \dfrac{4a^2 + 4a - 24}{4} = a^2 + a - 6 = (a + 3)(a - 2)

⟹ n = ( a + 3 ) ( a − 2 ) \implies n = \sqrt{(a + 3)(a - 2)}

⟹ E F ‾ = m + n = ( a + 3 ) ( a + 1 ) + ( a + 3 ) ( a − 2 ) = a + 3 ⟹ \implies \overline{EF} = m + n = \sqrt{(a + 3)(a + 1)} + \sqrt{(a + 3)(a - 2)} = a + 3 \implies

( a + 3 ) ( a + 1 ) + 2 ( a + 3 ) ( a + 1 ) ( a − 2 ) + ( a + 3 ) ( a − 2 ) = a + 3 (a + 3)(a + 1) + 2(a + 3)\sqrt{(a + 1)(a - 2)} + (a + 3)(a - 2) = a + 3

⟹ 2 a − 1 + 2 ( a + 1 ) ( a − 2 ) = a + 3 ⟹ 2 ( a + 1 ) ( a − 2 ) = 4 − a ⟹ \implies 2a - 1 + 2\sqrt{(a + 1)(a - 2)} = a + 3 \implies 2\sqrt{(a + 1)(a - 2)} = 4 - a \implies

4 a 2 − 4 a − 8 = 16 − 8 a + a 2 ⟹ 3 a 2 + 4 a − 24 = 0 ⟹ a = − 2 + 2 19 3 4a^2 - 4a - 8 = 16 - 8a + a^2 \implies 3a^2 + 4a - 24 = 0 \implies a = \dfrac{-2 + 2\sqrt{19}}{3}

dropping the negative root ⟹ a = 2 19 − 2 3 = β α − β γ \implies a = \dfrac{2\sqrt{19} - 2}{3} = \dfrac{\beta\sqrt{\alpha} - \beta}{\gamma}

⟹ α + β + γ = 24 \implies \alpha + \beta + \gamma = \boxed{24} .

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