Bundeswettbewerb Mathematik 1974 Problem 2.3

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Bundeswettbewerb Mathematik 1974 Problem 2.3

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Source: Bundeswettbewerb Mathematik 1974 Round 2

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#1 h Oct 13, 2022, 10:04 PM

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A circle $K_1$ of radius $r_1 = 1\slash 2$ is inscribed in a semi-circle $H$ with diameter $AB$ and radius $1.$ A sequence of different circles $K_2, K_3, \ldots$ with radii $r_2, r_3, \ldots$ respectively are drawn so that for each $n\geq 1$ , the circle $K_{n+1}$ is tangent to $H$ , $K_n$ and $AB.$ Prove that $a_n = 1\slash r_n$ is an integer for each $n$ , and that it is a perfect square for $n$ even and twice a perfect square for $n$ odd.

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#2 h Nov 17, 2024, 7:19 PM

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Given $A(-1,0),O(0,0),A(1,0)$ and the circle, midpoint $O$ and radius $1$ .
Given the circle, midpoint $O_{1}(0,\frac{1}{2}$ and radius $r_{1}=\frac{1}{2}$ .
To find: the circle, midpoint $O_{2}(t,r_{2})$ and radius $r_{2}$ , satisfying $t^{2}+r_{2}^{2}=(1-r_{2})^{2}$ and $t^{2}+(r_{2}-\frac{1}{2})^{2}=(r_{2}+\frac{1}{2})^{2}$ ,
solving $t=\frac{\sqrt{2}}{2}$ and $r_{2}=\frac{1}{4}$ .

To find: the circle, midpoint $O_{3}(s,r_{3})$ and radius $r_{3}$ , satisfying $s^{2}+r_{3}^{2}=(1-r_{3})^{2}$ and $(s-\frac{\sqrt{2}}{2})^{2}+(r_{3}-\frac{1}{4})^{2}=(r_{3}+\frac{1}{4})^{2}$ ,
solving $s=\frac{2\sqrt{2}}{3}$ and $r_{3}=\frac{1}{18}$ .

To find: the circle, midpoint $O_{4}(v,r_{4})$ and radius $r_{4}$ , satisfying $v^{2}+r_{4}^{2}=(1-r_{4})^{2}$ and $(v-\frac{2\sqrt{2}}{3})^{2}+(r_{4}-\frac{1}{18})^{2}=(r_{4}+\frac{1}{18})^{2}$ ,
solving $v=\frac{7\sqrt{2}}{10}$ and $r_{4}=\frac{1}{100}$ .

See OEIS, sequence $2,4,18,100,\cdots$

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