<p>Eight circles of radius <img alt="$34$" height="13" src="https://latex.artofproblemsolving.com/d/0/3/d037f1aa313d30fe7ade693f1b672ed2a68880fa.png" width="18" /> are sequentially tangent, and two of the circles are tangent to <img alt="$AB$" height="13" src="https://latex.artofproblemsolving.com/5/7/e/57ee5125358c0606c9b588580ddfa66f83e607b7.png" width="27" /> and <img alt="$BC$" height="12" src="https://latex.artofproblemsolving.com/6/c/5/6c52a41dcbd739f1d026c5d4f181438b75b76976.png" width="28" /> of triangle <img alt="$ABC$" height="13" src="https://latex.artofproblemsolving.com/e/2/a/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png" width="42" />, respectively. <img alt="$2024$" height="13" src="https://latex.artofproblemsolving.com/e/5/7/e5711c8d31d3bc3119aba377c9bda0f9dfb14e4b.png" width="36" /> circles of radius <img alt="$1$" height="14" src="https://latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" width="11" /> can be arranged in the same manner. The inradius of triangle <img alt="$ABC$" height="13" src="https://latex.artofproblemsolving.com/e/2/a/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png" width="42" /> can be expressed as <img alt="$\frac{m}{n}$" height="33" src="https://latex.artofproblemsolving.com/2/7/f/27f583697295fc6f723862c13228ee55107c33ca.png" width="18" />, where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime positive integers. Find <img alt="$m+n$" height="12" src="https://latex.artofproblemsolving.com/0/2/d/02dbab3601fa6cce490d54263017048f69e1a35d.png" width="48" />.</p> <p><img alt="[asy] pair A = (2,1); pair B = (0,0); pair C = (3,0); dot(A^^B^^C); label("$A$", A, N); label("$B$", B, S); label("$C$", C, S); draw(A--B--C--cycle); for(real i=0.62; i<2.7; i+=0.29){ draw(circle((i,0.145), 0.145)); } [/asy]" height="122" src="https://latex.artofproblemsolving.com/e/c/5/ec5d6f1f995c862b1609e69add5733cb092f1f3f.png" width="252" /></p>

Question No: 422

Eight circles of radius $34$ are sequentially tangent, and two of the circles are tangent to $AB$ and $BC$ of triangle $ABC$ , respectively. $2024$ circles of radius $1$ can be arranged in the same manner. The inradius of triangle $ABC$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

$[asy] pair A = (2,1); pair B = (0,0); pair C = (3,0); dot(A^^B^^C); label("$A$", A, N); label("$B$", B, S); label("$C$", C, S); draw(A--B--C--cycle); for(real i=0.62; i<2.7; i+=0.29){ draw(circle((i,0.145), 0.145)); } [/asy]$

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Tags: Geometry

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