Art of Problem Solving
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Tangent Spheres and Tangents to Spheres
Math-Problem-Solving   2
N Apr 2, 2025 by Math-Problem-Solving
Source: 2002 British Mathematical Olympiad Round 2
Prove this.
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Math-Problem-Solving
Apr 2, 2025
Math-Problem-Solving
Apr 2, 2025
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Source: 2002 British Mathematical Olympiad Round 2
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Math-Problem-Solving
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Math-Problem-Solving
#1 h  Apr 2, 2025, 5:26 AM
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Prove this.
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kiyoras_2001
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kiyoras_2001
#2 h  Apr 2, 2025, 8:37 AM • 1 Y
Y by MS_asdfgzxcvb
Let $d_i$ be the distance from $P$ to the center of $B_i$. Then $t_i^2=d_i^2-1$ and by the median length formula $PC_i^2 = \dfrac{2d_i^2+2d_{i+1}^2-4}{4} = \dfrac{t_i^2+t_{i+1}^2}{2}\ge {t_it_{i+1}}$. Hence $\prod PC_i \ge \prod t_i$.
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Math-Problem-Solving
#3 h  Apr 2, 2025, 9:17 AM
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kiyoras_2001 wrote:
Let $d_i$ be the distance from $P$ to the center of $B_i$. Then $t_i^2=d_i^2-1$ and by the median length formula $PC_i^2 = \dfrac{2d_i^2+2d_{i+1}^2-4}{4} = \dfrac{t_i^2+t_{i+1}^2}{2}\ge {t_it_{i+1}}$. Hence $\prod PC_i \ge \prod t_i$.

Thank you for this beautiful short solution.
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