<p>A convex polyhedron <img alt="$Q$" height="16" src="https://latex.artofproblemsolving.com/9/8/6/9866e3a998d628ba0941eb4fea0666ac391d149a.png" width="13" /> has vertices <img alt="$V_1,V_2,\ldots,V_n$" height="16" src="https://latex.artofproblemsolving.com/4/e/3/4e30c2d7dac2dcc50dca9fff6e71d80c372edb0a.png" width="103" />, and <img alt="$100$" height="13" src="https://latex.artofproblemsolving.com/e/5/9/e59e2c6e83eb78cca610a5fd4070ae01c8d4ae60.png" width="26" /> edges. The polyhedron is cut by planes <img alt="$P_1,P_2,\ldots,P_n$" height="16" src="https://latex.artofproblemsolving.com/0/b/9/0b9824caec77ced15e97fb784e77ee597b609eed.png" width="106" /> in such a way that plane <img alt="$P_k$" height="15" src="https://latex.artofproblemsolving.com/7/6/e/76ebaba4ae3d5c911d6e31ad855d502ad88d48a6.png" width="18" /> cuts only those edges that meet at vertex <img alt="$V_k$" height="15" src="https://latex.artofproblemsolving.com/4/0/b/40b514f3d50ab12888fa4886e266bb6a1e8a7e9f.png" width="17" />. In addition, no two planes intersect inside or on <img alt="$Q$" height="16" src="https://latex.artofproblemsolving.com/9/8/6/9866e3a998d628ba0941eb4fea0666ac391d149a.png" width="13" />. The cuts produce <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> pyramids and a new polyhedron <img alt="$R$" height="12" src="https://latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png" width="14" />. How many edges does <img alt="$R$" height="12" src="https://latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png" width="14" /> have?</p>

Question No: 303

A convex polyhedron $Q$ has vertices $V_1,V_2,\ldots,V_n$ , and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$ . In addition, no two planes intersect inside or on $Q$ . The cuts produce $n$ pyramids and a new polyhedron $R$ . How many edges does $R$ have?

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Tags: Algebra Probability

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