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Difference between revisions of "1971 IMO Problems/Problem 2"

(Created page with "==Problem== Consider a convex polyhedron <math>P_1</math> with nine vertices <math>A_1, A_2, \cdots, A_9;</math> let <math>P_i</math> be the polyhedron obtained from <math>P_1...")
 
 
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==Solution==
 
==Solution==
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{{solution}}
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WLOG let <math>A_1</math> be the origin <math>0</math>.
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Take any point <math>A_i</math>, then <math>P_i=A_i+P_1</math>, lies in <math>2 P_1</math>, the polyhedron <math>P_1</math> stretched by the factor <math>2</math> on <math>P_1=0</math>.
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More general: take any <math>p,q</math> in any convex shape <math>S</math>. Then <math>p+q \in 2S</math>.
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Prove: since <math>S</math> is convex, <math>\frac{p+q}{2} \in S</math>, thus <math>p+q \in 2S</math>.
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Now all these nine polyhedrons lie inside <math>2 P_1</math>. Let <math>V</math> be the volume of <math>P_1</math>.
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Then some polyhedrons with total sum of volumes <math>9V</math> lie in a shape of volume <math>8V</math>, thus they must overlap, meaning that they have an interior point in common.
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The above solution was posted by ZetaX. The original thread for this problem can be found here: [https://aops.com/community/p417224]
  
 
==See Also==
 
==See Also==

Latest revision as of 12:55, 29 January 2021

Problem

Consider a convex polyhedron $P_1$ with nine vertices $A_1, A_2, \cdots, A_9;$ let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves vertex $A_1$ to $A_i(i=2,3,\cdots, 9).$ Prove that at least two of the polyhedra $P_1, P_2,\cdots, P_9$ have an interior point in common.

Solution

WLOG let $A_1$ be the origin $0$. Take any point $A_i$, then $P_i=A_i+P_1$, lies in $2 P_1$, the polyhedron $P_1$ stretched by the factor $2$ on $P_1=0$. More general: take any $p,q$ in any convex shape $S$. Then $p+q \in 2S$. Prove: since $S$ is convex, $\frac{p+q}{2} \in S$, thus $p+q \in 2S$.

Now all these nine polyhedrons lie inside $2 P_1$. Let $V$ be the volume of $P_1$. Then some polyhedrons with total sum of volumes $9V$ lie in a shape of volume $8V$, thus they must overlap, meaning that they have an interior point in common.

The above solution was posted by ZetaX. The original thread for this problem can be found here: [1]

See Also

1971 IMO (Problems) • Resources
Preceded by
Problem 1 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 3
All IMO Problems and Solutions
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