Difference between revisions of "1971 IMO Problems/Problem 2"
Durianaops (talk | contribs) (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== | ||
− | {{solution | + | WLOG let <math>A_1</math> be the origin <math>0</math>. |
+ | 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>. | ||
+ | More general: take any <math>p,q</math> in any convex shape <math>S</math>. Then <math>p+q \in 2S</math>. | ||
+ | Prove: since <math>S</math> is convex, <math>\frac{p+q}{2} \in S</math>, thus <math>p+q \in 2S</math>. | ||
+ | |||
+ | Now all these nine polyhedrons lie inside <math>2 P_1</math>. Let <math>V</math> be the volume of <math>P_1</math>. | ||
+ | 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. | ||
+ | |||
+ | 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 13:55, 29 January 2021
Problem
Consider a convex polyhedron with nine vertices
let
be the polyhedron obtained from
by a translation that moves vertex
to
Prove that at least two of the polyhedra
have an interior point in common.
Solution
WLOG let be the origin
.
Take any point
, then
, lies in
, the polyhedron
stretched by the factor
on
.
More general: take any
in any convex shape
. Then
.
Prove: since
is convex,
, thus
.
Now all these nine polyhedrons lie inside . Let
be the volume of
.
Then some polyhedrons with total sum of volumes
lie in a shape of volume
, 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 |