Spheres and a point source of light

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Spheres and a point source of light

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mofidy

#1 h Mar 11, 2025, 8:27 AM • 1 Y

Y by Creativename27

How many spheres are needed to shield a point source of light?
Unfortunately, I didn't find a suitable solution for it on the page below:
https://artofproblemsolving.com/community/c4h1469498p8521602
Here too, two different solutions are given:
https://math.stackexchange.com/questions/2791186

This post has been edited 3 times. Last edited by mofidy, Mar 11, 2025, 1:08 PM

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greenturtle3141

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greenturtle3141

#2 h Mar 11, 2025, 5:40 PM

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The answer is Click to reveal hidden text. The solution is very beautiful. A hint is to Click to reveal hidden text

This post has been edited 1 time. Last edited by greenturtle3141, Mar 11, 2025, 5:41 PM

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mofidy

#3 h Mar 12, 2025, 2:36 PM

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greenturtle3141 wrote:

The answer is Click to reveal hidden text. The solution is very beautiful. A hint is to Click to reveal hidden text

Please state the argument more precisely and in the form of a mathematical proof.
Thanks

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#4 h Mar 14, 2025, 9:22 PM • 1 Y

Y by MS_asdfgzxcvb

Note that $4$ spheres suffice. Indeed, let $O$ be the light source, $ABCD$ be a regular tetrahedron inscribed in the unit sphere $S$ with center $O$ . Define the sphere $S_A$ with center $A$ passing through a point very close to $O$ , but not containing $O$ . Similarly define spheres $S_B, S_C, S_D$ . Then $S_A, S_B, S_C, S_D$ shield $O$ . If you need them to be non intersecting just dilate appropriatetly with respect to $O$ .

Now suppose that three spheres are given with centers $A, B, C$ . We show that they do not shield $O$ . Indeed, if $A, B, C, O$ are coplanar, then the line through $O$ perpendicular to $ABC$ does not cut any sphere. Otherwise define $D=\frac{A}{\|A\|} + \frac{B}{\|B\|} + \frac{C}{\|C\|}$ and $E=-D$ . Then the ray $OE$ does not cut any sphere since $OD$ makes acute and equal angles with $OA, OB, OC$ .

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