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Although this was not a competition problem, two correct solutions have been submitted: by Dinis Cheian and Ivan Ivashkovskiy.

The Ivan's solution is very thorough – with error analysis (as should be for such a semi-experimental problem) and is presented below on two images. Note that he defines n as the ratio of two lengths, him and l, which he can measure from the photo and thus calculate n. On the other hand, he finds expressions for him and l, and using these expressions shows that n =H/D. Knowing the object lenth H, he can find now D.

In addition to that solution, I have two comments. First, the usage of the Newton formula would have simplified slightly the mathematics: x1x2 = F2, where x1 = F and x2 = f F are the distances of the object and of the image from the respective focal planes (using Ivan's notations, d and f are the respective distances from the lens). So, the diameter of the circle of confusion l = Dx2 / F = FD / x1 = FD / (d F). Second, pay attention to the main result: the diameter of the lens equals to the diameter of the circle of confusion created by a far-away dot-source, if measured by the image of an in-focus ruler. So, if you have a subject for your photo (eg. friends face for a portrait) and you want to blur the (far-away) background, the degree of the background blur is defined purely by the diameter of the lens: a tele-lens 600mm/4 creates circles of confusion of the size of the face, leaving no visible details at the background; a point-and-shoot camera with a small sensor and 8mm/4 lens creates almost no background blur: the diameter of the circles of confusion are smaller than the pupils of the portrait.

Finally, let us analyse the benefits of a large camera: you have a full-frame DSLR, and your friend has a small point-and-shoot camera of a 4 times smaller sensor (in linear size).  Your friend takes a photo of something, zooming to the focal length of 13 mm and is using the full aperture of F/4 (ie. the focal-length-to-diameter ratio equals to 4). You want to obtain exactly the same result: in order to have the same angle of view (and perspective), you need to take the focal length equal to 13*4 = 52 (your 50mm/1.4 standard prime lens works well). In order to have the same blur, you need to shut down the lens — down to the aperture F/16 (using the diaphragm you decrease the effective diameter). On the other hand, if you take a photo at the full aperture of F/1.4, your friend would need the aperture of F/0.35, which is theoretically impossible (would violate the second law of thermodynamics). Meanwhile, if you want to have both sharp foreground and sharp background, the point-and-shoot is better: you can use F/22, which would correspond to F/88 for the DSLR. While theoretically this is possible, the smallest aperture is typically only F/32 (starting from ca F/16, diffraction starts degrading the image).

– Jaan Kalda – Academic Committe –

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