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I have a clock app that shows not only the time and date, but also shows a polar projection map (specifically azimuthal equidistant centered on the north pole) that includes all of the world that's north of 25° South. The portions of the world that are lit at that time by the sun are shown in light versions of its colors, while nighttime is shown in darker shades.

That app was written so that the sunlight is assumed to come directly from the left so the center of the arc defining the left edge of daylight is at would normally be called the "9 o'clock" position, though this is obviously where noon is. The 25° South limit allows the location of where the sun is directly overhead to be shown, but it also is near enough to the equator to allow every horizontal line through the map to be all daytime pixels, all nighttime pixels, or a straight run of daylight pixels followed by nighttime pixels.

If the map were to have been made to extend all the way to the south pole, not only would there be areas of great distortion, but the programming would also have to consider horizontal rows of pixels that went from day to night and back to day, or even the reverse at certain times of year: night, day and night again. When the map goes only as far out as 25° S this doesn't happen. Such would happen though before the outer limit got all the way to the south pole.

The question is: To what south latitude could the map be extended without this happening? ... so any line of pixels would still have at most two parts, not three.

Assume the Sun can be anywhere from 23.4° S to 23.4° N, and that daylight is where the geographic location is less than 90° from where the sun is directly overhead.

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On a circumference, points A and B are on opposite arcs of diameter CD. Line segments CE and DF are perpendicular to AB such that A, E, F and B are collinear in this order. Given that AE=1, find the length of BF.

Positive numbers x, y satisfy equality x³ + y³ = x - y. Prove that x² + y² < 1.

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Submitted on 2025-05-28 by Danish Ahmed Khan