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A sphere is inscribed in a regular octahedron.
What is the ratio of the sphere’s radius to the octahedron’s edge length?
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Find the sum of all primes of the form p=nn+1, where p<1019 and n is a positive integer.
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Here is how to make the ideal mathematical cannoli:
What is the volume of the filling in the cannoli?
Note: Top, front, and side views of the cannoli are shown. Since the excess filling has been scraped away, no filling is visible in the side view. Bon appétit!
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Given the areas of the triangles, find the area of this rectangle.
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From the AM-GM inequality, we know that the arithmetic mean (AM) of a list of non-negative real numbers is always greater than or equal to the geometric mean (GM). But the inequality doesn't tell us just how much larger the AM is.
Given a list of n random real numbers chosen uniformly and independently in the range [a0,a1], where 0≤a0<a1, find E[GMAM] in terms of n,a0,a1.
Then find a1→∞a0→0limE[GMAM].
If the formula is of the form (An−B)(n−1)n−Cnn, where A,B,C are positive integers, give your answer as A+B+C.
Note: The notation E[X] is the expected value of the random variable X.
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