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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 $p=n^{n} + 1$, where $p<10^{19}$ 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 $[a_0,a_1],$ where $0\le a_0<a_1,$ find $\text{E}\left[\frac{\text{AM}}{\text{GM}}\right]$ in terms of $n,a_0,a_1.$
Then find $\displaystyle \lim\limits_{\overset{a_0 \to 0}{a_1 \to \infty}} \text{E}\left[\frac{\text{AM}}{\text{GM}}\right]$.
If the formula is of the form $\large\frac{n^n}{\left(An-B\right){\left(n-1\right)}^{n-C}},$ where $A,B,C$ are positive integers, give your answer as $A+B+C.$
Note: The notation $\text{E}[X]$ is the expected value of the random variable $X.$
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