Maximum Length of an Antichain
Let $n$ be an integer and $S(n)$ be the set of factors of $n$. A subset $A$ of $S(n)$ is called an antichain of $S(n)$ if $A$...
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John Rhodes
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August 15, 2023
Ellipses Inside Triangles
For any triangle $T$ in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside $T$. For a...
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John Rhodes
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August 15, 2023
Rudin-Shapiro Sequence
Define the sequence $a(n)$ as the number of adjacent pairs of ones in the binary expansion of $n$ (possibly overlapping). E.g.: $a(5) = a(101_2) = 0$, $a(6)...
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John Rhodes
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August 15, 2023
Divisibility Comparison Between Factorials
Let $f_5(n)$ be the largest integer $x$ for which $5^x$ divides $n$. For example, $f_5(625000) = 7$. Let $T_5(n)$ be the number of integers $i$ which satisfy...
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John Rhodes
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August 15, 2023
Generating Polygons
A polygon is a flat shape consisting of straight line segments that are joined to form a closed chain or circuit. A polygon consists of at least...
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John Rhodes
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August 15, 2023
$(\text{prime}-k)$ Factorial
For a prime $p$ let $S(p) = (\sum (p-k)!) \bmod (p)$ for $1 \le k \le 5$. For example, if $p=7$, $(7-1)! + (7-2)! + (7-3)! +...
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John Rhodes
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August 15, 2023
Amazing Mazes!
An $m \times n$ maze is an $m \times n$ rectangular grid with walls placed between grid cells such that there is exactly one path from the...
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John Rhodes
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August 15, 2023
Least Common Multiple Count
Let $f(n)$ be the number of couples $(x, y)$ with $x$ and $y$ positive integers, $x \le y$ and the least common multiple of $x$ and $y$...