Rolling Ellipse
An ellipse $E(a, b)$ is given at its initial position by equation: $\frac {x^2} {a^2} + \frac {(y – b)^2} {b^2} = 1$ The ellipse rolls without...
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John Rhodes
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August 15, 2023
First Sort I
Consider the following algorithm for sorting a list: 1. Starting from the beginning of the list, check each pair of adjacent elements in turn. 2. If the...
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John Rhodes
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August 15, 2023
Hilbert’s Blackout
Despite the popularity of Hilbert’s infinite hotel, Hilbert decided to try managing extremely large finite hotels, instead. To cut costs, Hilbert wished to power the new hotel...
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John Rhodes
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August 15, 2023
Smallest Prime Factor
Let $\operatorname{smpf}(n)$ be the smallest prime factor of $n$. $\operatorname{smpf}(91)=7$ because $91=7\times 13$ and $\operatorname{smpf}(45)=3$ because $45=3\times 3\times 5$. Let $S(n)$ be the sum of $\operatorname{smpf}(i)$ for...
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John Rhodes
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August 15, 2023
Simbers
We define a simber to be a positive integer in which any odd digit, if present, occurs an odd number of times, and any even digit, if...
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John Rhodes
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August 15, 2023
Tricoloured Coin Fountains
An arrangement of coins in one or more rows with the bottom row being a block without gaps and every coin in a higher row touching exactly...
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John Rhodes
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August 15, 2023
Prime Triples and Geometric Sequences
Let $S(n) = \sum a + b + c$ over all triples $(a, b, c)$ such that: $a$, $b$ and $c$ are prime numbers. $a \lt b...
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John Rhodes
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August 15, 2023
A Real Recursion
For every real number $a \gt 1$ is given the sequence $g_a$ by: $g_{a}(x)=1$ for $x \lt a$ $g_{a}(x)=g_{a}(x-1)+g_a(x-a)$ for $x \ge a$ $G(n)=g_{\sqrt {n}}(n)$ $G(90)=7564511$. Find...