Hypocycloid and Lattice Points
A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the...
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John Rhodes
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August 15, 2023
Chocolate Covered Candy
Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation: $b^2...
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John Rhodes
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August 15, 2023
Average Least Common Multiple
The function $\operatorname{\mathbf{lcm}}(a,b)$ denotes the least common multiple of $a$ and $b$. Let $A(n)$ be the average of the values of $\operatorname{lcm}(n,i)$ for $1 \le i \le...
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John Rhodes
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August 15, 2023
Retractions C
For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined by $f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a 1$,...
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John Rhodes
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August 15, 2023
Retractions B
For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined by $f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a 1$,...
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John Rhodes
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August 15, 2023
Eleven-free Integers
An integer is called eleven-free if its decimal expansion does not contain any substring representing a power of $11$ except $1$. For example, $2404$ and $13431$ are...
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John Rhodes
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August 15, 2023
The Inverse Summation of Coprime Couples
For an integer $M$, we define $R(M)$ as the sum of $1/(p \cdot q)$ for all the integer pairs $p$ and $q$ which satisfy all of these...
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John Rhodes
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August 15, 2023
GCD and Tiling
We want to tile a board of length $n$ and height $1$ completely, with either $1 \times 2$ blocks or $1 \times 1$ blocks with a single...