Polynomials of Fibonacci Numbers
The Fibonacci numbers $\{f_n, n \ge 0\}$ are defined recursively as $f_n = f_{n-1} + f_{n-2}$ with base cases $f_0 = 0$ and $f_1 = 1$. Define...
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John Rhodes
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August 15, 2023
Rigid Graphs
Recall that a graph is a collection of vertices and edges connecting the vertices, and that two vertices connected by an edge are called adjacent. Graphs can...
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John Rhodes
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August 15, 2023
Steps in Euclid’s Algorithm
Let $E(x_0, y_0)$ be the number of steps it takes to determine the greatest common divisor of $x_0$ and $y_0$ with Euclid’s algorithm. More formally:$x_1 = y_0$,...
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John Rhodes
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August 15, 2023
Totient Sum
Let $S(n,m) = \sum\phi(n \times i)$ for $1 \leq i \leq m$. ($\phi$ is Euler’s totient function) You are given that $S(510510,10^6)= 45480596821125120$. Find $S(510510,10^{11})$. Give the...
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John Rhodes
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August 15, 2023
Square Space Silo
Fred the farmer arranges to have a new storage silo installed on his farm and having an obsession for all things square he is absolutely devastated when...
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John Rhodes
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August 15, 2023
Range Flips
$N$ disks are placed in a row, indexed $1$ to $N$ from left to right. Each disk has a black side and white side. Initially all disks...
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John Rhodes
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August 15, 2023
Sum of Squares of Unitary Divisors
A unitary divisor $d$ of a number $n$ is a divisor of $n$ that has the property $\gcd(d, n/d) = 1$. The unitary divisors of $4! =...
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John Rhodes
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August 15, 2023
Necklace of Circles
Let $a$, $b$ and $c$ be positive numbers. Let $W, X, Y, Z$ be four collinear points where $|WX| = a$, $|XY| = b$, $|YZ| = c$...