Sum of Sum of Divisors
Let $d(k)$ be the sum of all divisors of $k$. We define the function $S(N) = \sum_{i=1}^N \sum_{j=1}^Nd(i \cdot j)$. For example, $S(3) = d(1) + d(2)...
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John Rhodes
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August 15, 2023
Integer Part of Polynomial Equation’s Solutions
For an $n$-tuple of integers $t = (a_1, \dots, a_n)$, let $(x_1, \dots, x_n)$ be the solutions of the polynomial equation $x^n + a_1 x^{n-1} + a_2...
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John Rhodes
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August 15, 2023
Unfair Wager
Julie proposes the following wager to her sister Louise. She suggests they play a game of chance to determine who will wash the dishes. For this game,...
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John Rhodes
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August 15, 2023
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...