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...
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John Rhodes
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August 15, 2023
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,...