Iterated Composition
Let $\Bbb R^2$ be the set of pairs of real numbers $(x, y)$. Let $\pi = 3.14159\cdots\ $. Consider the function $f$ from $\Bbb R^2$ to $\Bbb...
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John Rhodes
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August 16, 2023
Hybrid Integers
An integer of the form $p^q q^p$ with prime numbers $p \neq q$ is called a hybrid-integer. For example, $800 = 2^5 5^2$ is a hybrid-integer. We...
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John Rhodes
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August 16, 2023
Pentagonal Puzzle
Pentagonal numbers are generated by the formula: $P_n = \tfrac 12n(3n-1)$ giving the sequence: $$1,5,12,22,35, 51,70,92,\ldots $$ Some pentagonal numbers can be expressed as the sum of...
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John Rhodes
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August 16, 2023
Card Stacking Game
Two players play a game with a deck of cards which contains $s$ suits with each suit containing $n$ cards numbered from $1$ to $n$. Before the...
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John Rhodes
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August 16, 2023
Cyclogenic Polynomials
A monic polynomial is a single-variable polynomial in which the coefficient of highest degree is equal to $1$. Define $\mathcal{F}$ to be the set of all monic...
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John Rhodes
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August 16, 2023
A Grand Shuffle
A standard $52$ card deck comprises thirteen ranks in four suits. However, modern decks have two additional Jokers, which neither have a suit nor a rank, for...
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John Rhodes
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August 16, 2023
Alternating GCD Sum
For a positive integer $n$, the function $g(n)$ is defined as $$\displaystyle g(n)=\sum_{i=1}^{n} (-1)^i \gcd \left(n,i^2\right)$$ For example, $g(4) = -\gcd \left(4,1^2\right) + \gcd \left(4,2^2\right) – \gcd...
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John Rhodes
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August 16, 2023
Seventeen Points
This problem uses half open interval notation where $[a,b)$ represents $a \le x < b$. A real number, $x_1$, is chosen in the interval $[0,1)$. A second...