Supernatural Triangles
A Pythagorean triangle is called supernatural if two of its three sides are consecutive integers. Let $S(N)$ be the sum of the perimeters of all distinct supernatural...
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John Rhodes
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August 16, 2023
Add and Divide
A sequence is created by starting with a positive integer $n$ and incrementing by $(n+m)$ at the $m^{th}$ step. If $n=10$, the resulting sequence will be $21,33,46,60,75,91,108,126,\ldots$....
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John Rhodes
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August 16, 2023
Square Triangle Products
Triangle numbers $T_k$ are integers of the form $\frac{k(k+1)} 2$. A few triangle numbers happen to be perfect squares like $T_1=1$ and $T_8=36$, but more can be...
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John Rhodes
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August 16, 2023
Mex Sequence
In this problem $\oplus$ is used to represent the bitwise exclusive or of two numbers. Starting with blank paper repeatedly do the following: Write down the smallest...
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John Rhodes
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August 16, 2023
Triple Product
Let $g(m)$ be the integer defined by the following double sum of products of binomial coefficients: $$\sum_{j=0}^m\sum_{i = 0}^j (-1)^{j-i}\binom mj \binom ji \binom{j+5+6i}{j+5}.$$ You are given...
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John Rhodes
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August 16, 2023
Binomials and Powers
Let $\displaystyle S(n)=\sum\limits_{k=0}^{n}\binom{n}{k}k^n$. You are given, $S(10)=142469423360$. Find $S(10^{18})$. Submit your answer modulo $83^3 89^3 97^3$. This is a Combinatorial Mathematics problem and surely involves some use...
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John Rhodes
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August 16, 2023
Integral Fusion
Given any integer $n \gt 1$ a binary factor tree $T(n)$ is defined to be: A tree with the single node $n$ when $n$ is prime. A...
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John Rhodes
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August 16, 2023
Numbers Challenge
It is a common recreational problem to make a target number using a selection of other numbers. In this problem you will be given six numbers and...