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...
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John Rhodes
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August 16, 2023
Pythagorean Triple Occurrence
Define $Q(n)$ to be the smallest number that occurs in exactly $n$ Pythagorean triples $(a,b,c)$ where $a \lt b \lt c$. For example, $15$ is the smallest...
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John Rhodes
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August 16, 2023
Birds on a Wire
Consider a wire of length 1 unit between two posts. Every morning $n$ birds land on it randomly with every point on the wire equally likely to...
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John Rhodes
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August 16, 2023
Chasing Game
Two cars are on a circular track of total length $2n$, facing the same direction, initially distance $n$ apart. They move in turn. At each turn, the...