Maximum Number of Divisors
Let $d(n)$ be the number of divisors of $n$. Let $M(n,k)$ be the maximum value of $d(j)$ for $n \le j \le n+k-1$. Let $S(u,k)$ be the...
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John Rhodes
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August 15, 2023
Arithmetic Derivative
The arithmetic derivative is defined by $p^\prime = 1$ for any prime $p$ $(ab)^\prime = a^\prime b + ab^\prime$ for all integers $a, b$ (Leibniz rule) For...
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John Rhodes
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August 15, 2023
Repeated Permutation
We define a permutation as an operation that rearranges the order of the elements $\{1, 2, 3, …, n\}$. There are $n!$ such permutations, one of which...
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John Rhodes
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August 15, 2023
The Incenter of a Triangle
$ABC$ is an integer sided triangle with incenter $I$ and perimeter $p$. The segments $IA$, $IB$ and $IC$ have integral length as well. Let $L = p...
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John Rhodes
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August 15, 2023
Chef Showdown
A group of chefs (numbered #$1$, #$2$, etc) participate in a turn-based strategic cooking competition. On each chef’s turn, he/she cooks up a dish to the best...
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John Rhodes
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August 15, 2023
The Last Question
Consider all the words which can be formed by selecting letters, in any order, from the phrase: thereisasyetinsufficientdataforameaningfulanswer Suppose those with 15 letters or less are listed...
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John Rhodes
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August 15, 2023
Roots on the Rise
Let $a_k$, $b_k$, and $c_k$ represent the three solutions (real or complex numbers) to the equation $\frac 1 x = (\frac k x)^2(k+x^2)-k x$. For instance, for...
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John Rhodes
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August 15, 2023
Number Sequence Game
The number sequence game starts with a sequence $S$ of $N$ numbers written on a line. Two players alternate turns. The players on their respective turns must...