Roman Numerals II
A random generator produces a sequence of symbols drawn from the set {I, V, X, L, C, D, M, #}. Each item in the sequence is determined...
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John Rhodes
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August 16, 2023
$\pi$ Sequences
For every $n \ge 1$ the prime-counting function $\pi(n)$ is equal to the number of primes not exceeding $n$. E.g. $\pi(6)=3$ and $\pi(100)=25$. We say that a...
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John Rhodes
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August 16, 2023
Divisor Sums
Let $D(m,n)=\displaystyle\sum_{d\mid m}\sum_{k=1}^n\sigma_0(kd)$ where $d$ runs through all divisors of $m$ and $\sigma_0(n)$ is the number of divisors of $n$. You are given $D(3!,10^2)=3398$ and $D(4!,10^6)=268882292$. Find...
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John Rhodes
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August 16, 2023
Marsh Crossing
Frodo and Sam need to travel 100 leagues due East from point A to point B. On normal terrain, they can cover 10 leagues per day, and...
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John Rhodes
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August 16, 2023
Pairwise Coin-Tossing Game
Consider an $n$-player game played in consecutive pairs: Round $1$ takes place between players $1$ and $2$, round $2$ takes place between players $2$ and $3$, and...
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John Rhodes
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August 16, 2023
Convex Path in Square
Let $F(N)$ be the maximum number of lattice points in an axis-aligned $N\times N$ square that the graph of a single strictly convex increasing function can pass...
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John Rhodes
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August 16, 2023
Substring Sums of Prime Concatenations
Let $S(n)$ be the sum of all contiguous integer-substrings that can be formed from the integer $n$. The substrings need not be distinct. For example, $S(2024) =...
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John Rhodes
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August 16, 2023
Product of Head Counts
Alice enlists the help of some friends to generate a random number, using a single unfair coin. She and her friends sit around a table and, starting...