$5$-smooth Totients
$5$-smooth numbers are numbers whose largest prime factor doesn’t exceed $5$. $5$-smooth numbers are also called Hamming numbers. Let $S(L)$ be the sum of the numbers $n$...
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John Rhodes
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August 15, 2023
Dissonant Numbers
Let $d(p, n, 0)$ be the multiplicative inverse of $n$ modulo prime $p$, defined as $n \times d(p, n, 0) = 1 \bmod p$. Let $d(p, n,...
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John Rhodes
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August 15, 2023
Geoboard Shapes
A geoboard (of order $N$) is a square board with equally-spaced pins protruding from the surface, representing an integer point lattice for coordinates $0 \le x, y...
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John Rhodes
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August 15, 2023
Integral Median
$ABC$ is an integral sided triangle with sides $a \le b \le c$. $m_C$ is the median connecting $C$ and the midpoint of $AB$. $F(n)$ is the...
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John Rhodes
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August 15, 2023
Sums of Totients of Powers
Let $\varphi(n)$ be Euler’s totient function. Let $f(n)=(\sum_{i=1}^{n}\varphi(n^i)) \bmod (n+1)$. Let $g(n)=\sum_{i=1}^{n} f(i)$. $g(100)=2007$. Find $g(5 \times 10^8)$. To solve this problem, we can use dynamic programming...
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John Rhodes
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August 15, 2023
Sequences with Nice Divisibility Properties
Let $Seq(n,k)$ be the number of positive-integer sequences $\{a_i\}_{1 \le i \le n}$ of length $n$ such that: $n$ is divisible by $a_i$ for $1 \le i...
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John Rhodes
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August 15, 2023
Tangent Circles
Circles $A$ and $B$ are tangent to each other and to line $L$ at three distinct points. Circle $C$ is inside the space between $A$, $B$ and...
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John Rhodes
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August 15, 2023
Divisor Nim
Anton and Bertrand love to play three pile Nim. However, after a lot of games of Nim they got bored and changed the rules somewhat. They may...