Tatami-Free Rooms
Tatami are rectangular mats, used to completely cover the floor of a room, without overlap. Assuming that the only type of available tatami has dimensions $1 \times...
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John Rhodes
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August 15, 2023
Rounded Square Roots
We define the rounded-square-root of a positive integer $n$ as the square root of $n$ rounded to the nearest integer. The following procedure (essentially Heron’s method adapted...
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John Rhodes
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August 15, 2023
Sums of Digit Factorials
Define $f(n)$ as the sum of the factorials of the digits of $n$. For example, $f(342) = 3! + 4! + 2! = 32$. Define $sf(n)$ as...
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John Rhodes
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August 15, 2023
Tidying Up A
A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the...
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John Rhodes
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August 15, 2023
Convex Holes
Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given points and...
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John Rhodes
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August 15, 2023
Cardano Triplets
A triplet of positive integers $(a, b, c)$ is called a Cardano Triplet if it satisfies the condition: $$\sqrt[3]{a + b \sqrt{c}} + \sqrt[3]{a – b \sqrt{c}}...
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John Rhodes
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August 15, 2023
$250250$
Find the number of non-empty subsets of $\{1^1, 2^2, 3^3,\dots, 250250^{250250}\}$, the sum of whose elements is divisible by $250$. Enter the rightmost $16$ digits as your...
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John Rhodes
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August 15, 2023
Prime Subset Sums
Let $S = \{2, 3, 5, \dots, 4999\}$ be the set of prime numbers less than $5000$. Find the number of subsets of $S$, the sum of...