Heron Envelopes
A standard envelope shape is a convex figure consisting of an isosceles triangle (the flap) placed on top of a rectangle. An example of an envelope with...
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John Rhodes
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August 16, 2023
Nearly Isosceles $120$ Degree Triangles
Let $a, b$ and $c$ be the sides of an integer sided triangle with one angle of $120$ degrees, $a \le b \le c$ and $b-a \le...
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John Rhodes
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August 16, 2023
$47$-smooth Triangular Numbers
A number is $p$-smooth if it has no prime factors larger than $p$. Let $T$ be the sequence of triangular numbers, i.e. $T(n)=n(n+1)/2$. Find the sum of...
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John Rhodes
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August 16, 2023
Squarefree Hilbert Numbers
A Hilbert number is any positive integer of the form $4k+1$ for integer $k\geq 0$. We shall define a squarefree Hilbert number as a Hilbert number which...
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John Rhodes
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August 16, 2023
Lattice Points in Lattice Cubes
A lattice cube is a cube in which all vertices have integer coordinates. Let $C(n)$ be the number of different lattice cubes in which the coordinates of...
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John Rhodes
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August 16, 2023
Integers with Decreasing Prime Powers
Any positive integer can be written as a product of prime powers: $p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$, where $p_i$ are distinct prime integers, $a_i \gt...
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John Rhodes
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August 16, 2023
Counting Hexagons
An equilateral triangle with integer side length $n \ge 3$ is divided into $n^2$ equilateral triangles with side length 1 as shown in the diagram below. The...
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John Rhodes
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August 15, 2023
Irrational Jumps
A bouncing point moves counterclockwise along a circle with circumference $1$ with jumps of constant length $l \lt 1$, until it hits a gap of length $g...