A Rectangular Tiling
We wish to tile a rectangle whose length is twice its width. Let $T(0)$ be the tiling consisting of a single rectangle. For $n \gt 0$, let...
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John Rhodes
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August 15, 2023
Crisscross Ellipses
$E_a$ is an ellipse with an equation of the form $x^2 + 4y^2 = 4a^2$. $E_a^\prime$ is the rotated image of $E_a$ by $\theta$ degrees counterclockwise around...
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John Rhodes
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August 15, 2023
Lattice Points Enclosed by Parabola and Line
For integers $a$ and $b$, we define $D(a, b)$ as the domain enclosed by the parabola $y = x^2$ and the line $y = a\cdot x +...
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John Rhodes
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August 15, 2023
Integer-valued Polynomials
It can be shown that the polynomial $n^4 + 4n^3 + 2n^2 + 5n$ is a multiple of $6$ for every integer $n$. It can also be...
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John Rhodes
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August 15, 2023
Sum of Squares of Divisors
The divisors of $6$ are $1,2,3$ and $6$. The sum of the squares of these numbers is $1+4+9+36=50$. Let $\operatorname{sigma}_2(n)$ represent the sum of the squares of...
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John Rhodes
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August 15, 2023
Fibonacci Tree Game
A Fibonacci tree is a binary tree recursively defined as:$T(0)$ is the empty tree. $T(1)$ is the binary tree with only one node. $T(k)$ consists of a...
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John Rhodes
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August 15, 2023
Cutting Rope
Inside a rope of length $n$, $n – 1$ points are placed with distance $1$ from each other and from the endpoints. Among these points, we choose...
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John Rhodes
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August 15, 2023
Triangle on Parabola
On the parabola $y = x^2/k$, three points $A(a, a^2/k)$, $B(b, b^2/k)$ and $C(c, c^2/k)$ are chosen. Let $F(K, X)$ be the number of the integer quadruplets...