Triangle of Circular Arcs
Let $r_a$, $r_b$ and $r_c$ be the radii of three circles that are mutually and externally tangent to each other. The three circles then form a triangle...
-
John Rhodes
-
August 16, 2023
Falling Bottles
Consider a stack of bottles of wine. There are $n$ layers in the stack with the top layer containing only one bottle and the bottom layer containing...
-
John Rhodes
-
August 16, 2023
Digit Sum Numbers
A number where one digit is the sum of the other digits is called a digit sum number or DS-number for short. For example, $352$, $3003$ and...
-
John Rhodes
-
August 16, 2023
Drone Delivery
A depot uses $n$ drones to disperse packages containing essential supplies along a long straight road. Initially all drones are stationary, loaded with a supply package. Every...
-
John Rhodes
-
August 16, 2023
Slowly Converging Series
For a non-negative integer $k$, define \[ E_k(q) = \sum\limits_{n = 1}^\infty \sigma_k(n)q^n \] where $\sigma_k(n) = \sum_{d \mid n} d^k$ is the sum of the $k$-th...
-
John Rhodes
-
August 16, 2023
High Powers of Irrational Numbers
Given is the function $f(a,n)=\lfloor (\lceil \sqrt a \rceil + \sqrt a)^n \rfloor$. $\lfloor \cdot \rfloor$ denotes the floor function and $\lceil \cdot \rceil$ denotes the ceiling...
-
John Rhodes
-
August 16, 2023
Unpredictable Permutations
Consider all permutations of $\{1, 2, \ldots N\}$, listed in lexicographic order.For example, for $N=4$, the list starts as follows: $$\displaylines{ (1, 2, 3, 4) \\ (1,...
-
John Rhodes
-
August 16, 2023
Number Splitting
We define an $S$-number to be a natural number, $n$, that is a perfect square and its square root can be obtained by splitting the decimal representation...