Ascending Subsequences
Let $a_i$ be the sequence defined by $a_i=153^i \bmod 10\,000\,019$ for $i \ge 1$. The first terms of $a_i$ are: $153, 23409, 3581577, 7980255, 976697, 9434375, \dots$...
-
John Rhodes
-
August 16, 2023
A Stoneham Number
$$A=\sum_{i=1}^{\infty} \frac{1}{3^i 10^{3^i}}$$ Define $A(n)$ to be the $10$ decimal digits from the $n$th digit onward. For example, $A(100) = 4938271604$ and $A(10^8)=2584642393$. Find $A(10^{16})$. To find...
-
John Rhodes
-
August 16, 2023
Shifted Pythagorean Triples
For a non-negative integer $k$, the triple $(p,q,r)$ of positive integers is called a $k$-shifted Pythagorean triple if $$p^2 + q^2 + k = r^2$$ $(p, q,...
-
John Rhodes
-
August 16, 2023
Range of Periodic Sequence
Consider the sequence of real numbers $a_n$ defined by the starting value $a_0$ and the recurrence $\displaystyle a_{n+1}=a_n-\frac 1 {a_n}$ for any $n \ge 0$. For some...
-
John Rhodes
-
August 16, 2023
Circle of Coins
Consider $n$ coins arranged in a circle where each coin shows heads or tails. A move consists of turning over $k$ consecutive coins: tail-head or head-tail. Using...
-
John Rhodes
-
August 16, 2023
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...