Expert Solutions | Senioritis

John Rhodes
August 16, 2023
Computational Mathematics

Paths to Equality

Define two functions on lattice points: $r(x,y) = (x+1,2y)$ $s(x,y) = (2x,y+1)$ A path to equality of length $n$ for a pair $(a,b)$ is a sequence $\Big((a_1,b_1),(a_2,b_2),\ldots,(a_n,b_n)\Big)$,...

John Rhodes
August 16, 2023
Computational Mathematics

Divisors of $2n^2$

Let $f(n)$ be the number of divisors of $2n^2$ that are no greater than n. For example, $f(15)=8$ because there are 8 such divisors: 1,2,3,5,6,9,10,15. Note that...

John Rhodes
August 16, 2023
Computational Mathematics

A Bit of Prime

The logical-OR of two bits is $0$ if both bits are $0$, otherwise it is $1$. The bitwise-OR of two positive integers performs a logical-OR operation on...

John Rhodes
August 16, 2023
Computational Mathematics

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
Computational Mathematics

$$A=\sum_{i=1}^{\infty} \frac{1}{3^i 10^{3^i}}$$ 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
Computational Mathematics

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
Computational Mathematics

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
Computational Mathematics

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...