Expert Solutions | Senioritis

John Rhodes
August 16, 2023
Computational Mathematics

Quintinomial Coefficients

The coefficients in the expansion of $(x+1)^k$ are called binomial coefficients. Analoguously the coefficients in the expansion of $(x^4+x^3+x^2+x+1)^k$ are called quintinomial coefficients. (quintus= Latin for fifth)....

John Rhodes
August 16, 2023
Computational Mathematics

Nested Square Roots

Consider the term $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$ that is representing a nested square root. $x$, $y$ and $z$ are positive integers and $y$ and $z$ are not allowed to...

John Rhodes
August 16, 2023
Computational Mathematics

Birthday Problem Revisited

A long long time ago in a galaxy far far away, the Wimwians, inhabitants of planet WimWi, discovered an unmanned drone that had landed on their planet....

John Rhodes
August 16, 2023
Computational Mathematics

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...

John Rhodes
August 16, 2023
Computational Mathematics

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...

John Rhodes
August 16, 2023
Computational Mathematics

$47$-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...

John Rhodes
August 16, 2023
Computational Mathematics

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...

John Rhodes
August 16, 2023
Computational Mathematics

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...