Math Thread

$2 coin

Three integers

x, y and z are integers. Given the two equations:

x² + y² + z² = 2(yz + 1)

x + y + z = 4030

Find all possible values of x, y, and z.

10 bucks from coins

How many ways can you make exactly £10 using only 5p, 10p and 20p pieces?

Recursive Polynomial Sequence

A sequence of polynomials an(x) are defined recursively by

a₀( x ) = 1
a₁( x ) = x² + x + 1
a_n( x ) = (xⁿ + 1)a_n-1( x ) - a_n-2( x ), for all n > 1

For example,

a₂( x ) = (x² + 1)(x² + x + 1) - 1 = x⁴ + x³ + 2x² + x
a₃( x ) = (x³ + 1)(x⁴ + x³ + 2x² + x) - (x² + x + 1)
= x⁷ + x⁶ + 2x⁵ + 2x⁴ + x³ + x² - 1

Evaluate a₂₀₁₅(1)

Ninth Degree Polynomial

The polynomial p(x) is of degree 9 and p(x) − 1 is exactly divisible by (x − 1)⁵.

1. Find the value of p(1).
2. Show that p'(x) is exactly divisible by (x − 1)⁴.
3. Given also that p(x) + 1 is exactly divisible by (x + 1)⁵, find p(x)

Rocket Engine Repairs

Satellites are launched using two different types of rocket: the Andover and the Basingstoke. The Andover has four engines and the Basingstoke has six. Each engine has a probability p of failing during any given launch. After the launch, the rockets are retrieved and repaired by replacing some or all of the engines. The cost of replacing each engine is K.For the Andover, if more than one engine fails, all four engines are replaced. Otherwise, only the failed engine (if there is one) is replaced.

a: Show that the expected repair cost for a single launch using the Andover is:

4Kp(1 + q + q2 − 2q3) (*)

Note: (q = 1 − p)

For the Basingstoke, if more than two engines fail, all six engines are replaced. Otherwise only the failed engines (if there are any) are replaced.

b: Find, in a form similar to (*), the expected repair cost for a single launch using the Basingstoke.

c: Find the values of p for which the expected repair cost for the Andover is 2/3 of the expected repair cost for the Basingstoke.