Algebra

# Algebra Warmups: Level 4 Challenges

Given that

$\frac{ \color{red}{a}}{\color{blue}{b}+\color{orange}{c}} + \frac{\color{blue}{b}} {\color{red}{a}+\color{orange}{c}} + \frac{ \color{orange}{c}}{\color{red}{a}+\color{blue}{b}} = 1,$

find the value of

$\large \frac{ \color{red}{a}^2}{\color{blue}{b}+\color{orange}{c}} + \frac{ \color{blue}{b}^2}{\color{red}{a}+\color{orange}{c}} + \frac{ \color{orange}{c}^2} { \color{red}{a}+\color{blue}{b}}.$

Find the sum of all solutions to the equation

$\large (x^2+5x+5)^{x^2-10x+21}=1 .$

A polynomial $$f(x)$$ satisfies the equation $$f(x)+(x+1)^3=2f(x+1)$$. Find $$f(10)$$.

Let $$f(x)$$ be a quintic polynomial such that

$\begin{array} { r l } f(1) & = 1 \\ f(2) & = 1 \\ f(3) & = 2 \\ f(4) & = 3 \\ f(5) & = 5 \\ f(6) & = 8. \\ \end{array}$

Determine $$f(7)$$.


Note: Many people are answering this incorrectly because they think it is the Fibonacci sequence, but this problem is asking about a quintic polynomial that passes through those points. That does not necessarily mean the next term behaves as the Fibonacci sequence would.

$\begin{eqnarray} |a - b | &=& 2 \\ |b - c | &=& 3 \\ |c - d | &=& 4 \\ \end{eqnarray}$

Given that $$a,b,c,d$$ are real numbers that satisfy the system of equations above, what is the sum of all distinct values of $$|a-d|$$?

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