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# Algebra Warmups

From cracking cryptograms to calculating the top speed of a rocket, algebra gives you tools to apply mathematical reasoning to a wide range of problems. Dive in and see what you already know!

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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