Algebra

Algebra Warmups

Algebra Warmups: Level 4 Challenges

         

Given that

ab+c+ba+c+ca+b=1, \frac{ \color{#D61F06}{a}}{\color{#3D99F6}{b}+\color{#EC7300}{c}} + \frac{\color{#3D99F6}{b}} {\color{#D61F06}{a}+\color{#EC7300}{c}} + \frac{ \color{#EC7300}{c}}{\color{#D61F06}{a}+\color{#3D99F6}{b}} = 1,

find the value of

a2b+c+b2a+c+c2a+b.\large \frac{ \color{#D61F06}{a}^2}{\color{#3D99F6}{b}+\color{#EC7300}{c}} + \frac{ \color{#3D99F6}{b}^2}{\color{#D61F06}{a}+\color{#EC7300}{c}} + \frac{ \color{#EC7300}{c}^2} { \color{#D61F06}{a}+\color{#3D99F6}{b}}.

Find the sum of all solutions to the equation

(x2+5x+5)x210x+21=1. \large (x^2+5x+5)^{x^2-10x+21}=1 .

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

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

f(1)=1f(2)=1f(3)=2f(4)=3f(5)=5f(6)=8. \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) 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.

ab=2bc=3cd=4 \begin{aligned} |a - b | &=& 2 \\ |b - c | &=& 3 \\ |c - d | &=& 4 \\ \end{aligned}

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

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