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MathMagic Cycle 3 Problems - GOOD LUCK!

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**** MathMagic Problem Cycle 3: Level 10-12 Regular ***** 

A diophantine equation is an equation in which only integer
solutions are desired. Example: a solution to 3x + 4y = 11 is 
x= 1 and y = 2 since  3(1) + 4(2) = 11 

1)  Find TWO sets of solutions to the diophantine equations and 
    explain your solution strategy.     

    a)   5x + 4y = 32     b)   6x + 4y = 18 

2)  A school ordered books and calculators. The books cost $8 
    each and the calculators cost $10 each.     

    a)  Write an equation which can be used to represent the cost 
    of the books and calculators is $74 was spent.     

    b)  How many books and how many calculators could the school 
    have purchased?
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***** MathMagic Problem Cycle 3: Level 10-12 Advanced ***** 

Radical Numbers 
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1)  Solve for x: x^2 + x - 1 = 0.  (x^2 means x to the power of 2) 
    The positive result is called the Golden Mean. What is the value 
    of the reciprocal of this number? Compare the numbers in decimal 
    form using a calculator. What do you notice? 

2)  What is the value of  \| (2 + \| (2 + \| (2 +... ))) 
    where \| means square root and they are nested, one square root 
    inside the other? 

    How about the value of  \| (6 + \| (6 + \| (6 + ... ))) ?