--------------------------------------------------------- MathMagic Cycle 3 Problems - GOOD LUCK! --------------------------------------------------------- **** 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? ---------------------------------------------------------------------- ***** MathMagic Problem Cycle 3: Level 10-12 Advanced ***** Radical Numbers --------------- 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 + ... ))) ?