M135F08A1 - MATH 135 Assignment #1 Fall 2008 Due: Wednesday...

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MATH 135 Fall 2008 Assignment #1 Due: Wednesday 17 September 2008, 8:20 a.m. N.B. Assignments 3 to 9 will not be distributed in class. You must download them from the course Web site. Hand-In Problems 1. Disprove the statement “There is no positive integer n > 3 such that n 2 + ( n + 1) 2 is a perfect square”. 2. (a) Treating the equation 4 x 2 + 4 xy + 2 y 2 - 4 x - 2 y + 1 = 0 as a quadratic equation in x with coefficients in terms of y , solve for x . (b) Disprove the statement “For all real numbers x and y , 4 x 2 +4 xy +2 y 2 - 4 x - 2 y +1 6 = 0”. 3. Prove that sin 4 x + 2 cos 2 x - cos 4 x = 1 for all x . 4. Prove that if 2 x 5, then - 2 x 2 - 3 x 10. 5. Any positive integer n with units digit 5 can be written in the form n = 10 k + 5, for some non-negative integer k . Prove that if n is a positive integer with units digit 5, then n 2 ends with 25. 6. Prove that if n is a positive integer with n 2, then n 3 - n is always divisible by 6. 7. To prove a statement of the form “
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M135F08A1 - MATH 135 Assignment #1 Fall 2008 Due: Wednesday...

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