hw8 - T = ( x-y,y-z,z-x ), where x , y and z are complex...

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Putnam Exam Seminar Assignment 8 Fall 2010 Due November 1 Exercise 1. Find all pairs of real numbers ( x,y ) satisfying the system of equations 1 x + 1 2 y = ( x 2 + 3 y 2 )(3 x 2 + y 2 ) 1 x - 1 2 y = 2( y 4 - x 4 ) . [Putnam Exam, 2001, B-2] Exercise 2. Assume that x , y and z are all positive real numbers that satisfy the system of equations x + y + xy = 8 y + z + yz = 15 z + x + xz = 35 . Determine the value of x + y + z + xyz . Exercise 3. Find all quadruples of real numbers ( x 1 ,x 2 ,x 3 ,x 4 ) such that the sum of any one and the product of the other three is equal to 2. Exercise 4. Prove that there are only a finite number of possibilities for the ordered triple
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Unformatted text preview: T = ( x-y,y-z,z-x ), where x , y and z are complex numbers satisfying the simultaneous equations x ( x-1) + 2 yz = y ( y-1) + 2 zx = z ( z-1) + 2 xy, and list all such triples T . [Putnam Exam, 1986, B-2] Exercise 5. Find all positive integers n,k 1 ,...k n such that k 1 + k n = 5 n-4 1 k 1 + 1 k n = 1 . [Putnam Exam, 2005, B-2]...
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