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# quiz6 - x,y that satisfy the equation(4 x 2 6 x 4(4 y 2-12...

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Putnam Exam Seminar Quiz 6 Fall 2010 October 25 Problem 1. Find all ordered pairs of real numbers ( x,y ) which satisfy the system of equations x + y = 3 3 x + 2 y = 14 . Problem 2. Curves A , B , C and D are deﬁned in the plane as follows: A = ± ( x,y ) : x 2 - y 2 = x x 2 + y 2 ² , B = ± ( x,y ) : 2 xy + y x 2 + y 2 = 3 ² , C = ³ ( x,y ) : x 3 - 3 xy 2 + 3 y = 1 ´ , D = ³ ( x,y ) : 3 x 2 y - 3 x - y 3 = 0 ´ . Prove that A B = C D . [Putnam Exam, 1987, A-1] Problem 3. Show that there is a unique pair of real numbers (
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Unformatted text preview: x,y ) that satisfy the equation (4 x 2 + 6 x + 4)(4 y 2-12 y + 25) = 28 . Problem 4. Find all values of α for which the curves y = αx 2 + αx + 1 24 and x = αy 2 + αy + 1 24 are tangent to each other. [Putnam Exam, 2007, A-1]...
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