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Unformatted text preview: language: English day: 1 13 July 2006 Problem 4. Determine all pairs ( x, y ) of integers such that 1 + 2 x + 2 2 x +1 = y 2 . Problem 5. Let P ( x ) be a polynomial of degree n > 1 with integer coeﬃcients and let k be a positive integer. Consider the polynomial Q ( x ) = P ( P ( . . . P ( P ( x )) . . . )), where P occurs k times. Prove that there are at most n integers t such that Q ( t ) = t . Problem 6. Assign to each side b of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P . Show that the sum of the areas assigned to the sides of P is at least twice the area of P . Time allowed: 4 hours 30 minutes Each problem is worth 7 points language: English day: 2...
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This note was uploaded on 09/07/2011 for the course RESEARCH R 101 taught by Professor T.s. during the Fall '11 term at Research College.
 Fall '11
 T.S.

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