2006_eng - language: English day: 1 13 July 2006 Problem 4....

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12 July 2006 Problem 1. Let ABC be a triangle with incentre I . A point P in the interior of the triangle satisﬁes 6 PBA + 6 PCA = 6 PBC + 6 PCB. Show that AP AI , and that equality holds if and only if P = I . Problem 2. Let P be a regular 2006-gon. A diagonal of P is called good if its endpoints divide the boundary of P into two parts, each composed of an odd number of sides of P . The sides of P are also called good . Suppose P has been dissected into triangles by 2003 diagonals, no two of which have a common point in the interior of P . Find the maximum number of isosceles triangles having two good sides that could appear in such a conﬁguration. Problem 3. Determine the least real number M such that the inequality ± ± ± ab ( a 2 - b 2 ) + bc ( b 2 - c 2 ) + ca ( c 2 - a 2 ) ± ± ± M ( a 2 + b 2 + c 2 ) 2 holds for all real numbers a , b and c . Time allowed: 4 hours 30 minutes Each problem is worth 7 points

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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.

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2006_eng - language: English day: 1 13 July 2006 Problem 4....

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