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2004_eng - Problem 4 Let n ≥ 3 be an integer Let t 1,t...

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45rd IMO 2004 Problem 1 . Let ABC be an acute-angled triangle with AB = AC . The circle with diameter BC intersects the sides AB and AC at M and N respectively. Denote by O the midpoint of the side BC . The bisectors of the angles BAC and MON intersect at R . Prove that the circumcircles of the triangles BMR and CNR have a common point lying on the side BC . Problem 2 . Find all polynomials f with real coefficients such that for all reals a , b , c such that ab + bc + ca = 0 we have the following relations f ( a - b ) + f ( b - c ) + f ( c - a ) = 2 f ( a + b + c ) . Problem 3 . Define a ”hook” to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. Determine all m × n rectangles that can be covered without gaps and without overlaps with hooks such that the rectangle is covered without gaps and without overlaps no part of a hook covers area outside the rectagle.
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Unformatted text preview: Problem 4 . Let n ≥ 3 be an integer. Let t 1 ,t 2 ,...,t n be positive real numbers such that n 2 + 1 > ( t 1 + t 2 + ... + t n ) ± 1 t 1 + 1 t 2 + ... + 1 t n ² . Show that t i ,t j ,t k are side lengths of a triangle for all i , j , k with 1 ≤ i < j < k ≤ n . Problem 5 . In a convex quadrilateral ABCD the diagonal BD does not bisect the angles ABC and CDA . The point P lies inside ABCD and satisfies 6 PBC = 6 DBA and 6 PDC = 6 BDA. Prove that ABCD is a cyclic quadrilateral if and only if AP = CP . 1 Problem 6 . We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity. Find all positive integers n such that n has a multiple which is alternating. 2...
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