Soal Olimpiade Matematika Internasional 2003

Posted: March 27, 2008 in Tak Berkategori
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44th IMO 2003

A1. S is the set {1, 2, 3, … , 1000000}. Show that for any subset A of S with 101 elements we can find 100 distinct elements xi of S, such that the sets xi + A are all pairwise disjoint. [Note that xi + A is the set {a + xi | a is in A} ].

A2. Find all pairs (m, n) of positive integers such that m2/(2mn2 – n3 + 1) is a positive integer.

A3. A convex hexagon has the property that for any pair of opposite sides the distance between their midpoints is (v3)/2 times the sum of their lengths. Show that all the hexagon’s angles are equal.

B1. ABCD is cyclic. The feet of the perpendicular from D to the lines AB, BC, CA are P, Q, R respectively. Show that the angle bisectors of ABC and CDA meet on the line AC iff RP = RQ.

B2. Given n > 2 and reals x1 = x2 = … = xn, show that (?i,j |xi – xj| )2 = (2/3) (n2 – 1) ?i,j (xi – xj)2. Show that we have equality iff the sequence is an arithmetic progression.

B3. Show that for each prime p, there exists a prime q such that np – p is not divisible by q for any positive integer n.

by: Denli

  1. Klik kata OlimpiadeMatematika di atas untuk mendapat soal-soal Olimpiade Matematika terbaru.

  2. terjemahin dongggg//upaya qta dapat mengertiiiiiiii……z minat banget 2 dgan soal2 yang kaya ginieeeee

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