MathDB

6

Part of 2001 IMO Shortlist

Problems(3)

IMO ShortList 2001, geometry problem 6

Source: IMO ShortList 2001, geometry problem 6

9/30/2004
Let ABCABC be a triangle and PP an exterior point in the plane of the triangle. Suppose the lines APAP, BPBP, CPCP meet the sides BCBC, CACA, ABAB (or extensions thereof) in DD, EE, FF, respectively. Suppose further that the areas of triangles PBDPBD, PCEPCE, PAFPAF are all equal. Prove that each of these areas is equal to the area of triangle ABCABC itself.
geometrylinear algebraareaTriangleIMO Shortlist
IMO ShortList 2001, number theory problem 6

Source: IMO ShortList 2001, number theory problem 6

9/30/2004
Is it possible to find 100100 positive integers not exceeding 25,00025,000, such that all pairwise sums of them are different?
modular arithmeticnumber theoryAdditive Number TheorysumsIMO Shortlist
IMO ShortList 2001, combinatorics problem 6

Source: IMO ShortList 2001, combinatorics problem 6

9/30/2004
For a positive integer nn define a sequence of zeros and ones to be balanced if it contains nn zeros and nn ones. Two balanced sequences aa and bb are neighbors if you can move one of the 2n2n symbols of aa to another position to form bb. For instance, when n=4n = 4, the balanced sequences 0110100101101001 and 0011010100110101 are neighbors because the third (or fourth) zero in the first sequence can be moved to the first or second position to form the second sequence. Prove that there is a set SS of at most 1n+1(2nn)\frac{1}{n+1} \binom{2n}{n} balanced sequences such that every balanced sequence is equal to or is a neighbor of at least one sequence in SS.
modular arithmeticIMO ShortlistcombinatoricsSequencegraph theory