MathDB

7

Part of 2001 IMO Shortlist

Problems(2)

IMO ShortList 2001, geometry problem 7

Source: IMO ShortList 2001, geometry problem 7

9/30/2004
Let OO be an interior point of acute triangle ABCABC. Let A1A_1 lie on BCBC with OA1OA_1 perpendicular to BCBC. Define B1B_1 on CACA and C1C_1 on ABAB similarly. Prove that OO is the circumcenter of ABCABC if and only if the perimeter of A1B1C1A_1B_1C_1 is not less than any one of the perimeters of AB1C1,BC1A1AB_1C_1, BC_1A_1, and CA1B1CA_1B_1.
geometrycircumcircleperimeterperpendicularIMO Shortlist
IMO ShortList 2001, combinatorics problem 7

Source: IMO ShortList 2001, combinatorics problem 7

9/30/2004
A pile of nn pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a final configuration. For each nn, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of nn. [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=119189]IMO ShortList 2001, combinatorics problem 7, alternative
IMO ShortlistcombinatoricsgameinvariantHi