MathDB

7

Part of 1998 IMO Shortlist

Problems(3)

calculation !

Source: IMO ShortList 1998, geometry problem 7

10/15/2004
Let ABCABC be a triangle such that ∠ACB=2∠ABC\angle ACB=2\angle ABC. Let DD be the point on the side BCBC such that CD=2BDCD=2BD. The segment ADAD is extended to EE so that AD=DEAD=DE. Prove that ∠ECB+180∘=2∠EBC. \angle ECB+180^{\circ }=2\angle EBC.
trigonometrygeometrytrapezoidparallelogramIMO Shortlist
IMO ShortList 1998, number theory problem 7

Source: IMO ShortList 1998, number theory problem 7

10/22/2004
Prove that for each positive integer nn, there exists a positive integer with the following properties: It has exactly nn digits. None of the digits is 0. It is divisible by the sum of its digits.
number theorysum of digitsDigitsDivisibilityIMO Shortlist
IMO ShortList 1998, combinatorics theory problem 7

Source: IMO ShortList 1998, combinatorics theory problem 7

10/22/2004
A solitaire game is played on an m×nm\times n rectangular board, using mnmn markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs (m,n)(m,n) of positive integers such that all markers can be removed from the board.
rectanglecombinatoricsmatrixIMO Shortlistgameinvariant