MathDB

5

Part of 2006 IMO Shortlist

Problems(4)

Italian WinterCamp test07 problem3

Source:

1/29/2007
If a,b,ca,b,c are the sides of a triangle, prove that b+c−ab+c−a+c+a−bc+a−b+a+b−ca+b−c≤3\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3
Proposed by Hojoo Lee, Korea
inequalitiesalgebraIMO ShortlistHi
(n,k) tournament

Source: IMO Shortlist 2006, Combinatorics 5, AIMO 2007, TST 7, P2

6/28/2007
An (n, k) \minus{} tournament is a contest with n n players held in k k rounds such that:
(i) (i) Each player plays in each round, and every two players meet at most once. (ii) (ii) If player A A meets player B B in round i i, player C C meets player D D in round i i, and player A A meets player C C in round j j, then player B B meets player D D in round j j.
Determine all pairs (n,k) (n, k) for which there exists an (n, k) \minus{} tournament.
Proposed by Carlos di Fiore, Argentina
combinatoricsgraph theoryExtremal combinatoricsTournament graphsIMO Shortlist
an excircle and determination of two angles

Source: IMO Shortlist 2006, Geometry 5

6/28/2007
In triangle ABCABC, let JJ be the center of the excircle tangent to side BCBC at A1A_{1} and to the extensions of the sides ACAC and ABAB at B1B_{1} and C1C_{1} respectively. Suppose that the lines A1B1A_{1}B_{1} and ABAB are perpendicular and intersect at DD. Let EE be the foot of the perpendicular from C1C_{1} to line DJDJ. Determine the angles ∠BEA1\angle{BEA_{1}} and ∠AEB1\angle{AEB_{1}}.
Proposed by Dimitris Kontogiannis, Greece
geometrycircumcircleperpendicular bisectorIMO ShortlistAngle Chasing
I Brazilian TST 2007 - Problem 4

Source: 2007 Brazil TST, Russia TST, and AIMO; also SL 2006 N5

3/11/2007
Find all integer solutions of the equation \frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.
number theoryIMO Shortlist