MathDB

1

Part of 1996 IMO Shortlist

Problems(3)

Inequality 15

Source: IMO 1996 Shortlist

9/13/2003
Suppose that a,b,c>0a, b, c > 0 such that abc=1abc = 1. Prove that abab+a5+b5+bcbc+b5+c5+caca+c5+a5≤1. \frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1.
inequalitiesthree variable inequalityalgebraIMO Shortlist
Four integers are marked on a circle

Source: IMO Shortlist 1996, N1

8/9/2008
Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers a,b,c,d a,b,c,d are replaced by a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a. Is it possible after 1996 such to have numbers a,b,c,d a,b,c,d such the numbers |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd| are primes?
number theoryprime numbersgameinvariantIMO Shortlist
Triangle, orthocenter, parallels - prove that EX || AP

Source: IMO Shortlist 1996, G1

12/10/2005
Let ABC ABC be a triangle, and H H its orthocenter. Let P P be a point on the circumcircle of triangle ABC ABC (distinct from the vertices A A, B B, C C), and let E E be the foot of the altitude of triangle ABC ABC from the vertex B B. Let the parallel to the line BP BP through the point A A meet the parallel to the line AP AP through the point B B at a point Q Q. Let the parallel to the line CP CP through the point A A meet the parallel to the line AP AP through the point C C at a point R R. The lines HR HR and AQ AQ intersect at some point X X. Prove that the lines EX EX and AP AP are parallel.
geometrycircumcirclereflectionvectorparallelogramIMO Shortlistmoving points