MathDB

3

Part of 2002 All-Russian Olympiad

Problems(5)

18 colored points

Source: 2002 All-Russian MO, Grade 9, Problem 3

1/2/2012
On a plane are given 66 red, 66 blue, and 66 green points, such that no three of the given points lie on a line. Prove that the sum of the areas of the triangles whose vertices are of the same color does not exceed quarter the sum of the areas of all triangles with vertices in the given points.
geometrycombinatorics unsolvedcombinatorics
Geometric inequality

Source: All Rusian math olymp. 2002

1/21/2009
Let O be the circumcenter of a triangle ABC. Points M and N are choosen on the sides AB and BC respectively so that the angle AOC is two times greater than angle MON. Prove that the perimeter of triangle MBN is not less than the lenght of side AC
inequalitiesgeometrycircumcircleRussia
Sum of two squares for n>10000

Source: All-Russian MO 2002 Grade 10 #3

1/2/2012
Prove that for every integer n>10000n > 10000 there exists an integer mm such that it can be written as the sum of two squares, and 0<m−n<3n40<m-n<3\sqrt[4]n.
algebra unsolvedalgebra
Concurrent point through Excircle tangency points

Source: All-Russian MO 2002 Grade 10 #7

1/2/2012
Let A′A^\prime be the point of tangency of the excircle of a triangle ABCABC (corrsponding to AA) with the side BCBC. The line aa through A′A^\prime is parallel to the bisector of ∠BAC\angle BAC. Lines bb and cc are analogously defined. Prove that a,b,ca, b, c have a common point.
geometry unsolvedgeometry
Trig inequality for n&gt;m natural numbers

Source: All-Russian MO 2002 Grade 11 #3

1/2/2012
Prove that if 0<x<π20<x<\frac{\pi}{2} and n>mn>m, where nn,mm are natural numbers, 2∣sin⁡nx−cos⁡nx∣≤3∣sin⁡mx−cos⁡mx∣. 2 \left| \sin^n x - \cos^n x \right| \le 3 \left| \sin^m x - \cos^m x \right|.
trigonometryinequalitiesfunctioninequalities unsolved