MathDB

1

Part of 2007 Moldova Team Selection Test

Problems(4)

Inequality with areas

Source: Moldova 2007 IMO-BMO TST I problem 1

3/5/2007
Let ABCABC be a triangle and M,N,PM,N,P be the midpoints of sides BC,CA,ABBC, CA, AB. The lines AM,BN,CPAM, BN, CP meet the circumcircle of ABCABC in the points A1,B1,C1A_{1}, B_{1}, C_{1}. Show that the area of triangle ABCABC is at most the sum of areas of triangles BCA1,CAB1,ABC1BCA_{1}, CAB_{1}, ABC_{1}.
inequalitiesgeometrycircumcircletrigonometryfunctiontriangle inequalitygeometry proposed
it's quite hard for me to characterize this..

Source: Moldova 2007 IMO-BMO TST III problem 1

3/24/2007
Let a1,a2,…,an∈[0;1]a_{1}, a_{2}, \ldots, a_{n}\in [0;1]. If S=a13+a23+…+an3S=a_{1}^{3}+a_{2}^{3}+\ldots+a_{n}^{3} then prove that a12n+1+S−a13+a22n+1+S−a23+…+an2n+1+S−an3≤13\frac{a_{1}}{2n+1+S-a_{1}^{3}}+\frac{a_{2}}{2n+1+S-a_{2}^{3}}+\ldots+\frac{a_{n}}{2n+1+S-a_{n}^{3}}\leq \frac{1}{3}
inequalitiesinequalities proposed
Sum of consecutive squares or cubes is a square or cube

Source: Moldova 2007 IMO-BMO TST II problem 1

3/23/2007
Find the least positive integers m,km,k such that a) There exist 2m+12m+1 consecutive natural numbers whose sum of cubes is also a cube. b) There exist 2k+12k+1 consecutive natural numbers whose sum of squares is also a square. The author is Vasile Suceveanu
geometry3D geometryalgebrafactorizationsum of cubesnumber theory proposednumber theory
A finite number of parabolas do not cover the entire plane

Source: Moldova 2007 IMO-BMO TST IV problem 1

3/25/2007
Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.
conicsparabolasymmetryellipsehyperbolageometry proposedgeometry