MathDB

4

Part of 2011 JBMO Shortlist

Problems(4)

Maximum of xy

Source: JBMO 2011 Shortlist A4

5/15/2016
A4\boxed{\text{A4}} Let x,yx,y be positive reals satisfying the condition x3+y3≤x2+y2x^3+y^3\leq x^2+y^2.Find the maximum value of xyxy.
inequalities
swaps with balls

Source: JBMO 2011 Shortlist C4

10/14/2017
In a group of nn people, each one had a different ball. They performed a sequence of swaps, in each swap, two people swapped the ball they had at that moment. Each pair of people performed at least one swap. In the end each person had the ball he/she had at the start. Find the least possible number of swaps, if: a) n=5n = 5, b) n=6n = 6.
JBMOcombinatorics
2011 JBMO Shortlist G4

Source: 2011 JBMO Shortlist G4

10/8/2017
Point D{D} lies on the side BC{BC} of △ABC\vartriangle ABC. The circumcenters of △ADC\vartriangle ADC and △BAD\vartriangle BAD are O1{O_1} and O2{O_2}, respectively and O1O2∥AB{O_1O_2\parallel AB}. The orthocenter of △ADC\vartriangle ADCis H{H} and AH=O1O2.{AH=O_1O_2}. Find the angles of △ABC\vartriangle ABC if 2m(∠C)=3m(∠B).2m\left( \angle C \right)=3m\left( \angle B \right).
geometryJBMO
Prime equation

Source: JBMO 2011 Shortlist

5/15/2016
N4\boxed{\text{N4}} Find all primes p,qp,q such that 2p3−q2=2(p+q)22p^3-q^2=2(p+q)^2.
number theory