MathDB

4

Part of 2016 JBMO Shortlist

Problems(3)

Inequality

Source: JBMO 2016 shortlist

6/25/2017
If the non-negative reals x,y,zx,y,z satisfy x2+y2+z2=x+y+zx^2+y^2+z^2=x+y+z. Prove that x+1x5+x+1+y+1y5+y+1+z+1z5+z+1≥3.\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3. When does the equality occur?
Proposed by Dorlir Ahmeti, Albania
JBMO ShortlistInequality
splitting of planar polygons, largest integer of triangles

Source: JBMO 2016 Shortlist C4

10/14/2017
A splitting of a planar polygon is a fi nite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer n≥3n \ge 3, determine the largest integer mm such that no planar nn-gon splits into less than mm triangles.
JBMOcombinatoricsgraph theoryplanar graphtriangulation
2016 JBMO Shortlist G4

Source: 2016 JBMO Shortlist G4

10/8/2017
Let ABC{ABC} be an acute angled triangle whose shortest side is BC{BC}. Consider a variable point P{P} on the side BC{BC}, and let D{D} and E{E} be points on AB{AB} and AC{AC}, respectively, such that BD=BP{BD=BP} and CP=CE{CP=CE}. Prove that, as P{P} traces BC{BC}, the circumcircle of the triangle ADE{ADE} passes through a fixed point.
geometryJBMO