MathDB

2

Part of 2011 JBMO Shortlist

Problems(3)

Σ(x + 2y)(z + 2x + 3y) <= 3/2

Source: JBMO 2011 Shortlist A2

10/14/2017
Let x,y,zx, y, z be positive real numbers. Prove that: x+2yz+2x+3y+y+2zx+2y+3z+z+2xy+2z+3x≤32\frac{x + 2y}{z + 2x + 3y}+\frac{y + 2z}{x + 2y + 3z}+\frac{z + 2x}{y + 2z + 3x} \le \frac{3}{2}
JBMOalgebrainequalities
2011 JBMO Shortlist G2

Source: 2011 JBMO Shortlist G2

10/8/2017
Let AD,BFAD,BF and CE{CE} be the altitudes of △ABC\vartriangle ABC. A line passing through D{D} and parallel to AB{AB}intersects the line EF{EF}at the point G{G}. If H{H} is the orthocenter of △ABC\vartriangle ABC, find the angle ∠CGH{\angle{CGH}}.
geometryJBMO
splitting an equilateral into 2011 triangles with 122 lines

Source: JBMO 2011 Shortlist C2

10/14/2017
Can we divide an equilateral triangle △ABC\vartriangle ABC into 20112011 small triangles using 122122 straight lines? (there should be 20112011 triangles that are not themselves divided into smaller parts and there should be no polygons which are not triangles)
JBMOcombinatorics