MathDB

3

Part of 1989 Polish MO Finals

Problems(2)

Labeling edges of a cube

Source: Problem 3, Polish NO 1989

10/1/2005
The edges of a cube are labeled from 11 to 1212. Show that there must exist at least eight triples (i,j,k)(i, j, k) with 1≤i<j<k≤121 \leq i < j < k \leq 12 so that the edges i,j,ki, j, k are consecutive edges of a path. Also show that there exists labeling in which we cannot find nine such triples.
geometry3D geometrycombinatorics unsolvedcombinatorics
Inequality in 4 variables

Source: Problem 6, Polish NO 1989

10/1/2005
Show that for positive reals a,b,c,da, b, c, d we have (ab+ac+ad+bc+bd+cd6)3≥(abc+abd+acd+bcd4)2 \left(\dfrac{ab + ac + ad + bc + bd + cd}{6} \right)^3 \geq \left(\dfrac{abc + abd + acd + bcd}{4}\right)^2
inequalitiesinequalities unsolved