MathDB

3

Part of 2014 Iran MO (2nd Round)

Problems(2)

Old Inequality on Iran NMO

Source: Iran National Olympiad - 2014 Second Round - D1P3

5/1/2014
Let x,y,z x,y,z be three non-negative real numbers such that x2+y2+z2=2(xy+yz+zx).x^2+y^2+z^2=2(xy+yz+zx). Prove that x+y+z3≥2xyz3.\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.
inequalitiesquadraticsfunctionalgebraquadratic formulainequalities unsolved
Professionous Riddlous, society of 1+\sqrt{2n} members

Source: Iran National Olympiad - 2014 Second Round - D2P3

5/8/2014
Members of "Professionous Riddlous" society have been divided into some groups, and groups are changed in a special way each weekend: In each group, one of the members is specified as the best member, and the best members of all groups separate from their previous group and form a new group. If a group has only one member, that member joins the new group and the previous group will be removed. Suppose that the society has nn members at first, and all the members are in one group. Prove that a week will come, after which number of members of each group will be at most 1+2n1+\sqrt{2n}.
floor functionfunctioncombinatorics unsolvedcombinatorics