5

Part of 2008 Tuymaada Olympiad

Problems(2)

A loader has a waggon and a little cart

Source: Tuymaada 2008, Junior League, Second Day, Problem 5.

A loader has a waggon and a little cart. The waggon can carry up to 1000 kg, and the cart can carry only up to 1 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 1001 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry in the waggon and the cart, regardless of particular weights of sacks? Author: M.Ivanov, D.Rostovsky, V.Frank

algebra unsolvedalgebra

Every street in Hamiltonville connects two squares

Source: Tuymaada 2008, Senior League, Second Day, Problem 5.

Every street in the city of Hamiltonville connects two squares, and every square may be reached by streets from every other. The governor discovered that if he closed all squares of any route not passing any square more than once, every remained square would be reachable from each other. Prove that there exists a circular route passing every square of the city exactly once. Author: S. Berlov

graph theorycombinatorics unsolvedcombinatorics