7

Part of 2008 Tuymaada Olympiad

Problems(2)

Exactly one of the numbers a+b and |a-b| belongs to X

Source: Tuymaada 2008, Senior League, Second Day, Problem 6.

X

of positive integers is called nice if for each pair

a

b\in X

exactly one of the numbers a \plus{} b and |a \minus{} b| belongs to

X

a

b

may be equal). Determine the number of nice sets containing the number 2008. Author: Fedor Petrov

combinatorics unsolvedcombinatorics

A loader has two carts

Source: Tuymaada 2008, Senior League, Second Day, Problem 7.

A loader has two carts. One of them can carry up to 8 kg, and another can carry up to 9 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 17 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry on his two carts, regardless of particular weights of sacks? Author: M.Ivanov, D.Rostovsky, V.Frank

algebra unsolvedalgebra