Problems(2)
\phi(n)
Source: 2nd international zhautykov olympiad of mathematics and phys, Problem 1
1/22/2006
Solve in positive integers the equation
n \equal{} \varphi(n) \plus{} 402 ,
where is the number of positive integers less than having no common prime factors with .
inequalitiesfunctionnumber theorynumber theory unsolved
Pile with 100 stones
Source: Kazakhstan international contest 2006, Problem 4
1/22/2006
In a pile you have 100 stones. A partition of the pile in piles is good if:
1) the small piles have different numbers of stones;
2) for any partition of one of the small piles in 2 smaller piles, among the k \plus{} 1 piles you get 2 with the same number of stones (any pile has at least 1 stone).
Find the maximum and minimal values of for which this is possible.
combinatorics proposedcombinatorics