MathDB

1

Part of 2014 Iran MO (2nd Round)

Problems(2)

Stuff Baskets everyone.

Source: Iran National Olympiad - 2014 Second Round - D1P1

5/1/2014
A basket is called "Stuff Basket" if it includes 1010 kilograms of rice and 3030 number of eggs. A market is to distribute 100100 Stuff Baskets. We know that there is totally 10001000 kilograms of rice and 30003000 number of eggs in the baskets, but some of market's baskets include either more or less amount of rice or eggs. In each step, market workers can select two baskets and move an arbitrary amount of rice or eggs between selected baskets. Starting from an arbitrary situation, what's the minimum number of steps that workers provide 100100 Stuff Baskets?
combinatorics unsolvedcombinatorics
Equation n^{n^{n}}=m^{m}

Source: Iran National Olympiad - 2014 Second Round - D2P1

5/2/2014
Find all positive integers (m,n)(m,n) such that nnn=mm.n^{n^{n}}=m^{m}.
number theory proposednumber theory