3

Part of 1992 Cono Sur Olympiad

Problems(2)

Find all values

Source: Cono Sur 1992-problem 3

Consider the set

S

100

1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}

. Any two numbers,

a

b

, are eliminated in

S

, and the number

a+b+ab

is added. Now, there are

99

S

. After doing this operation

99

times, there's only

1

S

. What values can this number take?

algebra unsolvedalgebra

Find conditions

Source: Cono Sur 1992-problem 6

m*n

board. On each box there's a non-negative integrer number assigned. An operation consists on choosing any two boxes with

1

side in common, and add to this

2

numbers the same integrer number (it can be negative), so that both results are non-negatives. What conditions must be satisfied initially on the assignment of the boxes, in order to have, after some operations, the number

0

combinatorics unsolvedcombinatorics