1

Part of 1993 Cono Sur Olympiad

Problems(2)

On a table there is a pile with 2001 tokens

Source: Cono Sur 1993-problem 1, Germany Bundeswettbewerb Mathematik 2001, Round 1, Problem 1

On a table there is a pile with

T

tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process? a.) T \equal{} 1000 (Cono Sur) b.) T \equal{} 2001 (BWM)

algebra unsolvedalgebra

Source: Cono Sur 1993-problem 4

On a chess board (

8*8

) there are written the numbers

1

64

: on the first line, from left to right, there are the numbers

1, 2, 3, ... , 8

; on the second line, from left to right, there are the numbers

9, 10, 11, ... , 16

;etc. The \"+\" and \"-\" signs are put to each number such that, in each line and in each column, there are

4

\"+\" signs and

4

\"-\" signs. Then, the

64

numbers are added. Find all the possible values of this sum.

combinatorics unsolvedcombinatorics