3

Part of 2006 May Olympiad

Problems(2)

6 positive integers wanted

Source: XII May Olympiad (Olimpiada de Mayo) 2006 L2 P3

Write a positive integer in each box so that: All six numbers are different. The sum of the six numbers is

100

. If each number is multiplied by its neighbor (in a clockwise direction) and the six results of those six multiplications are added, the smallest possible value is obtained. Explain why a lower value cannot be obtained. https://cdn.artofproblemsolving.com/attachments/7/1/6fdadd6618f91aa03cdd6720cc2d6ee296f82b.gif

combinatoricsnumber theory

red points among 20 collinear points

Source: XII May Olympiad (Olimpiada de Mayo) 2006 L1 P3

20

collinear points, separated by the same distance:

. \,\,\, . \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\, .

Miguel has to paint three or more of these points red, in such a way that the red points are separated by the same distance and it is impossible to paint exactly one more point red without violating the previous condition. Determine in how many ways Miguel can do his homework.

Coloringcombinatorics