MathDB

3

Part of 1999 Vietnam Team Selection Test

Problems(2)

clockwise sides of convex polygon

Source: Vietnam TST 1999 for the 40th IMO, problem 3

6/26/2005
Let a convex polygon HH be given. Show that for every real number a∈(0,1)a \in (0, 1) there exist 6 distinct points on the sides of HH, denoted by A1,A2,…,A6A_1, A_2, \ldots, A_6 clockwise, satisfying the conditions: I. (A1A2)=(A5A4)=a⋅(A6A3)(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3). II. Lines A1A2,A5A4A_1A_2, A_5A_4 are equidistant from A6A3A_6A_3. (By (AB)(AB) we denote vector ABAB)
vectorgeometryparallelogramrotationgeometry unsolved
Monkeys are going to pick up peanuts

Source: Vietnam TST 1999 for the 40th IMO, problem 6

6/26/2005
Let a regular polygon with pp vertices be given, where pp is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes pp peanuts, goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanut for the leader, skips the two next vertices and gives the second peanut to the monkey at the next vertex; skip four next vertices gives the second peanut for the monkey at the next vertex ... after giving the kk-th peanut, he skips the 2⋅k2 \cdot k next vertices and gives k+1k+1-th for the monkey at the next vertex. He does so until all pp peanuts are delivered. I. How many monkeys are there which does not receive peanuts? II. How many edges of polygon are there which satisfying condition: both two monkey at its vertex received peanut(s)?
geometryperimetercombinatorics unsolvedcombinatorics