Problemset 02

Prepared by Chams Eddine Abdelali Derreche

Problem 1

For each positive integer , find the number of -digit positive integers that satisfy both of the following conditions:

• no two consecutive digits are equal, and
• the last digit is a prime.

Problem 2

Determine the smallest positive integer with the following property. For any subset of the set with , there are two distinct elements such that is a perfect square.

Problem 3

We call a tuple of integers arrangeable if its elements can be labeled in some order so that . Determine all tuples of integers such that if we place them in a circle in clockwise order, then any tuple of numbers in consecutive positions on the circle is arrangeable.

Problem 4

Let be a positive integer such that is divisible by 6. An board is given. Prove that it is possible to place non-overlapping right triangles on the board with the lengths of .

Problem 5

In each square of a garden shaped like a board, there is initially a tree of height 0 (A real tree dear programmers !). A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:

• The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller.
• The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter.

We say that a tree is majestic if its height is at least . Determine the largest such that the gardener can ensure that there are eventually majestic trees on the board, no matter how the lumberjack plays.