▪ Italian Mathematical Olympiad (ITAMO) 2012

1. On the sides of a triangle right angled at three points and (respectively

and ) are chosen so that the quadrilateral is a square. If is the length of the side of the square, show that

2. Determine all positive integers that are equal to times the sum of their digits. 

3. Let be an integer greater than or equal to . There are people in one line, each of which is either a scoundrel (who always lie) or a knight (who always tells the truth). Every person, except the first, indicates a person in front of him/her and says "This person is a scoundrel" or "This person is a knight." Knowing that there are strictly more scoundrel than knights, seeing the statements show that it is possible to determine each person whether he/she is a scoundrel or a knight.

4. Let be a sequence defined by the following recurrence relation:

The first few terms of the sequence are

Find all pairs of positive integers such that is a perfect square. 

5. is a square. Describe the locus of points , different from , on that plane for which

6. Determine all pairs of positive integers with the property that, in whatever manner you color the positive integers with two colors and , there always exist two positive integers of color having their difference equal to or of color having their difference equal to . 

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