**A. 596.** Find all integers *k*3 with the property that for every infinite sequence *P*_{1},*P*_{2},... of points in the plane such that no three of them are collinear, there exists a sequence of integers for which the points , in that order, form a convex *k*-gon.

Proposed by: *László Surányi,* Budapest

(5 points)

**A. 597.** The circles *k*_{0}, *k*_{1}, *k*_{2}, *k*_{3} and *k*_{4} lie in the plane in such a way that for *i*=1,2,3,4 the circle *k*_{i} is externally tangent to *k*_{0} at point *T*_{i}, and *k*_{i} is externally tangent to *k*_{i+1} at point *S*_{i} (*k*_{5}=*k*_{1}). Let *O* be the center of *k*_{0}. Let the lines *T*_{1}*T*_{3} and *T*_{2}*T*_{4} meet at *T*, and let the lines *S*_{1}*S*_{3} and *S*_{2}*S*_{4} meet at *S*. Prove that the points *O*, *T* and *S* are collinear.

Proposed by: *Márton Mester,* Cambridge

(5 points)

**A. 598.** Denote by *u*_{n} the *n*th Fibonacci number (*u*_{1}=*u*_{2}=1, *u*_{n+1}=*u*_{n}+*u*_{n-1}). Prove that if *a*,*b*,*c*>1 are integers such that *a* divides *u*_{b}, *b* divides *u*_{c} and *c* divides *u*_{a}, then 5 divides *a*, *b* and *c*, or 12 divides *a*, *b *and *c*.

(5 points)

**B. 4566.** The squares *ABDE*, *BCFG* and *CAHI* are drawn over the sides of a triangle *ABC*, on the outside. The triangles *DBG*, *FCI* and *HAE* are completed to form the parallelograms *DBGJ*, *FCIK* and *HAEL*. Prove that .

Suggested by *Sz. Miklós,* Herceghalom

(5 points)

**B. 4568.** There are *n* prisoners in a prison. Since the guards are bored, they invent the following game: Either a red hat or a blue hat is placed on the head of each prisoner such that no one can see the colour of the hat on their own heads. Then the prisoners are allowed to look at one another (everyone can see everyone else's hat). Finally, each of them guesses the colour of their own hat, and writes it down on a sheet of paper. If all answers are correct, the prisoners may go for a walk in the courtyard. What strategy should they agree on, so that the probability of a walk in the courtyard is a maximum?

(5 points)

**K. 385.** A square sheet of paper is folded, and then a piece of the folded sheet is cut off along a straight line. When the sheet is unfolded, a square hole is found at the centre, as shown in the *figure.* (The centres of the two squares coincide, and the sides of the small square are parallel to the diagonals of the large square.)

Find a possible combination of folding and cutting that produces the results described. Find a possible combination of folding and cutting in which the part removed is a rectangle, but not a square. (The sheet is folded and a piece is cut off along a straight line. The centres of the square and rectangle should coincide, and the sides of the rectangle are parallel to the diagonals of the square.)

(6 points)

This problem is for grade 9 students only.

**K. 388.** The letters of the English alphabet (ABCDEFGHIJKLMNOPQRSTUVWXYZ) are arranged in a pyramid, such that each row contains one more letter than the previous row. When the letter Z is reached, it is followed by the letters A, B, C, ...again. In which rows will two consecutive rows first end with the letter M? In which row will a letter M first occur at the end of the row?

(6 points)

This problem is for grade 9 students only.

**K. 389.** Consider the figure forming a letter F in the coordinate plane, whose vertices are the points with the following coordinates: (0,2), (3,2), (3,1), (1,1), (1,0), (2,0), (2,-1), (1,-1), (1,-4), (0,-4). Find the rule of assignment of the first-degree function whose graph halves the area of the letter F.

(6 points)

This problem is for grade 9 students only.