**B. 4678.** Ann and Bill take turns writing digits on a sheet of paper, left to right. Ann starts with a nonzero digit, and they continue until a 100-digit number is formed. Bill wins if the resulting number divided by 11 leaves a remainder of 5, otherwise Ann wins. Both players are good at mathematics. Who will win the game?

Suggested by *Gy. Károlyi,* Budajenő

(4 points)

**B. 4682.** For a given positive integer \(\displaystyle k\), find the largest positive integer \(\displaystyle m\) such that the following statement should be true: If at most \(\displaystyle m\) of \(\displaystyle 3k\) different points in the plane are collinear, then the points can be divided into \(\displaystyle k\) groups of three such that the points in each group form a triangle.

Suggested by *A. Frank, *Nagykovácsi

(5 points)

**B. 4684.** The diagonals of a quadrilateral \(\displaystyle ABCD\) are perpendicular, they intersect at \(\displaystyle E\). From point \(\displaystyle E\), drop a perpendicular onto the line of each side. Consider the intersection of each perpendicular with the opposite side. Prove that the four points all lie on a circle centred at a point of the line segment connecting the midpoints of the diagonals.

Suggested by *Sz. Miklós,* Herceghalom

(5 points)

**C. 1266.** Solve the equation \(\displaystyle 5(2n+1)(2n+3)(2n+5) =\overline{ababab}\), where \(\displaystyle n\) denotes a positive integer, \(\displaystyle a\) and \(\displaystyle b\) stand for different digits, and \(\displaystyle \overline{ababab}\) is a six-digit number.

Suggested by *L. Számadó,* Budapest

(5 points)

This problem is for grade 1 - 10 students only.

**C. 1271.** Consider circumscribed circle of a right-angled triangle. Draw the semicircle containing the triangle, and draw tangents to it, parallel to the legs. The parallels together with the line of the hypotenuse form a similar right-angled triangle.

Find the angles of the triangle if the area of the outer triangle is 6 times the area of the inner triangle.

Based on the idea of *I. Légrádi,* Sopron

(5 points)

This problem is for grade 11 - 12 students only.

**K. 448.** There are four discs on a carousel, arranged as shown in the *diagram.* In how many ways is it possible to colour the four discs with four colours, given that colourings obtained from each other by rotating the carousel around its midpoint in its plane are not considered different? Any colour may be chosen for any disk, but each disc is only allowed to have a single colour.

(6 points)

This problem is for grade 9 students only.