**A. 602.** Let *ABC* be a non-equilateral triangle. Consider those equilateral triangles *XYZ *whose vertices *X*, *Y* and *Z* lie on the lines *BC*, *CA* and *AB*, respectively. Show that the locus of centers of such triangles *XYZ* is a pair of parallel lines which are perpendicular to the Euler line of the triangle *ABC*.

Proposed by: *András Hraskó,* Budapest

(5 points)

**B. 4583.** The points *D* and *E* lie on the line segment *AB*. In the same half plane, a regular triangle is drawn over each of the line segments *AD*, *DB*, *AE* and *EB*. The third vertices are *F*, *G*, *H* and *I*, respectively. Prove that if the lines *FI* and *GH* are not parallel, then their intersection lies on the line *AB*.

Suggested by *Sz. Miklós,* Herceghalom

(3 points)

**B. 4588.** The point *D* lies in the interior of a triangle *ABC*. Lines *CD*, *AD* and *BD *intersect sides *AB*, *BC* and *CA* at the points *E*, *F* and *G*, respectively. The intersection of lines *EG* and *AF* is *H*, and that of lines *EF* and *BG *is *I*. Show that the lines *AB*, *FG* and *HI* are concurrent.

Suggested by *Sz. Miklós,* Herceghalom

(5 points)

**B. 4591.** Let be an irrational number. For every positive integer *q*, let , that is, the distance from the closest fraction that can be represented with a denominator of *q* (not necessarily cancelled to lowest terms). Show that there exists a *k* such that .

Suggested by *P. Maga,* Budapest

(6 points)

**C. 1199.** The accompanying *figure* shows a sequence of designs made up of floor tiles. The number of dark grey tiles in the designs is 1,6,13,24,37,..., respectively.

Sophie proved that the number of dark grey tiles in the designs with odd indices in the series is a quadratic function of the index. Determine what number of dark grey tiles there are in the ninety-ninth design according to Sophie's formula.

(5 points)

**K. 397.** A certain island that is also a country has its own currency, the wooden knut. Everyone pays tax to the state (independently of the size of his or her income). The tax is a certain percentage of the income. However, the government supports the raising of children, therefore every family receives a tax reduction of a certain number of wooden knuts per child (for example, the reduction is twice as many wooden knuts for two children as for a single child). One particular family has an annual income of 1500000 wooden knuts and one child. They pay a tax of 150000 wooden knuts. Another family has an annual income of 2500000 wooden knuts, two children and a tax of 225000 wooden knuts to pay. What percentage of the income is the tax on the island, and what is the tax allowance per child?

(6 points)

This problem is for grade 9 students only.

**K. 398.** In the six-digit number 135726, the digit in the thousands' place (the 5) is equal to the double of the number of hundred thousands plus the number of ten thousands (that is, 2^{.}1+3). The digit in the hundreds' place (the 7) is equal to the double of the number of ten thousands plus the number of hundred thousands (that is, 2^{.}3+1). The digit in the tens' place is twice the number of hundred thousands, and the digit in the units' place is twice the number of ten thousands. The number given as an example above is divisible by six. Is it true that all numbers of this property are divisible by six?

(6 points)

This problem is for grade 9 students only.

**K. 399.** *a*) How many numbers *A* are there for which the least common multiple of 6^{6}, 8^{8} and *A* is 12^{12}?

*b*) How many numbers *B* less than 1000 are there for which the greatest common divisor of 6^{6}, 30^{3} and *B* is 3^{3}?

(6 points)

This problem is for grade 9 students only.

**K. 400.** Steve and Kate are running around the football field, and measuring their times. Steve's stopwatch is set to lap times, that is, whenever he presses the button the watch will store the actual time reading and start the measurement from zero again. Kate's stopwatch is set differently: whenever she presses the button, the watch stores the actual time reading, but it will then continue the measurement without resetting zero. The two of them ran four laps together. Unfortunately, Kate forgot to press the button at the ends of the first and third laps. Thus she only had two readings at the end: 164 seconds and 340 seconds. Based on his own watch, Steve told her that they had run the first lap 5 seconds faster than the mean of the four lap times, and that the mean of the second and third lap times was 4 seconds more than the fourth lap time. How long did they take to cover the fourth lap?

(6 points)

This problem is for grade 9 students only.