**A. 559.** The incircle of triangle *ABC* is *k*. The circle *k*_{A} touches *k* and the segments *AB* and *AC* at at *A*', *A*_{B} and *A*_{C} respectively. The circles *k*_{B}, *k*_{C} and the points *B*', *C*' are defined analogously. The second intersection point of the circles *A*'*B*'*A*_{B} and *A*'*C*'*A*_{C}, other than *A*', is *K*. The line *A*'*K* meets *k* at *R*, other than *A*'. Prove that *R* lies on the radical axis of the circles *k*_{B} and *k*_{C}.

(Kolmogorov's Cup, 2011; a problem by *F. Ivlev*)

(5 points)

**B. 4440.** On a winter's day, an absent minded mathematician went on a walk with his old poodle along a long straight alley. He was so absorbed in his thoughts that when he finally thought of his dog, he had to realize that it was not near him. In the snowfall, he did not see further than 5 metres. He did not see the dog in front of him, he did not see it behind either, and he did not know which direction it had gone. After a little thinking, he went to look for his poodle. The old poodle can only walk half as fast as its master. The mathematician chose a searching strategy, such that the following condition should hold for the smallest possible value of the constant *c*: if the dog is at a distance *x* from him then he needs to cover a distance of at most *cx* to find it. What is this smallest value of *c*?

(5 points)

**K. 331.** The task is to cross a desert with a car. The width of the desert is 600 km. The capacity of the petrol tank of the car is only enough for 400 km, and the car cannot carry fuel in any other way. The car can travel with an average speed of 50--60 km/h. It needs to start out at 8 a.m. and it is to arrive on the other side by 8 p.m. What is the minimum number of such cars needed, for one of them to be able to cross the desert, while the others return back? It is possible to transfer fuel from one car to another.

(6 points)

This problem is for grade 9 students only.

**K. 333.** The three interior angles of a triangle followed by the three exterior angles, in the appropriate order, form six consecutive terms of an arithmetic sequence. Find the measures of the angles of the triangle in degrees. (In an arithmetic sequence, the difference of the consecutive terms is constant, that is, the sequence increases uniformly.)

(6 points)

This problem is for grade 9 students only.

**K. 335.** A peculiar calculator only has four buttons on it: (eight plus root seven), (addition), (reciprocal) and (equals) (see the *figure*).

The calculator always carries out the operations with exact values, and it can also store the current value as a constant if the button is pressed twice. That is, in subsequent calculations whenever the button is pressed, the number is incremented by that value any number of times. (E.g. if the buttons are pressed, it will display ). Prove that the result of the following sequence of operations is 1:

... (56 times)

... (15 times)

(6 points)

This problem is for grade 9 students only.