**A. 472.** Call a finite sequence (*p*_{1}(*x*),...,*p*_{k}(*x*)) of polynomials with integer coefficients *Euclidean* if there exist polynomials with integer coefficients such that *d*(*x*)=*q*_{1}(*x*)*p*_{1}(*x*)+...+*q*_{k}(*x*)*p*_{k}(*x*) is a common divisor of *p*_{1}(*x*),...,*p*_{k}(*x*), i.e. there are polynomials with integer coefficients for which *p*_{i}(*x*)=*r*_{i}(*x*)*d*(*x*) for every 1*i**k*. Prove that whenever are polynomials with integer coefficients and every two of them form a Euclidean pair, then the sequence is Euclidean as well.

(5 points)

**B. 4149.** The orthogonal projections of a point onto the lines of the altitudes drawn from vertices *A*, *B*, *C* of a triangle are denoted by *A*_{1}, *B*_{1}, *C*_{1}, respectively. Prove that there is exactly one point in the plane of the triangle for which the line segments *AA*_{1}, *BB*_{1}, *CC*_{1} are equal, and then the length of these line segments is equal to the diameter of the inscribed circle of triangle *ABC*.

Based on a problem in *Kvant*

(5 points)

**C. 970.** Rose, Ivy and Violet decided to solve all problems in their book of mathematics problems. Rose solves *a* problems, Ivy solves *b* problems and Violet solves *c* problems a day. (Each problem is only solved by one of the girls.) If Rose solved 11 times as many problems a day, Ivy solved 7 times as many and Violet solved 9 times as many, they would finish the job in 5 days. If Rose solved 4 times as many, Ivy solved 2 times as many and Violet solved 3 times as many problems a day, they would take 16 days. How many days does it take them to get done with the solutions?

(5 points)

**K. 193.** Four girls and four boys went dancing together. The first dance was a waltz. Each boy asked a girl to dance (each girl was only asked by one boy). The next dance was a tango. This time, it was the girls who asked the boys to dance. Each girl asked a boy (and each boy was asked by only one girl). There was no couple that danced both dances together. Given the following information, find out who danced each dance with whom.

*a*) Charles waltzed with the girl who danced tango with Daniel.

*b*) Andrew asked that girl to waltz who danced tango with the boy who waltzed with Emily.

*c*) Bernard waltzed with the girl who danced tango with Mary's waltz partner.

*d*) Gillian did not dance tango with Bernard.

*e*) Helen danced tango with a boy who did not waltz with Gillian.

After the two dances, when the dance instructor asked everyone to stand behind the back of the partner whom he or she asked to dance, the result was one big circle of people.

(6 points)

This problem is for grade 9 students only.

**K. 198.** The inhabitants of a certain planet use not four but five basic arithmetic operations. The operations of addition, multiplication, subtraction and division are the same as ours, but they also have a special operation denoted by the operation sign @. We do not know exactly how this operation works, but we have found out that the following properties are valid for all *x* and *y*:

(*a*) ,

(*b*) ,

(*c*) .

What is the value of on the planet in question?

(6 points)

This problem is for grade 9 students only.