**A. 512.** We have coins, each one indistinguishable from the other in terms of size and appearance, but only *n*-1 of them have the same weight, whilst one of them is a counterfeit, being either lighter or heavier than the rest. Using a balance we have to find the counterfeit coin. Show that by *k* measurements it is possible to find the counterfeit coin and to determine whether it is heavier or lighter than the rest; moreover, it is possible to arrange the measurements in advance, without knowing the results of the measurements.

(5 points)

**A. 514.** There are given three circles in the plane, *k*_{0}, *k*_{1} and *k*_{2}, being externally tangent to each other. The center of *k*_{0} is *O* and one of its diameters is *A*_{1}*A*_{2}. Denote by *B *the point of tangency between *k*_{1} and *k*_{2}, by *C*_{1} the point of tangency between *k*_{0} and *k*_{1} and by *C*_{2} the point of tangency between *k*_{0} and *k*_{2}. The line segments *A*_{1}*C*_{2} and *A*_{2}*C*_{1} meet at point *D* in the interior of the circle *k*_{0}. Let *t*_{1} and *t*_{2} be the tangent lines to the circle *k*_{0} at *A*_{1} and *A*_{2}, respectively. Prove that if *t*_{1} is tangent to *k*_{1} and *t*_{2} is tangent to *k*_{2} then the line segment *OB* passes through point *D*.

(5 points)

**B. 4289.** The diagonals of a trapezium *A*_{1}*A*_{2}*A*_{3}*A*_{4} are *A*_{1}*A*_{3}=*e* and *A*_{2}*A*_{4}=*f*. Let *r*_{i} denote the radius of the circumscribed circle of triangle *A*_{j}*A*_{k}*A*_{l}, where {1,2,3,4}={*i*,*j*,*k*,*l*}. Show that .

(4 points)

**K. 256.** In a three-digit number, the digit in the ones' place is 3 smaller than the digit in the hundreds' place.

*a*) Find the largest number that meets the condition.

*b*) How many such numbers are there altogether?

*c*) Find all numbers that meet the above condition, for which by subtracting the number obtained by reversing the order of the digits the result is 297.

(6 points)

This problem is for grade 9 students only.