Third Millennium International Mathematical Olympiad 2003
Grade 8
A toy car stands on one of the 16 squares of a 4×4 board. The car can move horizontally or vertically, 2 or 3 squares at a time, jumping over 1 or 2 squares, respectively. Pick any square on the board and move the car according to the rules. Your goal is to pass through as many squares as possible without stepping into any square more than once. Mark the order of your moves with numbers 1,2,3…
There is a point P inside the square ABCD such that AP = AB. Straight line AP intersects the line segment BC at point K. Straight line BP intersects the line segment CD at point L. Prove that BK > 2·CL.)
13 boys and d girls participated in the Olympiad. Together they were awarded d² + 10d + 17 points. It turned out that every participant received the same number of points. How many students participated in the Olympiad?
Bob, Alex and Donna are friends. One of them has a bicycle and can give a ride to one person at a time. Can three friends travel the distance 21 km in 3 hours if each of them can walk at the speed 5 km/h, ride a bike alone at the speed 15 km/h, or ride a bike with one passenger at the speed 10 km/h?
Peter wrote down several polynomials, then squared each one and added them all up. Could he end up with the algebraic expression x² + y² + z² + 3y + 4x + 2003xz + 1?
If you write the three last digits of the number 2003 backwards, you get 300. In May of 2003 Saint-Petersburg will be exactly 300 years old. Were there in the history of Saint-Petersburg the years with the same property?
