Third Millennium International Mathematical Olympiad 2003
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…
Misha picked 5 different digits out of 10 and composed a 5-digit number. Sasha composed another 5-digit number out of 5 remaining digits. George added those two numbers. Could George get a number consisting of three ones and three twos?
At a fleamarket, a crook asked Mary to sell him an antique teapot for $30 and gave her a fake $100 bill. Mary had no change and went to another seller Bill to exchange the $100 bill. Later Bill figured out that $100 bill was fake. Mary had to pay him $100 back. What was Mary's loss?
Anya multiplied the number of her apartment either by 6 or by 7. Bob added either 6 or 7 to Anya's resulting number. Vania substracted either 6 or 7 from Bob's number. After all these operations the final number was 2003. What is the number of Anya's apartment?
Leopold the cat wants to give 9 mice a box of chocolates as a gift. He plans to find a box with such number of chocolates that the mice could divide all the chocolates without breaking them and that each mouse would get a different number of chocolates. Leopold found boxes with 40, 45, 50 and 55 chocolates. Which boxes do not fit his requirements?
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. What is the next year that has the same property?
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