Third Millennium International Mathematical Olympiad 2003 Grade 7

1. 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…

2. 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. Masha added those two numbers. Could George get a number consisting of just two different digits and such that each digit is used 3 times? (Example: 434343, 444333 etc.)

3. A math teacher told her students to simplify the fraction

 111112003111113002
.

In a few minutes industrious Natasha says that she simplified it, but lazy Andrey says that it is impossible. Who is correct?

4. Is it possible to build a right-angle triangle from two noncongruent isosceles triangles in such a way that one acute angle of the right-angle triangle is 6 degrees larger than the other acute angle?

5. All squares of an 8×8 board are initially white. The squares are painted blue one by one in such a way that each time the new resulting blue shape on the board has a line of symmetry. Color the 8×8 white board according to this rule. Your goal is to color as many squares as possible. Mark the order in which you color the squares with numbers 1, 2, 3…?

6. 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?