|
|
Third Millennium International Mathematical Olympiad 2003
Grade 6
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?
In the sequence 1, 1, 2, 3, 5, 8, 13…, every number starting from the third one is a sum of two previous numbers (2 = 1 + 1, 3 = 2 + 1 etc.). Is it possible that two consecutive numbers of this sequence be divisible by 2003?
George owns a souvenir shop. He bought some nice clay cups from a local artist. He sells either one cup for $5 or three cups for $10 because he wants to get the same profit from each customer. How much, then, should George charge for 5 cups?
Kathy Corner draws several angles at random. The measure of each angle is bigger than 10 degrees and smaller than 90 degrees. All the measures turn out to be whole numbers. What minimum number of angles should Kathy draw to make sure that at least three angles are congruent?
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. How many other years in the XXI century share the same property?
Copyright © 1996-2010 Alexander Bogomolny
|
| 35596553 |  |
|
|
