Third Millennium International Mathematical Olympiad 2008
Grade 7

1. Two players take turns placing counters on a table. On each turn, a player puts on the table either 1 counter or as many counters as their are on the table. The player who makes the last move wins. At the beginning there are no counters on the table. Assuming both players found an optimal strategy, who will win the game if the initial number of counters is 7.

2. Is it possible to draw 7 straight line segments in the plane so all but one intersects all the remaining segments.

3. A regiment has less than 1000 soldiers. The colonel plans to arrange the soldiers in the form of a rectangle. When he places 17 soldiers in a row, one row comes 1 soldier short. The same happened when he tried to place 19 soldiers in a row. Could he achieve his goal by placing 20 soldiers in a row?

4. Is it possible to draw on a square grid a broken line of length 2008 that encloses 2008 cells.

5. The language of the δεβαγ tribe contains two vowels (α and ε) and three consonants (β, γ, and δ), The words in the language are formed according to the three rules: (1) no two consecutive letters may be equal, (2) no three consonants in a row, (3) no two identical syllables. A syllable includes a vowel and a preceding consonant (if such exists). In addition, two successive consonants between two vowels are split between two syllables. What is the largest number of letters may be in the δεβαγ's alphabet?

6. The chess board is an arrangement of 8×8 squares in a traditional pattern of interchanging white and black squares. A novel piece - a dinosaur - hits all the squares of the opposite color, except for those on the same vertical, horizontal or diagonal lines. Find a position for a dinosaur from which it may hit the greatest number of squares.

   

   

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