Third Millennium International Mathematical Olympiad 2008
Grade 8
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 8.
An Architect plans a city with 2008 straight streets and a circular beltway around the city. The streets and the road in the city are required to meet at T-intersection. What may be the number of the intersections?
Given a parallelogram with marked center and the midpoints of the sides, consider all the triangles with all three vertices marked. In each of these triangles mark in addition the endpoints and the midpoints of the medians. How many marked points will there be altogether?
A series of bus tickets includes all 6-digit number from 000000 through 999999. A girl collects the tickets whose numbers are divisible by 78 and a boy collects the tickets whose numbers are divisible by 77, but not by 78. Which kind of tickets in the series is more numerous: those of intersect to the girl or to the boy?
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 least number of squares.
Each of the three natural number is multiplied by the difference of the other two. The sum of the products happened to be 2008. Find a triple of such numbers with the minimum possible sum.
