▪International Mathematics Tournament of Towns - 2003

Junior A-Level Paper

1.Johnny writes down quadratic equation  with positive integer coefficients . Then Pete changes one, two, or none “” signs to “”. Johnny wins, if both roots of the (changed) equation are integers. Otherwise (if there are no real roots or at least one of them is not an integer), Pete wins. Can Johnny choose the coefficients in such a way that he will always win? 

2. Triangle is given. Prove that , where is the radius of the circumscribed circle, is the radius of the inscribed circle, is the length of the longest side, is the length of the shortest altitude.

3. In a tournament, each of teams played with each other exactly once. Let us call the game “odd” if the total number of games previously played by both competing teams was odd.

(a) Prove that there was at least one “odd” game.

(b) Could it happen that there was exactly one “odd” game? 

4. A chocolate bar in the shape of an equilateral triangle with side of the length , consists of triangular chips with sides of the length , parallel to sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up and passes leftovers to the other player (as long as bar contains more than one chip, the player is not allowed to eat it completely).

A player who has no move or leaves exactly one chip to the opponent, loses. For each , find who has a winning strategy.

5. What is the largest number of squares on square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?

6. A trapezoid with bases and is circumscribed about a circle, is the intersection point of the diagonals. Prove that is not acute.

Junior O-Level Paper

1. dollars are placed into purses, and the purses are placed into pockets. It is known that is greater than the number of dollars in any pocket. Is it true that there is a purse with less than dollars in it?

2. Two players in turns color the sides of an -gon. The first player colors any side that has or common vertices with already colored sides. The second player colors any side that has exactly common vertex with already colored sides. The player who cannot move, loses. For which the second player has a winning strategy?

3. Points and are chosen on the sides and of the isosceles () so that . A line parallel to is drawn through midpoint of the segment , intersecting side at point . Find the value of .

4. Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

5. Is it possible to tile board by dominoes placed horizontally and rectangles placed vertically?

Senior A-Level Paper

1. A triangular pyramid is given. Prove that , where is the radius of the circumscribed sphere, is the radius of the inscribed sphere, is the length of the longest edge, is the length of the shortest altitude (from a vertex to the opposite face).

2. is a polynomial with real coefficients such that () where is an infinite sequence of distinct natural numbers. Determine the possible values of degree of .

3. Can one cover a cube by three paper triangles (without overlapping)?

4. A right triangle with hypotenuse is inscribed in a circle. Let be the midpoint of the arc not containing the midpoint of side , and a point of intersection of ray with the circle. Let be a point of intersection of tangents to the circle at points and . Prove that .

5. Prior to the game John selects an integer greater than .

Then Mary calls out an integer greater than . If John's integer is divisible by , then Mary wins. Otherwise, John subtracts from his number and the game continues (with the new number). Mary is not allowed to call out any number twice. When John's number becomes negative, Mary loses. Does Mary have a winning strategy?

6. The signs "" or "" are placed in all cells of a square table. It is allowed to change a sign of any cell altogether with signs of all its adjacent cells (i.e. cells having a common side with it). Find the number of different tables that could be obtained by iterating this procedure.

7. A square is triangulated in such way that no three vertices are collinear. For every vertex (including vertices of the square) the number of sides issuing from it is counted. Can it happen that all these numbers are even?

Senior O-Level Paper

1. Two players in turns color the sides of an -gon. The first player colors any side that has or common vertices with already colored sides. The second player colors any side that has exactly common vertex with already colored sides. The player who cannot move, loses. For which the second player has a winning strategy?

2. -gon made of sticks. Could it happen that it is not possible to construct a polygon from any lesser number of these sticks?

3. Point is chosen in triangle so that the radii of the circumcircles of triangles , and are no smaller than the radius of the circumcircle of . Prove that all four radii are equal.

4. In the sequence the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by (for example, can be followed by , or , but not by or ). What is the maximal number of terms that could remain on their places? 

5. Prove that one can cut rectangle, , into three pieces and rearrange them into a square (without overlaps and holes).