Your total score is based on the three problems for which you earn the most points; the scores for the individual parts of a single problem are summed. Points for each problem are shown in brackets [ ].
1. [3] A square is placed on the plane and a point P is marked on this plane by invisible ink; but a certain person can see it through special glasses. You may draw any straight line and this person will tell you whether P lies on one or another side of the line (or exactly on the line). What minimal number of the questions you need to find out if P lies inside the square or not?
2. [3] On each of 100 cards a positive integeris written. Is it possible that their sum is equal to their least common multiple?
3. [3] A paper rectangle ABCD of area 1 is folded through a straight line so that C coincides with A. Prove that the area of the pentagon obtained is less than 3/4.
4. [5] From the vertex A of triangle ABC three segments are drawn: the bisectors AM and AN of its interior and exterior angles and the tangent line AK to the circumcircle of the triangle (the points M, K, N lie on the line BC). Prove that MK = KN.
1. [3] A square is placed on the plane and a point P is marked on this plane by invisible ink; but a certain person can see it through special glasses. You may draw any straight line and this person will tell you whether P lies on one or another side of the line (or exactly on the line). What minimal number of the questions you need to find out if P lies inside the square or not?
2. Do there exist
a) [2] 4 positive integers,
b) [2] 5 positive integers such that the sum of
any three of them is a prime number?
3. [3] The first digit of a 6-digit number is 5. Is it always possible to attach 6 more digits to it (from the right) so that the resulting 12-digit number will be a perfect square?
4. [5] Three different points A, B, C are given on a plane. Draw a line m through C so that the product of distances from A and B to m takes the maximal value. Is m defined uniquely for any triple A, B, C?