▪ Russia All-Russian Mathematical Olympiad 2012

Grade 9

Ημέρα 1η

1. Let be distinct positive integers, all at least and with sum . Does there exist an integer such that the sum of the remainders after the division of by is ? 

2. A regular -gon is inscribed in a circle. Find the maximal such that we can choose vertices from given and construct a convex -gon without parallel sides. 

3. Consider the parallelogram with obtuse angle . Let be the feet of perpendicular from to the side . The median from in triangle meets the circumcircle of triangle at the point . Prove that points lie on the same circle. 

4. The positive real numbers and are such that , and . Prove that the difference between any two of is greater than .

Ημέρα 2η

1. wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions. 

2. The points lie on the sides sides and of the triangle respectively. Suppose that . Let be the incentres of triangles and respectively. Prove that the circumcentre of triangle is the incentre of triangle . 

3. Initially, ten consecutive natural numbers are written on the board. In one turn, you may pick any two numbers from the board (call them and ) and replace them with the numbers and . After several turns, there were no initial numbers left on the board. Could there, at this point, be again, ten consecutive natural numbers? 

4. In a city's bus route system, any two routes share exactly one stop, and every route includes at least four stops. Prove that the stops can be classified into two groups such that each route includes stops from each group. 

Grade 10

Ημέρα 1η

1. Let be distinct positive integers, all at least and with sum . Does there exist a positive integer such that the sum of the remainders of after division by is ? 

2. The inscribed circle of the non-isosceles acute-angled triangle touches the side at the point . Suppose that and are the centres of inscribed circle and circumcircle of triangle respectively. The circumcircle of triangle intersects at the points and . Prove that is equal to the radius of . 

3. Any two of the real numbers differ by no less than . There exists some real number satisfying

Prove that . 

4. Initially there are monomials on the blackboard: . Every minute each of boys simultaneously write on the blackboard the sum of some two polynomials that were written before. After minutes among others there are the polynomials on the blackboard. Prove that

  . 

Ημέρα 2η

1. wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions. 

2. Do there exist natural numbers all greater than such that their product is divisible by each of these numbers increased by ? 

3. On a Cartesian plane, parabolas are drawn, all of which are graphs of quadratic trinomials. No two of them are tangent. They divide the plane into many areas, one of which is above all the parabolas. Prove that the border of this area has no more than corners.(i.e. the intersections of a pair of parabolas). 

4. The point is the midpoint of the segment connecting the orthocentre of the scalene triangle and the point . The incircle of triangle incircle is tangent to and at points and respectively. Prove that point , the point symmetric to point with respect to line , lies on the line that passes through both the circumcentre and the incentre of triangle . 

Grade 11

Ημέρα 1η

1. Initially, there are pieces of clay on the table of equal mass. In one turn, you can choose several groups of an equal number of pieces and push the pieces into one big piece for each group. What is the least number of turns after which you can end up with pieces no two of which have the same mass? 

2. Any two of the real numbers differ by no less than . There exists some real number satisfying

Prove that . 

3. A plane is coloured into black and white squares in a chessboard pattern. Then, all the white squares are coloured red and blue such that any two initially white squares that share a corner are different colours. (One is red and the other is blue.) Let be a line not parallel to the sides of any squares. For every line segment that is parallel to , we can count the difference between the length of its red and its blue areas. Prove that for every such line there exists a number that exceeds all those differences that we can calculate.

4. Given is a pyramid whose base is convex polygon . For every there is a triangle congruent to triangle that lies on the same side from as the base of that pyramid. (You can assume is the same as .) Prove that these triangles together cover the entire base. 

Ημέρα 2η

1. Given is the polynomial and the numbers such that . Suppose that for every , we have

Prove that the polynomial has at least one real root. 

2. The points lie on the sides and of the triangle respectively. Suppose that . Let and be the circumcentres of triangles and respectively. Prove that the incentre of triangle is the incentre of triangle too. 

3. On a circle there are points, dividing it into equal arcs (). Two players take turns to erase one point. If after one player's turn, it turned out that all the triangles formed by the remaining points on the circle were obtuse, then the player wins and the game ends.

Who has a winning strategy: the starting player or his opponent? 

4. For a positive integer define . Prove that there exists an integer such that has a prime divisor greater than .

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