28th Russian 2002

28th Russian 2002 problems


1. Can the cells of a 2002 x 2002 table be filled with the numbers from 1 to 2002² (one per cell) so that for any cell we can find three numbers a, b, c in the same row or column (or the cell itself) with a = bc?
2. ABC is a triangle. D is a point on the side BC. A is equidistant from the incenter of ABD and the excenter of ABC which lies on the internal angle bisector of B. Show that AC = AD.
3. Given 18 points in the plane, no three collinear, so that they form 816 triangles. The sum of the area of these triangles is A. Six are colored red, six green and six blue. Show that the sum of the areas of the triangles whose vertices are the same color does not exceed A/4.
4. A graph has n points and 100 edges. A move is to pick a point, remove all its edges and join it to any points which it was not joined to immediately before the move. What is the smallest number of moves required to get a graph which has two points with no path between them?
5. The real polynomials p(x), q(x), r(x) have degree 2, 3, 3 respectively and satisfy p(x)² + q(x)² = r(x)². Show that either q(x) or r(x) has all its roots real.
6. ABCD is a cyclic quadrilateral. The tangent at A meets the ray CB at K, and the tangent at B meets the ray DA at M, so that BK = BC and AM = AD. Show that the quadrilateral has two sides parallel.
7. Show that for any integer n > 10000, there are integers a, b such that n < a² + b² < n + 3 n^1/4.
8. A graph has 2002 points. Given any three distinct points A, B, C there is a path from A to B that does not involve C. A move is to take any cycle (a set of distinct points P₁, P₂, ... , P_n such that P₁ is joined to P₂, P₂ is joined to P₃, ... , P_n-1 is joined to P_n, and P_n is joined to P₁) remove its edges and add a new point X and join it to each point of the cycle. After a series of moves the graph has no cycles. Show that at least 2002 points have only one edge.
9. n points in the plane are such that for any three points we can find a cartesian coordinate system in which the points have integral coordinates. Show that there is a cartesian coordinate system in which all n points have integral coordinates.
10. Show that for n > m > 0 and 0 < x < π/2 we have \| sinⁿx - cosⁿx \| ≤ 3/2 \| sin^mx - cos^mx \|.
11. [unclear]
12. Eight rooks are placed on an 8 x 8 chessboard, so that there is just one rook in each row and column. Show that we can find four rooks, A, B, C, D, so that the distance between the centers of the squares containing A and B equals the distance between the centers of the squares containing C and D.
13. Given k+1 cells. A stack of 2n cards, numbered from 1 to 2n, is in arbitrary order on one of the cells. A move is to take the top card from any cell and place it either on an unoccupied cell or on top of the top card of another cell. The latter is only allowed if the card being moved has number m and it is placed on top of card m+1. What is the largest n for which it is always possible to make a series of moves which result in the cards ending up in a single stack on a different cell.
14. O is the circumcenter of ABC. Points M, N are taken on the sides AB, BC respectively so that ∠MON = ∠B. Show that the perimeter of MBN is at least AC.
15. 2^2n-1 odd numbers are chosen from {2²ⁿ + 1, 2²ⁿ + 2, 2²ⁿ + 3, ... , 2³ⁿ}. Show that we can find two of them such that neither has its square divisible by any of the other chosen numbers.
16. Show that √x + √y + √z ≥ xy + yz + zx for positive reals x, y, z with sum 3.
17. In the triangle ABC, the excircle touches the side BC at A' and a line is drawn through A' parallel to the internal bisector of angle A. Similar lines are drawn for the other two sides. Show that the three lines are concurrent.
18. There are a finite number of red and blue lines in the plane, no two parallel. There is always a third line of the opposite color through the point of intersection of two lines of the same color. Show that all the lines have a common point.
19. Find the smallest positive integer which can be represented both as a sum of 2002 positive integers each with the same sum of digits, and as a sum of 2003 positive integers each with the same sum of digits.
20. ABCD is a cyclic quadrilateral. The diagonals AC and BD meet at X. The circumcircles of ABX and CDX meet again at Y. Z is taken so that the triangles BZC and AYD are similar. Show that if BZCY is convex, then it has an inscribed circle.
21. Show that for infinitely many n the if 1 + 1/2 + 1/3 + ... + 1/n = r/s in lowest terms, then r is not a prime power.