ࡱ> &(%7 !bjbjUU "87|7|l,,$,2  x          F ^  ,h\ 02, : ,,Berkeley Math Circle BAMO 2002 practice problems by Maksim Maydanskiy Note: The problems below appeared at George Soros Mathematical Olympiads of Ukraine during 1995 1999. I have selected and translated them, occasionally making slight modifications. Simply patterns: How many zeroes are in the end of  EMBED Equation.3 ? Is it possible to tile the plane with regular pentagons of side b and rhombs of side b and acute angle 36(the tiles can not lie one over other), using: a) only rhombuses b) only pentagons c) both pentagons and rhombuses? 3. Find all ordered triples a, b, c, d of positive integers such that da+db=dc. 4. Solve [x]+[2x]++[2002x] = 2002 ([x] denotes the greatest integer not exceeding x). 5. Find all functions f: R EMBED Equation.3 R such that for all real x and y  EMBED Equation.3 . Some geometry: 6. Triangles MAB and MA1B1 are similar and oriented the same way (that is, without loss of generality, MAB and MA1B1 are both names of the vertices read in clockwise direction). Prove that circles circumscribed around these triangles and lines AA1 and BB1 pass through the same point. 7.Circles  EMBED Equation.3  and  EMBED Equation.3 intersect in two points A and B. Chord BC of  EMBED Equation.3  intersects  EMBED Equation.3 in point E (B EMBED Equation.3 E), chord BD of  EMBED Equation.3  intersects  EMBED Equation.3 in point F (B EMBED Equation.3 F), so that DF = CE. Prove that distances from A to segments DF and CE are equal. 8. Find all triples of positive integers (p, r, q) such that  EMBED Equation.3  9. What maximum number of circles of radius  EMBED Equation.3 is it possible to place inside of the regular hexagon of side 1?(The circles may touch the sides of the hexagon.) 10. Prove that for all real  EMBED Equation.3 x the inequality  EMBED Equation.3  holds. 11. The 2001x2001 board is colored in black and white in a pattern of a chessboard (black and white squares alternating), so that corner squares are black. A spider sits on a black square and in one move can move to a square having a common side with the square it is on. Prove that the spider can always get to the fly, that sits motionless on some black square, visiting all squares of the board exactly once. Almost not geometry: 9. In triangle ABC a point of intersections of altitude H lies on the inscribed circle. Is ABC necessarily isosceles? And in similar fashion: 10. Does there exist a pentagon each diagonal of which is equal to some side of it? More geometry (work backward and forward) 11. In a trapezium ABCD (BC||AD, BC1 exists a integer positive integer m, such that for some integers  EMBED Equation.3 , different from zero, the we have:  EMBED Equation.3  19. I have cut from a piece of cardboard two equal regular n-gons ( EMBED Equation.3 ), have covered them with glue and am about to glue them together to make one n-gon. For which n can I number the vertices of these n-gons with integers 1 through n, so that for arbitrary way of gluing, vertices in at least one of the pairs glued together have the same numbers? Harder: 20. Prove, that for any positive integer m exist integers a and b such that  EMBED Equation.3 and  EMBED Equation.3 . Think outside of the board: 21. Given a table 2002x2002, in each cell of which a real number is written. Two players, Inna and Neil, take turns choosing rows of the table (at each step any player can choose any row among those not already chosen). The game ends when all rows have been chosen. After that, Inna computes the square of the sum of all numbers in the first column of her 1001 rows, and then she computes the square of the sum of all numbers in the second column of her 1001 rows etc. Then Inna computes the end result the sum of those 2002 squares. Similarly, Neil finds his end result, using his 1001 rows. The one who gets a bigger number wins. If the numbers are equal the game is considered a draw. Inna starts the game. Prove that she can play so that she wont loose. More geometry: 22. Let ABC be an isosceles triangle with AB=AC. Denote as A1, B1, C1 points of tangency of the inscribed circle with sides BC, AC, AB respectively. Prove that the acute angle between lines BB1 and CC1 does not exceed 60 . Try smaller sizes, see the pattern: 23. A square of size (n-1)x(n-1) is divided into (n-1)2 one by one squares in a regular manner. Each of n2 vertices of these squares is to be painted blue or gold. Find the number of different colorings, for which each unit square has exactly two blue vertices. (Colorings are called different if they differ in the color of at least one vertex). Simpler one: 24. Find the biggest possible number of consecutive positive integers each of which has exactly 4 positive integer divisors. 25. Prove that in an arbitrary n-gon there exist two sides with lengths a and b, so that [a / b] = 1 (where [x] denotes the greatest integer not exceeding x). Yet more geometry: 26. In a rhombus ABCD angle A is equal 60 . On the rays CD and AD points P and Q, respectively, are chosen so that AP||CQ. Find the angle PBQ. 27. In the coordinate plane, the circle centered at the origin and radius 2002 is drawn. In each point of the plain with integer coordinates and inside the circle sits a gastropod. At some instant of time all gastropods crawl distance 1 up, down, left or right, staying inside of the circle. Is it necessary that two gastropods will end up at the same point? 28. Prove that base10 expansion of at least 300 numbers form the set 1, 2, 4, ..., 21000 begins with 1. (Follow-up for die-hards: what is the probability that a number from the sequence 1, 2, , 2k, starts with 1?) 29. (For die-hards again) Points B1 and B2 lie on the median BD of a triangle ABC. Lines AB2 and AB1, CB2 and CB1 cross BC and BA at the points A2, A1 and C2, C1 respectively. M and K are points of intersection of AA2 and CC1, CC2 and AA1 respectively. 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