Đang tải... (xem toàn văn)
Đề thi toán học bulgarian năm 2003 2006 bulgarian mathematical competitions 2003 2006
Peter Boalenkov 0 leg M ushkarov Emil Kolev Nikolai Nikolov 2003-2006 [...]... are not coprime Prove that if the of the elements of a good set C equals 2003, then there exists c E C for which the set C \ { c} good Problem 6 A set · :r 2 z z2 = ,23++yy+ + = 3:ry3:rzz :r + y3 + = 3y a, b is sum k k b, b k is Ale:rander Ivanov, Emil Kolev 6 52 Bulgarian Mathematical Olympiad National round, Sofia, May 17-18, 2003 Problem 1 Find the least positive integer n "'• distinct sums of the... incenter on the diagonal AC Prove that 1 APB = 1APD m and n such that m(m + 1)(m + 2)(m + 3) = n(n + 1)2(n + 2)3 (n 3)4 Problem 6 Prove that there are no positive integers + 9 Bulgarian Mathematical Competitions 2004 Wmter Mathematical Competition Rousse, January 30 - February 1, 2004 a such that the equation 2 - a- 9)x2 - 6x - a 0 (a Problem 9.1 Find all values of = has two distinct positive roots... lines and columns contain white pieces The game is over if it is not possible to make a move Find the maximum possible number of the white pieces that can be put on the table 20 Bulgarian Mathematical Competitions 2005 Winter Mathematical Competition Bourgas, January 28-30, 2005 Find all values of the real parameter a for which the equations x2 - (2a+ 1)x + a = 0 and x2 + (a - 4)x + a-1 = 0 have real... all written numbers are equal Mediteronian Mathematical Competition n set A n = {j: 1 � j � n, (j,n) = 1} Problem 4 Fbr any positive integer Find all n such that the polynomial Pn(x) is = L ,;-t jEAn irreducible over Z[x] 8 Team selection test for 44 IMO Sofia, May 29-30, 2003 !:: ABC such that any of them AB and the sum of their areas is maximal Problem 1 Cut 2003 rectangles from an acute has a side... f : lR ., lR with a) f(x) =ax +1- a for any x b) f(!(x)) = 3- 2x for any x > 0 such that the following two properties: [2,3); E E IR 5 Oleg Mushkarov, Nikolai Nikolov 52 Bulgarian Mathematical Olympiad Regional round, April 19-20, 2003 Problem 1 A right-angled trapezoid with area 10 and altitude 4 is divided into two circumscribed trapezoids by a line parallel to its bases Find their inradii Oleg Mushkarov... selection test for 20 BMO Kazanlak, March 3, 2003 Problem 1 Let D be a point on the side AC of L:.ABC with AC = BC, and E be a point on the segment BD Prove that 'I:EDC = 2-l:CED if BD = 2AD = 4BE Mediteronian Mathematical Competition Problem 2 Prove that if a, b and c are positive numbers with sum a b2 + 1 + b c2 + 1 + c a2 + 3 1 � 2· 3, then Mediteronian Mathematical Competition Problem 3 At any lattice... the sides BC, CA and AB form (in this order) an arithmetic progression Oleg Mushkarov c 4n+l c n m m = = = x Problem 12.3 that Find the number of the sequences {an}::' l of integers such = for every n Problem 12.4 Nikolai Nikolov Let a, b1 , c1 , , bn, c, be real numbers such that for every real number x Prove that c1 = · · · = c, = 26 1 Nikolai Nikolov 54 Bulgarian Mathematical Olympiad Regional... groups, there are at least two familiar tourists in some of the groups Prove that there 2n is a tourist who is familiar with at most 5" tourists Problem 3 Ivan Landjev aba b, b aba, bba a a a bba.b b� �b? 2003 2003 Emil Kolev Problem 5 Let a, b, and d be positive integers such that there are exactly 2004 ordered pairs (x,y), x,y E (0,1}, for which ax + by and ex + dy are integers If (a, ) 6, find (b, d) Oleg... excircles to the sides H are tangent to the line at points M and N Find 1 if the points M, N, P and Q are concyclic Problem 12.2 The incircle of AC # BC, BC AB Problem 12.3 See Problem 11.3 13 53 Bulgarian Mathematical Olympiad Regional round, April 17-18, 2004 a such that the equation J(4a2 4a l)x2 2ax + l - 1 - ax - x2 Problem 9.1 Find all values of has exactly two solutions Sava Grozdev, Svetlozar... Mushkarov, Nikolai Nikolov P(x) any x P(x)Q(x 1) = P(x 2004)Q(x) forand Q(x) with real Find all non-constant polynomials coefficients such that + + Problem 12.6 Oleg Mushkarov, Nikolai Nikolov 16 53 Bulgarian Mathematical Olympiad National Round, Sofia, May 15-16, 2004 �ABC At, B1 C1 A2 , B2 C2 Problem 1 Let I be the incenter of and let and be points on the segments AI, and CI The perpendicular bisectors . ushkarov Emil Kolev Nikolai Nikolov 2003- 2006