54th Polish Mathematical Olympiad Problems 2003

54th Polish Mathematical Olympiad Problems 2003A1.  ABC is acute-angled. M is the midpoint of AB. A line through M meets the lines CA, CB at K, L with CK = CL. O is the circumcenter of CKL and CD is an altitude of ABC. Show that OD = OM. A2.  P is a point inside a regular tetrahedron with

51st Polish Mathematical Olympiad Problems 2000

51st Polish Mathematical Olympiad Problems 2000A1.  How many solutions in non-negative reals are there to the equations: x1 + xn2 = 4xn x2 + x12 = 4x1 ... xn + xn-12 = 4xn-1? A2.  The triangle ABC has AC

50th Polish Mathematical Olympiad Problems 1999

50th Polish Mathematical Olympiad Problems 1999A1.  D is a point on the side BC of the triangle ABC such that AD > BC. E is a point on the side AC such that AE/EC = BD/(AD-BC). Show that AD > BE. A2.  Given 101

49th Polish Mathematical Olympiad Problems 1998

49th Polish Mathematical Olympiad Problems 1998A1.  Find all solutions in positive integers to a + b + c = xyz, x + y + z = abc.A2.  Fn is the Fibonacci sequence F0 = F1 = 1, Fn+2 = Fn+1 + Fn. Find all pairs m > k ≥ 0 such that the sequence x0, x1, x2, ... defined by x0 = Fk/Fm and xn+1 = (2xn - 1)/(1 - xn) for xn ≠ 1, or 1 if xn = 1, contains the

48th Polish Mathematical Olympiad Problems 1997

48th Polish Mathematical Olympiad Problems 1997A1.  The positive integers x1, x2, ... , x7 satisfy x6 = 144, xn+3 = xn+2(xn+1+xn) for n = 1, 2, 3, 4. Find x7. A2.  Find all real solutions to 3(x2 + y2 + z2) = 1, x2y2 + y2z2 + z2x2 = xyz(x + y + z)3. A2.  P is a point inside the triangle ABC such that ∠PBC = ∠PCA <

46th Polish Mathematical Olympiad Problems 1995

46th Polish Mathematical Olympiad Problems 1995 A1.  How many subsets of {1, 2, ... , 2n} do not contain two numbers with sum 2n+1?A2.  The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area? A2.  L, L' are parallel lines. C is a circle that does not

44th Polish Mathematical Olympiad Problems 1993

44th Polish Mathematical Olympiad Problems 1993A1.  Find all rational solutions to: t2 - w2 + z2 = 2xy t2 - y2 + w2 = 2xz t2 - w2 + x2 = 2yz. A2.  A circle center O is inscribed in the quadrilateral ABCD

43rd Polish Mathematical Olympiad Problems 1992

43rd Polish Mathematical Olympiad Problems 1992A1.  Segments AC and BD meet at P, and |PA| = |PD|, |PB| = |PC|. O is the circumcenter of the triangle PAB. Show that OP and CD are perpendicular.A2.  Find all functions f : Q

42nd Polish Mathematical Olympiad Problems 1991

42nd Polish Mathematical Olympiad Problems 1991A1.  Do there exist tetrahedra T1, T2 such that (1) vol T1 > vol T2, and (2) every face of T2 has larger area than any face of T1? A2.  Let F(n) be the number of paths P0, P1, ... , Pn of length n that go from P0 = (0,0) to a lattice point Pn on the line y = 0, such that each Pi is a lattice point and

41st Polish Mathematical Olympiad Problems 1990

41st Polish Mathematical Olympiad Problems 1990A1.  Find all real-valued functions f on the reals such that (x-y)f(x+y) - (x+y)f(x-y) = 4xy(x2-y2) for all x, y. A2.  For n > 1 and positive reals x1, x2, ... , xn, show that x12/(x12+x2x3) + x22/(x22+x3x4) + ... + xn2/(xn2+x1x2) ≤ n-1.