Tổng hợp các đề thi Toán Olympiads từ 1981-1995 tổ chức Ba Lan và Úc
[...]... 1) Compute the probability that the sum of those four numbers is divisible by 3 15 For what natural numbers n is it possible to tile the n x n-chess board with 2 x 2 and 3 x 3-squares? 16 A triangular prism is a pentahedron whose two parallel faces ( "top base" and "bottom base" ) are congruent triangles and the remain ing three faces are parallelograms We are given four non-coplanar points in space... 1 {12) On the complex plane, the numbers 0, and a represent the vertices of an equilateral triangle, which is completed to a parallelogram (rhombus) Solutions 20 by the vertex 1 + a Thus 1 + a =J3 (cos( 71"/6) + i sin( 71"/6)) , and by de Moivre's Theorem (1 + a t=3n l2 (cos( n1r (6) + i sin(n7r/6)) (13) Comparing the real parts of (12) and (13), · An= 2 3n/2-1 cos(n1r /6); this is the explicit formula... (1) is either 3 or 2 (mod 5), and this is obviously a contradiction because a perfect square y 2 can only be 0, 1, or 4 (mod 5) P roblem 5, Solution 2 Assume equation (1) and transform the product under examination as follows: x(x + 3) · (x + 1) (x + 2) = (x 2 + 3x) (x 2 + 3x + 2) = (z - 1) (z + 1) = z 2 - 1, where we have denoted by z the expression x 2 + 3x + 1; this quadratic trinomial has the minimum... by b, and the third by c, and add the resulting equations, we obtain an equation in which the coefficients of x 2 and z 2 are a - b + c and -a + 2b + 8c, respectively Setting these expressions to be zero, we find that e.g a = 10, b = 9 and c = -1 do the job, producing the equation Problem 10 i.e., · ( -3xy + 3y 2 ) + 9 · 6yz - xy · · = 10 3 1 + 9 44 - 100, y ( -31x + 30y + 54z) = 606 This yields the... the second and the third equation of the system by suitable factors a, b, c; now we need that a - b + c and - 3a + c ( the coefficients xy in the resulting equation ) should be zero When we take 1, b = 4, c = 3, we obtain (3y 2 - z 2 ) + 4(6yz + 2z 2 ) + 3 8z 2 = 31 + 4 · 44 + 3 · 100, A rithmetic a nd Combinatorics 25 of x2 and a= · i.e., 31z 2 + 24yz + (3y 2 - 507) = 0 Viewing this as a quadratic... values of I YI this expression takes values 15768, 15921, 16176, 17553, 535968, 2096721, 4697976, 18744753, no one of which is a ZI, perfect square So the system has no integer solutions P roblem 7, Solut ion 2 An astonishingly simple proof results from examination of the two outer equations modulo 5 ( the middle equation is not needed! ) Multiplying the first equation by 8 and adding the third equation... the other to 1 or -1 (respectively) This means that the sum u + v and the product uv have to satisfy one of the four equation systems: u +v uv 1 0 u+v uv = u+v uv -3 u+v uv = -5 = -6 4 -9 -2 Accordingly, the numbers u and v have to be the roots of one of the four quadratic trinomials: t 2 + 9t - 2 j t2 + 5t -6 The two trinomials in the middle (the second and the third) have no integer roots The first... - 2) < 0, showing that D is strictly comprised between the squares of two skip consecutive integers x 2 + 2x - 4 and x 2 + 2x - 2 Therefore D has to be the square of x 2 + 2x - 3 This, however, cannot be the case, since this last number is of different parity than D (see (6)) The only possibility that remains is that x 2 Equation (5) then be comes y 2 + 4y - 5 0; equivalently; (y - 1)(y + 5) 0, and... recurrence equation of the (8) An+2 3(An+l An) for n 1 (with the initial data A1 A2 1); we invite the reader to do that Readers familiar with linear recurrences may like to work out an explicit formula (on the basis of the system (5), (6), (7) or of the single equation (8); compare Problem 10, Solution 3) We now show how to find that formula by a different method = - � = = Problem 3, Solution 2 Notation ) together... day, no truck would follow the same truck that it followed on the first day How many such rearrangements are possible? 20 We are considering paths (Po, Pt , Pn) of length n over lattice points in the plane (i.e., points (x, y) with integer coordinates); for each i, the points Pi-l and Pi are assumed to be adjacent on the lattice grid Let F(n) be the number of those paths that begin in Po = (0, 0) and