1998年罗马尼亚数学国家集训队测试题

Romania Team Selection Tests 1998 Day 1 1 A word of length n is an ordered sequence x1x2 . . . xn where xi is a letter from the set {a, b, c}. Denote by An the set of words of length n which do not contain any block xixi+1, i = 1, 2, . . . , n − 1, of the form aa or bb and by Bn the set of words of length n in which none of the subsequences xixi+1xi+2, i = 1, 2, . . . n − 2, contains all the letters a, b, c. Prove that |Bn+1| = 3|An|. Vasile Pop 2 A parallelepiped has surface area 216 and volume 216. Show that it is a cube. 3 Let m ≥ 2 be an integer. Find the smallest positive integer n > m such that for any partition with two classes of the set {m,m + 1, . . . , n} at least one of these classes contains three numbers a, b, c (not necessarily different) such that ab = c. Ciprian Manolescu 4 Consider in the plane a finite set of segments such that the sum of their lengths is less than√ 2. Prove that there exists an infinite unit square grid covering the plane such that the lines defining the grid do not intersect any of the segments. Vasile Pop http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 1 Romania Team Selection Tests 1998 Day 2 1 We are given an isosceles triangle ABC such that BC = a and AB = BC = b. The variable points M ∈ (AC) and N ∈ (AB) satisfy a2 · AM · AN = b2 · BN · CM . The straight lines BM and CN intersect in P . Find the locus of the variable point P . Dan Branzei 2 All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least 52 . Bogdan Enescu 3 Find all positive integers (x, n) such that xn + 2n + 1 divides xn+1 + 2n+1 + 1. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 2 Romania Team Selection Tests 1998 Day 3 1 Let n ≥ 2 be an integer. Show that there exists a subset A ∈ {1, 2, . . . , n} such that: i) The number of elements of A is at most 2b√nc+ 1 ii) {|x− y| | x, y ∈ A, x 6= y} = {1, 2, . . . n− 1} Radu Todor 2 An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer. 3 Show that for any positive integer n the polynomial f(x) = (x2 + x)2 n + 1 cannot be decom- posed into the product of two integer non-constant polynomials. Marius Cavachi http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 3 Romania Team Selection Tests 1998 Day 4 1 Let ABC be an equilateral triangle and n ≥ 2 be an integer. Denote by A the set of n − 1 straight lines which are parallel to BC and divide the surface [ABC] into n polygons having the same area and denote by P the set of n− 1 straight lines parallel to BC which divide the surface [ABC] into n polygons having the same perimeter. Prove that the intersection A∩P is empty. Laurentiu Panaitopol 2 Let n ≥ 3 be a prime number and a1 < a2 < · · · < an be integers. Prove that a1, · · · , an is an arithmetic progression if and only if there exists a partition of {0, 1, 2, · · · } into sets A1, A2, · · · , An such that a1 +A1 = a2 +A2 = · · · = an +An, where x+A denotes the set {x+ a|a ∈ A}. 3 Let n be a positive integer and Pn be the set of integer polynomials of the form a0 + a1x + . . .+ anx n where |ai| ≤ 2 for i = 0, 1, . . . , n. Find, for each positive integer k, the number of elements of the set An(k) = {f(k)|f ∈ Pn}. Marian Andronache http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 4 Romania Team Selection Tests 1998 Day 5 1 Find all monotonic functions u : R → R which have the property that there exists a strictly monotonic function f : R→ R such that f(x+ y) = f(x)u(x) + f(y) for all x, y ∈ R. Vasile Pop 2 Find all positive integers k for which the following statement is true: If F (x) is a polynomial with integer coefficients satisfying the condition 0 ≤ F (c) ≤ k for each c ∈ {0, 1, . . . , k + 1}, then F (0) = F (1) = . . . = F (k + 1). 3 The lateral surface of a cylinder of revolution is divided by n−1 planes parallel to the base and m parallel generators into mn cases (n ≥ 1,m ≥ 3). Two cases will be called neighbouring cases if they have a common side. Prove that it is possible to write a real number in each case such that each number is equal to the sum of the numbers of the neighbouring cases and not all the numbers are zero if and only if there exist integers k, l such that n + 1 does not divide k and cos 2lpi m + cos kpi n+ 1 = 1 2 Ciprian Manolescu http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 5