2007年罗马尼亚数学国家集训队测试题

Romania Team Selection Tests Pitesti and Bucuresti 2007 Day 1 - 13 April 2007 1 If a1, a2, . . ., an ≥ 0 are such that a21 + · · ·+ a2n = 1, then find the maximum value of the product (1− a1) · · · (1− an). 2 Let f : Q→ R be a function such that |f(x)− f(y)| ≤ (x− y)2 for all x, y ∈ Q. Prove that f is constant. 3 Let A1A2 . . . A2n be a convex polygon and let P be a point in its interior such that it doesn’t lie on any of the diagonals of the polygon. Prove that there is a side of the polygon such that none of the lines PA1, . . ., PA2n intersects it in its interior. 4 Let O1 and O2 two exterior circles. Let A, B, C be points on O1 and D, E, F points on O1 such that AD and BE are the common exterior tangents to these two circles and CF is one of the interior tangents to these two circles, and such that C, F are in the interior of the quadrilateral ABED. If CO1 ∩AB = {M} and FO2 ∩DE = {N} then prove that MN passes through the middle of CF . http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 1 Romania Team Selection Tests Pitesti and Bucuresti 2007 Day 2 - 14 April 2007 1 Let f = Xn + an−1Xn−1 + . . .+ a1X + a0 be an integer polynomial of degree n ≥ 3 such that ak + an−k is even for all k ∈ 1, n− 1 and a0 is even. Suppose that f = gh, where g, h are integer polynomials and deg g ≤ deg h and all the coefficients of h are odd. Prove that f has an integer root. 2 Let ABC be a triangle, E and F the points where the incircle and A-excircle touch AB, and D the point on BC such that the triangles ABD and ACD have equal in-radii. The lines DB and DE intersect the circumcircle of triangle ADF again in the points X and Y . Prove that XY ‖ AB if and only if AB = AC. 3 Find all subsets A of {1, 2, 3, 4, . . .}, with |A| ≥ 2, such that for all x, y ∈ A, x 6= y, we have that x+ygcd(x,y) ∈ A. Dan Schwarz 4 Let S be the set of n-uples (x1, x2, . . . , xn) such that xi ∈ {0, 1} for all i ∈ 1, n, where n ≥ 3. Let M(n) be the smallest integer with the property that any subset of S with at least M(n) elements contains at least three n-uples (x1, . . . , xn) , (y1, . . . , yn) , (z1, . . . , zn) such that n∑ i=1 (xi − yi)2 = n∑ i=1 (yi − zi)2 = n∑ i=1 (zi − xi)2 . (a) Prove that M(n) ≤ ⌊ 2n+1 n ⌋ + 1. (b) Compute M(3) and M(4). http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 2 Romania Team Selection Tests Pitesti and Bucuresti 2007 Day 3 1 Let F be the set of all the functions f : P(S) −→ R such that for all X,Y ⊆ S, we have f(X∩Y ) = min(f(X), f(Y )), where S is a finite set (and P(S) is the set of its subsets). Find max f∈F |Im(f)|. 2 Prove that for n, p integers, n ≥ 4 and p ≥ 4, the proposition P(n, p) n∑ i=1 1 xip ≥ n∑ i=1 xi p for xi ∈ R, xi > 0, i = 1, . . . , n , n∑ i=1 xi = n, is false. Dan Schwarz [hide=”Remark”]In the competition, the students were informed (fact that doesn’t actually relate to the problem’s solution) that the propositions P(4, 3) are P(3, 4) true. 3 Let ai, i = 1, 2, . . . , n, n ≥ 3, be positive integers, having the greatest common divisor 1, such that aj divide n∑ i=1 ai for all j = 1, 2, . . . , n. Prove that n∏ i=1 ai divides ( n∑ i=1 ai )n−2 . 4 The points M,N,P are chosen on the sides BC,CA,AB of a triangle ∆ABC, such that the triangle ∆MNP is acute-angled. We denote with x the length of the shortest altitude of the triangle ∆ABC, and with X the length of the longest altitudes of the triangle ∆MNP . Prove that x ≤ 2X. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 3 Romania Team Selection Tests Pitesti and Bucuresti 2007 Day 4 1 Prove that the function f : N −→ Z defined by f(n) = n2007 − n!, is injective. 2 Let A1A2A3A4A5 be a convex pentagon, such that [A1A2A3] = [A2A3A4] = [A3A4A5] = [A4A5A1] = [A5A1A2]. Prove that there exists a point M in the plane of the pentagon such that [A1MA2] = [A2MA3] = [A3MA4] = [A4MA5] = [A5MA1]. Here [XY Z] stands for the area of the triangle ∆XY Z. 3 Consider the set E = {1, 2, . . . , 2n}. Prove that an element c ∈ E can belong to a subset A ⊂ E, having n elements, and such that any two distinct elements in A do not divide one each other, if and only if c > n ( 2 3 )k+1 , where k is the exponent of 2 in the factoring of c. 4 i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number N ∈ N, such that for any prime p, p > N , the p-th term of the progression is also prime. ii) Find all polynomials f(X) ∈ Z[X], such that there exist N ∈ N, such that for any prime p, p > N , |f(p)| is also prime. Dan Schwarz http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 4 Romania Team Selection Tests Pitesti and Bucuresti 2007 Day 5 1 In a circle with center O is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from O to the incenters of the formed triangles is independent of the triangulation. 2 Let ABC be a triangle, and ωa, ωb, ωc be circles inside ABC, that are tangent (externally) one to each other, such that ωa is tangent to AB and AC, ωb is tangent to BA and BC, and ωc is tangent to CA and CB. Let D be the common point of ωb and ωc, E the common point of ωc and ωa, and F the common point of ωa and ωb. Show that the lines AD, BE and CF have a common point. 3 Let ABCDE be a convex pentagon, such that AB = BC, CD = DE, ∠B+∠D = 180◦, and it’s area is √ 2. a) If ∠B = 135◦, find the length of [BD]. b) Find the minimum of the length of [BD]. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 5 Romania Team Selection Tests Pitesti and Bucuresti 2007 Day 6 1 Let ABCD be a parallelogram with no angle equal to 60o. Find all pairs of points E,F , in the plane of ABCD, such that triangles AEB and BFC are isosceles, of basis AB, respectively BC, and triangle DEF is equilateral. Valentin Vornicu 2 The world-renowned Marxist theorist Joric is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer n, he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the defect of the number n. Determine the average value of the defect (over all positive integers), that is, if we denote by δ(n) the defect of n, compute lim n→∞ ∑n k=1 δ(k) n . Iurie Boreico 3 Three travel companies provide transportation between n cities, such that each connection between a pair of cities is covered by one company only. Prove that, for n ≥ 11, there must exist a round-trip through some four cities, using the services of a same company, while for n < 11 this is not anymore necessarily true. Dan Schwarz http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 6 Romania Team Selection Tests Pitesti and Bucuresti 2007 Day 7 1 For n ∈ N, n ≥ 2, ai, bi ∈ R, 1 ≤ i ≤ n, such that n∑ i=1 a2i = n∑ i=1 b2i = 1, n∑ i=1 aibi = 0. Prove that ( n∑ i=1 ai )2 + ( n∑ i=1 bi )2 ≤ n. Cezar Lupu Tudorel Lupu 2 Let ABC be a triangle, let E,F be the tangency points of the incircle Γ(I) to the sides AC, respectively AB, and let M be the midpoint of the side BC. Let N = AM ∩ EF , let γ(M) be the circle of diameter BC, and let X,Y be the other (than B,C) intersection points of BI, respectively CI, with γ. Prove that NX NY = AC AB . Cosmin Pohoata 3 The problem is about real polynomial functions, denoted by f , of degree deg f . a) Prove that a polynomial function f can‘t be wrriten as sum of at most deg f periodic functions. b) Show that if a polynomial function of degree 1 is written as sum of two periodic functions, then they are unbounded on every interval (thus, they are ”wild”). c) Show that every polynomial function of degree 1 can be written as sum of two periodic functions. d) Show that every polynomial function f can be written as sum of deg f+1 periodic functions. e) Give an example of a function that can‘t be written as a finite sum of periodic functions. Dan Schwarz http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 7