Các bài toán vô địch Bankan

11-th Balkan Mathematical Olympiad Novi Sad, Yugoslavia – May 10, 1994 An acute angle XAY and a point P inside it are given Construct (by a ruler and a compass) a line that passes through P and intersects the rays AX and AY at B and C such that the area of the triangle ABC equals AP (Cyprus) Let m be an integer Prove that the polynomial x4 − 1994x3 + (1993 + m)x2 − 11x + m has at most one integer zero (Greece) Let (a1 , a2 , , an ) be a permutation of the numbers 1, 2, , n, where n ≥ Determine the largest possible value of n−1 |ak+1 − ak | k=1 (Romania) Find the smallest number n > for which there can exist a set of n people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances (Acquaintance is a symmetric relation.) (Bulgaria) 12-th Balkan Mathematical Olympiad Plovdiv, Bulgaria – May 9, 1995 Define x ∗ y = x+y Evaluate ( (((2 ∗ 3) ∗ 4) ∗ 5) ∗ ) ∗ 1995 + xy (FYR Macedonia) Circles c1 (O1 , r1 ) and c2 (O2 , r2 ), r2 > r1 , intersect at A and B so that ∠O1 AO2 = 90◦ The line O1 O2 meets c1 at C and D, and c2 at E and F (in the order C − E − D − F ) The line BE meets c1 at K and AC at M , and the line BD meets c2 at L and AF at N Prove that KE LN r2 · = r1 KM LD (Greece) Let a and b be natural numbers with a > b and | a + b Prove that the solutions of the equation x2 − (a2 − a + 1)(x − b2 − 1) − (b2 + 1)2 = are natural numbers, none of which is a perfect square (Albania) Let n be a natural number and S be the set of points (x, y) with x, y ∈ {1, 2, , n} Let T be the set of all squares with the verticesw in the set S We denote by ak (k ≥ 0) the number of (unordered) pairs of points for which there are exactly k squares in T having these two points as vertices Show that a0 = a2 + 2a3 (Yugoslavia) 13-th Balkan Mathematical Olympiad Bacau, Romania – April 30, 1996 Let O be the circumcenter and G be the centroid of a triangle ABC If R and r are the circumcenter and incenter of the triangle, respectively, prove that OG ≤ R(R − 2r) (Greece) Let p > be a prime Consider X = {p − n2 | n ∈ N} Prove that there are two distinct elements x, y ∈ X such that x = and x | y (Albania) In a convex pentagon ABCDE, M, N, P, Q, R are the midpoints of the sides AB, BC, CD, DE, EA, respectively If the segments AP, BQ, CR, DM pass through a single point, prove that EN contains that point as well (Yugoslavia) Show that there exists a subset A of the set {1, 2, , 21996 − 1} with the following properties: (i) ∈ A and 21996 − ∈ A; (ii) Every element of A \ {1} is the sum of two (possibly equal) elements of A; (iii) A contains at most 2012 elements (Romania) 14-th Balkan Mathematical Olympiad Kalabaka, Greece – April 30, 1997 Suppose that O is a point inside a convex quadrilateral ABCD such that OA2 + OB + OC + OD2 = 2SABCD , where SABCD denotes the area of ABCD Prove that ABCD is a square and O its center (Yugoslavia) Let A = {A1 , A2 , , Ak } be a collection of subsets of an n-element set S If for any two elements x, y ∈ S there is a subset Ai ∈ A containing exactly one of the two elements x, y, prove that 2k ≥ n (Yugoslavia) Circles C1 and C2 touch each other externally at D, and touch a circle Γ internally at B and C, respectively Let A be an intersection point of Γ and the common tangent to C1 and C2 at D Lines AB and AC meet C1 and C2 again at K and L, respectively, and the line BC meets C1 again at M and C2 again at N Show that the lines AD, KM, LN are concurrent (Greece) Determine all functions f : R → R that satisfy f (xf (x) + f (y)) = f (x)2 + y for all x, y (Bulgaria) 15-th Balkan Mathematical Olympiad Nicosia, Cyprus – May 5, 1998 k Consider the finite sequence 1998 , k = 1, 2, , 1997 How many distinct terms are there in this sequence? (Greece) Let n ≥ be an integer, and let < a1 < a2 < · · · < a2n+1 be real numbers Prove the inequality √ n a1 − √ n a2 + √ n a3 − · · · + √ a2n+1 < n a1 − a2 + a3 − · · · + a2n+1 (Romania) Let S denote the set of points inside or on the border of a triangle ABC, without a fixed point T inside the triangle Show that S can be partitioned into disjoint closed segemnts (Yugoslavia) Prove that the equation y = x5 − has no integer solutions (Bulgaria) 16-th Balkan Mathematical Olympiad Ohrid, Macedonia – May 8, 1999 Let D be the midpoint of the shorter arc BC of the circumcircle of an acute-angled triangle ABC The points symmetric to D with respect to BC and the circumcenter are denoted by E and F , respectively Let K be the midpoint of EA (a) Prove that the circle passing through the midpoints of the sides of ABC also passes through K (b) The line through K and the midpoint of BC is perpendicular to AF Let p > be a prime number with | p − Consider the set S = {y − x3 − | x, y ∈ Z, ≤ x, y ≤ p − 1} Prove that at most p − elements of S are divisible by p Let M, N, P be the orthogonal projections of the centroid G of an acuteangled triangle ABC onto AB, BC, CA, respectively Prove that SM N P < ≤ 27 SABC 4 Let ≤ x0 ≤ x1 ≤ x2 ≤ · · · be a sequence of nonnegative integers such that for every k ≥ the number of terms of the sequence which not exceed k is finite, say yk Prove that for all positive integers m, n, n m yj ≥ (n + 1)(m + 1) xi + i=0 j=0 17-th Balkan Mathematical Olympiad Chi¸sinˇau, Moldova – May 5, 2000 [BMO 1997#4] Determine all functions f : R → R that satisfy f (xf (x) + f (y)) = f (x)2 + y for all x, y (Albania) Let ABC be a scalene triangle and E be a point on the median AD Point F is the orthogonal projection of E onto BC Let M be a point on the segment EF , and N, P be the orthogonal projections of M onto AC and AB respectively Prove that the bisectors of the angles P M N and P EN are parallel √ Find the maximal number of rectangles × 10 that can be cut off from a rectangle 50 × 90 by using cuts parallel to the edges of the big rectangle (Yugoslavia) A positive integer is a power if it is of the form ts for some integers t, s ≥ Prove that for any natural number n there exists a set A of positive integers with the following properties: (i) A has n elements; (ii) Every element of A is a power; (iii) For any ≤ k ≤ n and any r1 , , rk ∈ A, r1 + · · · + rk is a power k 18-th Balkan Mathematical Olympiad Belgrade, Yugoslavia – May 5, 2001 Let n be a positive integer Prove that if a, b are integers greater than such that ab = 2n − 1, then the number ab − (a − b) − is of the form k · 22m , where k is odd and m a positive integer Prove that a convex pentagon that satisfies the following two conditions must be regular: (i) All its interior angles are equal; (ii) The lengths of all its sides are rational numbers Let a, b, c be positive real numbers such that a + b + c ≥ abc Prove that √ a2 + b2 + c2 ≥ abc A cube of edge is divided into 27 unit cube cells One of these cells is empty, while in the other cells there are unit cubes which are arbitrarily denoted by 1, 2, , 26 An legal move consists of moving a unit cube into a neighboring empty cell (two cells are neighboring if they share a face) Does there exist a finite sequence of legal moves after which any two cubes denoted by k and 27 − k (k = 1, 2, , 13) will exchange their positions? 19-th Balkan Mathematical Olympiad Antalya, Turkey – April 27, 2002 Points A1 , A2 , , An (n ≥ 4), no three of which are collinear, are given on the plane Some pairs of distinct points among them are connected by segments such that every point is connected to at least three other points Prove that there exist an integer k > and distinct points X1 , X2 , , X2k from the set {A1 , , An } such that Xi is connected to Xi+1 for i = 1, 2, , 2k, where X2k+1 ≡ X1 The sequence (an ) is defined by a1 = 20, a2 = 30 and an+2 = 3an+1 − an for every n ≥ Find all positive integers n for which + 5an an+1 is a perfect square Two circles with different radii intersect at A and B Their common tangents M N and ST touch the first circle at M and S and the second circle at N and T Show that the orthocenters of triangles AM N , AST , BM N , and BST are the vertices of a rectangle Determine all functions f : N → N such that for all positive integers n 2n + 2001 ≤ f (f (n)) + f (n) ≤ 2n + 2002 20-th Balkan Mathematical Olympiad Tirana, Albania – May 4, 2003 Does there exist a set B of 4004 distinct natural numbers, such that for any subset A of B containing 2003 elements, the sum of the elements of A is not divisible by 2003? (FYR Macedonia) Let ABC be a triangle with AB = AC The tangent at A to the circumcircle of the triangle ABC meets the line BC at D Let E and F be the points on the perpendicular bisectors of the segments AB and AC respectively, such that BE and CF are both perpendicular to BC Prove that the points D, E, and F are collinear (Romania) Find all functions f : Q → R which satisfy the following conditions: (i) f (x + y) − yf (x) − xf (y) = f (x)f (y) − x − y + xy for all x, y ∈ Q; (ii) f (x) = 2f (x + 1) + + X for all x ∈ Q; (iii) f (1) + > (Cyprus) Let m and n be coprime odd positive integers A rectangle ABCD with AB = m and AD = n is divided into mn unit squares Let A1 , A2 , , Ak be the consecutive points of intersection of the diagonal AC with the sides of the unit squares (where A1 = A and Ak = C) Prove that √ k−1 j+1 (−1) j=1 Aj Aj+1 = m2 + n mn (Bulgaria) 21-st Balkan Mathematical Olympiad Pleven, Bulgaria – May 7, 2004 A sequence of real numbers a0 , a2 , a2 , satisfies the condition am+n + am−n − m + n − = a2m + a2n for all m, n ∈ N with m ≥ n If a1 = 3, determine a2004 (Cyprus) Find all solutions in the set of prime numbers of the equation xy − y x = xy − 19 (Albania) Let O be an interior point of an acute-angled triangle ABC The circles centered at the midpoints of the sides of the triangle ABC and passing through point O, meet in points K, L, M different from O Prove that O is the incenter of the triangle KLM if and only if O is the circumcenter of the triangle ABC (Romania) A plane is divided into regions by a finite number of lines, no three of which are concurrent We call two regions neighboring if their common boundary is either a segment, a ray, or a line One should write an integer in each of the regions so as to fulfil the following two conditions: (a) The product of the numbers from two neighboring regions is less than their sum; (b) The sum of all the numbers in the halfplane determined by any of the lines is equal to zero Prove that this can be done if and only if not all the lines are parallel (Serbia and Montenegro) 22-nd Balkan Mathematical Olympiad Ia¸si, Romania – May 6, 2005 The incircle of an acute-angled triangle ABC touches AB at D and AC at E Let the bisectors of the angles ∠ACB and ∠ABC intersect the line DE at X and Y respectively, and let Z be the midpoint of BC Prove that the triangle XY Z is equilateral if and only if ∠A = 60◦ (Bulgaria) Find all primes p such that p2 − p + is a perfect cube (Albania) If a, b, c are positive real numbers, prove the inequality a2 b2 c2 4(a − b)2 + + ≥a+b+c+ b c a a+b+c When does equality occur? (Serbia and Montenegro) Let n ≥ be an integer, and let S be a subset of {1, 2, , n} such that S neither contains two coprime elements, nor does it contain two elements, one of which divides the other What is the maximum possible number of elements of S? (Romania) 23-rd Balkan Mathematical Olympiad Agros, Cyprus – April 29, 2006 If a, b, c are positive numbers, prove the inequality 1 + + ≥ a(1 + b) b(1 + c) c(1 + a) + abc A line m intersects the sides AB, AC and the extension of BC beyond C of the triangle ABC at points D, F, E, respectively The lines through points A, B, C which are parallel to m meet the circumcircle of triangle ABC again at points A1 , B1 , C1 , respectively Show that the lines A1 E, B1 F , C1 D are concurrent Determine all triples (m, n, p) of positive rational numbers such that the numbers 1 m+ , n+ , p+ np pm mn are integers Given a positive integer m, consider the sequence (an ) of positive integers defined by the initial term a0 = a and the recurrent relation an+1 = an if an is even, an + m if an is odd Find all values of a for which this sequence is periodic (i.e there exists d > such that an+d = an for all n) 24-th Balkan Mathematical Olympiad Rhodes, Greece – April 28, 2007 In a convex quadrilateral ABCD with AB = BC = CD, the diagonals AC and BD are of different length and intersect at point E Prove that AE = DE if and only if ∠BAD + ∠ADC = 120◦ (Albania) Find all functions f : R → R such that for all real x, y, f (f (x) + y) = f (f (x) − y) + 4f (x)y (Bulgaria) Determine all natural numbers n for which there exists a permutation σ of numbers 1, 2, , n such that the number σ(1) + is rational σ(2) + ··· + σ(n) (Serbia) Let n ≥ be an integer Let C1 , C2 , C3 be the circumferences of three convex n-gons in a plane such that the intersection of any two of them is a finite set of points Find the maximum possible number of points in C1 ∩ C2 ∩ C3 (Turkey) 25-th Balkan Mathematical Olympiad Ohrid, FYR Macedonia – May 6, 2008 An acute-angled scalene triangle ABC with AC > BC is given Let O be its circumcenter, H its orthocenter, and F the foot of the altitude from C Let P be the point (other than A) on the line AB such that AF = P F , and M be the midpoint of AC We denote the intersection of P H and BC by X, the intersection of OM and F X by Y , and the intersection of OF and AC by Z Prove that the points F, M, Y and Z are concyclic Does there exist a sequence a1 , a2 , of positive numbers satisfying both of the following conditions: (i) (ii) n i=1 ≤ n2 for every positive integer n; n i=1 ≤ 2008 for every positive integer n? Let n be a positive integer The rectangle ABCD with side lengths 90n + and 90n + is partitioned into unit squares with sides parallel to the sides of ABCD Let S be the set of all points which are vertices of these unit squares Prove that the number of lines which pass through at least two points from S is divisible by 4 Let c be a positive integer The sequence a1 , a2 , is defined by a1 = c and an+1 = a2n + an + c3 for every positive integer n Find all values of c for which there exist some integers k ≥ and m ≥ such that a2k + c3 is the m-th power of some positive integer Time allowed: 4.5 hours Each problem is worth 10 points 26-th Balkan Mathematical Olympiad Kragujevac, Serbia – April 30, 2009 Find all integer solutions of the equation 3x − 5y = z (Greece) In a triangle ABC, points M and N on the sides AB and AC respectively are such that M N BC Let BN and CM intersect at point P The circumcircles of triangles BM P and CN P intersect at two distinct points P and Q Prove that ∠BAQ = ∠CAP (Moldova) A 9×12 rectangle is divided into unit squares The centers of all the unit squares, except the four corner squares and the eight squares adjacent (by side) to them, are colored red Is it possible to numerate the red centers by C1 , C2 , , C96 so that the following two conditions are fulfilled: √ 1◦ All segments C1 C2 , C2 C3 , C95 C96 , C96 C1 have the length 13; 2◦ The polygonal line C1 C2 C96 C1 is centrally symmetric? (Bulgaria) Determine all functions f : N → N satisfying f f (m)2 + 2f (n)2 = m2 + 2n2 for all m, n ∈ N Each problem is worth 10 points Time allowed: 21 hours (Bulgaria)