1975~2002匈牙利奥林匹克数学竞赛(英文版)

匈牙利数学奥林匹克试题(1975-2002)75th Kürschák Competition 1975Problem1.Transform the equation ab2(1/(a + c)2 + 1/(a - c)2) = (a - b) into a simpler form, given that a > c ≥ 0, b > 0.Problem2.Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral Problem3.Let x0 = 5, x n+1 = x n + 1/x n. Prove that 45 < x1000 < 45.1.76th Kürschák Competition 1976Problem1 ABCD is a parallelogram. P is a point outside the parallelogram such that angles PAB and PCB have the same value but opposite orientation. Show that angle APB = angle DPC. Problem2 A lottery ticket is a choice of 5 distinct numbers from 1, 2, 3, ... , 90. Suppose that 55 distinct lottery tickets are such that any two of them have a common number. Prove that one can find four numbers such that every ticket contains at least one of the fourProblem3 Prove that if the quadratic x2 + ax + b is always positive (for all real x) then it can be written as the quotient of two polynomials whose coefficients are all positive.77th Kürschák Competition 1977Problem1 Show that there are no integers n such that n4 + 4n is a prime greater than 5.Problem2ABC is a triangle with orthocenter H.The median from A meets the circumcircle again at A1,and A2 is the reflection of A1 in the midpoint of BC. Thepoints B2 and C2 are defined similarly. Show that H, A2,B2 and C2 lie on a circle.Problem3 Three schools each have n students. Eachstudent knows a total of n+1 students at the other twoschools. Show that there must be three students, onefrom each school, who know each other.78th Kürschák Competition 1978Problem1 a and b are rationals. Show that if ax2 + by2 =1 has a rational solution (in x and y), then it must have infinitely many.Problem2 The vertices of a convex n-gon are colored so that adjacent vertices have different colors. Prove that if n is odd, then the polygon can be divided into triangles with non-intersecting diagonals such that no diagonal has its endpoints the same color.Problem3 A triangle has inradius r and circumradius R. Its longest altitude has length H. Show that if the triangle does not have an obtuse angle, then H ≥ r + R. When does equality hold?79th Kürschák Competition 1979Problem1 The base of a convex pyramid has an odd number of edges. The lateral edges of the pyramid are all equal, and the angles between neighbouring faces are all equal. Show that the base must be a regular polygon.Problem 2f is a real-valued function defined on the reals such that f(x) ≤ x and f(x+y) ≤ f(x) + f(y) for all x, y. Prove that f(x) = x for all x.Solution Suppose f(0) < 0. Then f(y) ≤ f(0) + f(y) < f(y). Contradiction. So f(0) ≥ 0 and hence f(0) = 0. Now suppose f(x) < x. Then f(0) ≤ f(x) + f(-x) < x + f(-x) ≤ x - x = 0. Contradiction. Hencef(x) = x.Problem3 An n x n array of letters is such that no two rows are the same. Show that it must be possible to omit a column, so that the remaining table has no two rows the same.80th Kürschák Competition 1980Problem1 Every point in space is colored with one of 5 colors. Prove that there are four coplanar points with different colors.Problem2 n > 1 is an odd integer. Show that there are positive integers a and b such that 4/n = 1/a + 1/b iff n has a prime divisor of the form 4k-1.Problem3 There are two groups of tennis players, one of 1000 players and the other of 1001 players. The players can ranked according to their ability. A higher ranking player always beats a lower ranking player (and the ranking never changes). We know the ranking within each group. Show how it is possible in 11 games to find the player who is 1001st out of 2001.81st Kürschák Competition 1981Problem1 Given any 5 points A, B, P, Q, R (in the plane) show that AB + PQ + QR + RP <= AP + AQ + AR + BP + BQ + BRProblem2 n > 2 is even. The squares of an n x n chessboard are painted with n2/2 colors so that there are exactly two squares of each color. Prove that one can always place n rooks on squares of different colors so that no two are in the same row or column.Problem3 Divide the positive integer n by the numbers 1, 2, 3, ... , n and denote the sum of the remainders by r(n). Prove that for infinitely many n we have r(n) = r(n+1).82nd Kürschák Competition 1982Problem1 A cube has all 4 vertices of one face at lattice points and integral side-length. Prove that the other vertices are also lattice points.Problem2 Show that for any integer k > 2, there are infinitely many positive integers n such that the lowest common multiple of n, n+1, ... , n+k-1 is greater than the lowest common multiple of n+1, n+2, ... , n+k.Problem3 The integers are colored with 100 colors, so that all the colors are used and given any integers a < b and A < B such that b - a = B - A, with a and A the same color and b and B the same color, we have that the whole intervals [a, b] and [A, B] are identically colored. Prove that -1982 and 1982 are different colors.83rd Kürschák Competition 1983Problem1 Show that the only rational solution to x3 + 3y3 + 9z3 - 9xyz = 0 is x = y = z = 0. Problem2 The polynomial x n + a1x n-1 + ... + a n-1x + 1 has non-negative coefficients and n real roots. Show that its value at 2 is at least 3n.Problem3 The n+1 points P1, P2, ... , P n, Q lie in the plane and no 3 are collinear. Given any two distinct points P i and P j, there is a third point P k such that Q lies inside the triangle P i P j P k. Prove that n must be odd.84th Kürschák Competition 1984Problem1 If we write out the first four rows of the Pascal triangle and add up the columns we get: 11 11 2 11 3 3 11 1 4 3 4 1 1If we write out the first 1024 rows of the triangle and add up the columns, how many of theresulting 2047 totals will be odd?Problem2 A1B1A2, B1A2B2, A2B2A3, B2A3B3, ... , A13B13A14, B13A14B14, A14B14A1, B14A1B1 are thin triangular plates with all their edges equal length, joined along their common edges. Can the network of plates be folded (along the edges A i B i) so that all 28 plates lie in the same plane? (They are allowed to overlap).Problem3 A and B are positive integers. We are given a collection of n integers, not all of which are different. We wish to derive a collection of n distinct integers. The allowed move is to take any two integers in the collection which are the same (m and m) and to replace them by m + A and m - B. Show that we can always derive a collection of n distinct integers by a finite sequence of moves.85th Kürschák Competition 1985Problem1 The convex polygon P0P1 ... P n is divided into triangles by drawing non-intersecting diagonals. Show that the triangles can be labeled with the numbers 1, 2, ... , n-1 so that the triangle labeled i contains the vertex P i (for each i).Problem2 For each prime dividing a positive integer n, take the largest power of the prime not exceeding n and form the sum of these prime powers. For example, if n = 100, the sum is 26 + 52 = 89. Show that there are infinitely many n for which the sum exceeds n.Problem3 Vertex A of the triangle ABC is reflected in theopposite side to give A'. The points B' and C' are definedsimilarly. Show that the area of A'B'C' is less than 5 times thearea of ABC.86th Kürschák Competition 1986Problem1 Prove that three half-lines from a given pointcontain three face diagonalsof a cuboid iff the half-linesmake with each other threeacute angles whose sum is180o.Problem2 Given n > 2, find the largest h and the smallest H such that h < x1/(x1 + x2) + x2/(x2 + x3) + ... + x n/(x n + x1) < H for all positive real x1, x2, ... , x n.Problem3 k numbers are chosen at random from the set {1, 2, ... , 100}. For what values of k is the probability ½ that the sum of the chosen numbers is even?87th Kürschák Competition 1987Problem1 Find all quadruples (a, b, c, d) of distinct positive integers satisfying a + b = cd and c + d = ab.Problem2 Does there exist an infinite set of points in space such that at least one, but only finitely many, points of the set belong to each plane?Problem3 A club has 3n+1 members. Every two members play just one of tennis, chess and table-tennis with each other. Each member has n tennis partners, n chess partners and n table-tennis partners. Show that there must be three members of the club, A, B and C such that A and B play chess together, B and C play tennis together and C and A play table-tennis together.88th Kürschák Competition 1988Problem1 P is a point inside a convex quadrilateral ABCD such that the areas of the triangles PAB, PBC, PCD and PDA are all equal. Show that one of its diagonals must bisect the area of thequadrilateral.Problem2 What is the largest possible number of triples a < b < c that can be chosen from 1, 2, 3, ... , n such that for any two triples a < b < c and a' < b' < c' at most one of the equations a = a', b = b', c = c' holds?Problem3 PQRS is a convex quadrilateral whose vertices are lattice points. The diagonals of the quadrilateral intersect at E. Prove that if the sum of the angles at P and Q is less than 180o then the triangle PQE contains a lattice point apart from P and Q either on its boundary or in its interior.89th Kürschák Competition 1989Problem1 Given two non-parallel lines e and f anda circle C which does not meet either line. Constructthe line parallel to f such that the length of its segmentinside C divided by the length of its segment from Cto e (and outside C) is as large as possible.Problem2 Let S(n) denote the sum of the decimaldigits of the positive integer n. Find the set of allpositive integers m such that s(km) = s(m) for k = 1,2, ... , m.Problem3 Walking in the plane, we are allowed to move from (x, y) to one of the four points (x, y ± 2x), (x ± 2y, y). Prove that if we start at (1, √2), then we cannot return there after finitely many moves.90th Kürschák Competition 1990Problem1 Show that for p an odd prime and n a positive integer, there is at most one divisor d of n2p such that d + n2 is a square.Problem2 I is the incenter of the triangle ABC and A' is the center of the excircle opposite A. The bisector of angle BIC meets the side BC at A". The points B', C', B", C" are defined similarly. Show that the lines A'A", B'B", C'C" are concurrent.Problem3 A coin has probability p of heads and probability 1-p of tails. The outcome of each toss is independent of the others. Show that it is possibleto choose p and k, so that if we toss the coin k times we can assign the 2k possible outcomes amongst 100 children, so that each has the same 1/100 chance of winning. [A child wins if one of its outcomes occurs.]91st Kürschák Competition 1991Problem1 a >= 1, b >= 1 and c > 0 are reals and n is a positive integer. Show that ( (ab + c)n - c) <= a n ( (b + c)n - c).Problem2 ABC is a face of a convex irregular triangularprism(the triangular faces are not necessarily congruent orparallel). The diagonals of the quadrilateral face opposite Ameet at A'. The points B' and C' are defined similarly. Showthat the lines AA', BB' and CC' are concurrent.Problem3 There are 998 red points in the plane, no threecollinear. What is the smallest k for which we can alwayschoose k blue points such that each triangle with red vertices has a blue point inside?92nd Kürschák Competition 1992Problem1 Given n positive integers a i, define S k = Σ a i k, A = S2/S1, and C = ( S3/n)1/3. For each of n = 2, 3 which of the following is true: (1) A >= C; (2) A <= C; or (3) A may be > C or < C,depending on the choice of a i?Problem2 Let f1(k) be the sum of the (base 10) digits of k. Define f n(k) = f1(f n-1(k) ). Find f1992(21991).Problem3 A finite number of points are given in the plane, no three collinear. Show that it is possible to color the points with two colors so that it is impossible to draw a line in the plane with exactly three points of the same color on one side of the line.93rd Kürschák Competition 1993Problem1 a and b are positive integers. Show that there are at most a finite number of integers n such that an2 + b and a(n + 1)2 + b are both squares.Problem2 The triangle ABC is not isosceles. The incircle touches BC, CA, AB at K, L, M respectively. N is the midpoint of LM. The line through B parallel to LM meets the line KL at D, and the line through C parallel to LM meets the line MK at E. Show that D, E and N are collinear. Problem3 Find the minimum value of x2n + 2 x2n-1 + 3 x2n-2 + ... + 2n x + (2n+1) for real x.94th Kürschák Competition 1994Problem1 Let r > 1 denote the ratio of two adjacent sides of a parallelogram. Determine how the largest possible value of the acute angle included by the diagonals depends on r.Problem2 Prove that after removing any n-3 diagonals of a convex n-gon, it is always possible to choose n-3 non-intersecting diagonals amongst those remaining, but that n-2 diagonals can be removed so that it is not possible to find n-3 non-intersecting diagonals amongst those remaining. Problem3 For k = 1, 2, ... , n, H k is a disjoint union of k intervals of the real line. Show that one can find [(n + 1)/2] disjoint intervals which belong to different H k.95th Kürschák Competition 1995Problem1 A rectangle has its vertices at lattice points and its sides parallel to the axes. Its smaller side has length k. It is divided into triangles whose vertices are all lattice points, such that each triangle has area ½. Prove that the number of the triangles which are right-angled is at least 2k. Problem2 A polynomial in n variables has the property that if each variable is given one of the values 1 and -1, then the result is positive whenever the number of variables set to -1 is even and negative when it is odd. Prove that the degree of the polynomial is at least n.Problem3 A, B, C, D are points in the plane, no three collinear. The lines AB and CD meet at E, and the lines BC and DA meet at F. Prove that the three circles with diameters AC, BD and EF either have a common point or are pairwise disjoint.96th Kürschák Competition 1996Problem1 The diagonals of a trapezium are perpendicular. Prove that the product of the two lateral sides is not less than the product of the two parallel sides.Problem2 Two delegations A and B, with the same number of delegates, arrived at a conference. Some of the delegates knew each other already. Prove that there is a non-empty subset A' of A such that either each member in B knew an odd number of members from A', or each member of B knew an even number of members from A'.Problem3 2kn+1 diagonals are drawn in a convex n-gon. Prove that among them there is a broken line having 2k+1 segments which does not go through any point more than once. Moreover, this is not necessarily true if kn diagonals are drawn.97th Kürschák Competition 1997Problem1 Let S be the set of points with coordinates (m, n), where 0 <= m, n < p. Show that we can find p points in S with no three collinearProblem2 A triangle ABC has incenter I and circumcenter O. The orthocenter of the three points at which the incircle touches its sides is X. Show that I, O and X are collinear.Problem3 Show that the edges of a planar graph can be colored with three colors so that there is no monochromatic circuit.98th Kürschák Competition 1998Problem1 Can you find an infinite set of positive integers such that each pair has a common divisor (greater than 1), no integer (greater than 1) divides all members of the set, and no member of the set divides any other member?Problem2 Show that there is a polynomial with integer coefficients whose values at 1, 2, ... , n are different powers of 2.Problem3 For which n > 2 can we find n points in the plane, no three collinear, so that for each triangle of the points which are in the convex hull, exactly one of the points belongs to its interior. 99th Kürschák Competition 1999Problem1 Let e(k) be the number of positive even divisors of k, and let o(k) be the number of positive odd divisors of k. Show that the difference between e(1) + e(2) + ... + e(n) and o(1) + o(2) + ... + o(n) does not exceed n.Problem2 ABC is an arbitrary triangle. Construct an interior point P such that if A' is the foot of the perpendicular from P to BC, and similarly for B' and C', then the centroid of A'B'C' is P. Problem3 Prove that every set of integers with more than 2k members has a subset B with k+2 members such that any two non-empty subsets of B with the same number of members have different sums.100th Kürschák Competition 2000Problem1 The square 0 ≤ x ≤ n, 0 ≤ y ≤ n has (n+1)2 lattice points. How many ways can each of these points be colored red or blue, so that each unit square has exactly two red vertices? Problem2 ABC is any non-equilateral triangle. P is any point in the plane different from the vertices. Let the line PA meet the circumcircle again at A'. Define B' and C' similarly. Show that there are exactly two points P for which the triangle A'B'C' is equilateral and that the line joining them passes through the circumcenter.Problem3 k is a non-negative integer and the integers a1, a2, ... , a n give at least 2k different remainders on division by n+k. Prove that among the a i there are some whose sum is divisible by n+k.101st Kürschák Competition 2001Problem 1Given any 3n-1 points in the plane, no three collinear, show that it is possible to find 2n whose convex hull is not a triangle.Solution The difficulty is that extra points can reduce thenumber of points in the convex hull. Consider for example theconfiguration above. If we take at least one point from each arc,then the convex hull is a triangle. So we can pick at most 2npoints to get a convex hull which is not a triangle.On the other hand, if we have N > 4 points with convex hull nota triangle, then it is easy to remove a point and still have theconvex hull not a triangle. If there is an interior point then wecan remove that. If there are no interior points, then the convex hull has N points and we can remove any point to get a convex hull of N-1 > 3 points.So if the result is false, then if we take any N ≥ 2n of thepoints the convex hull must be a triangle. Suppose the convexhull of the 3n-1 points is A1, B1, C1. If we remove A1, then theconvex hull is a triangle. Now B1 must be one of the verticesof this triangle, for if it belonged to a triangle XYZ of otherpoints, then it could not be part of the convex hull of thewhole set. Similarly for C1. So the convex hull after removingA1 must be A2B1C1 for some A2. We can now remove A2 and the convex hull must be A3B1C1 for some A3, and so on. Finally, we remove A n-1 to get A n B1C1 as the convex hull of the remaining 2n points.Similarly, we can define B2, B3, ... , B n, so that the convex hull after removing B1, B2, ... B i is A1B i+1C1, and we can define C2, C3, ... , C n, so that the convex hull after removing C1, C2, ... , C i is A1B1C i+1.But now we have 3n points A i, B j, C k chosen from 3n-1, so two must be the same. The A i are all distinct, and similarly the B j and the C k. So wlog we have A i = B j for some i,j. Now if we remove all the As, all the Bs and C1 from the original set we are left with at least n-1 points (because we are removing at most 2n distinct points) and these must belong to the interior of the blue triangle A i B1C1 and the yellow triangle A1B j C1 = A1A i C1. But the interiors are disjoint, so we have a contradiction.Problem 2k > 2 is an integer and n > kC3 (where aCb is the usual binomial coefficient a!/(b! (a-b)!) ). Show that given 3n distinct real numbers a i, b i, c i (where i = 1, 2, ... , n), there must be at least k+1 distinct numbers in the set {a i + b i, b i + c i, c i + a i | i = 1, 2, ... , n}. Show that the statement is not always true for n = kC3.Solution Suppose there are at most k distinct numbers. Then there are at most kC3 and hence <n distinct sets of 3 numbers chosen from them. So for some i ≠ j we must have {a i + b i, b i + c i, c i + a i} = {a j + b j, b j + c j, c j + a j}. But the set {a i, b i, c i} is uniquely determined by {a i + b i, b i + c i, c i + a i}, so {a i, b i, c i} = {a j, b j, c j}. Contradiction.Suppose we take S to be the set {30, 31, 32, ... , 3k-1}. Take all n = kC3 subsets of 3 elements and for each such subset A i take {a i, b i, c i} so that {a i + b i, b i + c i, c i + a i} = A i. Obviously a i, b i, c i are distinct, so we have to show that if (3a + 3b - 3c)/2 = (3r + 3s - 3t)/2, where a, b, c are distinct and r, s, t are distinct, then {a,b,c} = {r,s,t}. We have 3a + 3b + 3t = 3r + 3s + 3c. There are two cases. If a = t, then since the representation base 3 is unique, we must have one of r, s, c equal to b and the other two equal to a. Since c ≠ a, and c ≠ b that is impossible. So a, b, t must all be distinct and hence {a, b, t} = {r, s, c}. Since c ≠ a or b, we must have c = t and hence {a, b} = {r, s} and so {a, b, c} = {r, s, t} as required.For example, if k = 4, then n = 4 and we can take: a1, b1, c1 = -5/2, 7/2, 11/2 ;a2, b2, c2 = -23/2, 25/2, 29/2 ;a3, b3, c3 = -17/2, 19/2, 35/2 ;a4, b4, c4 = -15/2, 21/2, 33/2 .Problem 3The vertices of the triangle ABC are lattice points and there is no smaller triangle similar to ABC with its vertices at lattice points. Show that the circumcenter of ABC is not a lattice point.Solution Let the points A, B, C have coordinates A (0,0), B(a,b), C(c,d). Suppose the circumcenter D (x,y) is a lattice point. Then AD2 = AB2, so x2 + y2 = (x-a)2 + (y-b)2. Hence a2 + b2 is even. Hence a + b and a - b are even. Similarly, c + d and c - d are even. So the points X = ((a+b)/2, (a-b)/2) and Y = ((c+d)/2, (c-d)/2) are lattice points.But AX2 = (a+b)2/4 + (a-b)2/4 = (a2+b2)/2 = AB2/2. Similarly AY2 = AC2/2. XY2 = (a+b-c-d)2/4 + (a-b+c-d)2/4 = ((b-c)2 + (a-d)2)/2 = BC2/2. So AXY is similar to ABC and smaller. Contradiction. 102th Kürschák Competition 2002Problem1 ABC is an acute-angled non-isosceles triangle. H is the orthocenter, I is the incenter and O is the circumcenter. Show that if one of the vertices lies on the circle through H, I and O, then at least two vertices lie on it.Problem2 The Fibonacci numbers are defined by F1 = F2 = 1, F n = F n-1 + F n-2. Suppose that a rational a/b belongs to the open interval (F n/F n-1, F n+1/F n). Prove that b ≥ F n+1.Problem3 S is a convex 3n gon. Show that we can choose a set of triangles, such that the edges of each triangle are sides or diagonals of S, and every side or diagonal of S belongs to just one triangle.。