IMO预选题1967

IMO LongList 1967 IMO ShortList/LongList Project Group June 19, 2004 1. (Bulgaria 1) Prove that all numbers of the sequence 107811 3 , 110778111 3 , 111077781111 3 , . . . are exact cubes. 2. (Bulgaria 2) Prove that 1 3 n2 + 1 2 n+ 1 6 ≥ (n!) 2 n , and let n ≥ 1 be an integer. Prove that this inequality is only possible in the case n = 1. 3. (Bulgaria 3) Prove the trigonometric inequality cos(x) < 1− x 2 2 + x4 16 , when x ∈ ( 0, pi 2 ) . 4. (Bulgaria 4) Suppose medians ma and mb of a triangle are orthogonal. Prove that: a.) Using medians of that triangle it is possible to construct a rectangular triangle. b.) The following inequality: 5(a2 + b2 − c2) ≥ 8ab, is valid, where a, b and c are side length of the given triangle. 5. (Bulgaria 5) Solve the system of equations: x2 + x− 1 = y y2 + y − 1 = z z2 + z − 1 = x. 6. (Bulgaria 6) Solve the system of equations: |x+ y|+ |1− x| = 6 |x+ y + 1|+ |1− y| = 4. 1 7. (Great Britain 1) Let k,m, n be natural numbers such that m+ k + 1 is a prime greater than n+ 1. Let cs = s(s+ 1). Prove that (cm+1 − ck)(cm+2 − ck) . . . (cm+n − ck) is divisible by the product c1c2 . . . cn. Remark: This question was chosen as third question in the IMO. 8. (Great Britain 2) If x is a positive rational number show that x can be uniquely expressed in the form x = n∑ k=1 ak k! where a1, a2, . . . are integers, 0 ≤ an ≤ n − 1, for n > 1, and the series terminates. Show that x can be expressed as the sum of reciprocals of different integers, each of which is greater than 106. 9. (Great Britain 3) The n points P1, P2, . . . , Pn are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance dn between any two of these points has its largest possible value Dn. Calculate Dn for n = 2 to 7. and justify your answer. 10. (Hungary 1) In a sports meeting a total of m medals were awarded over n days. On the first day one medal and 17 of the remaining medals were awarded. On the second day two medals and 1 7 of the remaining medals were awarded, and so on. On the last day, the remaining n medals were awarded. How many medals did the meeting last, and what was the total number of medals ? Remark: This question was chosen as sixth question in the IMO. 11. (Hungary 2) In the space n ≥ 3 points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way. 12. (Hungary 3) Without using tables, find the exact value of the product: P = 7∏ k=1 cos ( kpi 15 ) . 13. (Hungary 4) The distance between centers of circles k1 and k2 with radii r is equal to r. Points A and B are on the circle k1, symmetric with respect to the line connecting the centers of circles. P is an arbitrary point on k2. Prove that PA2 + PB2 ≥ 2r2. 14. (Hungary 5) Prove that for an arbitrary pair of vectors f and g in the space the inequality af2 + bfg + cg2 ≥ 0 holds if and only if the following conditions are fulfilled: a ≥ 0, c ≥ 0, 4ac ≥ b2. Remark: This problem can also be found in one of the volumes by Morozova/Petrakov. c© by Orlando Do¨hring, member of the IMO ShortList/LongList Project Group, page 2/8 15. (Hungary 6) Three disks of diameter d are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius R of the sphere in order that axis of the whole figure has an angle of 60◦ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of 120◦ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis). 16. (Italy 1) A0B0C0 and A1B1C1 are acute-angled triangles. Describe, and prove, how to construct the triangle ABC with the largest possible area which is circumscribed about A0B0C0 (so BC contains B0, CA contains B0, and AB contains C0) and similar to A1B1C1. Remark: This question was chosen as fourth question in the IMO. 17. (Italy 2) Let ABCD be a regular tetrahedron. To an arbitrary point M on one edge, say CD, corresponds the point P = P (M) which is the intersection of two lines AH and BK, drawn from A orthogonally to BM and from B orthogonally to AM. What is the locus of P when M varies ? 18. (Italy 3) Which regular polygon can be obtained (and how) by cutting a cube with a plane ? 19. (Italy 4) Find values of the parameter u for which the expression y = tan(x− u) + tan(x) + tan(x+ u) tan(x− u)tan(x)tan(x+ u) does not depend on x. 20. (Mongolia 1) Givenm+n numbers: ai, i = 1, 2, . . . ,m, bj , j = 1, 2, . . . , n, determine the number of pairs (ai, bj) for which |i− j| ≥ k, where k is a non-negative integer. 21. (Mongolia 2) An urn contains balls of k different colors; there are ni balls of i− th color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls m balls of the same color appear. Find the greatest number of selections. 22. (Mongolia 3) Determine the volume of the body obtained by cutting the ball of radius R by the trihedron with vertex in the center of that ball, it its dihedral angles are α, β, γ. 23. (Mongolia 4) In what case does the system of equations x+ y +mz = a x+my + z = b mx+ y + z = c have a solution ? Find conditions under which the unique solution of the above system is an arithmetic progression. c© by Orlando Do¨hring, member of the IMO ShortList/LongList Project Group, page 3/8 24. (Mongolia 5) Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere. 25. (Mongolia 6) Prove the identity: n∑ k=1 Cnk ( tan (x 2 ))2k 1 + 2k( 1− tan2 (x 2 )k)  = sec2n (x2)+ secn(x). 26. (Poland 1) Prove that a tetrahedron with just one edge length greater than 1 has volume at most 1 8 . Remark: This question was chosen as second question in the IMO. 27. (Poland 2) Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal. 28. (Poland 3) Prove that for arbitrary positive numbers the following inequality holds 1 a + 1 b + 1 c ≤ a 8 + b8 + c8 a3b3c3 . 29. (Poland 4) Does there exist an integer such that its cube is equal to 3n2+3n+7, where n is an integer. 30. (Poland 5) Show that the triangle whose angles satisfy the equality sin2(A) + sin2(B) + sin2(C) cos2(A) + cos2(B) + cos2(C) = 2 is a rectangular triangle. 31. (Poland 6) A line l is drawn through the intersection pointH of altitudes of acute-angle triangles. Prove that symmetric images la, lb, lc of l with respect to the sides BC,CA,AB have one point in common, which lies on the circumcircle of ABC. 32. (Romania 1) Decompose the expression into real factors: E = 1− sin5(x)− cos5(x). 33. (Romania 2) The equation x5 + 5λx4 − x3 + (λα− 4)x2 − (8λ+ 3)x+ λα− 2 = 0 is given. Determine α so that the given equation has exactly (i) one root or (ii) two roots, respec- tively, independent from λ. c© by Orlando Do¨hring, member of the IMO ShortList/LongList Project Group, page 4/8 34. (Romania 3) Suppose that p and q are two different positive integers and x is a real number. Form the product (x + p)(x + q). Find the sum S(x, n) = ∑ (x + p)(x + q), where p and q take values from 1 to n. Does there exist integer values of x for which S(x, n) = 0. 35. (Romania 4) (i) Solve the equation: sin3(x) + sin3 ( 2pi 3 + x ) + sin3 ( 4pi 3 + x ) + 3 4 cos(2x) = 0. (ii) Supposing the solutions are in the form of arcs AB with one end at the point A, the beginning of the arcs of the trigonometric circle, and P a regular polygon inscribed in the circle with one vertex in A, find: 1.) The subsets of arcs having the other end in B in one of the vertices of the regular dodecagon. 2.) Prove that no solution can have the end B in one of the vertices of polygon P whose number of sides is prime or having factors other than 2 or 3. 36. (Romania 5) If x, y, z are real numbers satisfying relations x+ y + z = 1 and arctan(x) + arctan(y) + arctan(z) = pi 4 , prove that x2n+1 + y2n+1 + z2n+1 = 1 holds for all positive integers n. 37. (Romania 6) Prove the following inequality: k∏ i=1 xi · k∑ i=1 xn−1i ≤ k∑ i=1 xn+k−1i , where xi > 0, k ∈ N, n ∈ N. 38. (Socialists Republic Of Czechoslovakia 1) The parallelogram ABCD has AB = a,AD = 1, ∠BAD = A, and the triangle ABD has all angles acute. Prove that circles radius 1 and center A,B,C,D cover the parallelogram if and only a ≤ cosA+ √ 3 sinA. Remark: This question was chosen as first question in the IMO. 39. (Socialists Republic Of Czechoslovakia 2) Find all real solutions of the system of equations: n∑ k=1 xik = a i for i = 1, 2, . . . , n. 40. (Socialists Republic Of Czechoslovakia 3) Circle k and its diameter AB are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on AB and two other vertices on k. 41. (Socialists Republic Of Czechoslovakia 4) The square ABCD has to be decomposed into n triangles (which are not overlapping) and which have all angles acute. Find the smallest integer n for which there exist a solution of that problem and for such n construct at least one decom- position. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number. c© by Orlando Do¨hring, member of the IMO ShortList/LongList Project Group, page 5/8 42. (Socialists Republic Of Czechoslovakia 5) Let n be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers ≤ n. 43. (Socialists Republic Of Czechoslovakia 6) Given a segment AB of the length 1, define the set M of points in the following way: it contains two points A,B, and also all points obtained from A,B by iterating the following rule: With every pair of points X,Y the set M contains also the point Z of the segment XY for which Y Z = 3XZ. (i )Prove that the set M consists of points X from the segment AB for which the distance from the point A is either AX = 3k 4n or AX = 3k − 2 4n where n, k are non-negative integers. (ii) Prove that the point X0 : AX0 = 12 = X0B does not belong to the set M. 44. (Soviet Union 1) Let a1, . . . , a8 be reals, not all equal to zero. Let cn = 8∑ k=1 ank for n = 1, 2, 3, . . .. Given that among the numbers of the sequence (cn), there are infinitely many equal to zero, determine all the values of n for which cn = 0. Remark: This question was chosen as fifth question in the IMO. 45. (Soviet Union 2) Is it possible to put 100 (or 200) points on the wooden cube so that by all rotations of the cube the points map into themselves ? Justify your answer. (Darij Grinberg’s translation from Morozova/Petrakov:) Is it possible to find a set of 100 (or 200) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ? 46. (Soviet Union 3) Find all x for which, for all n, n∑ k=1 sin(kx) ≤ √ 3 2 . Remark: This problem can also be found in one of the volumes by Morozova/Petrakov. 47. (Soviet Union 4) In a group of interpreters each one speaks one of several foreign languages, 24 of them speak Japanese, 24 Malaysian, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malaysian and exactly 12 speak Farsi. (remark by Darij Grinberg:) However, it is not always possible to find such a subset from an odd number of translators. c© by Orlando Do¨hring, member of the IMO ShortList/LongList Project Group, page 6/8 48. (Soviet Union 5) A linear binomial l(z) = Az +B with complex coefficients A and B is given. It is known that the maximal value of |l(z)| on the segment −1 ≤ x ≤ 1 (y = 0) of the real line in the complex plane z = x+ iy is equal to M. Prove that for every z |l(z)| ≤Mρ, where ρ is the sum of distances from the point P = z to the points Q1 : z = 1 and Q3 : z = −1. (Darij Grinberg’s translation from Morozova/Petrakov:) Let z denote a variable complex number. Given the linear function l (z) = Az +B with complex coefficients A and B. Given that the maximum of |l (z)| on the interval −1 ≤ x ≤ 1 (with y = 0) of the real number axis of the complex plane z = x + iy is M. Show that for any z, the inequality |l (z)| ≤ Mρ holds, where ρ is the sum of the distances from the point P = z to the points Q1 (corresponding to the complex numer 1) and Q3 (corresponding to the complex number −1). 49. (Soviet Union 6) On the circle with center 0 and radius 1 the point A0 is fixed and points A1, A2, . . . , A999, A1000 are distributed in such a way that the angle ∠A00Ak = k (in radians). Cut the circle at points A0, A1, . . . , A1000. How many arcs with different lengths are obtained. ? Remark: This problem can also be found in one of the volumes by Morozova/Petrakov. 50. (Sweden 1) Determine all positive roots of the equation xx = 1√ 2 . 51. (Sweden 2) Let n and k be positive integers such that 1 ≤ n ≤ N + 1, 1 ≤ k ≤ N + 1. Show that: minn 6=k|sin(n)− sin(k)| < 2 N . 52. (Sweden 3) The function ϕ(x, y, z) defined for all triples (x, y, z) of real numbers, is such that there are two functions f and g defined for all pairs of real numbers, such that ϕ(x, y, z) = f(x+ y, z) = g(x, y + z) for all real numbers x, y and z. Show that there is a function h of one real variable, such that ϕ(x, y, z) = h(x+ y + z) for all real numbers x, y and z. 53. (Sweden 4) A subset S of the set of integers 0 - 99 is said to have property A if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in S (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set S with the property A. 54. (Sweden 5) In the plane a point O is and a sequence of points P1, P2, P3, . . . are given. The distances OP1, OP2, OP3, . . . are r1, r2, r3, . . . Let α satisfies 0 < α < 1. Suppose that for every n the distance from the point Pn to any other point of the sequence is ≥ rαn . Determine the exponent β, as large as possible such that for some C independent of n rn ≥ Cnβ , n = 1, 2, . . . c© by Orlando Do¨hring, member of the IMO ShortList/LongList Project Group, page 7/8 55. (Sweden 6) In making Euclidean constructions in geometry it is permitted to use a ruler and a pair of compasses. In the constructions considered in this question no compasses are permitted, but the ruler is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the rule. Then the distance between the parallel lines is equal to the breadth of the ruler. Carry through the following constructions with such a ruler. Construct: a.) The bisector of a given angle. b.) The midpoint of a given rectilinear line segment. c.) The center of a circle through three given non-collinear points. d.) A line through a given point parallel to a given line. 56. (The Democratic Republic Of Germany 1) Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius r, there exist one (or more) with maximum area. If so, determine their shape and area. 57. (The Democratic Republic Of Germany 2) Which fractions p q , where p, q are positive in- tegers < 100, is closest to √ 2? Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of √ 2 (without using any table). 58. (The Democratic Republic Of Germany 3) Suppose tan(α) = p q , where p and q are integers and q 6= 0. Prove that the number tan(β) for which tan(2β) = tan(3α) is rational only when p2 + q2 is the square of an integer. 59. (The Democratic Republic Of Germany 4) Prove the following statement: If r1 and r2 are real numbers whose quotient is irrational, then any real number x can be approximated arbitrarily well by the numbers of the form zk1,k2 = k1r1 + k2r2 integers, i.e. for every number x and every positive real number p two integers k1 and k2 can be found so that |x− (k1r1 + k2r2)| < p holds. Final Remark: This IMO Longlist is almost complete. Three problems got lost, two by Great Britain and another one, probably one by Italy. The problems where I (Orlando Do¨hring) provided a remark regarding the books by Morozova/Petrakov were also translated by Darij Grinberg. But I only used his translations where it is said explicitly. In this case the problems were also taken from the four volumes by Morozova/Petrakov. c© by Orlando Do¨hring, member of the IMO ShortList/LongList Project Group, page 8/8