IMO shortlist problems 1959-2009 (new)

IN THE NAME OF ALLAH IMO ShortList Problems 1959 – 2009 Collected by: Amir Hossein Parvardi (amparvardi) Problems from: http://www.artofproblemsolving.com/Forum/resources.php Published: 2010-10 Email: amir_ahp_2009@yahoo.com Contents Year Page Number of Problems 1959 5 6 ∗ 1960 7 7 1961 9 6 1962 11 7 1963 13 6 1964 15 6 1965 17 6 1966 19 63 1967 27 59 1968 38 25 1969 41 71 1970 48 12 1971 50 17 1972 53 12 1973 55 17 1974 58 12 1975 60 15 1976 62 12 1977 64 16 1978 67 17 1979 69 26 Year Page Number of Problems 1980 72 21 1981 75 19 1982 78 20 1983 81 25 1984 84 20 1985 87 22 1986 90 21 1987 93 23 1988 97 31 1989 99 32 1990 104 28 1991 112 30 1992 117 21 1993 120 26 1994 128 24 1995 133 28 1996 139 30 1997 146 26 1998 150 28 1999 156 27 2000 163 27 2001 169 28 2002 175 27 2003 180 27 Year Page Number of Problems 2004 188 30 2005 195 27 2006 201 30 2007 207 30 2008 214 26 2009 221 30 ∗ ShortListed Problems of the years 1959 to 1966 were the same, so I just added those problems to the year 1966 and used IMO problems for the years 1959 – 1965. Thanks Orlando (orl) for this suggestion. IMO 1959 Brasov and Bucharest, Romania Day 1 1 Prove that the fraction 21n + 4 14n + 3 is irreducible for every natural number n. 2 For what real values of x is  x + √ 2x − 1 +  x − √ 2x − 1 = A given a) A = √ 2; b) A = 1; c) A = 2, where only non-negative real numbers are admitted for square roots? 3 Let a, b, c be real numbers. Consider the quadratic equation in cos x a cos x 2 + b cos x + c = 0. Using the numbers a, b, c form a quadratic equation in cos 2x whose roots are the same as those of the original equation. Compare the equation in cos x and cos 2x for a = 4, b = 2, c = −1. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 1 IMO 1959 Brasov and Bucharest, Romania Day 2 4 Construct a right triangle with given hypotenuse c such that the median drawn to the hy- potenuse is the geometric mean of the two legs of the triangle. 5 An arbitrary point M is selected in the interior of the s egm ent AB. The square AMCD and M BEF are constructed on the same side of AB, with segments AM and M B as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and also at another point N . Let N  denote the point of intersection of the straight lines AF and BC. a) Prove that N and N  coincide; b) Prove that the straight lines M N pass through a fixed point S independent of the choice of M ; c) Find the locus of the midpoints of the segments P Q as M varies between A and B. 6 Two planes, P and Q, intersect along the line p. The point A is given in the plane P, and the point C in the plane Q; neither of these points lies on the straight line p. Construct an isosceles trapezoid ABCD (with AB  CD) in which a circle can be inscribed, and with vertices B and D lying in planes P and Q respectively. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 2 IMO 1960 Sinaia, Romania Day 1 1 Determine all three-digit numbers N having the property that N is divisible by 11, and N 11 is equal to the sum of the squares of the digits of N . 2 For what values of the variable x does the following inequality hold: 4x 2 (1 − √ 2x + 1) 2 < 2x + 9 ? 3 In a given right triangle ABC, the hypotenuse BC, of length a, is divided into n equal parts (n and odd integer). Let α be the acute angel subtending, from A, that segment which contains the mdipoint of the hypotenuse. Let h be the length of the altitude to the hypotenuse fo the triangle. Prove that: tan α = 4nh (n 2 − 1)a . http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 1 IMO 1960 Sinaia, Romania Day 2 4 Construct triangle ABC, given h a , h b (the altitudes from A and B), and m a , the median from vertex A. 5 Consider the cube ABCDA  B  C  D  (with face ABCD directly above face A  B  C  D  ). a) Find the locus of the midpoints of the segments XY , where X is any point of AC and Y is any piont of B  D  ; b) Find the locus of points Z which lie on the segment XY of part a) with ZY = 2XZ. 6 Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let V 1 be the volume of the cone and V 2 be the volume of the cylinder. a) Prove that V 1 = V 2 ; b) Find the smallest number k for which V 1 = kV 2 ; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone. 7 An isosceles trapezoid with bases a and c and altitude h is given. a) On the axis of symmetry of this trapezoid, find all points P such that both legs of the trapezoid subtend right angles at P ; b) Calculate the distance of p from either base; c) Determine under what conditions such points P actually exist. Discuss various cases that might arise. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 2 IMO 1961 Veszprem, Hungary Day 1 1 Solve the system of equations: x + y + z = a x 2 + y 2 + z 2 = b 2 xy = z 2 where a and b are constants. Give the conditions that a and b must satisfy so that x, y, z are distinct positive numbers. 2 Let a, b, c be the sides of a triangle, and S its area. Prove: a 2 + b 2 + c 2 ≥ 4S √ 3 In what case does equality hold? 3 Solve the equation cos n x − sin n x = 1 where n is a natural number. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 1 IMO 1961 Veszprem, Hungary Day 2 4 Consider triangle P 1 P 2 P 3 and a point p within the triangle. Lines P 1 P, P 2 P, P 3 P intersect the opposite sides in points Q 1 , Q 2 , Q 3 respectively. Prove that, of the numbers P 1 P P Q 1 , P 2 P P Q 2 , P 3 P P Q 3 at least one is ≤ 2 and at least one is ≥ 2 5 Construct a triangle ABC if AC = b, AB = c and ∠AMB = w, where M is the midpoint of the segment BC and w < 90. Prove that a solution exists if and only if b tan w 2 ≤ c < b In what case does the equality hold? 6 Consider a plane  and three non-collinear points A, B, C on the same side of ; suppose the plane determined by these three points is not parallel to . In plane  take three arbitrary points A  , B  , C  . Let L, M, N be the midpoints of segments AA  , BB  , CC  ; Let G be the centroid of the triangle LMN . (We will not consider positions of the points A  , B  , C  such that the points L, M, N do not form a triangle.) What is the locus of point G as A  , B  , C  range independently over the plane ? http://www.artofproblemsolving.com/ This file was downloaded from the AoPS Math Olympiad Resources Page Page 2 [...]... be any connecting segment of length d Prove that the number of diameters of the given set is at most n This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 IMO Shortlist 1966 1 Given n > 3 points in the plane such that no three of the points are collinear Does there exist a circle passing through (at least) 3 of the given points and not containing... the ”R-neighborhood” of a figure is defined as the locus of all points whose distance to the figure is ≤ R This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 1 IMO Shortlist 1966 7 For which arrangements of two infinite circular cylinders does their intersection lie in a plane? 8 We are given a bag of sugar, a two-pan balance, and a weight of 1 gram... triangle XY Z Describe the locus of points Z as X varies along K and Y varies along the boundary of Q This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 2 IMO Shortlist 1966 17 Let ABCD and A B C D be two arbitrary parallelograms in the space, and let M, N, P, Q be points dividing the segments AA , BB , CC , DD in equal ratios a.) Prove that... the tetrahedron equals one -sixth the product of the three smallest edges not belonging to the same face This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 3 IMO Shortlist 1966 24 There are n ≥ 2 people at a meeting Show that there exist two people at the meeting who have the same number of friends among the persons at the meeting (It is assumed... integer, prove that : 3 10n + log10 n; 1 1 1 2 + 3 + ··· + n − (a) log10 (n + 1) > (b) log n! > 3n 10 1 This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 4 IMO Shortlist 1966 31 Solve the equation |x2 − 1| + |x2 − 4| = mx as a function of the parameter m Which pairs (x, m) of integers satisfy this equation ? 32 The side lengths a, b, c of a triangle... and compute the distance between these two points, if the lengths OA = a, OB = b and AB = d are given This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 5 IMO Shortlist 1966 40 For a positive real number p, find all real solutions to the equation x2 + 2px − p2 − x2 − 2px − p2 = 1 41 Given a regular n-gon A1 A2 An (with n ≥ 3) in a plane How many... them, if the side lengths a, b, c of triangle ABC are given How many of these shortest segments exist ? This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 6 IMO Shortlist 1966 48 For which real numbers p does the equation x2 + px + 3p = 0 have integer solutions ? 49 Two mirror walls are placed to form an angle of measure α There is a candle inside... students who solved just one problem, half did not solve problem A How many students solved only problem B? This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 7 IMO Shortlist 1966 59 Let a, b, c be the lengths of the sides of a triangle, and α, β, γ respectively, the angles opposite these sides Prove that if γ a + b = tan (a tan α + b tan β) 2 the... where the abbreviation |P1 P2 P3 | denotes the (non-directed) area of an arbitrary triangle P1 P2 P3 This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 8 IMO Shortlist 1967 Bulgaria 1 Prove that all numbers of the sequence 107811 , 3 110778111 111077781111 , , 3 3 are exact cubes 2 Prove that 2 1 2 1 1 n + n + ≥ (n!) n , 3 2 6 and let n ≥ 1... 1 = z z 2 + z − 1 = x 6 Solve the system of equations: |x + y| + |1 − x| = 6 |x + y + 1| + |1 − y| = 4 This file was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/ Page 1 IMO Shortlist 1967 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 . IN THE NAME OF ALLAH IMO ShortList Problems 1959 – 2009 Collected by: Amir Hossein Parvardi (amparvardi) Problems from: http://www.artofproblemsolving.com/Forum/resources.php. were the same, so I just added those problems to the year 1966 and used IMO problems for the years 1959 – 1965. Thanks Orlando (orl) for this suggestion. IMO 1959 Brasov and Bucharest, Romania Day. 180 27 Year Page Number of Problems 2004 188 30 2005 195 27 2006 201 30 2007 207 30 2008 214 26 2009 221 30 ∗ ShortListed Problems of the years 1959 to