Math Olympiad Problems Collection v1

(Parte 1 de 9)

Contents

1.1 1st IMO, Romania, 1959	6
1.2 2nd IMO, Romania, 1960	7
1.3 3rd IMO, Hungary, 1961	8
1.4 4th IMO, Czechoslovakia, 1962	9
1.5 5th IMO, Poland, 1963	10
1.6 6th IMO, USSR, 1964	1
1.7 7th IMO, West Germany, 1965	1
1.8 8th IMO, Bulgaria, 1966	12
1.9 9th IMO, Yugoslavia, 1967	13
1.10 10th IMO, USSR, 1968	14
1.1 1th IMO, Romania, 1969	16
1.12 12th IMO, Hungary, 1970	16
1.13 13th IMO, Czechoslovakia, 1971	18

1 International Mathematics Olympiad 6 1

1.14 14th IMO, USSR, 1972	18
1.15 15th IMO, USSR, 1973	19
1.16 16th IMO, West Germany, 1974	20
1.17 17th IMO, Bulgaria, 1975	21
1.18 18th IMO, Austria, 1976	2
1.19 19th IMO, Yugoslavia, 1977	23
1.20 20th IMO, Romania, 1978	24
1.21 21st IMO, United Kingdom, 1979	25
1.2 2nd IMO, Washington, USA, 1981	26
1.23 23rd IMO, Budapest, Hungary, 1982	27
1.24 24th IMO, Paris, France, 1983	28
1.25 25th IMO, Prague, Czechoslovakia, 1984	29
1.26 26th IMO, Helsinki, Finland, 1985	30
1.27 27th IMO, Warsaw, Poland, 1986	31
1.28 28th IMO, Havana, Cuba , 1987	32
1.29 29th IMO, Camberra, Australia, 1988	3
1.30 30th IMO, Braunschweig, West Germany, 1989	34
1.31 31st IMO, Beijing, People’s Republic of China, 1990	36
1.32 32nd IMO, Sigtuna, Sweden, 1991	37

1.34 34th IMO, Istambul, Turkey, 1993	39
1.35 35th IMO, Hong Kong, 1994	40
1.36 36th IMO, Toronto, Canada, 1995	41
1.37 37th IMO, Mumbai, India, 1996	42
1.38 38th IMO, Mar del Plata, Argentina, 1997	43
1.39 39th IMO, Taipei, Taiwan, 1998	4
1.40 40th IMO, Bucharest, Romania, 1999	45
1.41 41st IMO, Taejon, South Korea, 2000	46
1.42 42nd IMO, Washington DC, USA, 2001	47
1.43 43rd IMO, Glascow, United Kingdom, 2002	48
1.4 4th IMO, Tokyo, Japan, 2003	49

CONTENTS 3

2.1 46th Anual William Lowell Putnam Competition, 1985	50
2.2 47th Anual William Lowell Putnam Competition, 1986	52
2.3 48th Anual William Lowell Putnam Competition, 1987	54
2.4 49th Anual William Lowell Putnam Competition, 1988	56
2.5 50th Anual William Lowell Putnam Competition, 1989	58
2.6 51th Anual William Lowell Putnam Competition, 1990	60
2.7 52th Anual William Lowell Putnam Competition, 1991	62

2.9 54th Anual William Lowell Putnam Competition, 1993	6
2.10 55th Anual William Lowell Putnam Competition, 1994	68
2.1 56th Anual William Lowell Putnam Competition, 1995	69
2.12 57th Anual William Lowell Putnam Competition, 1996	71
2.13 58th Anual William Lowell Putnam Competition, 1997	73
2.14 59th Anual William Lowell Putnam Competition, 1998	75
2.15 60th Anual William Lowell Putnam Competition, 1999	76
2.16 61st Anual William Lowell Putnam Competition, 2000	79
2.17 62nd Anual William Lowell Putnam Competition, 2001	80
2.18 63rd Anual William Lowell Putnam Competition, 2002	81

4 CONTENTS

3.1 1st Asiatic Pacic Mathematical Olympiad, 1989	84
3.2 2nd Asiatic Pacic Mathematical Olympiad, 1990	85
3.3 3rd Asiatic Pacic Mathematical Olympiad, 1991	86
3.4 4th Asiatic Pacic Mathematical Olympiad, 1992	86
3.5 5th Asiatic Pacic Mathematical Olympiad, 1993	87
3.6 6th Asiatic Pacic Mathematical Olympiad, 1994	8
3.7 7th Asiatic Pacic Mathematical Olympiad, 1995	89
3.8 8th Asiatic Pacic Mathematical Olympiad, 1996	90

3.10 10th Asiatic Pacic Mathematical Olympiad, 1998	92
3.1 1th Asiatic Pacic Mathematical Olympiad, 1999	93
3.12 12th Asiatic Pacic Mathematical Olympiad, 2000	93
3.13 13th Asiatic Pacic Mathematical Olympiad, 2001	94
3.14 14th Asiatic Pacic Mathematical Olympiad, 2002	95

Chapter 1 International Mathematics Olympiad

1.1 1st IMO, Romania, 1959

is irreductible for every natural number n

2. For what real values of x is√

given (a) A = p 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: acos2 x + bcosx + c = 0. Using the numbers a, b and c, form a quadratic ecuation in cos2x, whose roots are the same as those of the original ecuation. Compare the ecuations in cosx and cos2x for a = 4, b = 2 and c = 1

4. Construct a right triangle with hypotenuse c such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

5. An arbitrary point M is selected in the interior of the segment AB. The squares

AMCD and MBEF are constructed on the same side of AB, sith the segments AM and MB as their respective bases. The circles circumscribed abut these squares, with centers P and Q intersect at M and also at another point N. Let N′ denote the intersection of the straight lines AF and BC.

(a) Prove that the points N and N0 coinside.

(b) Prove that the straight lines MN pass throught a xed point S independent of the choice of M.

(c) Find the locus of the midpoints of the the segment PQ 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 parallel to CD) in which a circle can be inscribed, and with vertices B and D lying in the planes P and Q respectively.

1.2 2nd IMO, Romania, 1960

1. Determine all three-digit numbers N having the property that N is divisible by 1, and N11 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?

3. In a given right triangle 4ABC, the hypotenuse BC, of lenght a, is dividen into n equal parts (n an odd integer). Let be the acute angle subtending, from A, that segment which contains the middle point of the hypotenuse. Let h be the lenght of the altitude to the hypotenuse of the triangle. Prove:

4. Construct a triangle 4ABC, given ha, hb (the altitudes fron A and B) and ma, the median from vertex A.

5. Consider the cube ABCDA0B0C0D0 (whith face ABCD directly above face A0B0C0D0).

(a) Find the locus of the midpoints of segment XY , where X is any point of AC and Y is any point of B0D0.

8 CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD

(b) Find the locus of points Z which lie on the segment XY of part (a) with ZY = 2XZ.

6. Considere 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 V1 be the volume of the cone and V2 the volumen of the cilinder.

(b) Find the smallest number k for which V1 = kV2, for this case, construct the angle subtended by a diameter 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, nd all points P such that both legs of the trapezoid subtended right angles at P.

(b) Calculate the distance of P from either base.

(c) Determine under what conditions such points P actually exist. (Discuss varius case that might arise)

1. Solve the system of equations:

where a and b are constants. Give the conditions that a and b must satisfy so that x; y; z (the solutions of the system) are distinct positive numbers.

what case does the equality hold?

3. Solve the equation cosn x sinn x = 1, where n is a natural number.

PP3 intersect the opposite side in points Q1; Q2; Q3 respectively. Prove that, of the numbers P1P PQ1 ; P2P PQ2 ; P3P PQ3 at least one is less than or equal to 2 and at least one is grater than or equal to 2.

5. Construct triangle 4ABC if AC = b, AB = c and AMB = !, where M is the midpoint of the segment BC and ! < 90◦. Prove that a solution exists and only if btan !2 c < b. In what case does the equality hold?

6. Considere 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 a take three arbitrary points A0, B0, C0. Let L; M; N be the midpoints of segments AA0, BB0, CC0; let G the centroid of triangle 4LMN (We will not considere positions of A0, B0, C0 such that the points L, M, N do not form a triangle) What is the locus of point G as A0, B0, C0 range independently over the plane "?

1.4 4th IMO, Czechoslovakia, 1962

1. Find the smallest natural number n which has the following properties:

(a) Its decimal representation has 6 as the last digit.

(b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number n

2. Determine all real number x which satisfy the inequality:

3. Consider the cube ABCDA0B0C0D0 (ABCD and A0B0C0D0 are the upper and lower bases, respectively, and edges AA0; BB0; CC0;DD0 are parallel) The point X moves at constant speed along the perimeterof the square ABCD in the direction ABCDA, and the point Y moves at the same rate along the perimeter of the square B0C0CB in the direction B0C0CBB0. Points X and Y begin their motion at the same instant from the starting position A and B0, respectively. Determine and draw the locus of the midpoints of the segment XY .

5. On the circle K there are given three distinct points A, B, C. Construct (using only straightedge and compasses) a fourth point D on K such that a circle can be inscribed in the cuadrilateral thus obtained.

10 CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD

6. Considere an isosceles triangle. let r be the radius of its circumscribed circle and the radius of its inscribed circle. Prove that the distance d between the centers of these two circles is

7. The tetrahedon SABC has the following propoerty: there exists ve spheres, each tangent to the edges SA; SB; SC; BC; CA; AB or their extentions.

(Parte 1 de 9)