**UFRGS**

# Math Olympiad Problems Collection v1

(Parte **1** de 4)

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 Pacic Mathematical Olympiad, 1989 | 84 |

3.2 2nd Asiatic Pacic Mathematical Olympiad, 1990 | 85 |

3.3 3rd Asiatic Pacic Mathematical Olympiad, 1991 | 86 |

3.4 4th Asiatic Pacic Mathematical Olympiad, 1992 | 86 |

3.5 5th Asiatic Pacic Mathematical Olympiad, 1993 | 87 |

3.6 6th Asiatic Pacic Mathematical Olympiad, 1994 | 8 |

3.7 7th Asiatic Pacic Mathematical Olympiad, 1995 | 89 |

3.8 8th Asiatic Pacic Mathematical Olympiad, 1996 | 90 |

3.10 10th Asiatic Pacic Mathematical Olympiad, 1998 | 92 |

3.1 1th Asiatic Pacic Mathematical Olympiad, 1999 | 93 |

3.12 12th Asiatic Pacic Mathematical Olympiad, 2000 | 93 |

3.13 13th Asiatic Pacic Mathematical Olympiad, 2001 | 94 |

3.14 14th Asiatic Pacic 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.

(a) Prove that the tetrahedron SABC is regular. (b) Prove conversely that for every regular tetrahedron ve such spheres exist.

eter.

2. Point A and segment BC are given. Determine the locus of points in space which are vertices of right angles with one side passing throught A, and the other side intersecting the segment BC.

where y is a parameter

5. Prove that

6. Five students, A; B; C; D; E, took part in a contest. One prediction was that contestants would nish in the order ABCDE. This prediction was very poor. In fact no contestant nished in the position predicted, and no two contestants predicted to nish consecutively actually did so. A second prediction has the contestants nishing in the order DAECB. This prediction was better. Exactly two of the contestants nished in the places predicted, and two disjoint pairs of students predicted to nish consecutively actually did so. Determine the order in which the contestants nished.

1.6 6th IMO, USSR, 1964

1. (a) Find all positive integers n for which 2n − 1 is divisible by 7. (b) Prove that there is not positive integer n such that 2n + 1 is dibisible by 7.

3. A circle is inscribed in triangle 4ABC with sides a; b; c. Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts o a triangle from 4ABC. In each of these triangle, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of a; b; c)

4. Seventeen people correspond by mail with one another, each one with all the rest.

In their letters only three di erent topics are discussed. Each pair of correspondent deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

5. Suppose ve points in a plane are situated so that no two of the straight lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have.

6. In tetrahedron ABCD, vertex D is connected with D0 the centroid of 4ABC. Lines parallel to DD0 are drawn through A; B and C. These lines intersect the planes BCD; CAD and ABD in points A1, B1 and C1, respectively. Prove that the volume of ABCD is one third the volume of A1B1C1D0. Is the result true if point D0 is selected anywhere within 4ABC?

1. Determine all value x in the interval 0 x 2 which satisfy the inequality

12 CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD

2. Consider the system of equations

(a) a11; a22; a33 are positive numbers; (b) the remaining coe cients are negative numbers;

(c) in each equation, the sum of the coe cient is positive .

3. Given the tetrahedron ABCD whose edges AB and CD have lenght a and b respectively. The distance between the skew lines AB and CD is d, and the angle between them is !. Tetrahedron ABCD is divided into two solid by plane ", parallel to lines AB and CD. The ratio of the distances of " from AB and CD is equal to k. Compute the ratio of the volumes of the two solids obtained.

4. Find all sets of four real numbers x1; x2; x3; x4 such that the sum of any one and the product of the other three is equal to 2.

5. Consider 4OAB with acute angle AOB. Through a point M 6= O perpendiculars are dawn to OA and OB, the feet of which are P and Q respectively. The point of intersection of the altitudes of 4OPQ is H. What is the locus of H if M is permitted to range over

6. In a plane a set of n points (n 3) is given. Each pair of points is connected by a segment. Let d be the length of the longest of these segment. We de ne a diameter of the set to be any connecting segment of length d. Prove that the number of diameters of the given set is at most n.

1.8 8th IMO, Bulgaria, 1966

1. In a mathematical contest, three problems, A; B; C were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem A, the number who solved B was twice the number who solved C. The number of students who solved only problem A was one more than the number of students who solved A and at least one other problem. How many students solved only problem B?

2. Let a; b; c be the lengths of the sides of a triangle and ; ; , respectively, the is isosceles.

3. Prove: The sum of the distances of the vertices of a regular tetrahedron from the centre of its circumscribed sphere is less than the sum of the distances of these vertices from any other poin in space.

4. Prove that for every natural number n, and for every real number x 6= k 2t (t any non-negative integer and k any integer),

5. Solve the system of equations

6. In the interior of sides BC; CA; AB of triangle 4ABC, any points K; L; M, respectively, are selected. Prove that the area of at least one of the triangle 4AML; 4BKM; 4CLK is less than or equal to one quarter of the area of 4ABC

1.9 9th IMO, Yugoslavia, 1967

1. Let ABCD be a parallelogram with side lengths AB = a, AD = 1, and with BAD =

. If 4ABD is acute, prove that the four circles of radius 1 with centers A; B; C; D cover the parallelogram if and only if a cos + p 3sin .

2. Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is smaller than or equal to 1 8

14 CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD

3. Let k; m; n be natural numbers such that m + k + 1 is a prime greater than n + 1.

(where A0 lies on BC, B0 on CA and C0 on AB) Of all such triangles, determine the one with maximum area, and construct it.

5. Consider the sequence fcng, where in which a1;a2;:::; a8 are real numbers not all equal to zero. Suppose that an in nite number of terms of the sequence fcng are equal to zero. Find all natural numbers for which cn = 0.

6. In a sport contest, there were m medals awarded on n successive days (n > 1). On the rst day, one medal and 17 of the remaining medals were awarded. On the second day, two medals and 17 of the now remaining medals were awarded; and so on. On the n-th and last day, the remaining n medals were awarded. How many days did the contest last. and how many medals were awarded altogether?

1.10 10th IMO, USSR, 1968

1. Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.

2. Find all natural numbers x such that the product of their digits (in decimal notation) is equal to x2 10x 2.

1.10. 10TH IMO, USSR, 1968 15 3. Consider the system of equations:

(a) If 4 < 0, ther is no solution, (b) If 4 = 0, ther is exactly one solution, (c) If 4 > 0, ther is more than one solution.

4. Prove than in every tetrahedon there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

5. Let f be a real-valued function de ned for all real numbers x such that, for some positive constant a, the equation

holds for all x

(a) Prove that the function f is periodic (i:e: there exists a positive number b such that f (x + b) = f (x) for all x)

(b) For a = 1, give an example of a non-constant function with the requiered properties.

6. For every natural number n, evaluate the sum

(the symbol bxc denotes the greatest integer not exceding x).

16 CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD

1.1 1th IMO, Romania, 1969

1. Prove that there are in nitely many numbers a with the following property: the number z = n4 + a is not prime for any natural number n.

3. For each value of k = 1; 2; 3; 4; 5, nd necessary and su cient conditions on the number a > 0 so that there exist a tetrahedron with k edges of length a, and the remaining 6 k edges of lenght 1.

4. A semicircular arc is drawn on AB as diameter. C is a point on other than A and B, and D is the foot of the perpendicular from C to AB. We consider three circles

5. Given n > 4 points in the plane such that no three are collinear. Prove that there are at least (n 3 2 convex quadrilaterals whose vertices are four of the given points.

is satis ed. Give necessary and su cient conditions for equality.

1. Let M be a point on the sede AB of 4ABC. Let r1, r2 and r be the radii of the inscribed circles of the triangles 4AMC, 4BMC and 4ABC. Let q1, q2 and q be the radii of the excribed circles of the same triangles that lie in the angle 4ACB.

Prove that r1

2. Let a, b and n be integers greater than 1, and let a and b be the two bases of two number systems. An−1 and An are numbers in the system with base a and Bn 1 and Bn are numbers in the system with base b; these are related as follows:

ak 1p ak

4. Find the set of all positive integers n with the property that the set fn; n + 1; n + 2; n + 3; n + 4; n + 5g can be partitioned into sets such that the product of the numbers in one set equals the product of the numbers in the other set

5. In the tetrahedron ABCD, the angle BDC is a right angle. Suppose that the foot

H of the perpendicular from D to the plane ABC is the intersection of the altitudes of 4ABC. Prove that

For what tetrahedra does equality hold?

6. In the plane are 100 points, no three of them are collinear. Consider all posible triangles having these points as vertices. Prove that no more than 70% of these triangles are acute-angled.

(Parte **1** de 4)