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Contents
- 1 International Mathematics Olympiad
- 1.1 1 st IMO, Romania,
- 1.2 2 nd IMO, Romania,
- 1.3 3 rd IMO, Hungary,
- 1.4 4 th IMO, Czechoslovakia,
- 1.5 5 th IMO, Poland,
- 1.6 6 th IMO, USSR,
- 1.7 7 th IMO, West Germany,
- 1.8 8 th IMO, Bulgaria,
- 1.9 9 th IMO, Yugoslavia,
- 1.10 10th IMO, USSR,
- 1.11 11th IMO, Romania,
- 1.12 12th IMO, Hungary,
- 1.13 13th IMO, Czechoslovakia,
- 1.14 14th IMO, USSR, 2 CONTENTS
- 1.15 15th IMO, USSR,
- 1.16 16th IMO, West Germany,
- 1.17 17th IMO, Bulgaria,
- 1.18 18th IMO, Austria,
- 1.19 19th IMO, Yugoslavia,
- 1.20 20th IMO, Romania,
- 1.21 21st IMO, United Kingdom,
- 1.22 22nd IMO, Washington, USA,
- 1.23 23rd IMO, Budapest, Hungary,
- 1.24 24th IMO, Paris, France,
- 1.25 25th IMO, Prague, Czechoslovakia,
- 1.26 26th IMO, Helsinki, Finland,
- 1.27 27th IMO, Warsaw, Poland,
- 1.28 28th IMO, Havana, Cuba ,
- 1.29 29th IMO, Camberra, Australia,
- 1.30 30th IMO, Braunschweig, West Germany,
- 1.31 31st IMO, Beijing, People’s Republic of China,
- 1.32 32nd IMO, Sigtuna, Sweden,
- 1.33 33rd IMO, Moscow, Russia,
- CONTENTS
- 1.34 34th IMO, Istambul, Turkey,
- 1.35 35th IMO, Hong Kong,
- 1.36 36th IMO, Toronto, Canada,
- 1.37 37th IMO, Mumbai, India,
- 1.38 38th IMO, Mar del Plata, Argentina,
- 1.39 39th IMO, Taipei, Taiwan,
- 1.40 40th IMO, Bucharest, Romania,
- 1.41 41st IMO, Taejon, South Korea,
- 1.42 42nd IMO, Washington DC, USA,
- 1.43 43rd IMO, Glascow, United Kingdom,
- 1.44 44th IMO, Tokyo, Japan,
- 2 William Lowell Putnam Competition
- 2.1 46 th Anual William Lowell Putnam Competition,
- 2.2 47 th Anual William Lowell Putnam Competition,
- 2.3 48 th Anual William Lowell Putnam Competition,
- 2.4 49 th Anual William Lowell Putnam Competition,
- 2.5 50 th Anual William Lowell Putnam Competition,
- 2.6 51 th Anual William Lowell Putnam Competition,
- 2.7 52 th Anual William Lowell Putnam Competition,
- 2.8 53 th Anual William Lowell Putnam Competition,
- 2.9 54 th Anual William Lowell Putnam Competition, 4 CONTENTS
- 2.10 55th Anual William Lowell Putnam Competition,
- 2.11 56th Anual William Lowell Putnam Competition,
- 2.12 57th Anual William Lowell Putnam Competition,
- 2.13 58th Anual William Lowell Putnam Competition,
- 2.14 59th Anual William Lowell Putnam Competition,
- 2.15 60th Anual William Lowell Putnam Competition,
- 2.16 61st Anual William Lowell Putnam Competition,
- 2.17 62nd Anual William Lowell Putnam Competition,
- 2.18 63rd Anual William Lowell Putnam Competition,
- 3 Asiatic Pacific Mathematical Olympiads
- 3.1 1 st Asiatic Pacific Mathematical Olympiad,
- 3.2 2 nd Asiatic Pacific Mathematical Olympiad,
- 3.3 3 rd Asiatic Pacific Mathematical Olympiad,
- 3.4 4 th Asiatic Pacific Mathematical Olympiad,
- 3.5 5 th Asiatic Pacific Mathematical Olympiad,
- 3.6 6 th Asiatic Pacific Mathematical Olympiad,
- 3.7 7 th Asiatic Pacific Mathematical Olympiad,
- 3.8 8 th Asiatic Pacific Mathematical Olympiad,
- 3.9 9 th Asiatic Pacific Mathematical Olympiad,
- CONTENTS
- 3.10 10th Asiatic Pacific Mathematical Olympiad,
- 3.11 11th Asiatic Pacific Mathematical Olympiad,
- 3.12 12th Asiatic Pacific Mathematical Olympiad,
- 3.13 13th Asiatic Pacific Mathematical Olympiad,
- 3.14 14th Asiatic Pacific Mathematical Olympiad,
- 3.15 15th Asiatic Pacific Mathematical Olympiad,
1.2. 2 N D^ IMO, ROMANIA, 1960 7
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 N ′^ coinside. (b) Prove that the straight lines MN pass throught a fixed point S independent of the choice of M. (c) Find the locus of the midpoints of the the segment P Q as M varies between A and B.
- 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 2 nd^ IMO, Romania, 1960
- Determine all three-digit numbers N having the property that N is divisible by 11, and 11 N is equal to the sum of the squares of the digits of N.
- For what values of the variable x does the following inequality hold?
4 x^2 ( 1 −
1 + 2x
) 2 <^2 x^ + 9
- In a given right triangle 4 ABC, the hypotenuse BC, of lenght a, is dividen into n equal parts (n an odd integer). Let α 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:
tan α =
4 nh (n^2 − 1) a
- Construct a triangle 4 ABC, given ha, hb (the altitudes fron A and B) and ma, the median from vertex A.
- Consider the cube ABCDA′B′C′D′^ (whith face ABCD directly above face A′B′C′D′).
(a) Find the locus of the midpoints of segment XY , where X is any point of AC and Y is any point of B′D′.
8 CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD
(b) Find the locus of points Z which lie on the segment XY of part (a) with ZY = 2 XZ.
- 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 V 1 be the volume of the cone and V 2 the volumen of the cilinder.
(a) Prove that V 1 6 = V 2. (b) Find the smallest number k for which V 1 = kV 2 , for this case, construct the angle subtended by a diameter of the base of the cone at the vertex of the cone.
- 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 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)
1.3 3 rd^ IMO, Hungary, 1961
- 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 (the solutions of the system) are distinct positive numbers.
- Let a, b, c the sides of a triangle, and T its area. Prove: a^2 + b^2 + c^2 ≥ 4
3 T. In what case does the equality hold?
- Solve the equation cosn^ x − sinn^ x = 1, where n is a natural number.
- Consider the triangle 4 P 1 P 2 P 3 and a point P within the triangle. Lines P P 1 , P P 2 , P P 3 intersect the opposite side in points Q 1 , Q 2 , Q 3 respectively. Prove that, of the numbers P P Q^1 P 1 , (^) P QP^2 P 2 , (^) P QP^3 P 3 at least one is less than or equal to 2 and at least one is grater than or equal to 2.
10 CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD
- Considere an isosceles triangle. let r be the radius of its circumscribed circle and ρ the radius of its inscribed circle. Prove that the distance d between the centers of these two circles is d =
√ r (r − 2 ρ)
- The tetrahedon SABC has the following propoerty: there exists five 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 five such spheres exist.
1.5 5 th^ IMO, Poland, 1963
- Find all real roots of the equation
x^2 − p + 2
x^2 − 1 = x, where p is a real param- eter.
- 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.
- In an n−gon all of whose interior angles are equal, the lenght of consecutive sides satisfy the relation a 1 ≥ a 2 ≥ · ≥ an. Prove that a 1 = a 2 = · = an.
- Find all solution x 1 , x 2 , x 3 , x 4 , x 5 of the system
(1) x 5 + x 2 = yx 1 (2) x 1 + x 3 = yx 2 (3) x 2 + x 4 = yx 3 (4) x 3 + x 5 = yx 4 (5) x 4 + x 1 = yx 5
where y is a parameter
- Prove that cos
π 7
− cos
2 π 7
3 π 7
- Five students, A, B, C, D, E, took part in a contest. One prediction was that contestants would finish in the order ABCDE. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction has the contestants finishing
1.6. 6 T H^ IMO, USSR, 1964 11
in the order DAECB. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.
1.6 6 th^ IMO, USSR, 1964
- (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.
- Let a, b, c be the sides of a triangle. Prove that
a^2 (b + c − a) + b^2 (c + a − b) + c^2 (a + b − c) ≤ 3 abc
- A circle is inscribed in triangle 4 ABC with sides a, b, c. Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts off a triangle from 4 ABC. 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)
- Seventeen people correspond by mail with one another, each one with all the rest. In their letters only three different 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.
- Suppose five 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.
- In tetrahedron ABCD, vertex D is connected with D 0 the centroid of 4 ABC. Lines parallel to DD 0 are drawn through A, B and C. These lines intersect the planes BCD, CAD and ABD in points A 1 , B 1 and C 1 , respectively. Prove that the volume of ABCD is one third the volume of A 1 B 1 C 1 D 0. Is the result true if point D 0 is selected anywhere within 4 ABC?
th
IMO, West Germany, 1965
- Determine all value x in the interval 0 ≤ x ≤ 2 π which satisfy the inequality
2 cos x ≤
∣∣ ∣
1 + sin 2x −
1 − sin 2x
∣∣ ∣ ≤
1.9. 9 T H^ IMO, YUGOSLAVIA, 1967 13
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?
- Let a, b, c be the lengths of the sides of a triangle and α, β, γ, respectively, the angles opposite these sides. Prove tat if a + b = tan γ 2 (a tan α + b tan β), the triangle is isosceles.
- 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.
- Prove that for every natural number n, and for every real number x 6 = kπ 2 t (t any non-negative integer and k any integer),
1 sin 2x
sin 4x
sin 2nx
= cot x − cot 2nx
- Solve the system of equations
|a 1 − a 2 |x 2 + |a 1 − a 3 |x 3 + |a 1 − a 4 |x 4 = 1 |a 2 − a 1 |x 2 + |a 2 − a 3 |x 3 + |a 2 − a 4 |x 4 = 1 |a 3 − a 1 |x 1 + |a 3 − a 2 |x 2 + |a 3 − a 4 |x 4 = 1 |a 4 − a 1 |x 1 + |a 4 − a 2 |x 2 + |a 4 − a 3 |x 3 = 1
where a 1 , a 2 , a 3 , a 4 are four different real numbers.
- In the interior of sides BC, CA, AB of triangle 4 ABC, any points K, L, M, respec- tively, are selected. Prove that the area of at least one of the triangle 4 AML, 4 BKM , 4 CLK is less than or equal to one quarter of the area of 4 ABC
1.9 9 th^ IMO, Yugoslavia, 1967
- Let ABCD be a parallelogram with side lengths AB = a, AD = 1, and with ^BAD = α. If 4 ABD 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 α +
3 sin α.
- Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is smaller than or equal to (^18)
14 CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD
- 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 the product (cm+1 − ck) (cm+2 − ck) · · · (cm+n − ck) is divisible by the product c 1 c 2 · · · cn.
- Let 4 A 0 B 0 C 0 and 4 A 1 B 1 C 1 be any two acute-angled triangles. Consider all triangles 4 ABC that are similar to 4 A 1 B 1 C 1 and circumscribed about triangle 4 A 0 B 0 C 0 (where A 0 lies on BC, B 0 on CA and C 0 on AB) Of all such triangles, determine the one with maximum area, and construct it.
- Consider the sequence {cn}, where
c 1 = a 1 + a 2 + · · · + a 8 c 2 = a^21 + a^22 + · · · + a^28 .. . cn = an 1 + an 2 + · · · + an 8 .. .
in which a 1 , a 2 ,... , a 8 are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence {cn} are equal to zero. Find all natural numbers for which cn = 0.
- In a sport contest, there were m medals awarded on n successive days (n > 1). On the first 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 10 th^ IMO, USSR, 1968
- 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.
- Find all natural numbers x such that the product of their digits (in decimal notation) is equal to x^2 − 10 x − 22.
16 CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD
1.11 11 th^ IMO, Romania, 1969
- Prove that there are infinitely many numbers a with the following property: the number z = n^4 + a is not prime for any natural number n.
- Let a 1 , a 2 ,... , an be real variable, and
f (x) = cos (a 1 + x) +
cos (a 2 + x) +
cos (a 3 + x) + · · · +
2 n−^1
cos (an + x)
Given that f (x 1 ) = f (x 2 ) = 0, prove that x 2 − x 1 = mπ for some integer m.
- For each value of k = 1, 2 , 3 , 4 , 5, find necessary and sufficient 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.
- A semicircular arc γ is drawn on AB as diameter. C is a point on γ other than A and B, and D is the foot of the perpendicular from C to AB. We consider three circles γ 1 , γ 2 , γ 3 , all tangent to the line AB. Of these, γ 1 is inscrived in 4 ABC, while γ 2 and γ 3 are both tangent to CD and to γ, one on each side of CD. Prove that γ 1 , γ 2 and γ 3 have a second tangent in common.
- 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.
- Prove that for all real numbers x 1 , x 2 , y 1 , y 2 , z 1 , z 2 with x 1 > 0 , x 2 > 0 , x 1 y 1 − z 12 > 0 , x 2 y 2 − z 22 > 0, the inequality
8 (x 1 + x 2 ) (y 1 + y 2 ) − (z 1 + z 2 )^2
x 1 y 1 − z 12
x 2 y 2 − z 22
is satisfied. Give necessary and sufficient conditions for equality.
1.12 12 th^ IMO, Hungary, 1970
- Let M be a point on the sede AB of 4 ABC. Let r 1 , r 2 and r be the radii of the inscribed circles of the triangles 4 AMC, 4 BMC and 4 ABC. Let q 1 , q 2 and q be the radii of the excribed circles of the same triangles that lie in the angle 4 ACB. Prove that (^) r 1 q 1
r 2 q 2
r q
1.12. 12 T H^ IMO, HUNGARY, 1970 17
- 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:
An = xnxn− 1 · · · x 0 An− 1 = xn− 1 xn− 2 · · · x 0 Bn = xnxn− 1 · · · x 0 Bn− 1 = xn− 1 xn− 2 · · · x 0
such that xn 6 = 0 and xn− 1 6 = 0^1. Prove that
An− 1 An
Bn− 1 Bn
⇐ ⇒ a > b
- The real numbers a 0 , a 1 ,... , an,... satisfy the condition 1 = a 0 ≤ a 1 ≤ a 2 ≤ · · · ≤ an ≤ ·. The numbers b 1 , b 2 ,... , bn,... are defined by
bn =
∑^ n
k=
( 1 −
ak− 1 ak
) 1 √ ak
(a) Prove that 0 ≤ bn < 2 for all n.
(a) Given c with 0 ≤ c < 2, prove that there exist numbers a 0 , a 1 ,... such that bn > c for large enough n.
- Find the set of all positive integers n with the property that the set {n, n + 1, n + 2 , n + 3, n + 4, n + 5} 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
- 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 4 ABC. Prove that
(AB + BC + CA)^2 ≤ 6
( AD^2 + BD^2 + CD^2
)
For what tetrahedra does equality hold?
- 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.
(^1) The xi’s are the digits in the respective bases, and of course, all of them are lower than the lowest base
1.15. 15 T H^ IMO, USSR, 1973 19
- Prove that if n ≥ 4 , every quadrilateral that can be inscribed in acircle can be dissected into n quadrilaterals each of which is inscribablein a circle.
- Let m and n be arbitrary non-negative integers. Prove that
(2m)!(2n)! m!n!(m + n)!
is an integer. (0! = 1)
- Find all solutions (x 1 , x 2 , x 3 , x 4 , x 5 ) of the system of inequalities
(x^21 − x 3 x 5 )(x^22 − x 3 x 5 ) ≤ 0 (x^22 − x 4 x 1 )(x^23 − x 4 x 1 ) ≤ 0 (x^23 − x 5 x 2 )(x^24 − x 5 x 2 ) ≤ 0 (x^24 − x 1 x 3 )(x^25 − x 1 x 3 ) ≤ 0 (x^25 − x 2 x 4 )(x^21 − x 2 x 4 ) ≤ 0
where x 1 , x 2 , x 3 , x 4 , x 5 are positive real numbers.
- Let f and g be real-valued functions defined for all real values of xand y, and satisfying the equation f (x + y) + f (x − y) = 2f (x)g(y)
for all x, y. Prove that if f (x) is not identically zero, and if |f (x)| ≤ 1 for all x, then |g(y)| ≤ 1 for all y.
- Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
1.15 15 th^ IMO, USSR, 1973
- Point O lies on line g;
OP 1 ,
OP 2 ,... ,
OPn are unit vectors such that points P 1 , P 2 , ..., Pn all lie in a plane containing g and on one side of g. Prove that if n is odd, ∣∣ ∣
OP 1 +
OP 2 + · · · +
OPn
∣∣ ∣ ≥ 1
Here
∣∣ ∣
OM
∣∣ ∣ denotes the length of vector
OM.
20 CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD
- Determine whether or not there exists a finite set M of points in spacenot lying in the same plane such that, for any two points A and B of M,one can select two other points C and D of M so that lines AB and CD are parallel and not coincident.
- Let a and b be real numbers for which the equation
x^4 + ax^3 + bx^2 + ax + 1 = 0
has at least one real solution. For all such pairs (a, b), find the minimum value of a^2 + b^2.
- A soldier needs to check on the presence of mines in a region having theshape of an equilateral triangle. The radius of action of his detector isequal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path should he follow in order to travel the least possible distance and still accomplish his mission?
- G is a set of non-constant functions of the real variable x of the form f (x) = ax + b, a and b are real numbers, and G has the following properties: (a) If f and g are in G, then g ◦ f is in G; here (g ◦ f )(x) = g[f (x)]. (b) If f is in G, then its inverse f −^1 is in G; here the inverse of f (x) = ax + b is f −^1 (x) = (x − b)/a. (c) For every f in G, there exists a real number xf such that f (xf ) = xf. Prove that there exists a real number k such that f (k) = k for all f in G.
- Let a 1 , a 2 , ..., an be n positive numbers, and let q be a givenreal number such that 0 < q < 1. Find n numbers b 1 , b 2 , ..., bn forwhich (a) ak < bk for k = 1, 2 , · · · , n, (b) q < bk b+1k < (^1) q for k = 1, 2 , ..., n − 1 ,
(c) b 1 + b 2 + · · · + bn < 1+ 1 −qq (a 1 + a 2 + · · · + an).
1.16 16 th^ IMO, West Germany, 1974
- Three players A, B and C play the following game: On each of three cardsan integer is written. These three numbers p, q, r satisfy 0 < p < q < r. Thethree cards are shuffled and one is dealt to each player. Each then receivesthe number of counters indicated by the card he holds. Then the cards areshuffled again; the counters remain with the players.
