IN THE NAME OF
ALLAH





IMO ShortList Problems
1959 – 2009



Collected by: Amir Hossein Parvardi (amparvardi)

Problems from:


Published: 2010-10

Email:
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.
/>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.
/>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
.
/>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.
/>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.
/>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 ?
/>This file was downloaded from the AoPS Math Olympiad Resources Page Page 2
IMO 1962
Ceske Budejovice, Czechoslovakia
Day 1
1 Find the smallest natural number n which has the following prop e rties:
a) Its decimal representation has a 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 numbers x which satisfy the inequality:
√
3 − x −
√
x + 1 >
1
2

3 Consider the cube ABCDA

B

C

D

(ABCD and A

B

C

D

are the upper and lower bases,
repsectively, and edges AA

, BB

, CC

, DD

are parallel). The point X moves at a constant
speed along the perimeter of the square ABCD in the direction ABCDA, and the point Y
moves at the same rate along the perimiter of the square B

C


CB in the direction B

C

CBB

.
Points X and Y begin their motion at the same instant from the starting positions A and B

,
respectively. Determine and draw the locus of the midpionts of the segments XY .
/>This file was downloaded from the AoPS Math Olympiad Resources Page Page 1
IMO 1962
Ceske Budejovice, Czechoslovakia
Day 2
4 Solve the equation cos
2
x + cos
2
2x + cos
2
3x = 1
5 On the circle K there are given three distinct points A, B, C. Construct (using only a straight-
edge and a compass) a fourth point D on K such that a circle can be inscribed in the quadri-
lateral thus obtained.
6 Consider an isosceles triangle. let R be the radius of its circumscribed circle and r be the
radius of its inscribed circle. Prove that the distance d between the centers of these two circle
is
d =


R(R − 2r)
7 The tetrahedron SABC has the following property: there exist five spheres, each tangent to
the edges SA, SB, SC, BC, CA, AB, or to their extensions.
a) Prove that the tetrahedron SABC is regular.
b) Prove conversely that for every regular tetrahedron five such spheres exist.
/>This file was downloaded from the AoPS Math Olympiad Resources Page Page 2
IMO 1963
Warsaw, Poland
Day 1
1 Find all real roots of the equation

x
2
− p + 2

x
2
− 1 = x
where p is a real parameter.
2 Point A and segment BC are given. Determine the locus of p oints in space which are vertices
of right angles with one side passing through A, and the other side intersecting segment BC.
3 In an n-gon A
1
A
2
. . . A
n
, all of whose interior angles are equal, the lengths of c onsec utive
sides satisfy the relation

a
1
≥ a
2
≥ · · · ≥ a
n
.
Prove that a
1
= a
2
= . . . = a
n
.
/>This file was downloaded from the AoPS Math Olympiad Resources Page Page 1
IMO 1963
Warsaw, Poland
Day 2
4 Find all solutions x
1
, x
2
, x
3
, x
4
, x
5
of the system
x

5
+ x
2
= yx
1
x
1
+ x
3
= yx
2
x
2
+ x
4
= yx
3
x
3
+ x
5
= yx
4
x
4
+ x
1
= yx
5
where y is a parameter.

5 Prove that cos
π
7
− cos
2π
7
+ cos
3π
7
=
1
2
6 Five students A, B, C, D, E took part in a contest. One prediction was that the contestants
would finish in the order ABCD E. This prediction was ve ry poor. In fact, no contestant
finished in the position predicted, and no two contestants predicted to finish consecutively
actually did so. A second prediction had the contestants finishing 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.
/>This file was downloaded from the AoPS Math Olympiad Resources Page Page 2
IMO 1964
Moscow, USSR
Day 1
2 Suppose a, b, c are the sides of a triangle. Prove that
a
2
(b + c − a) + b
2
(a + c − b) + c
2

(a + b − c) ≤ 3abc
3 A circle is inscribed in a triangle ABC with sides a, b, c. Tangents to the circle parallel to the
sides of the triangle are contructe. Each of these tangents cuts off a triagnle from ABC.
In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed
circles (in terms of a, b, c).
/>This file was downloaded from the AoPS Math Olympiad Resources Page Page 1
IMO 1964
Moscow, USSR
Day 2
4 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 correspondents 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 Supppose five points in a plane are situated so that no two of the straight lines joining them
are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all
the lines joining the other four points. Determine the maxium number of intersections that
these perpendiculars can have.
6 In tetrahedron ABCD, vertex D is connected with D
0
, the centrod if ABC. Line parallel
to DD
0
are drawn through A, B and C. These lines intersect the planes BCD, CAD and
ABD in points A
2
, 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 if point D
o
is selected anywhere within ABC?
/>This file was downloaded from the AoPS Math Olympiad Resources Page Page 2
IMO 1965
Berlin, German Democratic Republic
Day 1
1 Determine all values of x in the interval 0 ≤ x ≤ 2π which satisfy the inequality
2 cos x ≤
√
1 + sin 2x −
√
1 − sin 2x ≤
√
2.
2 Consider the sytem of equations
a
11
x
1
+ a
12

x
2
+ a
13
x
3
= 0
a
21
x
1
+ a
22
x
2
+ a
23
x
3
= 0
a
31
x
1
+ a
32
x
2
+ a
33

x
3
= 0
with unknowns x
1
, x
2
, x
3
. The coefficients satisfy the conditions:
a) a
11
, a
22
, a
33
are positive numbers;
b) the remaining coefficients are negative numbers;
c) in each equation, the sum ofthe coefficients is positive.
Prove that the given system has only the solution x
1
= x
2
= x
3
= 0.
3 Given the tetrahedron ABCD whose edges AB and CD have lengths a and b respectively.
The distance between the skew lines AB and CD is d, and the angle between them is ω.
Tetrahedron ABCD is divided into two solids 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.
/>This file was downloaded from the AoPS Math Olympiad Resources Page Page 1
IMO 1965
Berlin, German Democratic Republic
Day 2
4 Find all sets of four real numbers x
1
, x
2
, x
3
, x
4
such that the sum of any one and the product
of the other three is equal to 2.
5 Consider O AB with acute angle AOB. Thorugh a point M = O perpendiculars are drawn
to OA and OB, the feet of which are P and Q respectively. The point of intersection of the
altitudes of OP Q is H. What is the locus of H if M is permitted to range over
a) the side AB;
b) the interior of OAB.
6 In a plane a set of n points (n ≥ 3) is give. Each pair of points is connected by a segment.
Let d be the length of the longest of these segments. We define 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.
/>This file was downloaded from the AoPS Math Olympiad Resources Page Page 2
IMO Shortlist 1966
1 Given n > 3 points in the plane s uch 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 any other of
the n points in its interior ?
2 Given n positive real numbers a

1
, a
2
, . . . , a
n
such that a
1
a
2
···a
n
= 1, prove that
(1 + a
1
)(1 + a
2
) ···(1 + a
n
) ≥ 2
n
.
3 A regular triangular prism has the altitude h, and the two bases of the prism are equilateral
triangles with side length a. Dream-holes are made in the centers of both bases, and the
three lateral faces are mirrors. Assume that a ray of light, entering the prism through the
dream-hole in the upper base, then being reflected once by any of the three mirrors, quits the
prism through the dream-hole in the lower base. Find the angle b e tween the upper base and
the light ray at the moment when the light ray entered the prism, and the length of the way
of the light ray in the interior of the prism.
4 Given 5 points in the plane, no three of them being collinear. Show that among these 5
points, we can always find 4 points forming a convex quadrilateral.

5 Prove the inequality
tan
π sin x
4 sin α
+ tan
π cos x
4 cos α
> 1
for any x, α with 0 ≤ x ≤
π
2
and
π
6
< y <
π
3
.
6 Let m be a convex polygon in a plane, l its perimeter and S its area. Let M (R) be the locus
of all points in the space whose distance to m is ≤ R, and V (R) is the volume of the solid
M (R) .
a.) Prove that
V (R) =
4
3
πR
3
+
π
2

lR
2
+ 2SR.
Hereby, we say that the distance of a point C to a figure m is ≤ R if there exists a point D
of the figure m such that the distance CD is ≤ R. (This point D may lie on the boundary of
the figure m and inside the figure.)
additional question:
b.) Find the area of the planar R-neighborhood of a convex or non-convex polygon m.
c.) Find the volume of the R-neighb orhood of a convex polyhedron, e. g. of a cube or of a
tetrahedron.
Note by Darij: I guess that 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 Page 1
IMO Shortlist 1966
7 For which arrangements of two infinite circular cylinders do e s their intersection lie in a plane?
8 We are given a bag of sugar, a two-pan balance, and a weight of 1 gram. How do we obtain
1 kilogram of sugar in the smallest possible number of weighings?
9 Find x such that trigonometric
sin 3x cos(60
◦
− x) + 1
sin(60
◦
− 7x) − cos(30
◦
+ x) + m
= 0
where m is a fixed real number.
10 How many real solutions are there to the equation x = 1964 sin x − 189 ?
11 Does there exist an integer z that can be written in two different ways as z = x! + y!, where

x, y are natural numbers with x ≤ y ?
12 Find digits x, y, z such that the equality

xx ···x
  
n times
−yy ···y
  
n times
= zz ···z
  
n times
holds for at least two values of n ∈ N, and in that case find all n for which this equality is
true.
13 Let a
1
, a
2
, . . . , a
n
be positive real numbers. Prove the inequality

n
2


i<j
1
a
i

a
j
≥ 4



i<j
1
a
i
+ a
j


2
14 What is the maximal number of regions a circle can be divided in by segments joining n
points on the boundary of the circle ?
Posted already on the board I think
15 Given four points A, B, C, D on a circle such that AB is a diamete r and CD is not a diameter.
Show that the line joining the point of intersection of the tangents to the circle at the points
C and D with the point of intersection of the lines AC and BD is perpendicular to the line
AB.
16 We are given a circle K with center S and radius 1 and a square Q with center M and side 2.
Let XY be the hypotenuse of an isosceles right 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 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 quadrilateral MNP Q is a parallelogram.
b.) What is the locus of the center of the parallelogram MNP Q, when the point M moves
on the segment AA

?
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
18 Solve the equation
1
sin x
+
1
cos x
=
1
p

where p is a real parameter. Discuss for which values of p
the equation has at least one real solution and determine the number of solutions in [0, 2π)
for a given p.
19 Construct a triangle given the radii of the excircles.
20 Given three congruent rec tangles in the space. Their centers coincide, but the planes they lie
in are mutually perpendicular. For any two of the three rectangles, the line of intersection of
the planes of these two rectangles contains one midparallel of one rectangle and one midparallel
of the other rectangle, and these two midparallels have different lengths. Consider the convex
polyhedron whose vertices are the vertices of the rectangles.
a.) What is the volume of this polyhedron ?
b.) Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition
for this polyhedron to be regular ?
21 Prove that the volume V and the lateral area S of a right circular cone satisfy the inequality

6V
π

2
≤

2S
π
√
3

3
When does equality occur?
22 Let P and P

be two parallelograms with equal area, and let their sidelengths be a, b and a


,
b

. Assume that a

≤ a ≤ b ≤ b

, and moreover, it is possible to place the segment b

such that
it completely lies in the interior of the parallelogram P.
Show that the parallelogram P can be partitioned into four polygons such that these four
polygons can be composed again to form the parallelogram P

.
23 Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle.
(a) Prove that a necessary and sufficient condition for the fourth face to be a right triangle
is that at some vertex exactly two angles are right.
(b) Prove that if all the faces are right triangles, then the volume of 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 Page 3
IMO Shortlist 1966
24 There are n ≥ 2 people at a mee ting. Show that there exist two people at the m ee ting who
have the same number of friends among the persons at the meeting. (It is assumed that if A
is a friend of B, then B is a friend of A; moreover, nobody is his own friend.)
25 Prove that
tan 730

=

√
6 +
√
2 −
√
3 − 2.
26 Prove the inequality
a.) (a
1
+ a
2
+ + a
k
)
2
≤ k

a
2
1
+ a
2
2
+ + a
2
k

,
where k ≥ 1 is a natural number and a
1

, a
2
, , a
k
are arbitrary real numbers.
b.) Using the inequality (1), show that if the real numbers a
1
, a
2
, , a
n
satisfy the inequality
a
1
+ a
2
+ + a
n
≥

(n − 1)

a
2
1
+ a
2
2
+ + a
2

n

,
then all of these numbers a
1
, a
2
, . . . , a
n
are non-negative.
27 Given a point P lying on a line g, and given a circle K. Construct a circle passing through
the point P and touching the circle K and the line g.
28 In the plane, consider a c ircle with center S and radius 1. Let ABC be an arbitrary triangle
having this circle as its incircle, and assume that SA ≤ SB ≤ SC . Find the locus of
a.) all vertices A of such triangles;
b.) all vertices B of such triangles;
c.) all vertices C of such triangles.
29 A given natural number N is being decomposed in a sum of some consecutive integers.
a.) Find all such decompositions for N = 500.
b.) How many such decompositions does the number N = 2
α
3
β
5
γ
(where α, β and γ are
natural numbers) have? Which of these decompositions contain natural summands only?
c.) Determine the number of such decompositions (= decompositions in a sum of consecutive
integers; these integers are not necessarily natural) for an arbitrary natural N.
Note by Darij: The 0 is not considered as a natural number.

30 Let n be a positive integer, prove that :
(a) log
10
(n + 1) >
3
10n
+ log
10
n;
(b) log n! >
3n
10

1
2
+
1
3
+ ··· +
1
n
− 1

.
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IMO Shortlist 1966
31 Solve the equation |x
2
− 1| + |x
2

− 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 ABC form an arithmetical progression (such that b−a =
c −b). The side lengths a
1
, b
1
, c
1
of a triangle A
1
B
1
C
1
also form an arithmetical progression
(with b
1
− a
1
= c
1
− b
1
). [Hereby, a = BC, b = CA, c = AB, a
1
= B
1
C
1

, b
1
= C
1
A
1
,
c
1
= A
1
B
1
.] Moreover, we know that CAB = C
1
A
1
B
1
.
Show that triangles ABC and A
1
B
1
C
1
are similar.
Note by Darij: I have changed the wording of the problem since the conditions given in the
original were not sufficient, at least not in the form they were written.
33 Given two internally tangent circles; in the bigger one we inscribe an equilateral triangle.

From each of the vertices of this triangle, we draw a tangent to the smaller circle. Prove that
the length of one of these tangents equals the sum of the lengths of the two other tangents.
34 Find all pairs of positive integers (x; y) satisfying the equation 2
x
= 3
y
+ 5.
35 Let ax
3
+ bx
2
+ cx + d be a polynomial with integer coefficients a, b, c, d such that ad is an
odd number and bc is an even number. Prove that (at least) one root of the polynomial is
irrational.
36 Let ABCD be a quadrilateral inscribed in a circle. Show that the centroids of triangles ABC,
CDA, BCD, DAB lie on one circle.
37 Show that the four perpendiculars dropp e d from the midpoints of the sides of a cyclic quadri-
lateral to the respective opposite sides are concurrent.
Note by Darij: A cyclic quadrilateral is a quadrilateral inscribed in a circle.
38 Two concentric circles have radii R and r respectively. Determine the greatest poss ible number
of circles that are tangent to both these circles and mutually nonintersecting. Prove that this
number lies between
3
2
·
√
R+
√
r
√

R−
√
r
− 1 and
63
20
·
R+r
R−r
.
39 Consider a circle with center O and radius R, and let A and B be two points in the plane of
this circle.
a.) Draw a chord CD of the circle such that CD is parallel to AB, and the point of the
intersection P of the lines AC and BD lies on the circle.
b.) Show that generally, one gets two possible points P (P
1
and P
2
) satisfying the condition
of the above problem, and compute the distance between these two points, if the lengths
OA = a, OB = b and AB = d are given.
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IMO Shortlist 1966
40 For a positive real number p, find all real s olutions to the e quation

x
2
+ 2px − p
2
−


x
2
− 2px − p
2
= 1.
41 Given a regular n-gon A
1
A
2
A
n
(with n ≥ 3) in a plane. How many triangles of the kind
A
i
A
j
A
k
are obtuse ?
42 Given a finite sequence of integers a
1
, a
2
, , a
n
for n ≥ 2. Show that there exists a subsequence
a
k
1

, a
k
2
, , a
k
m
, where 1 ≤ k
1
≤ k
2
≤ ≤ k
m
≤ n, such that the number a
2
k
1
+a
2
k
2
+ + a
2
k
m
is divisible by n.
Note by Darij: Of course, the 1 ≤ k
1
≤ k
2
≤ ≤ k

m
≤ n should be understood as
1 ≤ k
1
< k
2
< < k
m
≤ n; else, we could take m = n and k
1
= k
2
= = k
m
, so that the
number a
2
k
1
+ a
2
k
2
+ + a
2
k
m
= n
2
a

2
k
1
will surely be divisible by n.
43 Given 5 points in a plane, no three of them being collinear. Each two of these 5 points are
joined with a segment, and every of these segments is painted either red or blue; assume that
there is no triangle whose s ides are segments of equal color.
a.) Show that:
(1) Among the four segments originating at any of the 5 points, two are red and two are blue.
(2) The red segments form a closed way passing through all 5 given points. (Similarly for the
blue segments.)
b.) Give a plan how to paint the segments either red or blue in order to have the condition
(no triangle with equally colored sides) satisfied.
44 What is the greatest number of balls of radius 1/2 that can be placed within a rectangular
box of size 10 × 10 × 1 ?
45 An alphabet consists of n letters. What is the maximal length of a word if we know that any
two consecutive letters a, b of the word are different and that the word cannot be reduced to
a word of the kind abab with a = b by removing letters.
46 Let a, b, c be reals and
f(a, b, c) =




|b − a|
|ab|
+
b + a
ab
−

2
c




+
|b − a|
|ab|
+
b + a
ab
+
2
c
Prove that f(a, b, c) = 4 max{
1
a
,
1
b
,
1
c
}.
47 Consider all segments dividing the area of a triangle ABC in two equal parts. Find the length
of the shortest segment among them, if the side lengths a, b, c of triangle ABC are given.
How many of these shortest segments exist ?
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IMO Shortlist 1966

48 For which real numbers p does the equation x
2
+ px + 3p = 0 have integer solutions ?
49 Two mirror walls are placed to form an angle of measure α. There is a candle inside the
angle. How many reflections of the candle can an observer see?
50 Solve the equation
1
sin x
+
1
cos x
=
1
p
where p is a real parameter. Discuss for which values of p
the equation has at least one real solution and determine the number of solutions in [0, 2π)
for a given p.
51 Consider n students with numbers 1, 2, . . . , n standing in the order 1, 2, . . . , n. Upon a com-
mand, any of the students either remains on his place or switches his place with another
student. (Actually, if student A switches his place with student B, then B cannot switch his
place with any other student C any more until the next command comes.)
Is it possible to arrange the students in the order n, 1, 2, . . . , n − 1 after two commands ?
52 A figure with area 1 is cut out of paper. We divide this figure into 10 parts and color them
in 10 different colors. Now, we turn around the piece of paper, divide the same figure on the
other side of the paper in 10 parts again (in some different way). Show that we can color these
new parts in the same 10 colors again (hereby, different parts should have different colors)
such that the s um of the areas of all parts of the figure colored w ith the same color on both
sides is ≥
1
10

.
53 Prove that in every convex hexagon of area S one can draw a diagonal that cuts off a triangle
of area not exceeding
1
6
S.
54 We take 100 consecutive natural numbers a
1
, a
2
, , a
100
. Determine the last two digits of
the number a
8
1
+ a
8
2
+ + a
8
100
.
55 Given the vertex A and the centroid M of a triangle ABC, find the locus of vertices B such
that all the angles of the triangle lie in the interval [40
◦
, 70
◦
].
56 In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove

that the midpoints of the six edges of the tetrahedron lie on one sphere.
57 Is it possible to choose a set of 100 (or 200) points on the boundary of a cube such that this
set is fixed under each isometry of the cube into itself? Justify your answer.
58 In a mathematical contest, three problems, A, B, C were posed. Among the participants ther
were 25 students who solved at least one problem each. Of all the c ontestants 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. Of all students who solved just one problem, half did not
solve problem A. How many students solved only problem B?
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