1966 IMO Shortlist

Part of IMO Shortlist

Subcontests

(56)

1

Is it possible to choose a set of 100 points ?

Is it possible to choose a set of

100

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.

1

Find the locus of vertices B that angles lie in the interval

Given the vertex

A

and the centroid

M

ABC

, find the locus of vertices

B

such that all the angles of the triangle lie in the interval

[40^\circ, 70^\circ].

1

We can draw a diagonal

Prove that in every convex hexagon of area

S

one can draw a diagonal that cuts off a triangle of area not exceeding

\frac{1}{6}S.

1

Find number of reflections in mirrors

Two mirror walls are placed to form an angle of measure

\alpha

. There is a candle inside the angle. How many reflections of the candle can an observer see?

1

Prove that f(a, b, c) = 4 max {1/a, 1/b, 1/c}

a,b,c

f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c

f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.

1

The greatest number of balls

What is the greatest number of balls of radius

1/2

that can be placed within a rectangular box of size

10 \times 10 \times 1 \ ?

1

Find all real solutions to the equation for positive real p

For a positive real number

p

, find all real solutions to the equation

\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.

1

Geometric inequality on radii of concentric circles

Two concentric circles have radii

R

r

respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between

\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1

\frac{63}{20} \cdot \frac{R+r}{R-r}.

1

Solve the equation |x^2 -1|+ |x^2 - 4| = mx

Solve the equation

|x^2 -1|+ |x^2 - 4| = mx

as a function of the parameter

m

(x,m)

of integers satisfy this equation?

1

Logarithmic inequalities

n

be a positive integer, prove that :
(a)

\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;

\log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).

1

Faces of a tetrahedron

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.

1

Geometry inequality on volume and area

Prove that the volume

V

and the lateral area

S

of a right circular cone satisfy the inequality

\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3

When does equality occur?

1

Solve the equation for a real parameter p

Solve the equation

\frac{1}{\sin x}+\frac{1}{\cos x}=\frac 1p

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, 2\pi)

p.

1

Describe the locus of points Z

We are given a circle

K

S

1

Q

M

2

XY

be the hypotenuse of an isosceles right triangle

XY Z

. Describe the locus of points

Z

X

K

Y

varies along the boundary of

Q.

1

Prove the inequality on fractions

a_1, a_2, \ldots, a_n

be positive real numbers. Prove the inequality

\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2

1

Find digits x,y,z such that equality holds

x, y, z

such that the equality

\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}

holds for at least two values of

n \in \mathbb N

, and in that case find all

n

for which this equality is true.

1

Does there exist an integer z ?

Does there exist an integer

z

that can be written in two different ways as

z = x! + y!

x, y

are natural numbers with

x \le y

1

How many real solutions are there to the equation ?

How many real solutions are there to the equation

x = 1964 \sin x - 189

1

Solve the trigonometric equation

x

such that trigonometric

\frac{\sin 3x \cos (60^\circ -x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0

m

is a fixed real number.

1

What is the smallest number of wheighings ?

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?

1

Find arrangements - ILL 1966

For which arrangements of two infinite circular cylinders does their intersection lie in a plane?

1

Trigonometric inequality - ILL 1966

Prove the inequality

\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1

x, \alpha

0 \leq x \leq \frac{\pi }{2}

\frac{\pi}{6} < \alpha < \frac{\pi}{3}.

1

Prove that the product is not less than 2^n

n

positive real numbers

a_1, a_2, \ldots , a_n

a_1a_2 \cdots a_n = 1

(1 + a_1)(1 + a_2) \cdots (1 + a_n) \geq 2^n.

1

Poland 1

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.

1

Soviet Union 13

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.

1

Soviet Union 11

100

consecutive natural numbers

a_{1},

a_{2},

...,

a_{100}.

Determine the last two digits of the number

a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.

1

Soviet Union 9

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 sum of the areas of all parts of the figure colored with the same color on both sides is

\geq \frac{1}{10}.

1

Soviet Union 8

n

students with numbers

1, 2, \ldots, n

standing in the order

1, 2, \ldots, n.

Upon a command, 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,

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, \ldots, n-1

after two commands ?

1

Soviet Union 5

For which real numbers

p

does the equation

x^{2}+px+3p=0

have integer solutions ?

1

Romania 8

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

ABC

are given. How many of these shortest segments exist ?

1

Socialists Republic Of Czechoslovakia 10

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

a\neq b

by removing letters.

1

Socialists Republic Of Czechoslovakia 8

Given a finite sequence of integers

a_{1},

a_{2},

...,

a_{n}

n\geq 2.

Show that there exists a subsequence

a_{k_{1}},

a_{k_{2}},

...,

a_{k_{m}},

1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,

such that the number

a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}

is divisible by

n.

Note by Darij: Of course, the

1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n

should be understood as

1\leq k_{1}<k_{2}<...<k_{m}\leq n;

else, we could take

m=n

k_{1}=k_{2}=...=k_{m},

so that the number

a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}

will surely be divisible by

n.

1

Socialists Republic Of Czechoslovakia 9

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 sides 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.

1

Socialists Republic Of Czechoslovakia 7

Given a regular

n

A_{1}A_{2}...A_{n}

n\geq 3

) in a plane. How many triangles of the kind

A_{i}A_{j}A_{k}

1

Romania 7

Consider a circle with center

O

R,

A

B

be two points in the plane of this circle. a.) Draw a chord

CD

of the circle such that

CD

AB,

and the point of the intersection

P

AC

BD

lies on the circle. b.) Show that generally, one gets two possible points

P

P_{1}

P_{2}

) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths

OA=a,

OB=b

AB=d

1

Poland 7

Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent. Note by Darij: A cyclic quadrilateral is a quadrilateral inscribed in a circle.

1

Poland 6

ABCD

be a quadrilateral inscribed in a circle. Show that the centroids of triangles

ABC,

CDA,

BCD,

DAB

lie on one circle.

1

Poland 5

ax^{3}+bx^{2}+cx+d

be a polynomial with integer coefficients

a,

b,

c,

d

ad

is an odd number and

bc

is an even number. Prove that (at least) one root of the polynomial is irrational.

1

Bulgaria 7

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.

1

Bulgaria 8

Find all pairs of positive integers

\left( x;\;y\right)

satisfying the equation

2^{x}=3^{y}+5.

1

Bulgaria 6

The side lengths

a,

b,

c

ABC

form an arithmetical progression (such that

b-a=c-b

). The side lengths

a_{1},

b_{1},

c_{1}

A_{1}B_{1}C_{1}

also form an arithmetical progression (with

b_{1}-a_{1}=c_{1}-b_{1}

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

\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.

Show that triangles

ABC

A_{1}B_{1}C_{1}

1

Romania 3

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^{\alpha }3^{\beta }5^{\gamma }

\alpha ,

\beta

\gamma

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.

1

Socialists Republic Of Czechoslovakia 5

In the plane, consider a circle with center

S

1.

ABC

be an arbitrary triangle having this circle as its incircle, and assume that

SA\leq SB\leq 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.

1

The Democratic Republic Of Germany 4

P

lying on a line

g,

and given a circle

K.

Construct a circle passing through the point

P

and touching the circle

K

g.

1

Socialists Republic Of Czechoslovakia 4

Prove the inequality
a.)

\left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left( a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) ,

k\geq 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}\geq \sqrt{\left( n-1\right) \left( a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },

then all of these numbers

a_{1},

a_{2},

\ldots,

a_{n}

are non-negative.

1

The Democratic Republic Of Germany 3

\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.

1

Poland 4

n\geq 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 that if

A

B,

B

A;

moreover, nobody is his own friend.)

1

Bulgaria 4

P

P^{\prime }

be two parallelograms with equal area, and let their sidelengths be

a,

b

a^{\prime },

b^{\prime }.

a^{\prime }\leq a\leq b\leq b^{\prime },

and moreover, it is possible to place the segment

b^{\prime }

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^{\prime }.

1

Hungary 3

Given three congruent rectangles 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 ?

1

Hungary 2

Construct a triangle given the radii of the excircles.

1

Romania 2

ABCD

A^{\prime }B^{\prime}C^{\prime }D^{\prime }

be two arbitrary parallelograms in the space, and let

M,

N,

P,

Q

be points dividing the segments

AA^{\prime },

BB^{\prime },

CC^{\prime },

DD^{\prime }

in equal ratios. a.) Prove that the quadrilateral

MNPQ

is a parallelogram. b.) What is the locus of the center of the parallelogram

MNPQ,

M

moves on the segment

AA^{\prime }

? (Consecutive vertices of the parallelograms are labelled in alphabetical order.

1

Poland 3

Given four points

A,

B,

C,

D

on a circle such that

AB

is a diameter 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

D

with the point of intersection of the lines

AC

BD

is perpendicular to the line

AB.

1

Poland 2

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...

1

Soviet Union 2

m

be a convex polygon in a plane,

l

its perimeter and

S

M\left( R\right)

be the locus of all points in the space whose distance to

m

\leq R,

V\left(R\right)

is the volume of the solid

M\left( R\right) .

V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.

Hereby, we say that the distance of a point

C

m

\leq R

if there exists a point

D

m

such that the distance

CD

\leq R.

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

-neighborhood 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

\leq R.

1

Bulgaria 1

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 between 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.

1

Socialists Republic Of Czechoslovakia 1

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 any other of the

n

points in its interior ?