MathDB

1

Show that 2^147 − 1 is divisible by 343

2^{147} - 1

is divisible by

343

.

8

1

If S_k and S_(k+1) are integers, then all S_n are integers

Let

a

be a non-zero real number. For each integer

n

, we define

S_n = a^n + a^{-n}

. Prove that if for some integer

k

, the sums

S_k

and

S_{k+1}

are integers, then the sums

S_n

are integers for all integers

n

.

7

1

Triangle with sides x = AB · CD, y = AC · BD and z = AD · BC

Given a tetrahedron

ABCD

. Let

x = AB \cdot CD, y = AC \cdot BD

and

z = AD\cdot BC

. Prove that there exists a triangle with the side lengths

x, y

and

z

.

6

1

At least three triangles have integer areas

Let

P_i (x_i, y_i)

(with

i = 1, 2, 3, 4, 5

) be five points with integer coordinates, no three collinear. Show that among all triangles with vertices at these points, at least three have integer areas.

5

1

Locus of all vertices A of all parallelograms ABCD

Given a ball

K

. Find the locus of the vertices

A

of all parallelograms

ABCD

such that

AC \leq BD

, and the diagonal

BD

lies completely inside the ball

K

.

4

1

Find the locus of the center of the band

A circle of radius 1 is placed in a corner of a room (i.e., it touches the horizontal floor and two vertical walls perpendicular to each other). Find the locus of the center of the band for all of its possible positions.
Note. For the solution of this problem, it is useful to know the following Monge theorem: The locus of all points

P

, such that the two tangents from

P

to the ellipse with equation

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

are perpendicular to each other, is a circle − a so-called Monge circle − with equation

x^2 + y^2 = a^2 + b^2

.

3

1

Rational or Irrational? (ILL 1973)

Is the number

\sqrt[3]{\sqrt 5 + 2} + \sqrt[3]{\sqrt 5 - 2}

rational or irrational?

2

1

Construct the plane which minimizes the volume

Let

OX, OY

and

OZ

be three rays in the space, and

G

a point "between these rays" (i. e. in the interior of the part of the space bordered by the angles

Y OZ, ZOX

and

XOY

). Consider a plane passing through

G

and meeting the rays

OX, OY

and

OZ

in the points

A, B, C

, respectively. There are infinitely many such planes; construct the one which minimizes the volume of the tetrahedron

OABC

.

1

A sphere of radius r fits into the tetrahedron (ILL 1793)

Find the maximal positive number

r

with the following property: If all altitudes of a tetrahedron are

\geq 1

, then a sphere of radius

r

fits into the tetrahedron.

1973 IMO Longlists

Subcontests

Show that 2^147 − 1 is divisible by 343

If S_k and S_(k+1) are integers, then all S_n are integers

Triangle with sides x = AB · CD, y = AC · BD and z = AD · BC

At least three triangles have integer areas

Locus of all vertices A of all parallelograms ABCD

Find the locus of the center of the band

Rational or Irrational? (ILL 1973)

Construct the plane which minimizes the volume

A sphere of radius r fits into the tetrahedron (ILL 1793)