1972 IMO Shortlist

Part of IMO Shortlist

Subcontests

(6)

1

An infinite sequence of circles

Consider a sequence of circles

K_1,K_2,K_3,K_4, \ldots

r_1, r_2, r_3, r_4, \ldots

, respectively, situated inside a triangle

ABC

K_1

AB

AC

K_2

K_1

BA

BC

K_3

K_2

CA

CB

K_4

K_3

AB

AC

; etc. (a) Prove the relation

r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right)

r

is the radius of the incircle of the triangle

ABC

. Deduce the existence of a

t_1

r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1

(b) Prove that the sequence of circles

K_1,K_2, \ldots

1

Show that there exists a multiple of n

Show that for any

n \not \equiv 0 \pmod{10}

there exists a multiple of

n

not containing the digit

0

in its decimal expansion.

1

Altitudes of a tetrahedron intersect if and only if

Prove the following assertion: The four altitudes of a tetrahedron

ABCD

intersect in a point if and only if

AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.

1

Prove that there exists a straightline g -IMO ShortList 1973

n_1, n_2

be positive integers. Consider in a plane

E

two disjoint sets of points

M_1

M_2

2n_1

2n_2

points, respectively, and such that no three points of the union

M_1 \cup M_2

are collinear. Prove that there exists a straightline

g

with the following property: Each of the two half-planes determined by

g

E

g

not being included in either) contains exactly half of the points of

M_1

and exactly half of the points of

M_2.

1

HoW???????

The least number is

m

and the greatest number is

M

a_1 ,a_2 ,\ldots,a_n

satisfying a_1 \plus{}a_2 \plus{}...\plus{}a_n \equal{}0. Prove that a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM

1

extremal geometry

3n

A_1,A_2, \ldots , A_{3n}

in the plane, no three of them collinear. Prove that one can construct

n

disjoint triangles with vertices at the points

A_i.