MathDB

1

One can form a word of arbitrary length

In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of different lengths. Prove that one can form a word of arbitrary length that does not contain any non-allowed word.

11

1

Every rectangle is formed from a number of full squares

We consider the division of a chess board

8 \times 8

in p disjoint rectangles which satisfy the conditions:
a) every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares.
b) the numbers

\ a_{1}, \ldots, a_{p}

of white squares from

p

rectangles satisfy

a_1, , \ldots, a_p.

Find the greatest value of

p

for which there exists such a division and then for that value of

p,

all the sequences

a_{1}, \ldots, a_{p}

for which we can have such a division.
[color=#008000]Moderator says: see https://artofproblemsolving.com/community/c6h58591

9

1

Prove the identity if x+y+z=xyz

Let

x, y, z

be real numbers each of whose absolute value is different from

\frac{1}{\sqrt 3}

such that

x + y + z = xyz

. Prove that

\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}

7

1

Prove that the greatest common divisors are equal

Let

a_i, b_i

be coprime positive integers for

i = 1, 2, \ldots , k

, and

m

the least common multiple of

b_1, \ldots , b_k

. Prove that the greatest common divisor of

a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}

equals the greatest common divisor of

a_1, \ldots , a_k.

5

1

Prove that at least two triangle are congruent

Let

A_r,B_r, C_r

be points on the circumference of a given circle

S

. From the triangle

A_rB_rC_r

, called

\Delta_r

, the triangle

\Delta_{r+1}

is obtained by constructing the points

A_{r+1},B_{r+1}, C_{r+1}

on

S

such that

A_{r+1}A_r

is parallel to

B_rC_r

,

B_{r+1}B_r

is parallel to

C_rA_r

, and

C_{r+1}C_r

is parallel to

A_rB_r

. Each angle of

\Delta_1

is an integer number of degrees and those integers are not multiples of

45

. Prove that at least two of the triangles

\Delta_1,\Delta_2, \ldots ,\Delta_{15}

are congruent.

4

1

Prove that the least of the numbers does not exceed 1/10

The sum of the squares of five real numbers

a_1, a_2, a_3, a_4, a_5

equals

1

. Prove that the least of the numbers

(a_i - a_j)^2

, where

i, j = 1, 2, 3, 4,5

and

i \neq j

, does not exceed

\frac{1}{10}.

2

1

Squares may be put into the other square

Prove that the squares with sides

\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\ldots

may be put into the square with side

\frac{3}{2}

in such a way that no two of them have any interior point in common.

1974 IMO Shortlist

Subcontests

One can form a word of arbitrary length

Every rectangle is formed from a number of full squares

Prove the identity if x+y+z=xyz

Prove that the greatest common divisors are equal

Prove that at least two triangle are congruent

Prove that the least of the numbers does not exceed 1/10

Squares may be put into the other square