MathDB

1

ALPHABETA numbering system uses only digits 0 and 1

Have you ever found it strange that “almost the same” numbers can look very different? For example, in the decimal system

29

and

30

only differ by one unit but do not contain any common digits. The ALPHABETA numbering system uses only the digits

0

and

1

and avoids this situation:
https://cdn.artofproblemsolving.com/attachments/d/f/dcdf284b3baeea8775de56ece091c80d3449a8.png
In it, the rule for constructing the successor of a number is as follows: without repeating a previous number in the list, change the digit as far to the right as possible, otherwise a 1 is placed to the left. (a) What number in the decimal system is represented in the ALPHABETA code by the number

111111

?
(b) What is the next number in this code?
(c) Describe an algorithm to find, given any number in the ALPHABETA code, the next number in this code.

5

1

4 colors for each point of a right triangle

Consider a right-angled triangle whose legs are

1

cm long. Suppose that each point of the triangle was assigned a color from the set of Brown, Blue, Green and Orange colors. It proves that, whatever way this was done, there is at least one pair of points of the same color at a distance equal to or greater than

2-\sqrt 2

cm from each other.

6

1

67 diagonals in regular 134-gon

In a regular polygon with

134

sides,

67

diagonals are drawn so that exactly one diagonal emerges from each vertex. We call the length of a diagonal the number of sides of the polygon included between the vertices of the diagonal and which is less than or equal to

67

. If we order the lengths of the diagonals in ascending order, we obtain a succession of

67

numbers

(d_1,d_2,...,d_{67})

. It will be possible to draw diagonals such that
a)

(d_1,d_2,...,d_{67})=\underbrace{2 ... 2}_{6},\underbrace{3 ... 3}_{61}

?
b)

(d_1,d_2,...,d_{67}) =\underbrace{3 ... 3}_{8},\underbrace{6 ... 6}_{55}.\underbrace{8 ... 8}_{4}

?

3

1

900 cards numbered from 100 to 999

A box contains

900

cards numbered from

100

to

999

. Paulo randomly takes a certain number of cards from the box and calculates, for each card, the sum of the digits written on it. How many cards does Paulo need to take out of the box to be sure of finding at least three cards whose digit sums are the same?

2

1

(x-y) /(1 +xy )<1, for 2 of 3 positive numbers

Prove that given three positive numbers, we can choose two of them, say

x

and

y,

with

x >y

such that

\frac{x-y}{1 +xy }<1.

Prove also that if the number

1

that appears in the second member of the previous inequality is replaced by a lower number, even if very close to

1

, the previous proposition is false.

1

45^o wanted, square on hypotenuse of a right triangle 1996 Portugal p1

Consider a square on the hypotenuse of a right triangle

[ABC]

(right at

B

). Prove that the line segment that joins vertex

B

with the center of the square makes

45^o

angles with legs of the triangle.

1996 Portugal MO

Subcontests

ALPHABETA numbering system uses only digits 0 and 1

4 colors for each point of a right triangle

67 diagonals in regular 134-gon

900 cards numbered from 100 to 999

(x-y) /(1 +xy )<1, for 2 of 3 positive numbers

45^o wanted, square on hypotenuse of a right triangle 1996 Portugal p1

1996 Portugal MO

Subcontests

ALPHABETA numbering system uses only digits 0 and 1

4 colors for each point of a right triangle

67 diagonals in regular 134-gon

900 cards numbered from 100 to 999

(x-y) /(1 +xy )&lt;1, for 2 of 3 positive numbers

45^o wanted, square on hypotenuse of a right triangle 1996 Portugal p1

(x-y) /(1 +xy )<1, for 2 of 3 positive numbers