3

Part of 1993 IberoAmerican

Problems(2)

8th ibmo - mexico 1993/q3.

Source: Spanish Communities

\mathbb{N}^*=\{1,2,\ldots\}

. Find al the functions

f: \mathbb{N}^*\rightarrow \mathbb{N}^*

such that:
(1) If

x<y

f(x)<f(y)

f\left(yf(x)\right)=x^2f(xy)

x,y \in\mathbb{N}^*

functionalgebra proposedalgebra

8th ibmo - mexico 1993/q6.

Source: Spanish Communities

Two nonnegative integers

a

b

are tuanis if the decimal expression of

a+b

0

1

A

B

be two infinite sets of non negative integers such that

B

is the set of all the tuanis numbers to elements of the set

A

A

the set of all the tuanis numbers to elements of the set

B

. Show that in at least one of the sets

A

B

there is an infinite number of pairs

(x,y)

x-y=1

modular arithmeticnumber theory proposednumber theory