MathDB

1993 IberoAmerican

Part of IberoAmerican

Subcontests

(3)
3
2

8th ibmo - mexico 1993/q3.

Let N∗={1,2,…}\mathbb{N}^*=\{1,2,\ldots\}. Find al the functions f:N∗→N∗f: \mathbb{N}^*\rightarrow \mathbb{N}^* such that:
(1) If x<yx<y then f(x)<f(y)f(x)<f(y). (2) f(yf(x))=x2f(xy)f\left(yf(x)\right)=x^2f(xy) for all x,y∈N∗x,y \in\mathbb{N}^*.

8th ibmo - mexico 1993/q6.

Two nonnegative integers aa and bb are tuanis if the decimal expression of a+ba+b contains only 00 and 11 as digits. Let AA and BB be two infinite sets of non negative integers such that BB is the set of all the tuanis numbers to elements of the set AA and AA the set of all the tuanis numbers to elements of the set BB. Show that in at least one of the sets AA and BB there is an infinite number of pairs (x,y)(x,y) such that x−y=1x-y=1.
2
2

8th ibmo - mexico 1993/q2.

Show that for every convex polygon whose area is less than or equal to 11, there exists a parallelogram with area 22 containing the polygon.

8th ibmo - mexico 1993/q5.

Let PP and QQ be two distinct points in the plane. Let us denote by m(PQ)m(PQ) the segment bisector of PQPQ. Let SS be a finite subset of the plane, with more than one element, that satisfies the following properties: (i) If PP and QQ are in SS, then m(PQ)m(PQ) intersects SS. (ii) If P1Q1,P2Q2,P3Q3P_1Q_1, P_2Q_2, P_3Q_3 are three diferent segments such that its endpoints are points of SS, then, there is non point in SS such that it intersects the three lines m(P1Q1)m(P_1Q_1), m(P2Q2)m(P_2Q_2), and m(P3Q3)m(P_3Q_3). Find the number of points that SS may contain.
1
2