MathDB

1

Part of 2000 IberoAmerican

Problems(2)

15th ibmo - venezuela 2000/q1.

Source: Spanish Communities

4/15/2006
A regular polygon of n n sides (n≥3 n\geq3) has its vertex numbered from 1 to n n. One draws all the diagonals of the polygon. Show that if n n is odd, it is possible to assign to each side and to each diagonal an integer number between 1 and n n, such that the next two conditions are simultaneously satisfied: (a) The number assigned to each side or diagonal is different to the number assigned to any of the vertices that is endpoint of it. (b) For each vertex, all the sides and diagonals that have it as an endpoint, have different number assigned.
modular arithmeticcombinatorics unsolvedcombinatorics
15th ibmo - venezuela 2000/q4.

Source: Spanish Communities

4/15/2006
From an infinite arithmetic progression 1,a1,a2,… 1,a_1,a_2,\dots of real numbers some terms are deleted, obtaining an infinite geometric progression 1,b1,b2,… 1,b_1,b_2,\dots whose ratio is q q. Find all the possible values of q q.
ratioarithmetic sequencegeometric sequencealgebra unsolvedalgebra