3

Part of 2000 IberoAmerican

Problems(2)

15th ibmo - venezuela 2000/q3.

Source: Spanish Communities

Find all the solutions of the equation

\left(x+1\right)^y-x^z=1

x,y,z

integers greater than 1.

algebra unsolvedalgebra

15th ibmo - venezuela 2000/q6.

Source: Spanish Communities

A convex hexagon is called pretty if it has four diagonals of length 1, such that their endpoints are all the vertex of the hexagon. (

a

) Given any real number

k

0<k<1

find a pretty hexagon with area equal to

k

b

) Show that the area of any pretty hexagon is less than 1.

geometrycombinatorics unsolvedcombinatorics