3

Part of 1997 Vietnam National Olympiad

Problems(2)

numbers of functions

Source: Vietnam NMO 1997, Problem 3

Find the number of functions

f: \mathbb N\rightarrow\mathbb N

which satisfying: (i) f(1) \equal{} 1 (ii) f(n)f(n \plus{} 2) \equal{} f^2(n \plus{} 1) \plus{} 1997 for every natural numbers n.

functionnumber theoryrelatively primealgebra proposedalgebra

one's area is not greater than 7/72

Source: Vietnam NMO 1997, Problem 6

In the unit cube, given 75 points, no three of which are collinear. Prove that there exits a triangle whose vertices are among the given points and whose area is not greater than 7/72.

geometry3D geometrypigeonhole principlegeometry proposed