MathDB

4

Part of 1992 Hungary-Israel Binational

Problems(2)

Pentagon in the plane [lattice point in a pentagram]

Source: Russia 2000

5/16/2004
We are given a convex pentagon ABCDEABCDE in the coordinate plane such that AA, BB, CC, DD, EE are lattice points. Let QQ denote the convex pentagon bounded by the five diagonals of the pentagon ABCDEABCDE (so that the vertices of QQ are the interior points of intersection of diagonals of the pentagon ABCDEABCDE). Prove that there exists a lattice point inside of QQ or on the boundary of QQ.
analytic geometrygeometryparallelogramgeometry unsolved
(...) is not a perfect square (Fibonacci and Lucas numbers)

Source: 3-rd Hungary-Israel Binational Mathematical Competition 1992

5/24/2007
We examine the following two sequences: The Fibonacci sequence: F0=0,F1=1,Fn=Fn−1+Fn−2F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 } for n≥2n \geq 2; The Lucas sequence: L0=2,L1=1,Ln=Ln−1+Ln−2L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2} for n≥2n \geq 2. It is known that for all n≥0n \geq 0 Fn=αn−βn5,Ln=αn+βn,F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, where α=1+52,β=1−52\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}. These formulae can be used without proof. Prove that Fn−1FnFn+1Ln−1LnLn+1(n≥2)F_{n-1}F_{n}F_{n+1}L_{n-1}L_{n}L_{n+1}(n \geq 2) is not a perfect square.
inductionnumber theory proposednumber theory