MathDB

3

Part of 2011 Irish Math Olympiad

Problems(2)

irish lucas series

Source: irish 2011 paper 1

5/6/2012
The integers a0,a1,a2,a3,…a_0, a_1, a_2, a_3,\ldots are defined as follows: a0=1a_0 = 1, a1=3a_1 = 3, and an+1=an+an−1a_{n+1} = a_n + a_{n-1} for all n≥1n \ge 1. Find all integers n≥1n \ge 1 for which nan+1+anna_{n+1} + a_n and nan+an−1na_n + a_{n-1} share a common factor greater than 11.
inductionmodular arithmeticnumber theorygreatest common divisoralgebra unsolvedalgebra
Geometry question

Source:

4/12/2017
ABCDABCD is a rectangle. EE is a point on ABAB between AA and BB, and FF is a point on ADAD between AA and DD. The area of the triangle EBCEBC is 1616, the area of the triangle EAFEAF is 1212 and the area of the triangle FDCFDC is 30. Find the area of the triangle EFCEFC.
geometry