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4

Part of 2012 Junior Balkan Team Selection Tests - Moldova

Problems(2)

linear Sequence

Source:

8/9/2012
Let there be an infinite sequence ak a_{k} with k≥1 k\geq 1 defined by: ak+2=ak+14 a_{k+2} = a_{k} + 14 and a1=12 a_{1} = 12 , a2=24 a_{2} = 24 .
a) Does 20122012 belong to the sequence? b) Prove that the sequence doesn't contain perfect squares.
inductionnumber theory proposednumber theory
how many solutions

Source:

8/9/2012
How many solutions does the system have: {(3x+2y)∗(3x+1y)=2x2+y2≤2012 \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2\\ & x^2+y^2\leq 2012\\ \end{matrix} where x,y x,y are non-zero integers
linear algebramatrixalgebra proposedalgebra