The OP is studying for his local math competition (Australian), and when running through past papers I found some questions subtle to handle. I decide to buy some books to aid my study, but there are questions I totally don't have any idea in what books they might appear, can you guys please tell me what branches are these questions from, and what books can I use to attack them? (e.g. the question 'finding all the integer values of $\displaystyle \frac{7n+18}{2n+3}$' is classified as number theory problem.)
(answers to these questions are not required though.)
1.How many (disconnected) enclosures can 24 fences make if fences are not allowed to intersect and can only be jointed at the end points? (e.g. it's obvious that with 5 fences we can only make 2 enclosures, namely 2 triangles with a shared side.)
What is the maximum number of intervals that can be connected between 6 points, whose configuration can be arbitrary but on the same plane, where the intervals are not allowed to intersect each other except at endpoints. (e.g. it's also obvious that with 4 points we can connect 6 intervals, namely the configuration of a CONVEX quadrilateral and its 2 diagonals.)
2.If $a_1=2$, $a_{n+1}=a_{n}+\mbox{the largest prime factor of $a_n$}$ (namely, $a_2=4,a_3=6,a_4=9$...) after what $n$ will $a_n$ exceed $2014$(not quite sure which number it is, but some given number).
3.If $a_1=3$,$a_{n+1}=a_n+a_n^2$, what is the second last digit of $a_{2014}$? Note:I edited an absent-minded typo here.
(Probably 2 and 3 are from number theory, but from what book can I learn the recursion part?)
4.If we divide a cube into 8 pieces in the canonical way (it's too complicated to post pictures.)and an ant is going from one vertex to the opposite vertex along the traces of the cuts,and is only allowed to turn at the intersection points. If the distance to the terminal point is only monotonically decreasing, how many different ways can the ant get there?
5.If 8 teams are playing a tournament where every team plays each other team once. 2 points for a win, 1 point for a draw and 0 points for a lose, how many points must a team obtain to secure being in the top 4?
6.How many non-isosceles triangles can be drawn with its vertices on the grid points of a 3*3 grid?
7.Assign one number to each vertex of a hexagon, and one number in the middle. If every number is to be the average of the numbers surrounding it (i.e. the central number is the average of the brim 6 numbers, and the every brim point is the average of the central number and its adjacent 2 numbers.) If one brim point is 1 and the one opposite to it is 100, what are the others?
There are still others to be asked, but we can stop here. I only need to know what branches they are from, so that I can find books to aid myself. Thanks