

1. Let S be the system of equations (1) y(x^{4}  y^{2} + x^{2}) = x, (2) x(x^{4}  y^{2} + x^{2}) = 1. Take S' to be the system of equations (1) and x·(1)  y·(2) (or y = x^{2}). Show that S and S' do not have the same set of solutions and explain why.


2. Show that x_{1}/x_{n} + x_{2}/x_{n1} + x_{3}/x_{n2} + ... + x_{n}/x_{1} ≥ n for any positive reals x_{1}, x_{2}, ... , x_{n}.


3. For which n is it possible to put n identical candles in a candlestick and to light them as follows. For i = 1, 2, ... , n, exactly i candles are lit on day i and burn for exactly one hour. At the end of day n, all n candles must be burnt out. State a possible rule for deciding which candles to light on day i.


4. 288 points are placed inside a square ABCD of side 1. Show that one can draw a set S of lines length 1 parallel to AB joining AD and BC, and additional lines parallel to AD joining each of the 288 point to a line in S, so that the total length of all the lines is less than 24. Is there a stronger result?


5. n is a positive integer. Show that x^{6}/6 + x^{2}  nx has exactly one minimum a_{n}. Show that for some k, lim_{n→∞} a_{n}/n^{k} exists and is nonzero. Find k and the limit.

