1st Swedish 1961


1. Let S be the system of equations (1) y(x⁴ - y² + x²) = x, (2) x(x⁴ - y² + x²) = 1. Take S' to be the system of equations (1) and x·(1) - y·(2) (or y = x²). Show that S and S' do not have the same set of solutions and explain why.
2. Show that x₁/x_n + x₂/x_n-1 + x₃/x_n-2 + ... + x_n/x₁ ≥ n for any positive reals x₁, x₂, ... , 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 x⁶/6 + x² - nx has exactly one minimum a_n. Show that for some k, lim_n→∞ a_n/n^k exists and is non-zero. Find k and the limit.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.