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/xn + x2/xn-1 + x3/xn-2 + ... + xn/x1 ≥ n for any positive reals x1, x2, ... , xn.
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 an. Show that for some k, limn→∞ an/nk 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.

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© John Scholes
23 September 2003
Last corrected/updated 23 Sep 03