27th Swedish 1987

1.  In a 4 x 4 array of real numbers the sum of each row, column and main diagonal is k. Show that the sum of the four corner numbers is also k.
2.  A circular disk radius R is divided into two equal parts by a circle. Show that the arc of the circle inside the disk has length > 2R.
3.  10 closed intervals of length 1 lie in the interval [0,4]. Show that there is a point belonging to at least four of them.
4.  f(x) is differentiable on [0,1] and f(0) = f(1) = 0. Show that there is a point y in [0,1] such that |f '(y)| = 4 ∫01 |f(x)| dx.
5.  Show that for some t > 0, we have 1/(1+a) + 1/(1+b) + 1/(1+c) + 1/(1+d) > t for all positive a, b, c, d such that abcd = 1. Find the smallest such t.
6.  A baker uses n different spices. He bakes 10 loaves. Each loaf has a different combination of spices and uses more than n/2 spices. Show that there are three spices such that every loaf has at least one.

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

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© John Scholes
8 Oct 2003
Last corrected/updated 8 Oct 03