21st Swedish 1981

1.  Let N = 11 ... 122 ... 25, where there are n 1s and n+1 2s. Show that N is a perfect square.
2.  Does xy = z, yz = x, zx = y have any solutions in positive reals apart from x = y = z = 1?
3.  Find all polynomials p(x) of degree 5 such that p(x) + 1 is divisible by (x-1)3 and p(x) - 1 is divisible by (x+1)3.
4.  A cube side 5 is divided into 125 unit cubes. N of the small cubes are black and the rest white. Find the smallest N such that there must be a row of 5 black cubes parallel to one of the edges of the large cube.
5.  ABC is a triangle. X, Y, Z lie on BC, CA, AB respectively. Show that area XYZ cannot be smaller than each of area AYZ, area BZX, area CXY.
6.  Show that there are infinitely many triangles with side lengths a, b, c, where a is a prime, b is a power of 2 and c is the square of an odd integer.

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

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© John Scholes
29 September 2003
Last corrected/updated 30 Dec 03