22nd IMO 1981 shortlisted problems

1.  A 3 x 3 cube is assembled from 27 white unit cubes. The large cube is then painted black on the outside and then disassembled. If it is reassembled at random, what is the probability that the large cube is still completely black on the outside?
2.  Let Fn be the Fibonacci sequence defined by F0 = F1 = 1 and Fn+2 = Fn+1 + Fn for all n ≥ 0. Find all pairs (a, b) of real numbers such that for each n, aFn + bFn+1 is a member of the sequence. Find all pairs (u, v) of real numbers such that for each n, uFn2 + vFn+12 is a member of the sequence.
3.  A sequence an is defined as follows, a0 = 1, an+1 = (1 + 4an + √(1 + 24an))/16 for n ≥ 0. Find an explicit formula for an.
4.  A real sequence un is defined by u1 and 4un+1 = (64un + 15)1/3. Describe the behavior of the sequence as n → ∞.
5.  a+b+c+d+e+f+g = 1 and a, b, c, d, e, f, g are non-negative. Find the minimum value of max(a+b+c, b+c+d, c+d+e, d+e+f, e+f+g).
6.  p(k) is a polynomial of degree n such that p(k) = (n+1-k)!k!/(n+1)! for k = 0, 1, ... , n. Find p(n+1).
7.  AB, BC, CD, DE are consecutive chords on a semicircle of unit radius with lengths a, b, c, d. Prove that a2 + b2 + c2 + d2 + abc + bcd < 4.
8.  A convex pentagon ABCDE has equal sides and ∠A ≥ ∠B ≥ ∠C ≥ ∠D ≥ ∠E. Prove that it is regular.
9.  Find the smallest positive integer n such that for every integer m ≥ n, it is possible to partition a given square into m squares, not necessarily of the same size.
10.  A finite set of unit circular disks is given in a plane, such that the area of their union is S. Prove that there exists a subset of mutually disjoint disks whose union has area > 2S/9.
11.  Several equal spherical planets are in outer space. On the surface of each planet is a set of points which is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets equals the surface area of one planet.
12.  A sphere S is tangent to the edges AB, BC, CD, DA of a tetrahedron ABCD at the points E, F, G, H respectively. If EFGH is a square, prove that the sphere is tangent to the edge AC iff it is tangent to the edge BD.

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© John Scholes
1 Sep 2003
Last corrected/updated 4 Jan 04