23rd IMO 1982 shortlisted problems

1.  An urn contains w white balls and b black balls. In each move, two balls are drawn at random and removed from the urn, and one ball is added. If the two balls drawn have the same color, then a black ball is added. If they are opposite colors, then a white ball is added. Eventually, only one ball is left. What is the probability that it is white?
2.  p(x) is a cubic polynomial with integer coefficients and leading coefficient 1. One of its roots is the product of the other two. Prove that 2p(-1) is a multiple of p(1) + p(-1) - 2(1 + p(0)).
3.  Let a1, a2, ... , an be a permutation of 1, 2, ... , n. Find the permutation which maximises a1a2 + a2a3 + ... + an-1an + ana1 and the permutation which minimises it.
4.  Let M be the set of real numbers of the form (m+n)/√(m2+n2), where m and n are positive integers. Show that if x < y are two elements of M, then there is an element z of M such that x < z < y.
5.  Prove that (1 - sa)/(1 - s) ≤ (1 + s)a/(1 + s) for every positive real s ≠ 1 and every positive rational a ≤ 1.
6.  Find all real numbers a such that the equation 16x4 - ax3 + (2a + 17)x2 - ax + 16 = 0 has exactly four distinct real roots which form a geometric progression.
7.  P is a point inside the triangle ABC such that ∠PAC = ∠PBC. The perpendiculars from P meet the lines BC, CA at L, M respectively. Prove that DL = DM where D is the midpoint of AB.
8.  The triangles ABC and AB'C' have opposite orientation. ∠BCA = ∠B'C'A = 90o. BC' and B'C intersect at M. Prove that if the lines AM and CC' are well defined, then they are perpendicular.
9.  ABCD is a convex quadrilateral. A1, B1, C1, D1 are the circumcenters of BCD, CDA, DAB, ABC respectively. Prove that if two of A1, B1, C1, D1 coincide, then they all coincide. Prove that if they are distinct, then A1B1C1D1 is convex. In this case let A2, B2, C2, D2 be the circumcenters of B1C1D1, C1D1A1, D1A1B1, A1B1C1 respectively. Show that A2B22C2D2 is similar to ABCD.
10.  ABCD is a convex quadrilateral. ABM and CDP are equilateral triangles on the outside of the sides AB, CD. BCN and DAQ are equilateral triangles on the inside of sides BC and DA. Prove that MN = AC. What can be said about MNPQ?
11.  A convex figure lies inside a circle. The points on the boundary of the figure are considered to be in the figure. For each point on the circle, draw the smallest angle with this point as the vertex which contains the figure. If the angle is always a right angle, prove that the center of the circle is a center of symmetry of the figure.
12.  Exactly one quarter of the area of a convex polygon in the coordinate plane lies in each quadrant. If (0,0) is the only lattice point in or on the polygon, prove that its area is less than 4.
13.  S is a unit sphere with center at the origin. For any point P on S, the unit sphere with center P intersects the x-axis at O and X, the y-axis at O and Y and the z-axis at O and Z (where X, Y or Z may coincide with O). What is the locus of the centroid of XYZ as P varies over the sphere.

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© John Scholes
1 Nov 2003
Last corrected/updated 1 Nov 03