12th British Mathematical Olympiad 1976 Problems

12th British Mathematical Olympiad 1976 Problems

1.  ABC is a triangle area k. Let d be the length of the shortest line segment which bisects the area of the triangle. Find d. Give an example of a curve which bisects the area and has length < d.

2.  Prove that x/(y + z) + y/(z + x) + z/(x + y) ≥ 3/2 for any positive reals x, y, z.

3.  Given 50 distinct subsets of a finite set X, each containing more than | X |/2 elements, show that there is a subset of X with 5 elements which has at least one element in common with each of the 50 subsets.

4.  Show that 8ⁿ19 + 17 is not prime for any non-negative integer n.

5.  aCb represents the binomial coefficient a!/( (a - b)! b! ). Show that for n a positive integer, r ≤ n and odd, r' = (r - 1)/2 and x, y reals we have: ∑₀^r' nC(r-i) nCi (x^r-iyⁱ + xⁱy^r-i) = ∑₀^r' nC(r-i) (r-i)Ci xⁱyⁱ(x + y)^r-2i.

6.  A sphere has center O and radius r. A plane p, a distance r/2 from O, intersects the sphere in a circle C center O'. The part of the sphere on the opposite side of p to O is removed. V lies on the ray OO' a distance 2r from O'. A cone has vertex V and base C, so with the remaining part of the sphere it forms a surface S. XY is a diameter of C. Q is a point of the sphere in the plane through V, X and Y and in the plane through O parallel to p. P is a point on VY such that the shortest path from P to Q along the surface S cuts C at 45 deg. Show that VP = r√3 / √(1 + 1/√5). 

Problem 2

Prove that x/(y + z) + y/(z + x) + z/(x + y) ≥ 3/2 for any positive reals x, y, z.

Solution

If x + y + z = k, then (1/(k-x) + 1/(k-y) + 1/(k-z) )(k-x + k-y + k-z) ≥ 9 by Cauchy Schwartz. Hence 1/(y+z) + 1/(z+x) + 1/(x+y) ≥ 9/(2k). So k/(y+z) + k/(z+x) + k/(x+y) ≥ 9/2. Subtracting (y+z)/(y+z) + (z+x)/(z+x) + (x+y)/(x+y) = 3 gives the result.