11th British Mathematical Olympiad 1975 Problems

11th British Mathematical Olympiad 1975 Problems

1.  Find all positive integer solutions to [1^1/3] + [2^1/3] + ... + [(n³ - 1)^1/3] = 400.

2.  The first k primes are divided into two groups. n is the product of the first group and n is the product of the second group. M is any positive integer divisible only by primes in the first group and N is any positive integer divisible only by primes in the second group. If d > 1 divides Mm - Nn, show that d exceeds the kth prime.

3.  Show that if a disk radius 1 contains 7 points such that the distance between any two is at least 1, then one of the points must be at the center of the disk. [You may wish to use the pigeonhole principle.]

4.  ABC is a triangle. Parallel lines are drawn through A, B, C meeting the lines BC, CA, AB at D, E, F respectively. Collinear points P, Q, R are taken on the segments AD, BE, CF respectively such that AP/PD = BQ/CE = CR/RF = k. Find k.

5.  Let nCr represent the binomial coefficient n!/( (n-r)! r! ). Define f(x) = (2m)C0 + (2m)C1 cos x + (2m)C2 cos 2x + (2m)C3 cos 3x + ... + (2m)C(2m) cos 2mx. Let g(x) = (2m)C0 + (2m)C2 cos 2x + (2m)C4 cos 4x + ... + (2m)C(2m) cos 2mx. Find all x such that x/π is irrational and lim_m→∞ g(x)/f(x) = 1/2. You may use the identity: f(x) = (2 cos(x/2) )^2m cos mx.

6.  Show that for n > 1 and real numbers x > y > 1, (xⁿ⁺¹ - 1)/(xⁿ - x) > (yⁿ⁺¹ - 1)/(yⁿ - y).

7.  Show that for each n > 0 there is a unique set of real numbers x₁, x₂, ... , x_n such that (1 - x₁)² + (x₁ - x₂)² + ... + (x_n-1 - x_n)² + x_n² = 1/(n + 1).

8.  A wine glass has the shape of a right circular cone. It is partially filled with water so that when tilted the water just touches the lip at one end and extends halfway up at the other end. What proportion of the glass is filled with water?