Twenty-First Superbrain 2004

1. In the Irish National Lottery, "gamblers" are asked to choose six different numbers from the set {1, 2, 3,...., 42}. What precentage of choices contain at least two consecutive numbers.
2. Find all positive integers a and b such that a⁴ + (a+1)⁴ + (a+2)⁴ = b⁴
3. Prove or disprove that the enclosed map of the countries of Ireland can be coloured with three different colours in such a way that countries which touch each other have different colours.

4. Evaluate

5. The aera of an equilateral triangle OPQ is bisceted by a curve AB of minimal length. What is the equation of the curve with respect to the given axes?

6. Evaluate

7. Find the value of

   and justify your answer.

8. Show how to cut this figure into three pieces and reassemble them to form a square.

9. Solve the equation,
   x₁ + x₂ + ... + x_n = (x₁)(x₂)...(x_n)
   subject to the following conditions,
   (i) x_i is a natural number for all i.
   (ii) x_i not equal to x_j for i not equal to j.
   (iii) n > 2.
10. What are the coordinates of the point on the parabola y² = 4x which is nearest to the point with coordinates (-1, 4)?

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