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Published 2010 Revised 2019
1. Let ABC be an isosceles triangle with a right angle at B. Construct D inside ABC such that AB = AD and angle BAD = 30 degrees. Prove that BD = DC.
2. What is the algebraic expression in terms of x, y and z for the area of the triangle with vertices (x, 0, 0), (0, y, 0), and (0, 0, z)?
3. Let four equilateral triangles be sides of a square-based pyramid: find the ratio of the volume of this pyramid to a tetrahedron made of the same four triangles.
4. Prove that the square of any prime number greater than 3 is one more than a multiple of 24.
5. Prove that among n+1 numbers from the set {1,2, \dots ,2n} there are always two such that one divides the other.
6. Prove that n! is a divisor of the product of any n consecutive natural numbers.
7. Simplify \sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n}...}}}}}}