Irish Intervarsity Mathematics Competition 1993

University College Galway

Answer all ten questions

1. Evaluate ∑_{j=2}^{∞} (∑_{i=2}^{∞} \frac{1}{i^{j}} ) .

2. abcd is a square and e is a point inside the square such that |ae| = |be| and eb̂a = 15^{∘}. Show that dec is an equilateral triangle.

[figure: a; b; d; c; e]

3. If x, y and n are all natural numbers and t = x^{4n} + y^{4n} + x^{2n}y^{2n} is a prime number, find all possible values of t.

4. Construct, with proof, a non–constant arithmetic progression of positive integers which contains no squares, cubes or higher powers of integers.

5. If ^{n}C_{r} is the number of combinations of n objects r at a time, find the

value of ∑_{r=0}^{n} (\frac{1}{r + 1})^{n}C_{r}.

6. If x and y are real numbers, find all solutions of the equation (x^{4} + 1) = (x^{2})(2^{1−y^{2}}).

7. Find a formula for the nth derivative of the function f(x) = \frac{q}{x^{2} + x^{2}} where a is a constant.

8. If abc is an obtuse angled triangle with bâc > \frac{π}{2} prove that

54|ac|^{3}|ab|^{3}| cos bâc| ≤ |bc|^{6}

and find conditions under which equality holds.

9. Show that there do not exist polynomials f(x) and g(x) such that

e^{x} = \frac{f(x)}{g(x)} for all x.

10. Find the real factors of x^{4} + y^{4} + (x + y)^{4}.