Irish Intervarsity Mathematics Competition

University College Cork 1990

1. If x, y and z are real numbers solve the equations

x + y + z = \sqrt[3]{2}
x^{2} + y^{2} + z^{2} = \sqrt[3]{4}
x^{3} + y^{3} + z^{3} = 2

2. A hand of 13 cards is dealt from a pack of 52 cards containing 13 spades, 13 diamonds, 13 clubs and 13 hearts. If the hand has 3 clubs, 4 spades, 2 diamonds and 4 hearts, it is siad to have the distribution type 4 − 4 − 3 − 2, while if it has 6 hearts and 7 spades it is said to have distribution type 7 − 6. How many distribution types are there? List them all.

3. Evaluate

∑_{n=1}^{∞} \frac{n^{3}}{3^{n}} .

4. Find the minimum value of the real function

f(x) = x^{x}, for x > 0.

5. Prove that

tan^{2} (\frac{π}{7} ) + tan^{2} (\frac{2π}{7} ) + tan^{2} (\frac{3π}{7} ) = 21.

6. A ladder of constant length 2L extends from a horizontal floor to a wall which is inclined at 85^{∘} to the horizontal. If the ladder slips while remaining in contact with both the floor and the wall, find the locus of the mid-point of the ladder.

7. Prove that 3, 5 and 7 are the only 3 consecutive odd numbers all of which are prime.

8. If x, y and z are positive integers, all greater than 1, find any solution

of the equation

x^{x}y^{y} = z^{z}.

9. If a, b, c are the lengths of the side of a triangle prove that

3(ab + bc + ca) ≤ (a + b + c)^{2} ≤ 4(ab + bc + ca),

and in each case find the necessary and sufficient condition.

10. Let S be the set of all natural numbers, which can be written in the form x^{3} + y^{3} + z^{3} − 3xyz for some x, y, z ∈ N, where N is the set of natural numbers. if a ∈ S and b ∈ S prove that ab ∈ S.