Irish Intervarsity Mathematics Competition 1999

University College Cork

Answer all questions. Tables and calculators may be used.

1. Find the value of

lim_{n→∞} (∑_{r=0}^{n} \frac{1}{\binom{n}{r}}) ,

where \binom{n}{r} is the binomial coefficient, the number of combinations of n things taken r at a time.

2. The coordinates of four points in the plane are given by A(0, 0), B(1, 2), C(3, 3) and D(3, 0). What is the smallest possible value of |P A|+|P A|+|P C|+ |P D|, where P is any point in the plane.

3. Two real numbers a and b are chosen with 0 ≤ a ≤ 1 and 0 ≤ b ≤ 1. What is the probability that a^{2} + b^{2} ≤ 1.

4. If x, y, z, w, t and u are all prime numbers with x ≤ y ≤ z ≤ w ≤ t ≤ u, find all solutions of the equation

x^{2} + y^{2} + z^{2} + w^{2} + t^{2} = u^{2}.

5. Evaluate ∑_{r=1}^{n}(r + 1)^{2}(r!).

6. Find all positive integers less than 100 which have precisely seven distinct divisors (including 1 and n).

7. Let ABC be a triangle with a right angle at A. Show that the internal bisector of the angle BAC divides the square on the hypotenuse BCDE into two parts of equal area.

8. Evaluate

f(m, n) = ∫_{0}^{1} x^{m}(1 − x)^{n} dx

as a function of m and n only.

9. A window of total perimeter 200cm consists of a rectangle surmounted by a semicircle. Find the maximal area the window can have.

10. The lengths of the sides of a quadrilateral are 1, 2, 3 and 4. What is the maximal area the quadrilateral can have?