Irish Intervarsity Mathematical Examination 1999

Answer all questions. Tables and calculators may be used.

  1. Find the value of
    lim
    n->infinity
    [
    n
    S
    r = 0
    (
    1
    _
    ( n)
    r
    )] where
    ( 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 |PA| + |PB| + |PC| + |PD| 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 a2 + b2 ≤ 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
    x2 + y2 + z2 + w2 + t2 = u2.
  5. Evalute
    n
    S
    r=1
    (r + 1)2(r!).
  6. Find all positive integers n 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. Diagram as descriped in question 7
  8. Evaluate f(m, n) =
    the integral from 0 to 1 of 1
    0
    xn(1 - x)n dx as a funtion of m and n only.
  9. A window of total perimeter 200cm consists of a rectangle surmounted by a semicircle. Find the maximum area the window can have. rectangle surmounted by semicircle (no internal lines)
  10. The lengths of the sides of a plane quadrilateral are 1, 2, 3 and 4. What is the maximum area the quadrilateral can have?