Irish Intervarsity Mathematics Competition 1992

Trinity College Dublin

9.30–12.30 15^{th} February

Answer all questions.

Calculators permitted.

1. Solve the equation

(x − 2)(x − 3)(x + 4)(x + 5) = 44.

2. Find the greatest value of

\frac{x + 2}{2x^{2} + 3x + 6}.

3. Writing numbers to the base 8, show that there are infinitely many numbers which are doubled by reversing their ‘digits’.

4. Show that for any positive real numbers a, b with a > b,

a^{n} − b^{n} > n(a − b)(ab)^{(n−1)/2}.

5. Describe geometrically the points P, Q in an arbitrary triangle ABC that minimize

(a) AP + BP + CP

(b) AQ^{2} + BQ^{2} + CQ^{2}.

6. For m a positive integer, let k(m) denote the largest integer k such that 2^{k} divides m!. Let c(m) denote the number of 1’s in the binary representation of m. Show that k(m) = m − c(m).

7. If x_{1}, x_{2}, …, x_{n} are positive numbers and s is their sum, prove that

(1 + x_{1})(1 + x_{2}) ⋯ (1 + x_{n}) ≤ 1 + s + \frac{s^{2}}{2!} + ⋯ + \frac{s^{n}}{n!} .

8. A table tennis club with 20 members organises 14 singles games (2-player games) one Saturday morning in such a way that each member plays at least once. Show that there must be 6 games involving 12 different players.

9. Let a_{1}, a_{2}, … be the sequence of all positive integers with no 9’s in their decimal representation. Show that the series

\frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}} + ⋯

converges.

10. In a convex quadrilateral ABCD, let E and F be the midpoints of the sides BC and DA (respectively). Show that the sum of the areas of the triangles EDA and F BC is equal to the area of the quadrilateral.