1996 Irish Intervarsity Mathematics Contest

10am - 1pm March 2
Maynooth College
Answer all ten questions

Each question is worth 10 points. For full points, you will need not only a correct answer but a valid and correct justification. Calculators are permitted, but unlikely to be useful.
Good luck and HAVE FUN!!

1. The letters of CONTEST are to be placed at the corners of a regular heptagon (7-gon). How many ways can this be done if two arrangements are considered the same whenever one can be rotated to form the other.
2. For what values of n does n! terminate in exactly 36 zeros?
3. Find a positive integer solution of a³ + b³ + c³ + 2 = 1996 or prove that there are none.
4. You are handed a page on which there are written exactly twenty statements. For k = 1, . . . ,20, the k^th statement reads:
   There are exactly k false statements on this page. Which of the twenty statements are true and which are false?
5. Let a₁, . . . , a_n be a permutation of 1, . . . , n, and let P =

   P	n	(ai - i).
   	i = 1

   If n is odd, show that P is even.
6. Find a sequence of real numbers 0 < x₁ < x₂ < . . . < x_n < x_{n + 1} < . . . such that x_n but x_n² - x_n 0, as n .
7. For each integer n ≥ 10, we define an integer a_n by moving the last digit of n to the first position (e.g. a₁₂₃ = 312, a₁₉₉₆ = 6199). Find the smallest integer n 6 (mod 10) for which a_n = 4n.
8. Suppose f is a real valued function defined for all real x such that f(0) = 0 and f(y) - f(x) ≤ (y - x)(x² + y²) for all x, y. Prove that f(3) is less than 20.
9. On a plane there are n points A₁, . . . , A_n and a circle of unit radius. Prove that there exists a point M on the circle for which the sum of distances, |MA₁| + . . . + |MA_n|, is at least n
10. Let S = {m² - 5n² : m, n Z}. Show that S is closed under multiplication, i.e. if a, b S, then ab S.