Royal Life Ireland

Irish Intervarsity Mathematics Competition

Trinity College Dublin 1991

9.30–12.30 February 9

Answer all questions. Calculators permitted.

1. The prime factorisations of r + 1 positive integers (r ≥ 1) together involve only r primes. Prove that there is a subset of these integers whose product is a perfect square.

2. Consider a polynomial p(x) = x^{n} + nx^{n−1} + a_{2}x^{n−2} + ⋯ + a_{n} which has real roots

r_{1}, r_{2}, …, r_{n}. If

r_{1}^{16} + r_{2}^{16} + ⋯ + r_{n}^{16} = n,

find all the roots.

3. An n-inch cube (n a positive integer) is painted on all sides and then cut into 1-inch cubes. If the number of small cubes with one painted side is the same as the number with two painted sides, what could n have been?

4. How many ways are there of painting the 6 faces of a cube in 6 different colours, if two colourings are considered the same when one can be obtained from the other by rotating the cube?

5. How many positive integers x ≤ 1991 are such that 7 divides 2^{x} − x^{2}?

6. How many ways can 1,000,000 be expressed as a product of 3 positive integers? Factorisations different only in order are considered to be the same.

7. Prove that 2^{n} can begin with any sequence of digits.

8. Imagine a point P inside a square ABCD. If |P A| = 5, |P B| = 3 and |P C| = 7, what is the side of the square?

9. Let f(x) be a function such that f(1) = 1 and, for x ≥ 1

f′(x) = \frac{1}{x^{2} + f^{2}(x)}.

Prove that

lim_{x→∞} f(x)

exists and is less than 1 + π/4.

10. Prove that the number of odd binomial coefficients in each row of Pascal’s triangle

is a power of 2. [In Pascal’s triangle

1

1  1
1  2  1
1  3  3  1

⋮

each entry is the sum of the entries directly above it.]