Irish Intervarsity Mathematics Competition 2003

University College Dublin

Time allowed: Three hours

1. Let

f(x) = x^{3} + Ax^{2} + Bx + C,

where A, B, C are integers. Suppose the roots of f(x) = 0 (in the field of complex numbers) are α, β, γ. Prove that if

|α| = |β| = |γ| = 1

then

f(x) | (x^{12} − 1)^{3}.

2. Let n be a positive integer. Prove that when written in decimal form (in base 10),

(\sqrt{17} + 4)^{2n+1}

has at least n zeroes following the decimal point.

3. Find all integers n for which

n^{4} − 16n^{3} + 86n^{2} − 176n + 169

is the square of an integer.

4. Consider the sequence

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, …

in which each positive integer k is repeated k times. Prove that its n^{th} term is

[\frac{1 + \sqrt{8n − 7}}{2}] ,

where [x] denotes the greatest integer not exceeding x.

5. Let ABC be an acute angled triangle and a, b, c the lengths of the sides BC, CA, AB, respectively. Let P be a point inside ABC, and let x, y, z be the lengths P A, P B, P C, respectively. Prove that

(x + y + z)^{2} ≥ \frac{a^{2} + b^{2} + c^{2}}{2} .

6. Let ABCD be a convex quadrilateral with the lengths AB = AC, AD = CD and angles BÂC = 20^{∘}, AD̂C = 100^{∘}. Prove that the lengths AB = BC + CD.

7. Let S be a set of 30 positive integers less than 100. Prove that there exists a nonempty subset T of S such that the product of the elements of T is the square of an integer.

8. Let

f(x) = x^{5} + ax^{4} + bx^{3} + cx^{2} + dx + e,

where a, b, c, d, e are integers, and suppose that f(x) = 0 has no integer roots. Suppose also that f(x) = 0 has roots α, β (in the field of complex numbers) with α + β an integer. Show that αβ is an integer.

9. Let x be a real number with 0 < x < 1. Let {a_{n}} be a sequence of positive real numbers. Prove that the inequality

1 + xa_{n} ≥ x^{2}a_{n−1}

holds for infinitely many positive integers n.

10. Find the least positive integer n for which

m^{n} − 1 is divisible by 10^{2003}

for all integers m with greatest common divisor gcd(m, 10) = 1.