Irish Intervarsity Mathematics Competition 2006

University College Cork

9.30–12.30 Saturday 4^{th} March 2006

1. Let [x] denote the greatest integer not exceeding the real number x.

Prove that

[\sqrt{n} + \sqrt{n + 1} + \sqrt{n + 2}] = [\sqrt{9n + 8}]

for all positive integers n.

2. The sides BC, CA, AB of the triangle ABC have lengths a, b, c, respectively, and satisfy

b^{2} − a^{2} = ac, c^{2} − b^{2} = ab.

Determine the measure of the angles of ABC.

3. Prove that the polynomial

x^{100} + 33x^{67} + 67x^{33} + 101

cannot be factored as the product of two polynomials of lower degree with integer coefficients.

4. Let S be the set of all complex numbers of the form a + bi, where a and b are integers and i = \sqrt{−1}. Let A be a 2 × 2 matrix with entries in S and suppose A has determinant 1 and that

A^{n} = I.

for some positive integer n, where I is the identity matrix.

Prove that

A^{12} = I.

5. Evaluate

∫_{0}^{π/2} x cot(x) dx.

6. Let x and y be positive integers and suppose that

k = \frac{x^{2} + y^{2} + 6}{xy}

is also an integer. Prove that k is the cube of an integer.