Irish Intervarsity Mathematics Competition

Trinity College Dublin 1997

9.30–12.30 Saturday 22^{nd} February 1997

Answer as many questions as you can; all carry the same mark. Give

reasons in all cases. Tables and calculators are not allowed.

1. Compute

∑_{n=1}^{∞} \frac{n^{2}}{2^{n}} .

2. A stick is broken in random in 2 places (the 2 break-points being chosen independently). What is the probability that the 3 pieces form a triangle?

3. For which real numbers x > 0 is there a real number y > x such that

x^{y} = y^{x} ?

4. Show that there are an infinity of natural numbers n such that when the last digit of n is moved to the beginning (as eg 1234 ↦ 4123) n is multiplied by 3.

5. What is the whole number part of

1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + ⋯ + \frac{1}{\sqrt{1997}} ?

6. Prove that

n(n + 1)(n + 2) > (n + \frac{8}{9})^{3}

for any integer n ≥ 3.

7. Show that the determinant of the 3 × 3 matrix

A = (\begin{matrix}sin(x_{1} + y_{1}) & sin(x_{1} + y_{2}) & sin(x_{1} + y_{3}) \\ sin(x_{2} + y_{1}) & sin(x_{2} + y_{2}) & sin(x_{2} + y_{3}) \\ sin(x_{3} + y_{1}) & sin(x_{3} + y_{2}) & sin(x_{3} + y_{3}) \end{matrix})

is zero for all real numbers x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}.

8. Let

A(m, n) = \frac{m!(2m + 2n)!}{(2m)!n!(m + n)!}

for non-negative integers m and n. Show that

A(m, n) = 4A(m, n − 1) + A(m − 1, n)

for m ≥ 1, n ≥ 1. Hence or otherwise show that A(m, n) is always an integer.

9. Let P (x) = a_{0} + a_{1}x + ⋯ + a_{n}x^{n} be a real polynomial of degree n ≥ 2

such that

0 < a_{0} < − ∑_{k=1}^{[n/2]} \frac{1}{2k + 1}a_{2k}

(where [n/2] denotes the integer part of n/2). Prove that the equation P (x) = 0 has at least one solution in the range −1 < x < 1.

10. Suppose a_{1}, a_{2}, a_{3}, … is an infinite sequence of real numbers satisfying 0 < a_{n} ≤ 1 for all n. Let S_{n} = a_{1} + a_{2} + ⋯ + a_{n} and T_{n} = S_{1} + S_{2} + ⋯ + S_{n}. Show that

∑_{n=1}^{∞} \frac{a_{n}}{T_{n}} < ∞.