2010/11 AHSMC Part 1

The number of positive integers $n$ such that the number $4n$ has exactly two digits is

A $4 \times 6$ plot of land is divided into $1 \times 1$ lots by fences parallel to the edges of the plot, with fences along the edges as well. The total length of fences is:

The greatest common divisor and least common multiple of two positive integers are $1$ and $10$ respectively. If neither of them is equal to $10$, their sum is equal to

The number of pairs $(x, y)$ of non-negative integers such that $3x + 2y = 27$ is

In the sequence $1, 2, 3, 4, 6, 7, 8, 9, \ldots$, obtained by deleting the multiples of $5$ from the sequence of the positive integers, the $2010$th term is

Alice, Brian, Colin, Debra and Ethel are in a hotel. Their rooms are on floors $1$, $2$, $3$, $21$ and $40$ respectively. In order to minimize the total number of floors they have to cover to get together, the floor on which the get-together should be is

A square pigeon coop is divided by interior walls into $9$ square pigeonholes in a $3 \times 3$ configuration. Each of two pigeons chooses a pigeonhole at random, possibly the same one. The probability that they choose two holes on the opposite sides of an interior wall is

The set of all values of the real number $x$ such that $\frac{1}{x} \le -3 \le x$ is

In the quadrilateral $ABCD$, $AB$ is parallel to $DC$, $DC = 2AB$, $\angle ADC = 30^\circ$ and $\angle BCD = 50^\circ$. Let $M$ be the midpoint of $CD$. The measure of $\angle AMB$ is

We are constructing isosceles but non-equilateral triangles with positive areas and integral side lengths between $1$ and $9$ inclusive. The number of such triangles which are non-congruent is

In each of the following numbers, the exponents are to be evaluated from top down. For instance, $a^{b^c} = a^{(b^c)}$. The largest one of these five numbers is

A gold number is a positive integer which can be expressed in the form $ab + a + b$, where $a$ and $b$ are positive integers. The number of gold numbers between $1$ and $20$ inclusive is

The edges $DA$, $DB$ and $DC$ of a tetrahedron $ABCD$ are perpendicular to one another. If the length of $DA$ is $1$ cm and the length of each of $DB$ and $DC$ is $2$ cm, the radius, in cm, of the sphere passing through $A$, $B$, $C$ and $D$ is

Let $f(x) = x^2$ and $g(x) = x^4$. We apply $f$ and $g$ alternatively to form $$f(x) = x^2,\quad g(f(x)) = g(x^2) = (x^2)^4 = x^8,\quad f(g(f(x))) = f(x^8) = (x^8)^2 = x^{16},$$ and so on. After we have applied $f$ $50$ times and $g$ $49$ times, the answer is $x^n$ where $n$ is

Triangle $ABC$ has area $1$. $X$, $Y$ are points on the side $AB$ and $Z$ a point on the side $AC$ such that $XY = 2AX$, $XZ$ is parallel to $YC$ and $YZ$ is parallel to $BC$. The area of $XYZ$ is

The number of integers $n$ for which $n^3 - 3n + 2$ is divisible by $2n + 1$ is