2009/10 AHSMC Part 1

If $2^x = 3^y$, then $4^x$ is equal to

Caroline bought some bones for her $7$ dogs. Had she owned $8$ dogs, she could have given each the same number of bones. As it was, she needed two more bones to give each dog the same number of bones. The number of bones she could have bought was

Ace calculates the average of all the integers from $1$ to $100$. Bea calculates the average of all the integers from $1001$ to $1100$ and subtracts $1000$. Cec calculates the average of all the integers from $1000001$ to $1000100$ and subtracts $1000000$. The largest answer is given by

A large rectangular gymnasium floor is covered with unit square tiles, most of them blank, in the pattern shown in the diagram below. Of the following fractions, the one nearest to the fraction of tiles which are not blank is

The number of integers between $1$ and $2009$ inclusive which can be expressed as the difference of the squares of two integers is

Among the positive integers with six digits in their base-$10$ representation, the number of those whose digits are strictly increasing from left to right is

The number of arrangements of the letters AABBCC in a row such that no two identical letters are adjacent is

If $2^{2009}$ has $m$ digits and $5^{2009}$ has $n$ digits in their base-$10$ representations, then the value of $m + n$ is

An equilateral triangle has area $2\sqrt{3}$. From the midpoint of each side, perpendiculars are dropped to the other two sides. The area of the hexagon formed by these six lines is

Two sides of an obtuse triangle of positive area are of length $5$ and $11$. The number of possible integer lengths of the third side is

$Q(x)$ is a polynomial with integer coefficients such that $Q(9) = 2009$. If $p$ is a prime number such that $Q(p) = 392$, then $p$ can

A parallelogram has two opposite sides $5$ centimetres apart and the other two opposite sides $8$ centimetres apart. Then the area, in square centimetres, of the parallelogram

The number of positive integers $n$ such that $\sqrt{n + \sqrt{n + \cdots + \sqrt{n}}} < 10$ for any finite number of square root signs is

A chord of a circle divides the circle into two parts such that the squares inscribed in the two parts have areas $16$ and $144$ square centimetres. In centimetres, the radius of the circle is

The number of prime numbers $p$ such that $2^p + p^2$ is also a prime number is

Suppose that $2 - \sqrt{99}$ is a root of $x^2 + ax + b$ where $b$ is a negative real number and $a$ is an integer. The largest possible value of $a$ is