2011/12 AHSMC Part 1

If $2^{2012} + 4^{1006} = 2^n$, then $n$ is

Mini-marshmallows are cubes $1$ centimetre on each side, while giant marshmallows are cubes $3$ centimetres on each side. The number of mini-marshmallows whose combined surface area is the same as the surface area of one giant marshmallow is

The number of customers in a restaurant on Tuesday is $20\%$ more than the number on Monday, the number of customers on Wednesday is $50\%$ more than the number on Monday, and the number of customers on Wednesday is $n\%$ more than the number on Tuesday. The value of $n$ is

Sawa starts from point $S$ and walks $1$ kilometre north, $2$ kilometres east, $3$ kilometres south and $4$ kilometres west. At this point, her distance, in kilometres, from $S$ is

A millenium number is a positive integer such that the product of its digits is $1000$. The number of six-digit millenium numbers is

Let $x_1, x_2, \ldots$ be a sequence of positive rational numbers such that $x_1 = 16$, $x_2 = 32$ and $x_n = \frac{x_{n-1} + x_{n-2}}{2}$ for all positive integers $n \ge 3$. Then the value of $x_6$ is

The number of real solutions of the equation $2x^2 - 2x = 2x\sqrt{x^2 - 2x} + 1$ is

Let $f(x)$ be a quadratic polynomial. If $f(1) = 2$, $f(2) = 4$ and $f(3) = 8$, then the value of $f(4)$ is

A lucky number is a positive integer $n$ such that $7$ is the largest divisor of $n$ which is less than $n$. The number of lucky numbers is

The perimeter of a square lawn consists of four straight paths. Annabel and Bethany started at the same corner at the same time, running clockwise at constant speeds of $12$ and $10$ kilometres per hour respectively. Annabel finished one lap around the lawn in $1$ minute. During this minute, the number of seconds that Annabel and Bethany were on the same path is

$ABCD$ is a quadrilateral with $AD = BC$ and $AB$ parallel to $DC$. It is only given that the lengths of $AB$ and $DC$ are $20$ and $15$ centimetres respectively. Adrian puts $n$ copies of this tile together so that the edge $BC$ of each copy coincides with the edge $AD$ of the next, and the edges $DC$ of all copies together form a regular $n$-sided polygon. Then the value of $n$ is

The sum of twenty positive integers, not necessarily different, is $462$. The largest possible value of greatest common divisor of these numbers is

The largest real number $m$ such that $(x^2 + y^2)^3 > m(x^3 + y^3)^2$ for any positive real numbers $x$ and $y$ is

Of the $49$ squares of a $7 \times 7$ square sheet of paper, two are to be coloured black while the others remain white. Two colourings are called distinct if one cannot be obtained from the other by rotating the sheet of paper about its centre. The number of distinct colourings is

The lengths of the sides of triangle $ABC$ are consecutive positive integers. $D$ is the midpoint of $BC$ and $AD$ is perpendicular to the bisector of $\angle C$. The product of the lengths of the three sides is

For any real number $r$, $\lfloor r \rfloor$ is the largest integer less than or equal to $r$. For example, $\lfloor \pi \rfloor = 3$. Let $n$ be a positive integer. Let $a_1 = n$, $a_2 = \bigl\lfloor \frac{a_1}{3} \bigr\rfloor$, $a_3 = \bigl\lfloor \frac{a_2}{3} \bigr\rfloor$ and $a_4 = \bigl\lfloor \frac{a_3}{3} \bigr\rfloor$. The number of positive integers $n$ from $1$ to $1000$ inclusive such that none of $a_1$, $a_2$, $a_3$ and $a_4$ is divisible by $3$ is