2014/15 AHSMC Part 1

When the repeating decimal $0.\overline{6}$ is divided by the repeating decimal $0.\overline{3}$, the quotient is

The number of two-digit positive integers such that the sum of the two digits is $12$ is

Let $a > b > 0$ be prime numbers. Of the following five numbers, the one which cannot be equal to $a - b$ is

Chau and Matt are picking berries. On Monday Matt picks twice as many kilograms of berries as Chau does, and on Tuesday Chau picks twice as many kilograms of berries as Matt does. Between them, they pick a total of $30$ kilograms of berries over the two days. The number of kilograms of berries Chau picks over the two days is

For every set of five of the numbers $1$, $2$, …, $2014$, Lac writes down the smallest of the five numbers. The largest number she writes down is

The sum of four consecutive integers is $n$. The largest of these four numbers is

Let $f(n) = 2n^3$ for any positive integer $n$. For any odd number $m \ge 3$, the largest positive integer $k$ such that $2^k$ divides $f(f(f(m)))$ is

Let $f$ be a function such that $f(x) + 3f\bigl(\dfrac{1}{x}\bigr) = x^2$ for any nonzero real number $x$. Then the value of $f(-3)$ is

The number of integer pairs $(m, n)$ such that $mn = m + n$ is

The number of five-digit positive integers such that the digits are alternately odd and even (either odd-even-odd-even-odd or even-odd-even-odd-even) is

Let $a$, $b$ and $c$ be real numbers such that $a + b + c = 5$ and $\dfrac{1}{b+c} + \dfrac{1}{c+a} + \dfrac{1}{a+b} = 6$. The value of $\dfrac{a}{b+c} + \dfrac{b}{c+a} + \dfrac{c}{a+b}$ is

$P$ is a point inside an acute triangle $ABC$. $D$, $E$ and $F$ are the feet of the perpendiculars from $P$ on $BC$, $CA$ and $AB$ respectively. If $BD = 1$, $DC = 10$, $CE = 6$, $EA = 9$ and $AF = 13$, the length of $FB$ is

Three candles which can burn for $30$, $40$ and $50$ minutes respectively are lit at different times. All three candles are burning simultaneously for $10$ minutes, and there is a total of $20$ minutes in which exactly one of them is burning. The number of minutes in which exactly two of them are burning is

$P$ is a point inside a convex quadrilateral $ABCD$ of area $168$ such that $PA = 9$, $PB = PD = 12$ and $PC = 5$. The perimeter of the quadrilateral is

For any real numbers $x$ and $y$, the minimum value of $x^4 - 5x^2 + y^2 + 2x + 2y + 2xy + 6$ is

Four cages are arranged in a $2 \times 2$ formation. Each cage contains some chickens and some rabbits (some cages may contain only chickens or only rabbits). The total number of heads in the two cages in the first row is $60$. The total number of legs in the two cages in the second row is $240$. The total number of heads in the two cages in the first column is $70$. The total number of legs in the two cages in the second column is $230$. The minimum number of animals in all four cages is