2017/18 AHSMC Part 1

Peppa has twice as many apples as pears. After she eats $50$ pears, she now has $4$ times as many apples as pears. How many apples does Peppa have?

A math teacher is given a big bag of $300$ candies by a grateful parent. The teacher distributes as many of the candies as possible to her students so that each student gets the same number of candies. There are $14$ candies left over. How many students are there in her class, if each class in the school has at most $30$ students?

A vertical pole is $12$ m from a straight road. Mark is currently on the road $37$ m from the pole.

How many metres does Mark have to walk on the road before he is $20$ m from the pole?

Some $n \ge 3$ different positive real numbers are arranged on a circle such that each number is equal to the product of its two neighbours. The value of $n$ is

How many pairs $(n,p)$ are such that $n$ is a positive integer, $p$ is a prime number and $n + \dfrac{p}{n}$ is a square of a positive integer?

Ben and Cleo can paint a fence in four days. Anna and Ben can do it in two days, Anna and Cleo in three days. How many days, as a fraction, does it take all of them working if Cleo gets injured at the end of the first day and can't come back to work?

The number of ways to walk from $(0,0)$ to $(20,2)$ by using only up and right unit steps and such that the walk never visits the lines $y = x$ and $y = x - 18$, except at the beginning and end, is

Lan likes red or blue chopsticks, while Dan likes orange or green chopsticks. A drawer in a dark room contains $n$ chopsticks of each colour, where $n > 1$ is an integer. Lan goes into the room to get chopsticks for both her and Dan. What is the smallest number of these $4n$ chopsticks that Lan must pick, to be sure that both people will get a matched (i.e. same colour) pair that they like?

Let $x$, $y$, $z$ be any real numbers such that $3x + y + 2z \ge 3$ and $2y - x + 4z \ge 5$. The minimum possible value of $7x + 5y + 10z$ is

In a trapezoid $ABCD$, with $AD$ parallel to $BC$ the diagonals intersect at $O$. If the area of $\triangle AOD$ is $\dfrac{9}{16}$ of the area of the trapezoid then the ratio of the areas of triangles $AOD$ to $BOC$ is equal to

The area of the quadrilateral bounded by the graphs of the functions $y = |x - a|$, with $0 < a < 4$, and $y = 2 - |x - 2|$ is $\dfrac{15}{8}$. The smallest value of $a$ is

Let $a$, $b$ be positive integers. The number of quadratic equations $x^2 - ax - b = 0$ having the positive root less than $10$ is

Let $S$ be a subset of non-negative integers which contains $0$, and such that for any number $x$ in $S$, $3x$ and $3x + 1$ are also in $S$. The least possible number of elements of $S$ less than $2017$ is

Let $a$ be a real number so that the equation $x^4 - 2ax^2 - x - a + a^2 = 0$ has four different real solutions. Then which of the following must be true?

A positive integer is called a palindrome if it remains unchanged when written backwards. Find the number of $5$ digit palindromes which are divisible by $55$.

How many positive integers $n$ can be found such that the product of all divisors of $n$, including $n$, is $24^{240}$?