2018/19 AHSMC Part 1

The average of $1$, $a$ and $b$ is equal to $2c$. The average of $1$, $a$ and $c$ is equal to $2b$. The average of $1$, $b$ and $c$ is equal to $2a$. The average of $a$, $b$ and $c$ is equal to

Let $a, b > 0$ and suppose the line $y = b - ax$ has an $x$-intercept of $2$. If the triangle enclosed by the lines $x = 0$, $y = 0$ and $y = b - ax$ has area equal to $4$, the value of $a$ must be

Let $f$ be a function such that $f\bigl(\dfrac{x+1}{3}\bigr) = x^2 + x + 3$ for any real number $x$. The sum of all real numbers $a$ such that $f\bigl(\dfrac{a}{9}\bigr) = 10$ is

A sequence is defined by $a_0 = 0$ and $a_{n+1} = \dfrac{a_n - 3}{a_n - 2}$ for all integers $n \ge 0$. Then $a_{2018}$ is

Pete has $\$x$ in his pocket. If he buys two apples using his $\$x$ he will have $\$3$ in change remaining. If he buys three bananas using his $\$x$ he will have $\$2$ in change remaining. Similarly, if he buys four apples and one banana using his $\$x$ he will also have $\$2$ in change remaining. What is the maximum number of bananas Pete can afford to buy?

A student writes a multiple choice exam. The exam contains $16$ questions and the score is $5$ points for a correct answer, $2$ points for no answer and $0$ points for an incorrect answer. Each student also receives $20$ bonus points. If a student scores $78$ points, how many questions were answered correctly?

How many triplets $(m,n,k)$, of integers greater than one, exist such that $(m!) \cdot (n!) = (k!)$? (Notice that $n! = 1 \times 2 \times 3 \times \cdots \times n$.)

The sum of all positive integers smaller than $126$ which are not divisible by $2$ and not divisible by $3$ is

Let $k$ be a fixed positive real number. Suppose $kx^2 \le 10$ has exactly five positive integer solutions. The number of positive integer solutions to $kx \le 1$ must be

Consider polynomials $p(x)$ of degree $5$ with leading coefficient $1$ such that $p(-x) = -p(x)$ for all real $x$. If the roots of $p(x)$ include $-1$ and $\sqrt{2}$, the value of $p(2)$ is

A set $S$ of five different real numbers is given. When each pair of these numbers are added, you get the ten sums $-4,-3,-2,-1,0,1,4,5,6,8$. The ratio of the largest number in $S$ to the smallest number in $S$ is

Let $a_n = n^2 + 2n + 50$, $n = 1,2,\ldots$. Let $d_n$ be the largest positive integer that is a divisor of both $a_n$ and $a_{n+1}$. The maximum value of $d_n$, $n = 1,2,\ldots$ is

There are seven children belonging to three families. The first family has three children, and the other two families have two children each. In how many ways can these seven children be arranged in a circle so that no two children from the same family are next to each other? Notice that seating arrangements that differ by a rotation are considered the same.

The positive integers from $1$ to $999$ are written in a row (with no spaces) in increasing order: $$12345\cdots 998999.$$ How many times does the sequence of characters “$31$” occur?

In triangle $ABC$, $D$ is a point on $BC$ such that $\angle BAD = 30^\circ$ and $\angle DAC = 15^\circ$. If $AB = 3\sqrt{2}$ and $AC = 6$, the length of $AD$ is

Let $k \ge 1$ be an integer and $n_k = \left\lfloor \dfrac{2^k + 20182019}{2^{k+1}} \right\rfloor$ where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$. The value of the sum $n_1 + n_2 + n_3 + \cdots$ is