2022/23 AHSMC Part 1

Consider two identical beakers, beaker $A$ and beaker $B$. Beaker $B$ is full of liquid and beaker $A$ is empty. A scientist pours one third of the liquid from beaker $B$ into beaker $A$. She then pours one fourth of the liquid in beaker $A$ back into beaker $B$. Finally, she pours half of the liquid in beaker $B$ back into beaker $A$. After this process, what fraction of the liquid is in beaker $A$?

Suppose $r, s > 0$. A disk of radius $r$ cm is cut into nine pieces and a second disk of radius $(r+s)$ cm is cut into sixteen pieces. If each of the twenty-five pieces has equal area then $r/s$ is

The number of positive integers $n$ such that $5n$ has $4$ digits and $4n$ has $3$ digits is

For how many integers $n$ does the equation $$x^2 + nx + 10 = 0$$ have integer solutions?

A $3 \times 3$ table of zeros and ones is called $2$-avoiding if no row and no column has a sum of $2$. For example, the table on the left is $2$-avoiding (every row and column has a sum of $1$) but the table on the right is not (its first row has a sum of $2$). $$\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \qquad \begin{array}{ccc} 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}$$ The total number of $2$-avoiding tables with exactly $4$, $6$ or $8$ ones is

A square $ABCD$ with side $1$ is given. How many points $P$ in the plane of the square satisfy $$PA + PB + PC + PD = 2\sqrt{2}?$$

The faces of a die show distinct positive integers from one to six and the opposite sides sum to seven. You roll two dice and determine that the product of the two numbers on the top faces is equal to the sum of the two numbers on the bottom faces. What is the sum of the two numbers on the top faces?

Evan's living room has twice as much area as Victoria's, and three times as much area as Kyle's. Kyle mops floors half as fast as Victoria and one third as fast as Evan. If they all start mopping their floor at the same time, who will finish mopping first?

On a farm with several barns, there are pigs and ducks, altogether $40$ legs in each barn, in all possible combinations, and such that the total number of pigs and ducks in each barn is different. A random barn is chosen; what is the probability that the total number of pigs and ducks in that barn is a prime number?

Let $p(x) = 3x^4 + ax^2 + b$ where $a$, $b$ are real numbers. Suppose that $p(x)$ has exactly three distinct real roots $r$, $s$ and $t$ with $r < s < t$. Which one of the following statements must be true?

Let $N$ be the number of subsets of $\{-1, 1, 2, 3, 6, 9, 12, 15, 18, 21, 24, 26\}$ whose elements sum to $25$. For example, $\{-1, 2, 24\}$ is one such subset. Then $N$ is

What is the number of integers $m$, $1 \le m \le 300$ for which $m^m$ is a perfect cube?

In how many ways can we place six letters $A$, $A$, $B$, $B$, $C$, $C$ in the grid below, such that each square contains exactly one letter and no column contains $2$ identical letters?

The sum of $15$ positive integers, not necessarily distinct, is $425$. What is the smallest possible positive value of the least common multiple of these numbers?

Let $ABC$ be an isosceles right triangle with $AB = AC = 1$, and let $O$ be the centre of a circle drawn inside $ABC$ and tangent to all three of its sides. The line $BO$ intersects $AC$ at $D$. The length of the segment $DC$ is

Let $\mathbb{Z}_{\ge 0}$ be the set of non-negative integers and $f \colon \mathbb{Z}_{\ge 0} \to \mathbb{Z}_{\ge 0}$, $f(n) = \min |x+2y|$, where $x$, $y$ are integers such that $5x + 3y = n$. The value of the sum $f(0) + f(1) + \cdots + f(100)$ is