Australian Intermediate Mathematics Olympiad 2023

Positive integers $a,b$ satisfy $(a+b)(a-b)=2023$ and $a/b+b/a=25/12$. Find $a$.

Three squares lie inside a right-angled triangle as shown. The smallest and largest squares have side lengths $28$ and $63$. Find the side length of the middle square.

The ten pairwise sums of five distinct integers are $0,1,2,4,7,8,9,10,11,12$. Find the sum of the five integers.

Given that there is only one pair of numbers $\{a,b\}$ such that $16a^ab^b=81a^bb^a$, find $a+b$.

Arrange $1,2,3,4,5,6$ in a line as the base of a number pyramid. Each number above the base is the sum of the adjacent pair directly below it. The example shown has base $3,6,1,5,4,2$. How many base arrangements produce a top number divisible by $5$?

Two sports teams, the Tigers and the Lions, each have $10$ members. Each team’s members are numbered $1$ to $10$ in increasing order of height. Form $10$ Tiger–Lion pairs, using everyone once. A pair is allowed if and only if the two numbers differ by at most $1$; for example Tiger $4$ can pair with Lions $3,4,5$. How many allowed pairings are there?

The decimal $1/137=0.00729927\,00729927\ldots$ has shortest repeating period $8$. What is the smallest positive integer $n$ such that $1/n$ has shortest repeating period $8$?

The digits $0,1,\ldots,9$ are coded bijectively by letters $A$ to $J$. The integers $ABACDE$, $CAFDG$, $CHHBAED$ are side lengths of a triangle, with $A,C\ne0$. Find the three-digit number coded by $DEG$.

Let $n$ be the number of real ordered pairs $(x,y)$ solving
$$x^3-y^3=7(x-y),\qquad x^3+y^3=5(x+y).$$
Find the sum of the corresponding $n$ values of $x^2+y^2$, counting repeated values for different pairs.

Circles $C_1,C_2$ are tangent to the real number line and externally tangent to each other. Circle $C_1$ touches the line at $1/3$ and has diameter $1/9$; $C_2$ touches it at $3/8$. A third circle $C_3$ is tangent to $C_1,C_2$ and the real line at $x\lt 3/8$. Find $x$ and the radius of $C_3$.
Investigation (4 bonus marks, separate from the 5 base marks): starting with these $C_1,C_2$, define $C_n$ for $n\ge3$ to be tangent to $C_1,C_{n-1}$ and the real line at $x_n\lt x_{n-1}$. For $n\ge4$, find formulas for $x_n$ and its radius $r_n$.