The 2023 Indonesia Regional MO was held on 5th of June 2023. There are 8 short form problems and 5 essays. The 8 short form problems all have nonnegative integer answers. The time is 60 minutes. 1. Given two non constant arithmetic sequences $a_1,a_2,\ldots$ dan $b_1,b_2,\ldots$. If $a_{228} = b_{15}$ and $a_8 = b_5$, find $\dfrac{b_4-b_3}{a_2-a_1}$. 2. Find the number of positive integers $n\leq 221$ such that \[ \frac{1^2+2^2+\ldots+n^2}{1+2+\ldots+n}\]is an integer. 3. How many different lines on a cartesian coordinate system has equation $ax+by=0$ with $a,b \in \{0,1,2,3,6,7\}$ ? Note: $a$ and $b$ are not necessarily distinct 4. Given a rectangle $ABCD$ and equilateral triangles $BCP$ and $CDQ$ in the figure below. If $AB = 8$ and $AD = 10$, the sum of the area $ACP$ and $ACQ$ is $m\sqrt{3}+n$, for rational numbers $m,n$. Find $m+n$. 5. Find the number of three element subsets of $S = \{1,5,6,7,9,10,11,13,15,20,27,45 \}$ such that the multiplication of those three elements is divisible by $18$. 6. In the figure below, if $AP=22$, $CQ=14$, $RE=35$, with $PQR$ being an equilateral triangle, find the value of $BP+QD+RF$. 7. Find the sum of all positive integers $n$ such that $\sqrt{2n-12}+\sqrt{2n+40}$ is a positive integer. 8. Given positive real numbers $a$ and $b$ that satisfies \begin{align*} \frac{1}{a}+\frac{1}{b} &\leq 2\sqrt{\frac{3}{7}}\\ (a-b)^2 &= \frac{9}{49}(ab)^3 \end{align*}Find the maximum value of $a^2+b^2$.