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 $a^2+b^2$.
2023 Indonesia Regional
Each correct answer gets 2 points, blank or incorrect answers gets 0 points. The answer is always an integer. Time allowed is 60 minutes. - Short Answer
Each problem is worth a maximum of 7 points, time allowed is 150 minutes. - Essay
Let $ABCD$ be a square with side length $43$ and points $X$ and $Y$ lies on sides $AD$ and $BC$ respectively such that the ratio of the area of $ABYX$ to the area of $CDXY$ is $20 : 23$ . Find the maximum possible length of $XY$.
Let $K$ be a positive integer such that there exist a triple of positive integers $(x,y,z)$ such that \[x^3+Ky , y^3 + Kz, \text{and } z^3 + Kx\]are all perfect cubes. (a) Prove that $K \ne 2$ and $K \ne 4$ (b) Find the minimum value of $K$ that satisfies.
Find the maximum value of an integer $B$ such that for every 9 distinct natural number with the sum of $2023$, there must exist a sum of 4 of the number that is greater than or equal to $B$
Find all irrational real numbers $\alpha$ such that \[ \alpha^3 - 15 \alpha \text{ and } \alpha^4 - 56 \alpha \]are both rational numbers.
Given $\triangle ABC$ and points $D$ and $E$ at the line $BC$, furthermore there are points $X$ and $Y$ inside $\triangle ABC$. Let $P$ be the intersection of line $AD$ and $XE$, and $Q$ be the intersection of line $AE$ and $YD$. If there exist a circle that passes through $X, Y, D, E$, and $$\angle BXE + \angle BCA = \angle CYD + \angle CBA = 180^{\circ}$$Prove that the line $BP$, $CQ$, and the perpendicular bisector of $BC$ intersect at one point.