2024 Indonesia Regional MO Short Answer Section There are 8 problems, time allowed is 60 minutes. Answers are always in integer form. 1. It is known that $\overline{ab}$ and $\overline{cd}$ are both two-digit numbers whose product is $777$. If $\overline{ab}<\overline{cd}$, find the value of $a+b$. 2. Let $f$ and $g$ be linear functions that satisfy the equation \[f(x+g(y)) = 7x+2y+11 \text{ for every real number } x,y \]If $g(7)=3$, find the value of $ g(-11+f(4)) $. Note: A linear function is a function of the form $h(x)=ax+b$ with real constants $a,b$. 3. Given a triangle $ABC$ with side lengths $AB=15, AC=13, BC=4$. There exists an equilateral triangle $PQR$ with $P,Q,\text{ and } R$ lying on sides $BC,CA, \text{ and } AB$ respectively such that $PQ$ is parallel to $AB$. The value $\dfrac{PQ}{AB} $ can be expressed in the form $\dfrac{a }{b+c\sqrt{d} }$ with $a,b,c,d$ such that $a$ is a positive integer, $d$ is squarefree, and $\text{GCD}(a,b,c)=1 $. Find value of $a+b+c+d$. 4. Each tile on a board of size $2023 \times 3$ will be colored either black or white, such that each $2\times 2$ sub-board has an odd number of black tiles and an odd number of white tiles. Suppose the number of possible ways of such coloring is $A$. Find the remainder of $A$ when divided by $1000$. 5. Find the number of positive integers $a<209$ such that $\text{GCD}(a,209)=1 $ and $a^2-1$ is not a multiple of $209$. 6. In a square $ABCD$ with side length $\sqrt{2}+\sqrt{6}$, $X$ lies on the diagonal $AC$ such that $AX>XC$. The internal bisector of angle $AXB$ intersects side $AB$ at $U$. The internal bisector of angle $CXD$ intersects side $CD$ at $V$. If $\angle UXV = 150^{\circ} $, find the value of $\lfloor 3 \times UV^2 \rfloor $. Note: the notation $\lfloor x \rfloor $ represents the largest integer that is less than or equal to $x$. 7. Given the set $S = \{1,2,\ldots,18\} $. Let $N$ be the number of ordered pairs $(A,B)$ of subsets $A,B\subseteq S$ such that $| A \cap B | \le 2 $. Find the value of $\dfrac{N}{3^{16} }$. Note: $|X|$ is defined as the number of elements in the set $X$. 8. Let $a,b,c$ be real numbers that satisfy the inequality: \[ |ax^2+bx+c|\le (18x-5)^2 \text{ for all real numbers } x \]Find the smallest possible value of $a+2b+5c$ .

Given a real number $C\leqslant 2$. Prove that for every positive real number $x,y$ with $xy=1$, the following inequality holds: \[ \sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}.\] Proposed by Fajar Yuliawan, Indonesia

Given an $n \times n$ board which is divided into $n^2$ squares of size $1 \times 1$, all of which are white. Then, Aqua selects several squares from this board and colors them black. Ruby then places exactly one $1\times 2$ domino on the board, so that the domino covers exactly two squares on the board. Ruby can rotate the domino into a $2\times 1$ domino. After Aqua colors, it turns out there are exactly $2024$ ways for Ruby to place a domino on the board so that it covers exactly $1$ black square and $1$ white square. Determine the smallest possible value of $n$ so that Aqua and Ruby can do this. Proposed by Muhammad Afifurrahman, Indonesia

Given a triangle $ABC$, points $X,Y,$ and $Z$ are the midpoints of $BC,CA,$ and $AB$ respectively. The perpendicular bisector of $AB$ intersects line $XY$ and line $AC$ at $Z_1$ and $Z_2$ respectively. The perpendicular bisector of $AC$ intersects line $XZ$ and line $AB$ at $Y_1$ and $Y_2$ respectively. Let $K$ be a point such that $KZ_1 = KZ_2$ and $KY_1 = KY_2$. Prove that $KB=KC$.

Find the number of positive integer pairs $1\leqslant a,b \leqslant 2027$ that satisfy \[ 2027 \mid a^6+b^5+b^2.\](Note: For integers $a$ and $b$, the notation $a \mid b$ means that there is an integer $c$ such that $ac=b$.) Proposed by Valentio Iverson, Indonesia