Found problems: 5
2020 Thailand Mathematical Olympiad, 2
There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$ takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?
2018 Thailand Mathematical Olympiad, 4
Let $a, b, c$ be nonzero real numbers such that $a + b + c = 0$. Determine the maximum possible value of
$\frac{a^2b^2c^2}{ (a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}$
.
2022 Thailand TSTST, 3
An acute scalene triangle $ABC$ with circumcircle $\Omega$ is given. The altitude from $B$ intersects side $AC$ at $B_1$ and circle $\Omega$ at $B_2$. The circle with diameter $B_1B_2$ intersects circle $\Omega$ again at $B_3$. Similarly, the altitude from $C$ intersects side $AB$ at $C_1$ and circle $\Omega$ at $C_2$. The circle with diameter $C_1C_2$ intersects circle $\Omega$ again at $C_3$. Let $X$ be the intersection of lines $B_1B_3$ and $C_1C_3$, and let $Y$ be the intersection of lines $B_3C$ and $C_3B$. Prove that line $XY$ bisects side $BC$.
2024 Thailand TST, 2
Let $ABC$ be triangle with incenter $I$ . Let $AI$ intersect $BC$ at $D$. Point $P,Q$ lies inside triangle $ABC$ such that $\angle BPA + \angle CQA = 180^\circ$ and $B,Q,I,P,C$ concyclic in order . $BP$ intersect $CQ$ at $X$.
Prove that the intersection of $(ABC)$ and $(APQ)$ lies on line $XD$.
2022 Thailand Online MO, 5
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $M_B$ and $M_C$ be the midpoints of $AC$ and $AB$, respectively. Place points $X$ and $Y$ on line $BC$ such that $\angle HM_BX = \angle HM_CY = 90^{\circ}$. Prove that triangles $OXY$ and $HBC$ are similar.