This is the continuation of my previous post, i.e. part 2 of the same Mathematics Olympiad/Competition (Indonesia recently changed its name since 2020's competition). Each problem is worth 7 points and the same rules apply. Part 2 (Olympiad Round: 150 minutes) Problem 6. Suta writes 2021 of the first positive integers on a board, such that every number is written exactly once. She then circles some of them, then sums up all the numbers she's circled to get the value $K$. Then, Suta also adds up all the numbers she didn't circle to obtain that their sum is equal to $L$. Show that Suta is able to circle some numbers in the beginning, such that $K - L = 2021$. Problem 7. Determine all natural numbers $n > 3$ such that $\lfloor \sqrt{n} \rfloor - 1$ divides $n + 1$ and $\lfloor \sqrt{n} \rfloor + 1$ divides $n - 1$. Problem 8. Given a triangle $ABC$ with $G$ as its centroid. Point $D$ is the midpoint of $AC$. The line passing through $G$ and parallel to $BC$ cuts $AB$ at $E$. Prove that $\angle{AEC} = \angle{DGC}$ if and only if $\angle{ACB} = 90^{\circ}$. Problem 9. Let $X$ be the set containing rational positive numbers satisfying both criteria: (i) If $x$ is rational and $2021 \leq x \leq 2022$ then $x \in X$. (ii) If $x, y \in X$, then $\frac{x}{y}$ is also an element of $X$. Prove that all positive rational numbers are in $X$. Problem 10. Five unit squares from a $9 \times 9$ checkerboard are discarded as shown in the figure below (as an attachment for this post). The entire checkerboard will be covered with domino cards so that each domino covers exactly 2 unit squares, and every unit square is covered by exactly 1 domino. Can we tile the checkerboard with dominoes in such a way that every inner vertical and horizontal line (which are not coloured red) cuts at least 2 dominoes?

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