MathDB

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

E

. Prove that

\angle{AEC} = \angle{DGC}

if and only if

\angle{ACB} = 90^{\circ}

.
[url=https://artofproblemsolving.com/community/q2h2671443p23150906]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?

Problem Statement