MathDB

Indonesia Regional MO (more commonly known as the provincial level) is the selection for qualifying for the Indonesian MO (National Level) annually. It consists of some structured answers section (short answers) and 5 essay problems. Each short answer problem is worth 1 point, whereas each each essay problem is worth 7 points. The system varies often these days, with the 2021 test (held on 13 September 2021) being 10 essay problems, split into 2 tests.
Here I will only be posting the essay problems.
Indonesian Regional MO 2002 (Olimpiade Sains Nasional Tingkat Provinsi 2002) Problem 1. Five distinct natural numbers,

k, l, m, n

and

p

, will be chosen. The following five information turns out to be sufficient to order the five natural numbers (in ascending, or descending order). (i) Among any two numbers, one of the numbers must divide another, (ii) The integer

m

is either the largest or the smallest, (iii)

p

can't divide

m

and

k

simultaneously, (iv)

n \leq l - p

, and (v)

k

divides

n

p

divides

n

, but not both at the same time. Determine all possible orderings of

k, l, m, n

and

p

.
Problem 2. Determine all positive integers

p

so that

\frac{3p+25}{2p-5}

is also a positive integer.
Problem 3. Given a 6-digit number, prove that the six digits can be rearranged in such a way so that the (absolute) difference between the sum of the first and last three digits is no more than 9.
Problem 4. It is known the equilateral triangle

ABC

and a point

P

so that the distance between

P

A

and

C

is no more than the distance between

P

B

. Prove that

PB = PA + PC

if and only if

P

lies on the circumcircle of

\triangle{ABC}

.
Problem 5. Consider the tile with the shape of a T-tetromino. Each tile of the tetromino tiles exactly one cell of a checkerboard. We want to cover the checkerboard with tetrominoes so that each tile of the tetromino covers exactly one checkerboard cell, with no overlaps. (a) Prove that we can tile an

8 \times 8

checkerboard, with 16 T-tetrominoes. (b) Prove that we cannot tile a

10 \times 10

checkerboard with 25 T-tetrominoes.
(A tetromino is a tile with 4 cells. A T-tetromino is a tetromino with the shape "T".)

Problem Statement