MathDB

The 2023 Indonesia Regional MO was held on 5th of June 2023. There are 8 short form problems and 5 essays.
The 8 short form problems all have nonnegative integer answers. The time is 60 minutes.
1. Given two non constant arithmetic sequences

a_1,a_2,\ldots

dan

b_1,b_2,\ldots

. If

a_{228} = b_{15}

and

a_8 = b_5

, find

\dfrac{b_4-b_3}{a_2-a_1}

. 2. Find the number of positive integers

n\leq 221

such that

\frac{1^2+2^2+\ldots+n^2}{1+2+\ldots+n}

is an integer.
3. How many different lines on a cartesian coordinate system has equation

ax+by=0

with

a,b \in \{0,1,2,3,6,7\}

? Note:

a

and

b

are not necessarily distinct 4. Given a rectangle

ABCD

and equilateral triangles

BCP

and

CDQ

in the figure below. If

AB = 8

and

AD = 10

, the sum of the area

ACP

and

ACQ

m\sqrt{3}+n

, for rational numbers

m,n

. Find

m+n

. 5. Find the number of three element subsets of

S = \{1,5,6,7,9,10,11,13,15,20,27,45 \}

such that the multiplication of those three elements is divisible by

18

.
6. In the figure below, if

AP=22

CQ=14

RE=35

, with

PQR

being an equilateral triangle, find the value of

BP+QD+RF

. 7. Find the sum of all positive integers

n

such that

\sqrt{2n-12}+\sqrt{2n+40}

is a positive integer.
8. Given positive real numbers

a

and

b

that satisfies \begin{align*} \frac{1}{a}+\frac{1}{b} &\leq 2\sqrt{\frac{3}{7}}\\ (a-b)^2 &= \frac{9}{49}(ab)^3 \end{align*} Find the maximum value of

a^2+b^2

Problem Statement