MathDB

2024 Indonesia Regional MO Short Answer Section There are 8 problems, time allowed is 60 minutes. Answers are always in integer form.
1. It is known that

\overline{ab}

and

\overline{cd}

are both two-digit numbers whose product is

777

. If

\overline{ab}<\overline{cd}

, find the value of

a+b

.
2. Let

f

and

g

be linear functions that satisfy the equation

f(x+g(y)) = 7x+2y+11 \text{ for every real number } x,y

g(7)=3

, find the value of

g(-11+f(4))

.
Note: A linear function is a function of the form

h(x)=ax+b

with real constants

a,b

. 3. Given a triangle

ABC

with side lengths

AB=15, AC=13, BC=4

. There exists an equilateral triangle

PQR

with

P,Q,\text{ and } R

lying on sides

BC,CA, \text{ and } AB

respectively such that

PQ

is parallel to

AB

.
The value

\dfrac{PQ}{AB}

can be expressed in the form

\dfrac{a }{b+c\sqrt{d} }

with

a,b,c,d

such that

a

is a positive integer,

d

is squarefree, and

\text{GCD}(a,b,c)=1

. Find value of

a+b+c+d

. 4. Each tile on a board of size

2023 \times 3

will be colored either black or white, such that each

2\times 2

sub-board has an odd number of black tiles and an odd number of white tiles.
Suppose the number of possible ways of such coloring is

A

. Find the remainder of

A

when divided by

1000

. 5. Find the number of positive integers

a<209

such that

\text{GCD}(a,209)=1

and

a^2-1

is not a multiple of

209

.
6. In a square

ABCD

with side length

\sqrt{2}+\sqrt{6}

X

lies on the diagonal

AC

such that

AX>XC

. The internal bisector of angle

AXB

intersects side

AB

U

. The internal bisector of angle

CXD

intersects side

CD

V

. If

\angle UXV = 150^{\circ}

, find the value of

\lfloor 3 \times UV^2 \rfloor

.
Note: the notation

\lfloor x \rfloor

represents the largest integer that is less than or equal to

x

.
7. Given the set

S = \{1,2,\ldots,18\}

. Let

N

be the number of ordered pairs

(A,B)

of subsets

A,B\subseteq S

such that

| A \cap B | \le 2

. Find the value of

\dfrac{N}{3^{16} }

.
Note:

|X|

is defined as the number of elements in the set

X

. 8. Let

a,b,c

be real numbers that satisfy the inequality:

|ax^2+bx+c|\le (18x-5)^2 \text{ for all real numbers } x

Find the smallest possible value of

a+2b+5c

Problem Statement