2024 Indonesia MO

Part of Indonesia MO

Subcontests

(8)

1

Self-bounded Polynomial Inequalities

P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0

a_0, a_1, \ldots, a_{n-1}

n\geq 1

n

th-degree polynomial with real coefficients). If the inequality

3(P(x)+P(y)) \geq P(x+y)

holds for all reals

x,y

, determine the minimum possible value of

P(2024)

1

Geometric Angle Inequality

A_1 A_2 \ldots A_n

n

-sided polygon with

n \geq 3

\angle A_j \leq 180^{\circ}

j

(in other words, the polygon is convex or has fewer than

n

distinct sides).
For each

i \leq n

\alpha_i

is the smallest possible value of

\angle{A_i A_j A_{i+1}}

j

i

i+1

. (Here, we define

A_{n+1} = A_1

\alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ}

and determine all equality cases.

1

Divides lcm of any n pairs

n \ge 2

be a positive integer. Suppose

a_1, a_2, \dots, a_n

are distinct integers. For

k = 1, 2, \dots, n

s_k := \prod_{\substack{i \not= k, \\ 1 \le i \le n}} |a_k - a_i|,

s_k

is the product of all terms of the form

|a_k - a_i|

i \in \{ 1, 2, \dots, n \}

i \not= k

. Find the largest positive integer

M

M

divides the least common multiple of

s_1, s_2, \dots, s_n

for any choices of

a_1, a_2, \dots, a_n

1

Colorful Integers

Each integer is colored with exactly one of the following colors: red, blue, or orange, and all three colors are used in the coloring. The coloring also satisfies the following properties:
1. The sum of a red number and an orange number results in a blue-colored number, 2. The sum of an orange and blue number results in an orange-colored number; 3. The sum of a blue number and a red number results in a red-colored number.
(a) Prove that

0

1

must have distinct colors. (b) Determine all possible colorings of the integers which also satisfy the properties stated above.

1

Algebruh to Start the Day (Easy)

Determine all positive real solutions

(a,b)

to the following system of equations. \begin{align*} \sqrt{a} + \sqrt{b} &= 6 \\ \sqrt{a-5} + \sqrt{b-5} &= 4 \end{align*}

1

Indonesian Geometry Olympiad

ABC

O

as its circumcenter, and

H

as its orthocenter. The line

AH

BH

intersect the circumcircle of

ABC

for the second time at points

D

E

, respectively. Let

A'

B'

be the circumcenters of triangle

AHE

BHD

respectively. If

A', B', O, H

are not collinear, prove that

OH

intersects the midpoint of segment

A'B'

1

It is Just a Numbers Game

Kobar and Borah are playing on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board, either from

P

Q

2P-Q

2Q-P

P

Q

5P-4Q

5Q-4P

. The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner.
At the beginning of the game, the two numbers on the board are

2024

A

. If it is known that Kobar does not lose on his first move, determine the largest possible value of

A

so that Borah can win this game.

1

Fatal Numbers

The triplet of positive integers

(a,b,c)

a<b<c

is called a fatal triplet if there exist three nonzero integers

p,q,r

which satisfy the equation

a^p b^q c^r = 1

. As an example,

(2,3,12)

is a fatal triplet since

2^2 \cdot 3^1 \cdot (12)^{-1} = 1

. The positive integer

N

is called fatal if there exists a fatal triplet

(a,b,c)

N=a+b+c

. (a) Prove that 16 is not fatal. (b) Prove that all integers bigger than 16 which are not an integer multiple of 6 are fatal.