MathDB

1

configurational geometry as usual

\triangle ABC

with circumcircle

\ell

. Point

M

in

\triangle ABC

such that

AM

is the angle bisector of

\angle BAC

. Circle with center

M

and radius

MB

intersects

\ell

and

BC

at

D

and

E

respectively,

(B \not= D, B \not= E)

. Let

P

be the midpoint of arc

BC

in

\ell

that didn't have

A

. Prove that

AP

angle bisector of

\angle DPE

if and only if

\angle B = 90^{\circ}

.

8

1

Find the number of horizontal and vertical triminoes used to tile

On a

100 \times 100

chessboard, the plan is to place several

1 \times 3

boards and

3 \times 1

board, so that
[*] Each tile of the initial chessboard is covered by at most one small board. [*] The boards cover the entire chessboard tile, except for one tile. [*] The sides of the board are placed parallel to the chessboard.
Suppose that to carry out the instructions above, it takes

H

number of

1 \times 3

boards and

V

number of

3 \times 1

boards. Determine all possible pairs of

(H,V)

.
Proposed by Muhammad Afifurrahman, Indonesia

6

1

Produce same numbers on the board!

There are

n

natural numbers written on the board. Every move, we could erase

a,b

and change it to

\gcd(a,b)

and

\text{lcm}(a,b) - \gcd(a,b)

. Prove that in finite number of moves, all numbers in the board could be made to be equal.

5

1

simple polynomial -- prove an inequality

Let

P(x) = x^2 + rx + s

be a polynomial with real coefficients. Suppose

P(x)

has two distinct real roots, both of which are less than

-1

and the difference between the two is less than

2

. Prove that

P(P(x)) > 0

for all real

x

.

1

Add (+) or (-)?

On the whiteboard, the numbers are written sequentially:

1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9

. Andi has to paste a

+

(plus) sign or

-

(minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process.

2

1

An easy geometry about midpoints

Let

ABC

be an acute triangle. Let

D

and

E

be the midpoint of segment

AB

and

AC

respectively. Suppose

L_1

and

L_2

are circumcircle of triangle

ABC

and

ADE

respectively.

CD

intersects

L_1

and

L_2

at

M (M \not= C)

and

N (N \not= D)

. If

DM = DN

, prove that

\triangle ABC

is isosceles.

3

1

sequence of prime powers

A natural number is called a prime power if that number can be expressed as

p^n

for some prime

p

and natural number

n

. Determine the largest possible

n

such that there exists a sequence of prime powers

a_1, a_2, \dots, a_n

such that

a_i = a_{i - 1} + a_{i - 2}

for all

3 \le i \le n

.

4

1

"Can we use jensen for the test" The problem:

Let

x,y

and

z

be positive reals such that

x + y + z = 3

. Prove that

2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x}

2021 Indonesia MO

Subcontests

configurational geometry as usual

Find the number of horizontal and vertical triminoes used to tile

Produce same numbers on the board!

simple polynomial -- prove an inequality

Add (+) or (-)?

An easy geometry about midpoints

sequence of prime powers

"Can we use jensen for the test" The problem:

2021 Indonesia MO

Subcontests

configurational geometry as usual

Find the number of horizontal and vertical triminoes used to tile

Produce same numbers on the board!

simple polynomial -- prove an inequality

Add (+) or (-)?

An easy geometry about midpoints

sequence of prime powers

&quot;Can we use jensen for the test&quot; The problem:

"Can we use jensen for the test" The problem: