MathDB

1

Excircle tangency points lie on the circumcircle

I, O

be the incenter and circumcenter of the triangle

ABC

respectively. Let the excircle

\omega_A

of

ABC

be tangent to the side

BC

on

N

, and tangent to the extensions of the sides

AB, AC

on

K, M

respectively. If the midpoint of

KM

lies on the circumcircle of

ABC

, prove that

O, I, N

are collinear.

7

1

Not your usual subtraction game

Suppose there are three empty buckets and

n \ge 3

marbles. Ani and Budi play a game. For the first turn, Ani distributes all the marbles into the buckets so that each bucket has at least one marble. Then Budi and Ani alternate turns, starting with Budi. On a turn, the current player may choose a bucket and take 1, 2, or 3 marbles from it. The player that takes the last marble wins. Find all

n

such that Ani has a winning strategy, including what Ani's first move (distributing the marbles) should be for these

n

.

6

1

2^n*p^2+1 is a square

Find all prime numbers

p

such that there exists a positive integer

n

where

2^n p^2 + 1

is a square number.

5

1

A bit of a minimum, more of a maximum

Find all triples of reals

(x,y,z)

satisfying:

\begin{cases} \frac{1}{3} \min \{x,y\} + \frac{2}{3} \max \{x,y\} = 2017 \\ \frac{1}{3} \min \{y,z\} + \frac{2}{3} \max \{y,z\} = 2018 \\ \frac{1}{3} \min \{z,x\} + \frac{2}{3} \max \{z,x\} = 2019 \\ \end{cases}

4

1

Real game with a computer

In a game, Andi and a computer take turns. At the beginning, the computer shows a polynomial

x^2 + mx + n

where

m,n \in \mathbb{Z}

, such that it doesn't have real roots. Andi then begins the game. On his turn, Andi may change a polynomial in the form

x^2 + ax + b

into either

x^2 + (a+b)x + b

or

x^2 + ax + (a+b)

. However, Andi may only choose a polynomial that has real roots. On the computer's turn, it simply switches the coefficient of

x

and the constant of the polynomial. Andi loses if he can't continue to play. Find all

(m,n)

such that Andi always loses (in finitely many turns).

3

1

Guarantee a red equilateral triangle

Alzim and Badril are playing a game on a hexagonal lattice grid with 37 points (4 points a side), all of them uncolored. On his turn, Alzim colors one uncolored point with the color red, and Badril colors two uncolored points with the color blue. The game ends either when there is an equilateral triangle whose vertices are all red, or all points are colored. If the former happens, then Alzim wins, otherwise Badril wins. If Alzim starts the game, does Alzim have a strategy to guarantee victory?

2

1

Two touching circles

Let

\Gamma_1, \Gamma_2

be circles that touch at a point

A

, and

\Gamma_2

is inside

\Gamma_1

. Let

B

be on

\Gamma_2

, and let

AB

intersect

\Gamma_1

on

C

. Let

D

be on

\Gamma_1

and

P

be on the line

CD

(may be outside of the segment

CD

).

BP

intersects

\Gamma_2

at

Q

. Prove that

A,D,P,Q

lie on a circle.

1

2n+1 and an+1 are relatively prime

Let

a

be a positive integer such that

\gcd(an+1, 2n+1) = 1

for all integer

n

.
a) Prove that

\gcd(a-2, 2n+1) = 1

for all integer

n

. b) Find all possible

a

.

2018 Indonesia MO

Subcontests

Excircle tangency points lie on the circumcircle

Not your usual subtraction game

2^n*p^2+1 is a square

A bit of a minimum, more of a maximum

Real game with a computer

Guarantee a red equilateral triangle

Two touching circles

2n+1 and an+1 are relatively prime