MathDB

1

dividing a set into 2 sets with same sum each

2n

natural numbers, so that the average arithmetic of those

2n

number is

2

. If all the number is not more than

2n

. Prove we can divide those

2n

numbers into

2

sets, so that the sum of each set to be the same.

C6

1

fleas on the real line jumping over integers

Let

k

be a fixed natural number. In the infinite number of real line, each integer is colored with color ..., red, green, blue, red, green, blue, ... and so on. A number of flea settles at first at integer points. On each turn, a flea will jump over the other tick so that the distance

k

is the original distance. Formally, we may choose

2

tails

A, B

that are spaced

n

and move

A

to the different side of

B

so the current distance is

kn

. Some fleas may occupy the same point because we consider the size of fleas very small. Determine all the values of

k

so that, whatever the initial position of the ticks, we always get a position where all ticks land on the same color.

G2

1

externally tangent circles, perpendicular chords, collinearity wanted

Two circles that are not equal are tangent externally at point

R

. Suppose point

P

is the intersection of the external common tangents of the two circles. Let

A

and

B

are two points on different circles so that

RA

is perpendicular to

RB

. Show that the line

AB

passes through

P

.

G4

1

equal segments wanted, related to an isosceles triangle and 2 circumcircles

Given an isosceles triangle

ABC

with

AB = AC

, suppose

D

is the midpoint of the

AC

. The circumcircle of the

DBC

triangle intersects the altitude from

A

at point

E

inside the triangle

ABC

, and the circumcircle of the triangle

AEB

cuts the side

BD

at point

F

. If

CF

cuts

AE

at point

G

, prove that

AE = EG

.

G3

1

from 3 points of circumcircle, two tangents to incircle, means one more

Given

ABC

triangle with incircle

L_1

and circumcircle

L_2

. If points

X, Y, Z

lie on

L_2

, such that

XY, XZ

are tangent to

L_1

, then prove that

YZ

is also tangent to

L_1

.

G5

1

if A, P, Q are collinear, then AB = AC , two circles related

Let

ABC

be an acute triangle. Suppose that circle

\Gamma_1

has it's center on the side

AC

and is tangent to the sides

AB

and

BC

, and circle

\Gamma_2

has it's center on the side

AB

and is tangent to the sides

AC

and

BC

. The circles

\Gamma_1

and

\Gamma_2

intersect at two points

P

and

Q

. Show that if

A, P, Q

are collinear, then

AB = AC

.

C7

1

subset of {1,2, 3,... , 2014}, with 12 elements, with numbers of sum = 2015

Show that there is a subset of

A

from

\{1,2, 3,... , 2014\}

such that : (i)

|A| = 12

(ii) for each coloring number in

A

with red or white , we can always find some numbers colored in

A

whose sum is

2015

.

C5

1

max number of ways to choose representatives to speak

A meeting was attended by

n

people. They are welcome to occupy the

k

table provided

\left( k \le \frac{n}{2} \right)

. Each table is occupied by at least two people. When the meeting begins, the moderator selects two people from each table as representatives for talk to. Suppose that

A

is the number of ways to choose representatives to speak. Determine the maximum value of

A

that is possible.

C3

1

2015 marbles in a box, 3 colors at start, one color in the end

We have

2015

marbles in a box, where each marble has one color from red, green or blue. At each step, we are allowed to take

2

different colored marbles, then replace it with

2

marbles with the third color. For example, we take one blue marble and one green marble, and we fill with

2

red marbles. Prove that we can always do a series of steps so that all marbles in the box have the same color.

C1

1

2015 elephants on a chessboard of dimensions 2x 2015 without threats

Given natural number n. Suppose that

N

is the maximum number of elephants that can be placed on a chessboard measuring

2 \times n

so that no two elephants are mutually under attack. Determine the number of ways to put

N

elephants on a chessboard sized

2 \times n

so that no two elephants attack each other.
Alternative Formulation: Determine the number of ways to put

2015

elephants on a chessboard measuring

2 \times 2015

so there are no two elephants attacking each othe
PS. Elephant = Bishop

G8

1

circumcircle tangent to given circle

ABC

is an acute triangle with

AB> AC

.

\Gamma_B

is a circle that passes through

A,B

and is tangent to

AC

on

A

. Define similar for

\Gamma_C

. Let

D

be the intersection

\Gamma_B

and

\Gamma_C

and

M

be the midpoint of

BC

.

AM

cuts

\Gamma_C

at

E

. Let

O

be the center of the circumscibed circle of the triangle

ABC

. Prove that the circumscibed circle of the triangle

ODE

is tangent to

\Gamma_B

.

N8

1

infinite n such that exists naturals a,b: a + b = n and ab | n^2 + n + 1

The natural number

n

is said to be good if there are natural numbers

a

and

b

that satisfy

a + b = n

and

ab | n^2 + n + 1

. (a) Show that there are infinitely many good numbers. (b) Show that if

n

is a good number, then

7 \nmid n

.

N6

1

prove that each beautiful set must be charming (in non negative integers)

Defined as

N_0

as the set of all non-negative integers. Set

S \subset N_0

with not so many elements is called beautiful if for every

a, b \in S

with

a \ge b

(

a

and

b

do not have to be different), exactly one of

a + b

or

a - b

is in

S

. Set

T \subset N_0

with not so many elements is called charming if the largest number

k

such that up to 3

^k | a

is the same for each element

a \in T

. Prove that each beautiful set must be charming.

N5

1

ax+by=n, odd no. of non-negative integer solutions (x, y)

Given a prime number

n \ge 5

. Prove that for any natural number

a \le \frac{n}{2}

, we can search for natural number

b \le \frac{n}{2}

so the number of non-negative integer solutions

(x, y)

of the equation

ax+by=n

to be odd*.
Clarification: * For example when

n = 7, a = 3

, we can choose

b = 1

so that there number of solutions og

3x + y = 7

to be

3

(odd), namely:

(0, 7), (1, 4), (2, 1)

N4

1

a^ab^{a + b} = c^cd^{c + d}, gcd (a, b) = gcd $(c, d) = 1=>(a,b)=(c,d)

Suppose that the natural number

a, b, c, d

satisfy the equation

a^ab^{a + b} = c^cd^{c + d}

. (a) If gcd

(a, b) =

gcd

(c, d) = 1

, prove that

a = c

and

b = d

. (b) Does the conclusion

a = c

and

b = d

apply, without the condition gcd

(a, b) =

gcd

(c, d) = 1

?

N2

1

a,b in N: x^2+ax-b , x^2-ax+b have integer roots, existence of a right triangle

Suppose that

a, b

are natural numbers so that all the roots of

x^2 + ax - b

and

x^2 - ax + b

are integers. Show that exists a right triangle with integer sides, with

a

the length of the hypotenuse and

b

the area .

N1

1

min of abc when a>b>b, primes, a=b+2c, a + b + c is perfect square

A triple integer

(a, b, c)

is called brilliant when it satisfies: (i)

a> b> c

are prime numbers (ii)

a = b + 2c

(iii)

a + b + c

is a perfect square number Find the minimum value of

abc

if triple

(a, b, c)

is brilliant.

G1

1

rectangle ABCD so that AB = AD and AB + BC <CD, prove <ABC >120^o

Given a cyclic quadrilateral

ABCD

so that

AB = AD

and

AB + BC <CD

. Prove that the angle

ABC

is more than

120

degrees.

A2

1

\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}, prove a is a natural

Suppose

a

real number so that there is a non-constant polynomial

P (x)

such that

\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}

for each real number

x

, with

x+\pi \ne 0

and

P(x+\pi)\ne 0

. Show that

a

is a natural number.

A1

1

f_1 (x) f_2 (x) is periodic when f_1 (x), f_2 (x) are periodic

Function

f: R\to R

is said periodic , if

f

is not a constant function and there is a number real positive

p

with the property of

f (x) = f (x + p)

for every

x \in R

. The smallest positive real number p which satisfies the condition

f (x) = f (x + p)

for each

x \in R

is named period of

f

. Given

a

and

b

real positive numbers, show that there are periodic functions

f_1

and

f_2

, with periods

a

and

b

respectively, so that

f_1 (x)\cdot f_2 (x)

is also a periodic function.

A3

1

Inequality....

Let

a,b,c

positive reals such that

a^2+b^2+c^2=1

. Prove that

\frac{a+b}{\sqrt{ab+1}}+\frac{b+c}{\sqrt{bc+1}}+\frac{c+a}{\sqrt{ac+1}}\le 3

A7

1

Integer Polynomials!

Suppose

P(n)

is a nonconstant polynomial where all of its coefficients are nonnegative integers such that

\sum_{i=1}^n P(i) | nP(n+1)

for every

n \in \mathbb{N}

. Prove that there exists an integer

k \ge 0

such that

P(n) = \binom{n+k}{n-1} P(1)

for every

n \in \mathbb{N}

.

C9

1

A game but is this already set up?

Given 2015 balls. Astri and Budi will play a game. At first, Astri will choose two different numbers

a

and

b

from the set

S = \{ 1, 2, 3, \dots, 30 \}

. Budi will then choose another 2 different numbers

c

and

d

from the remaining 28 numbers in set

S

.
By taking turns, starting from Astri, they take balls with the following rules: (1) Astri could only take

a

or

b

balls. (2) Budi could only take

c

or

d

balls. until someone couldn't take any balls satisfying the condition given (and that person will lose).
Prove that Budi could choose

c,d

such that he has a strategy to ensure his victory on this game.

A4

1

Sorry, Just too many functions

Determine all functions

f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}

such that

f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \}

for every

x,y,z \in \mathbb{R}

and there exists some real

a

such that

f(x,a) = f(a,x)

for every

x \in \mathbb{R}

.

A6

1

I can't do two fe :")

Let functions

f, g: \mathbb{R}^+ \to \mathbb{R}^+

satisfy the following:

f(g(x)y + f(x)) = (y+2015)f(x)

for every

x,y \in \mathbb{R}^+

. (a) Prove that

g(x) = \frac{f(x)}{2015}

for every

x \in \mathbb{R}^+.

(b) State an example of function that satisfy the equation above and

f(x), g(x) \ge 1

for every

x \in \mathbb{R}^+

.

C8

1

colouring the floor's window of three 2025 floor buidings with 3 colors

It is known that there are

3

buildings in the same shape which are located in an equilateral triangle. Each building has a

2015

floor with each floor having one window. In all three buildings, every

1

st floor is uninhabited, while each floor of others have exactly one occupant. All windows will be colored with one of red, green or blue. The residents of each floor of a building can see the color of the window in the other buildings of the the same floor and one floor just below it, but they cannot see the colors of the other windows of the two buildings. Besides that, sresidents cannot see the color of the window from any floor in the building itself. For example, resident of the

10

th floor can see the colors of the

9

th and

10

th floor windows for the other buildings (a total of

4

windows) and he can't see the color of the other window. We want to color the windows so that each resident can see at lest

1

window of each color. How many ways are there to color those windows?

A5

1

sum \sqrt{\frac{a}{b+c}+\frac{b}{c+a}} >=3

Let

a,b,c

be positive real numbers. Prove that

\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3

G6

1

EQ = FQ iff BP = CP

Let

ABC

be an acute angled triangle with circumcircle

O

. Line

AO

intersects the circumcircle of triangle

ABC

again at point

D

. Let

P

be a point on the side

BC

. Line passing through

P

perpendicular to

AP

intersects lines

DB

and

DC

at

E

and

F

respectively . Line passing through

D

perpendicular to

BC

intersects

EF

at point

Q

. Prove that

EQ = FQ

if and only if

BP = CP

.

N3

1

Divisibility of power of integer

Given positive integers

a,b,c,d

such that

a\mid c^d

and

b\mid d^c

. Prove that

ab\mid (cd)^{max(a,b)}

G7

1

Geometry Collinearity

Given an acute triangle

ABC

.

\Gamma _{B}

is a circle that passes through

AB

, tangent to

AC

at

A

and centered at

O_{B}

. Define

\Gamma_C

and

O_C

the same way. Let the altitudes of

\triangle ABC

from

B

and

C

meets the circumcircle of

\triangle ABC

at

X

and

Y

, respectively. Prove that

A

, the midpoint of

XY

and the midpoint of

O_{B}O_{C}

is collinear.

N7

1

gcd and lcm

For every natural number

a

and

b

, define the notation

[a,b]

as the least common multiple of

a

and

b

and the notation

(a,b)

as the greatest common divisor of

a

and

b

. Find all

n \in \mathbb{N}

that satisfies

4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)}

2015 Indonesia MO Shortlist

Subcontests

dividing a set into 2 sets with same sum each

fleas on the real line jumping over integers

externally tangent circles, perpendicular chords, collinearity wanted

equal segments wanted, related to an isosceles triangle and 2 circumcircles

from 3 points of circumcircle, two tangents to incircle, means one more

if A, P, Q are collinear, then AB = AC , two circles related

subset of {1,2, 3,... , 2014}, with 12 elements, with numbers of sum = 2015

max number of ways to choose representatives to speak

2015 marbles in a box, 3 colors at start, one color in the end

2015 elephants on a chessboard of dimensions 2x 2015 without threats

circumcircle tangent to given circle

infinite n such that exists naturals a,b: a + b = n and ab | n^2 + n + 1

prove that each beautiful set must be charming (in non negative integers)

ax+by=n, odd no. of non-negative integer solutions (x, y)

a^ab^{a + b} = c^cd^{c + d}, gcd (a, b) = gcd $(c, d) = 1=>(a,b)=(c,d)

a,b in N: x^2+ax-b , x^2-ax+b have integer roots, existence of a right triangle

min of abc when a>b>b, primes, a=b+2c, a + b + c is perfect square

rectangle ABCD so that AB = AD and AB + BC <CD, prove <ABC >120^o

\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}, prove a is a natural

f_1 (x) f_2 (x) is periodic when f_1 (x), f_2 (x) are periodic

Inequality....

Integer Polynomials!

A game but is this already set up?

Sorry, Just too many functions

I can't do two fe :")

colouring the floor's window of three 2025 floor buidings with 3 colors

sum \sqrt{\frac{a}{b+c}+\frac{b}{c+a}} >=3

EQ = FQ iff BP = CP

Divisibility of power of integer

Geometry Collinearity

gcd and lcm

2015 Indonesia MO Shortlist

Subcontests

dividing a set into 2 sets with same sum each

fleas on the real line jumping over integers

externally tangent circles, perpendicular chords, collinearity wanted

equal segments wanted, related to an isosceles triangle and 2 circumcircles

from 3 points of circumcircle, two tangents to incircle, means one more

if A, P, Q are collinear, then AB = AC , two circles related

subset of {1,2, 3,... , 2014}, with 12 elements, with numbers of sum = 2015

max number of ways to choose representatives to speak

2015 marbles in a box, 3 colors at start, one color in the end

2015 elephants on a chessboard of dimensions 2x 2015 without threats

circumcircle tangent to given circle

infinite n such that exists naturals a,b: a + b = n and ab | n^2 + n + 1

prove that each beautiful set must be charming (in non negative integers)

ax+by=n, odd no. of non-negative integer solutions (x, y)

a^ab^{a + b} = c^cd^{c + d}, gcd (a, b) = gcd $(c, d) = 1=&gt;(a,b)=(c,d)

a,b in N: x^2+ax-b , x^2-ax+b have integer roots, existence of a right triangle

min of abc when a&gt;b&gt;b, primes, a=b+2c, a + b + c is perfect square

rectangle ABCD so that AB = AD and AB + BC &lt;CD, prove &lt;ABC &gt;120^o

\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}, prove a is a natural

f_1 (x) f_2 (x) is periodic when f_1 (x), f_2 (x) are periodic

Inequality....

Integer Polynomials!

A game but is this already set up?

Sorry, Just too many functions

I can't do two fe :&quot;)

colouring the floor's window of three 2025 floor buidings with 3 colors

sum \sqrt{\frac{a}{b+c}+\frac{b}{c+a}} &gt;=3

EQ = FQ iff BP = CP

Divisibility of power of integer

Geometry Collinearity

gcd and lcm

a^ab^{a + b} = c^cd^{c + d}, gcd (a, b) = gcd $(c, d) = 1=>(a,b)=(c,d)

min of abc when a>b>b, primes, a=b+2c, a + b + c is perfect square

rectangle ABCD so that AB = AD and AB + BC <CD, prove <ABC >120^o

I can't do two fe :")

sum \sqrt{\frac{a}{b+c}+\frac{b}{c+a}} >=3