2009 Japan MO Finals

Part of Japan MO Finals

Subcontests

(5)

1

Functional Equation for Non negative Reals

Find all functions

f

, defined on the non negative real numbers and taking non negative real numbers such that f(x^2)\plus{}f(y)\equal{}f(x^2\plus{}y\plus{}xf(4y)) for any non negative real numbers

x,\ y

1

If \angle BAO = \angle CAO, then \angle PAO = \angle QAO

\Gamma

be a circumcircle. A circle with center

O

touches to line segment

BC

P

and touches the arc

BC

\Gamma

which doesn't have

A

Q

\angle {BAO} = \angle {CAO}

, then prove that

\angle {PAO} = \angle {QAO}

1

(n1, n2, n3) such that b1 b2 = b3

k\geq 2

n_1,\ n_2,\ n_3

be positive integers and

a_1,\ a_2,\ a_3

be integers from 1,\ 2,\ \cdots ,\ k \minus{} 1. Let b_i \equal{} a_i\sum_{j \equal{} 0}^{n_i} k^{j}\ (i \equal{} 1,\ 2,\ 3). Find all possible pairs of integers

(n_1,\ n_2,\ n_3)

such that b_1b_2 \equal{} b_3.

1

Some Integers written in a black board

N

be postive integer. Some integers are written in a black board and those satisfy the following conditions. 1. Any numbers written are integers which are from 1 to

N

. 2. More than one integer which is from 1 to

N

is written. 3. The sum of numbers written is even. If we mark

X

to some numbers written and mark

Y

to all remaining numbers, then prove that we can set the sum of numbers marked

X

are equal to that of numbers marked

Y

1

2^n + n|8^n + n

Find all positive integers

n

such that 8^n \plus{} n is divisible by 2^n \plus{} n.