MathDB

1

Japan Mathematical Olympiad, finals Problem 2

n\geq 3

be an odd number. We will play a game using a

n

by

n

grid. The game is comprised of

n^2

turns, in every turn, we will perform the following operation sequentially.

\bullet

We will choose a square with an unwritten integer, and write down an integer among 1 through

n^2

. We can write down any integer only at once through the game.

\bullet

For each row, colum including the square, if the sum of integers is a multiple of

n

, then we will get 1 point (both of each sum is a multiple of

n

, we will get 2 points).
Determine the maximum possible value of the points as the total sum that we can obtain by the end of the game.

5

1

Japan Mathematical Olympiad, finals Problem 5

Let

S

be a set which is comprised of positive integers. We call

S

a

\it{beautiful\ number}

when the element belonging to

S

of which any two distinct elements

x,\ y,\ z

, at least of them will be a divisor of

x+y+z

. Show that there exists an integer

N

satisfying the following condition, and also determine the smallest

N

as such :
For any set S of

\it{beautiful\ number}

, there exists,

n_{s}\geq 2

being an integer, the number of the element belonging to

S

which is not a multiple of

n_s

, is less than or equal to

N

.

4

1

2019 Japan Mathematical Olympiad, Finals Problem 4

Let

ABC

be a triangle with its inceter

I

, incircle

w

, and let

M

be a midpoint of the side

BC

. A line through the point

A

perpendicular to the line

BC

and a line through the point

M

perpendicular to the line

AI

meet at

K

. Show that a circle with line segment

AK

as the diameter touches

w

.

3

1

2019 Japan Mathematical Olympiad, Finals Problem 3

Find all functions

f:\mathbb R^{+} \rightarrow \mathbb R^{+}

such that

f\left(\frac{f(y)}{f(x)}+1\right)=f\left(x+\frac{y}{x}+1\right)-f(x)

for all

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

.

1

2019 Japan Mathematical Olympiad, Finals Problem 1

Find all triples

(a,\ b,\ c)

of positive integers such that

a^2+b+3=(b^2-c^2)^2.

2019 Japan MO Finals

Subcontests

Japan Mathematical Olympiad, finals Problem 2

Japan Mathematical Olympiad, finals Problem 5

2019 Japan Mathematical Olympiad, Finals Problem 4

2019 Japan Mathematical Olympiad, Finals Problem 3

2019 Japan Mathematical Olympiad, Finals Problem 1