2024 Japan MO Finals

Part of Japan MO Finals

Subcontests

(5)

1

Simple NT

Prove that there are no pairs of positive integers

(a,b,c,d,n)

a^2+b^2+c^2+d^2-4\sqrt{abcd}=7\cdot 2^{2n-1}.

1

The circle tangent to the perpendicular bisector

ABC

be an acute triangle with

AB<AC

O

M

be the midpoint of the smaller arc

BC

of the circumcircle of triangle

ABC

D

lies on the extension of side

AB

B

BD=BM

E

AC

(excluding the endpoints) such that

CE=CM

X(\neq A)

be the intersection of circumcircles of triangles

ABE

ACD

. Prove that the perpendicular bisector of segment

DE

is tangent to the circumcircle of triangle

AOX

1

Z-Shaped Polyline

xy

-plane, points with integer coordinates ranging from

1

2000

x

y

coordinates are called good points. Moreover, for any four points

A(x_1, y_1), B(x_2, y_2), C(x_3, y_3), D(x_4, y_4)

satisfying the following conditions, the polyline

ABCD

\mathbb{Z}

-shaped polyline.
[*] All

A, B, C, D

are good points. [*]

x_1<x_2

y_1=y_2

x_2>x_3

y_2-x_2=y_3-x_3

x_3<x_4

y_3=y_4

Find the smallest possible positive integer

n

\mathbb{Z}

-shaped polylines

Z_1, Z_2,\dots, Z_n

satisfy the following condition: for any good point

P

, there exists an integer

i

1

n

inclusive such that

P

Z_i

1

Function about LCM

Find all functions

f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}

\text{lcm}(m, f(m+f(n)))=\text{lcm}(f(m), f(m)+n)

for any positive integers

m

n

1

Magical Set

n\geq 2

be an integer. Determine all sets of real numbers

(a_1, a_2,\dots, a_n)

a_1-2a_2, a_2-2a_3,\dots, a_{n-1}-2a_n, a_n-2a_1

are rearrangements of

a_1, a_2,\dots, a_n