2019 China Northern MO

Part of China Northern MO

Subcontests

(8)

1

2019 CNMO P8

For positive intenger

n

f(n)

: the smallest positive intenger that does not divide

n

. Consider sequence

(a_n): a_1=a_2=1, a_n=a_{f(n)}+1(n\geq3)

a_3=a_2+1=2,a_4=a_3+1=3

. (a) Prove that there exists a positive intenger

C

, for any positive intenger

n

a_n\leq C

. (b) Are there positive intengers

M

T

, satisfying that for any positive intenger

n\geq M

a_n=a_{n+T}

1

2019 CNMO P7

n

cities in Qingqiu Country. The distance between any two cities are different. The king of the country plans to number the cities and set up two-way air lines in such ways: The first time, set up a two-way air line between city 1 and the city nearest to it. The second time, set up a two-way air line between city 2 and the city second nearest to it. ... The

n-1

th time, set up a two-way air line between city

n-1

and the city farthest to it. Prove: The king can number the cities in a proper way so that he can go to any other city from any city by plane.

1

2019 CNMO P6

For nonnegative real numbers

a,b,c,x,y,z

a+b+c=x+y+z=1

, find the maximum value of

(a-x^2)(b-y^2)(c-z^2)

1

2019 CNMO P5

O_1

O_2

A,B

. Bisector of outer angle

\angle O_1AO_2

O_1

C

O_2

D

P

\odot(BCD)

CP\cap O_1=E,DP\cap O_2=F

PE=PF

1

2019 CNMO P4

A manager of a company has 8 workers. One day, he holds a few meetings. (1)Each meeting lasts 1 hour, no break between two meetings. (2)Three workers attend each meeting. (3)Any two workers have attended at least one common meeting. (4)Any worker cannot leave until he finishes all his meetings. Then, how long does the worker who works the longest work at least?

1

2019 CNMO P3

n(n\geq2)

is a given intenger, and

a_1,a_2,...,a_n

are real numbers. For any

i=1,2,\cdots ,n

a_i\neq -1,a_{i+2}=\frac{a_i^2+a_i}{a_{i+1}+1}.

a_1=a_2=\cdots=a_n

a_{n+1}=a_1,a_{n+2}=a_2

1

2019 CNMO P2

O_1

O_2

A,B

AC

\odot O_1

\odot O_2

E

AD

\odot O_2

\odot O_1

F

CF

O_2

H

DE

O_1

G,H

GH\cap O_1=P

PH=PK

1

2019 CNMO P1

Find all positive intengers

x,y

3^x+x^4=y!+2019.