MathDB

2

Arithmetic Sequence around a circle

2 players A and B play the following game with A going first: On each player's turn, he puts a number from 1 to 99 among 99 equally spaced points on a circle (numbers cannot be repeated), and the player who completes 3 consecutive numbers that forms an arithmetic sequence around the circle wins the game. Who has the winning strategy? Explain your reasoning.

2018 | ( 1+2^n+3^n+4^n)

Prove that there exist infinite positive integer

n,

such that

2018 | \left( 1+2^n+3^n+4^n \right).

7

1

Cyclic Inequality on Positive Reals

If

a

,

b

,

c

are positive reals, prove that

\frac{a+bc}{a+a^2}+\frac{b+ca}{b+b^2}+\frac{c+ab}{c+c^2} \geq 3

6

2

Equal Angles

Let

H

be the orthocenter of triangle

ABC

. Let

D

and

E

be points on

AB

and

AC

such that

DE

is parallel to

CH

. If the circumcircle of triangle

BDH

passes through

M

, the midpoint of

DE

, then prove that

\angle ABM=\angle ACM

na_{n+1}=2(n+1)a_n-n-2

For

a_1 = 3

, define the sequence

a_1, a_2, a_3, \ldots

for

n \geq 1

as

na_{n+1}=2(n+1)a_n-n-2.

Prove that for any odd prime

p

, there exist positive integer

m,

such that

p|a_m

and

p|a_{m+1}.

5

1

Right Traingle with Integral Sides

A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.

4

2

+1's and -1's in a 4x4 grid

In each square of a

4

by

4

grid, you put either a

+1

or a

-1

. If any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. What is the minimum number of

+1

's needed to be placed to be able to satisfy the conditions

The Missing Problem in 2018 CNMO

For

n(n\geq3)

positive intengers

a_1,a_2,\cdots,a_n

. Put the numbers on a circle. In each operation, calculate difference between two adjacent numbers and take its absolute value. Put the

n

numbers we get on another ciecle (do not change their order). Find all

n

, satisfying that no matter how

a_1,a_2,\cdots,a_n

are given, all numbers on the circle are equal after limited operations.

3

2

Divisibility by a prime

Let

p

be a prime such that

3|p+1

. Show that

p|a-b

if and only if

p|a^3-b^3

Five points are concyclic

A,B,C,D,E

lie on

\odot O

in that order,and

BD \cap CE=F,CE \cap AD=G,AD \cap BE=H,BE \cap AC=I,AC \cap BD=J.

Prove that

\frac{FG}{CE}=\frac{GH}{DA}=\frac{HI}{BE}=\frac{IJ}{AC}=\frac{JF}{BD}

when and only when

F,G,H,I,J

are concyclic.

2

Inequality on Nonnegative Reals

Let

a

,

b

,

c

be nonnegative reals such that

a^2+b^2+c^2+ab+\frac{2}{3}ac+\frac{4}{3}bc=1

Find the maximum and minimum value of

a+b+c

.

good prime

Let

p

be a prime. We say

p

is good if and only if for any positive integer

a,b,

such that

a\equiv b (\textup{mod}p)\Leftrightarrow a^3\equiv b^3 (\textup{mod}p).

Prove that (1)There are infinite primes

p

which are good; (2)There are infinite primes

p

which are not good.

1

Parallel Lines

In triangle

ABC

, let the circumcenter, incenter, and orthocenter be

O

,

I

, and

H

respectively. Segments

AO

,

AI

, and

AH

intersect the circumcircle of triangle

ABC

at

D

,

E

, and

F

.

CD

intersects

AE

at

M

and

CE

intersects

AF

at

N

. Prove that

MN

is parallel to

BC

.

2018 China Northern MO

Subcontests

Arithmetic Sequence around a circle

2018 | ( 1+2^n+3^n+4^n)

Cyclic Inequality on Positive Reals

Equal Angles

na_{n+1}=2(n+1)a_n-n-2

Right Traingle with Integral Sides

+1's and -1's in a 4x4 grid

The Missing Problem in 2018 CNMO

Divisibility by a prime

Five points are concyclic

Inequality on Nonnegative Reals

good prime

Parallel Lines