2020 China Second Round Olympiad

Part of (China) National High School Mathematics League

Subcontests

(4)

1

perfect matching of a triangluation diagram

Given a convex polygon with 20 vertexes, there are many ways of traingulation it (as 18 triangles). We call the diagram of triangulation, meaning the 20 vertexes, with 37 edges(17 triangluation edges and the original 20 edges), a T-diagram. And the subset of this T-diagram with 10 edges which covers all 20 vertexes(meaning any two edges in the subset doesn't cover the same vertex) calls a "perfect matching" of this T-diagram. Among all the T-diagrams, find the maximum number of "perfect matching" of a T-diagram.

1

Minimum of the sum of some nonnegative reals

n\geq3

be a given integer, and let

a_1,a_2,\cdots,a_{2n},b_1,b_2,\cdots,b_{2n}

4n

nonnegative reals, such that

a_1+a_2+\cdots+a_{2n}=b_1+b_2+\cdots+b_{2n}>0,

i=1,2,\cdots,2n,

a_ia_{i+2}\geq b_i+b_{i+1},

a_{2n+1}=a_1,

a_{2n+2}=a_2,

b_{2n+1}=b_1.

Detemine the minimum of

a_1+a_2+\cdots+a_{2n}.

1

Normal geometry

ABC,

AB=BC,

I

be the incentre of

\triangle ABC.

M

is the midpoint of segment

BI.

P

lies on segment

AC,

AP=3PC.

H

PI,

MH\perp PH.

Q

is the midpoint of the arc

AB

of the circumcircle of

\triangle ABC

BH\perp QH.

1

A new problem about linear recursive sequence

a_1=1,

a_2=2,

a_n=2a_{n-1}+a_{n-2},

n=3,4,\cdots.

Prove that for any integer

n\geq5,

a_n

has at least one prime factor

p,

p\equiv 1\pmod{4}.