MathDB

Midpoints connected to center are perpendicular

Source: China 2016 TST Day 2 Q5

3/16/2016

Refer to the diagram below. Let

ABCD

be a cyclic quadrilateral with center

O

. Let the internal angle bisectors of

\angle A

and

\angle C

intersect at

I

and let those of

\angle B

and

\angle D

intersect at

J

. Now extend

AB

and

CD

to intersect

IJ

and

P

and

R

respectively and let

IJ

intersect

BC

and

DA

at

Q

and

S

respectively. Let the midpoints of

PR

and

QS

be

M

and

N

respectively. Given that

O

does not lie on the line

IJ

, show that

OM

and

ON

are perpendicular.

geometrycyclic quadrilateral

Sum of product of elements from infinite sets

Source: China Team Selection Test 2016 Test 2 Day 2 Q5

3/21/2016

Does there exist two infinite positive integer sets

S,T

, such that any positive integer

n

can be uniquely expressed in the form

n=s_1t_1+s_2t_2+\ldots+s_kt_k

,where

k

is a positive integer dependent on

n

,

s_1<\ldots<s_k

are elements of

S

,

t_1,\ldots, t_k

are elements of

T

?

number theoryalgebra

Triangulation of a graph

Source: China Team Selection Test 2016 Test 3 Day 2 Q5

3/26/2016

Let

S

be a finite set of points on a plane, where no three points are collinear, and the convex hull of

S

,

\Omega

, is a

2016-

gon

A_1A_2\ldots A_{2016}

. Every point on

S

is labelled one of the four numbers

\pm 1,\pm 2

, such that for

i=1,2,\ldots , 1008,

the numbers labelled on points

A_i

and

A_{i+1008}

are the negative of each other. Draw triangles whose vertices are in

S

, such that any two triangles do not have any common interior points, and the union of these triangles is

\Omega

. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.

combinatoricsgraph theorycombinatorial geometry

5

Problems(3)