MathDB

question6

Source: iran tst 2014 first exam

4/15/2014

I

is the incenter of triangle

ABC

. perpendicular from

I

to

AI

meet

AB

and

AC

at

{B}'

and

{C}'

respectively . Suppose that

{B}''

and

{C}''

are points on half-line

BC

and

CB

such that

B{B}''=BA

and

C{C}''=CA

. Suppose that the second intersection of circumcircles of

A{B}'{B}''

and

A{C}'{C}''

is

T

. Prove that the circumcenter of

AIT

is on the

BC

.

geometryincentercircumcircletrigonometryprojective geometrygeometric transformationhomothety

Simple 2n-gon

Source: Iran TST 2014, second exam, day 2 ,problem 3

1/1/2015

Consider

n

segments in the plane which no two intersect and between their

2n

endpoints no three are collinear. Is the following statement true? Statement: There exists a simple

2n

-gon such that it's vertices are the

2n

endpoints of the segments and each segment is either completely inside the polygon or an edge of the polygon.

combinatorics unsolvedcombinatorics

question 6

Source: iran tst 2014 third exam

5/21/2014

The incircle of a non-isosceles triangle

ABC

with the center

I

touches the sides

BC

at

D

. let

X

is a point on arc

BC

from circumcircle of triangle

ABC

such that if

E,F

are feet of perpendicular from

X

on

BI,CI

and

M

is midpoint of

EF

we have

MB=MC

. prove that

\widehat{BAD}=\widehat{CAX}

geometrycircumcircletrigonometryratiosymmetrygeometry proposed

6

Problems(3)