MathDB

Hard geometry

Source: Iran TST 2015 ,exam 1, day 2 problem 3

5/11/2015

ABCD

is a circumscribed and inscribed quadrilateral.

O

is the circumcenter of the quadrilateral.

E,F

and

S

are the intersections of

AB,CD

,

AD,BC

and

AC,BD

respectively.

E'

and

F'

are points on

AD

and

AB

such that

A\hat{E}E'=E'\hat{E}D

and

A\hat{F}F'=F'\hat{F}B

.

X

and

Y

are points on

OE'

and

OF'

such that

\frac{XA}{XD}=\frac{EA}{ED}

and

\frac{YA}{YB}=\frac{FA}{FB}

.

M

is the midpoint of arc

BD

of

(O)

which contains

A

. Prove that the circumcircles of triangles

OXY

and

OAM

are coaxal with the circle with diameter

OS

.

geometrycircumcircle

Inequality

Source: Iran TST 2015, second exam, day 2, problem 3

5/17/2015

If

a,b,c

are positive real numbers such that

a+b+c=abc

prove that

\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}

inequalitiesinequalities proposedAM-GM

Iran TST 2015 Geometry

Source: Iran TST 2015,third exam,second day,problem 6

5/29/2015

AH

is the altitude of triangle

ABC

and

H^\prime

is the reflection of

H

trough the midpoint of

BC

. If the tangent lines to the circumcircle of

ABC

at

B

and

C

, intersect each other at

X

and the perpendicular line to

XH^\prime

at

H^\prime

, intersects

AB

and

AC

at

Y

and

Z

respectively, prove that

\angle ZXC=\angle YXB

.

geometrymoving points

6

Problems(3)