MathDB

Iran TST

Source: Iranian TST 2019, first exam day 2, problem 4

4/10/2019

Consider triangle

ABC

with orthocenter

H

. Let points

M

and

N

be the midpoints of segments

BC

and

AH

. Point

D

lies on line

MH

so that

AD\parallel BC

and point

K

lies on line

AH

so that

DNMK

is cyclic. Points

E

and

F

lie on lines

AC

and

AB

such that

\angle EHM=\angle C

and

\angle FHM=\angle B

. Prove that points

D,E,F

and

K

lie on a circle.
Proposed by Alireza Dadgarnia

geometry

Iran TST

Source: Iranian TST 2019, second exam day 2, problem 4

6/24/2019

Let

1<t<2

be a real number. Prove that for all sufficiently large positive integers like

d

, there is a monic polynomial

P(x)

of degree

d

, such that all of its coefficients are either

+1

or

-1

and

\left|P(t)-2019\right| <1.

Proposed by Navid Safaei

algebrapolynomial

Iran geometry

Source: Iranian TST 2019, third exam day 2, problem 4

4/15/2019

Given an acute-angled triangle

ABC

with orthocenter

H

. Reflection of nine-point circle about

AH

intersects circumcircle at points

X

and

Y

. Prove that

AH

is the external bisector of

\angle XHY

.
Proposed by Mohammad Javad Shabani

geometrygeometric transformationreflectioncircumcircle

4

Problems(3)