MathDB

Iran geometry

Source: Iran MO 3rd round 2017 mid-terms - Geometry P3

8/10/2017

Let

ABC

be an acute-angle triangle. Suppose that

M

be the midpoint of

BC

and

H

be the orthocenter of

ABC

. Let

F\equiv BH\cap AC

and

E\equiv CH\cap AB

. Suppose that

X

be a point on

EF

such that

\angle XMH=\angle HAM

and

A,X

are in the distinct side of

MH

. Prove that

AH

bisects

MX

.

geometry

Functional equation

Source: Iran 3rd round 2017 first Algebra exam

8/7/2017

Find all functions

f:\mathbb{R^+}\rightarrow\mathbb{R^+}

such that

\frac{x+f(y)}{xf(y)}=f(\frac{1}{y}+f(\frac{1}{x}))

for all positive real numbers

x

and

y

.

algebrafunctional equation

Number theory:function

Source: Iran 3rd round 2017 Number theory first exam-P3

8/9/2017

Let

k

be a positive integer. Find all functions

f:\mathbb{N}\to \mathbb{N}

satisfying the following two conditions:\\
• For infinitely many prime numbers

p

there exists a positve integer

c

such that

f(c)=p^k

.\\
• For all positive integers

m

and

n

,

f(m)+f(n)

divides

f(m+n)

.

number theoryfunction

P3- first combinatorics exam of 2017 Iran MO 3rd round

Source: 2017 Iran MO 3rd round, first combinatorics exam P3

9/12/2017

Ali has

6

types of

2\times2

squares with cells colored in white or black, and has presented them to Mohammad as forbidden tiles.

a)

Prove that Mohammad can color the cells of the infinite table (from each

4

sides.) in black or white such that there's no forbidden tiles in the table.

b)

Prove that Ali can present

7

forbidden tiles such that Mohammad cannot achieve his goal.

Irancombinatorics

Iran Geometry

Source: Iran MO 3rd round 2017 finals - Geometry P3

9/3/2017

In triangle

ABC

points

P

and

Q

lies on the external bisector of

\angle A

such that

B

and

P

lies on the same side of

AC

. Perpendicular from

P

to

AB

and

Q

to

AC

intersect at

X

. Points

P'

and

Q'

lies on

PB

and

QC

such that

PX=P'X

and

QX=Q'X

. Point

T

is the midpoint of arc

BC

(does not contain

A

) of the circumcircle of

ABC

. Prove that

P',Q'

and

T

are collinear if and only if

\angle PBA+\angle QCA=90^{\circ}

.

geometrycircumcircle

Inequality

Source:  Iran 3rd round-2017-Algebra final exam-P3

9/2/2017

Let

a,b

and

c

be positive real numbers. Prove that

\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3}

algebraInequalityAM-GMIraninequalities

P3- second combinatorics exam of 2017 Iran MO 3rd round

Source: 2017 Iran MO 3rd round, second combinatorics P3

9/12/2017

30

volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called unusual if at the end of the league, the winner of the match have a smaller amount of wins than the loser of the match. A team is called astonishing if all its matches are unusual matches. Find the maximum number of astonishing teams.

Irancombinatorics

3

Problems(7)