2018 Iran MO (2nd Round)

Part of Iran MO (2nd Round)

Subcontests

(6)

1

Geometry

\omega_1,\omega_2

P,Q

. An arbitrary line passing through

P

\omega_1 , \omega_2

A,B

respectively. Another line parallel to

AB

\omega_1

D,F

\omega_2

E,C

E,F

C,D

X\equiv AD\cap BE

Y\equiv BC\cap AF

R

be the reflection of

P

CD

. Prove that: a.

R

XY

. b. PR is the bisector of

\hat {XPY}

1

Combinatorics

Lamps of the hall switch by only five keys. Every key is connected to one or more lamp(s). By switching every key, all connected lamps will be switched too. We know that no two keys have same set of connected lamps with each other. At first all of the lamps are off. Prove that someone can switch just three keys to turn at least two lamps on.

1

Functional Equation

Find all functions

f:\Bbb {R} \rightarrow \Bbb {R}

f(x+y)f(x^2-xy+y^2)=x^3+y^3

x, y

1

easy geometry

P

be the intersection of

AC

BD

in isosceles trapezoid

ABCD

AB\parallel CD

BC=AD

) . The circumcircle of triangle

ABP

BC

for the second time at

X

Y

AX

DY\parallel BC

\hat {YDA} =2.\hat {YCA}

1

Iranian Second Round P2

n

be odd natural number and

x_1,x_2,\cdots,x_n

be pairwise distinct numbers. Prove that someone can divide the difference of these number into two sets with equal sum. (

X=\{\mid x_i-x_j \mid | i<j\}

1

Iran MO Number theory

a>k

be natural numbers and

r_1<r_2<\dots r_n,s_1<s_2<\dots <s_n

be sequences of natural numbers such that:

(a^{r_1}+k)(a^{r_2}+k)\dots (a^{r_n}+k)=(a^{s_1}+k)(a^{s_2}+k)\dots (a^{s_n}+k)

Prove that these sequences are equal.