MathDB

R.I.P synthetic geo

Source: Iran MO 3rd round 2023 ,Day 1 P1

8/16/2023

In triangle

\triangle ABC

,

M, N

are midpoints of

AC,AB

respectively. Assume that

BM,CN

cuts

(ABC)

at

M',N'

respectively. Let

X

be on the extention of

BC

from

B

st

\angle N'XB=\angle ACN

. And define

Y

similarly on the extention of

BC

from

C

. Prove that

AX=AY

.

geometry

The Appetizer of Iran NT2023

Source: Iran MO 3rd round 2023 NT exam , P1

8/17/2023

Find all integers

n > 4

st for every two subsets

A,B

of

\{0,1,....,n-1\}

, there exists a polynomial

f

with integer coefficients st either

f(A) = B

or

f(B) = A

where the equations are considered mod n. We say two subsets are equal mod n if they produce the same set of reminders mod n. and the set

f(X)

is the set of reminders of

f(x)

where

x \in X

mod n.

algebrapolynomial

Friendships on circle

Source: Iran MO 2023 3rd round , Combinatorics exam P1

8/19/2023

Let

n

and

a \leq n

be two positive integers. There's

2n

people sitting around a circle reqularly. Two people are friend iff one of their distance in the circle is

a

(that is ,

a-1

people are between them). Find all integers

a

in terms of

n

st we can choose

n

of these people , no two of them positioned in front of each other(means they're not antipodes of each other in the circle) and the total friendship between them is an odd number.

combinatorics

Maybe a P0

Source: Iran MO 2023 3rd round , geometry exam P1

8/23/2023

In triangle

\triangle ABC

,

I

is the incenter and

M

is the midpoint of arc

(BC)

in the circumcircle of

(ABC)

not containing

A

. Let

X

be an arbitrary point on the external angle bisector of

A

. Let

BX \cap (BIC) = T

.

Y

lies on

(AXC)

, different from

A

, st

MA=MY

. Prove that

TC || AY

(Assume that

X

is not on

(ABC)

or

BC

)

geometryincentercircumcircleangle bisector

Geometry in algebra exam

Source: Iran MO 2023 3rd round , Algebra exam P1

8/21/2023

Given

12

complex numbers

z_1,...,z_{12}

st for each

1 \leq i \leq 12

:

|z_i|=2 , |z_i - z_{i+1}| \geq 1

prove that :

\sum_{1 \leq i \leq 12} \frac{1}{|z_i\overline{z_{i+1}}+1|^2} \geq \frac{1}{2}

geometrycomplex numbers

1

Problems(5)