MathDB

24 robots with a 70degree field of view

Source: Iran MO 3rd round 2016 mid-terms - Combinatorics P3

9/6/2016

There are

24

robots on the plane. Each robot has a

70^{\circ}

field of view. What is the maximum number of observing relations? (Observing is a one-sided relation)

Irancombinatoricsgraph theoryDirected graphs

Sequences and polynomials

Source: Iranian 3rd round 2016 first Algebra exam

8/13/2016

Do there exists many infinitely points like

(x_1,y_1),(x_2,y_2),...

such that for any sequences like {

b_1,b_2,...

} of real numbers there exists a polynomial

P(x,y)\in R[x,y]

such that we have for all

i

:

P(x_{i},y_{i})=b_{i}

algebrapolynomial

Iran geometry

Source: Iranian 3rd round 2016 first geometry exam problem 3

8/14/2016

Let

ABC

be a triangle and let

AD,BE,CF

be its altitudes .

FA_{1},DB_{1},EC_{1}

are perpendicular segments to

BC,AC,AB

respectively. Prove that :

ABC

~

A_{1}B_{1}C_{1}

geometry

Golden Residue

Source: Iran MO 3rd round 2016 mid-terms - Number Theory P3

9/6/2016

Let

m

be a positive integer. The positive integer

a

is called a golden residue modulo

m

if

\gcd(a,m)=1

and

x^x \equiv a \pmod m

has a solution for

x

. Given a positive integer

n

, suppose that

a

is a golden residue modulo

n^n

. Show that

a

is also a golden residue modulo

n^{n^n}

.
Proposed by Mahyar Sefidgaran

number theorymodular arithmeticIran

Angle bisectors

Source: Iran MO 3rd round 2016 finals - Geometry P3

9/5/2016

Given triangle

\triangle ABC

and let

D,E,F

be the foot of angle bisectors of

A,B,C

,respectively.

M,N

lie on

EF

such that

AM=AN

. Let

H

be the foot of

A

-altitude on

BC

. Points

K,L

lie on

EF

such that triangles

\triangle AKL, \triangle HMN

are correspondingly similiar (with the given order of vertices) such that

AK \not\parallel HM

and

AK \not\parallel HN

.
Show that:

DK=DL

geometryangle bisectorIran

Functional Equation

Source: Iran MO 3rd round 2016 finals - Algebra P3

9/1/2016

Find all functions

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

such that for all positive real numbers

x,y:

f(y)f(x+f(y))=f(x)f(xy)

algebrafunctional equationfunctionIran

Number Theory

Source: Iran MO 3rd round 2016 finals - Number Theory P3

9/4/2016

A sequence

P=\left \{ a_{n} \right \}

is called a

\text{Permutation}

of natural numbers (positive integers) if for any natural number

m,

there exists a unique natural number

n

such that

a_n=m.

We also define

S_k(P)

as:

S_k(P)=a_{1}+a_{2}+\cdots +a_{k}

(the sum of the first

k

elements of the sequence).
Prove that there exists infinitely many distinct

\text{Permutations}

of natural numbers like

P_1,P_2, \cdots

such that

:

\forall k, \forall i<j: S_k(P_i)|S_k(P_j)

number theoryIran

Coloring a 30*30 table

Source: Iran MO 3rd round 2016 finals -Combinatorics P3

9/6/2016

A

30\times30

table is given. We want to color some of it's unit squares such that any colored square has at most

k

neighbors. ( Two squares

(i,j)

and

(x,y)

are called neighbors if

i-x,j-y\equiv0,-1,1 \pmod {30}

and

(i,j)\neq(x,y)

. Therefore, each square has exactly

8

neighbors) What is the maximum possible number of colored squares if

:

a) k=6

b)k=1

modular arithmeticcombinatoricstableColoring

3

Problems(8)