MathDB

question 2

Source: iran tst 2014 first exam

4/13/2014

find all polynomials with integer coefficients that

P(\mathbb{Z})=

{

p(a):a\in \mathbb{Z}

} has a Geometric progression.

algebrapolynomialnumber theoryalgebra unsolved

MN goes through fixed point

Source: Iran TST 2014, second exam, day 1, problem 2

1/1/2015

Point

D

is an arbitary point on side

BC

of triangle

ABC

.

I

,

I_1

and

I_2

are the incenters of triangles

ABC

,

ABD

and

ACD

respectively.

M\not=A

and

N\not=A

are the intersections of circumcircle of triangle

ABC

and circumcircles of triangles

IAI_1

and

IAI_2

respectively. Prove that regardless of point

D

, line

MN

goes through a fixed point.

geometryincentercircumcirclegeometric transformation

nice question 2

Source: iran tst 2014 third exam

5/22/2014

is there a function

f:\mathbb{N}\rightarrow \mathbb{N}

such that

i) \exists n\in \mathbb{N}:f(n)\neq n

ii)

the number of divisors of

m

is

f(n)

if and only if the number of divisors of

f(m)

is

n

functionnumber theoryprime numbersnumber theory proposed

2