MathDB

Kobayashi overkill

Source: Iranian third round midterm number theory exam problem 3

8/4/2019

Let

S

be an infinite set of positive integers and define:

T=\{ x+y|x,y \in S , x \neq y \}

Suppose that there are only finite primes

p

so that:
1.

p \equiv 1 \pmod 4

2.There exists a positive integer

s

so that

p|s,s \in T

.
Prove that there are infinity many primes that divide at least one term of

S

.

number theorykobayashi

Polynomial Algebra

Source: Iranian 3rd-Round MO 2019 ; mid-term Algabra Exam P3

7/31/2019

We are given a natural number

d

. Find all open intervals of maximum length

I \subseteq R

such that for all real numbers

a_0,a_1,...,a_{2d-1}

inside interval

I

, we have that the polynomial

P(x)=x^{2d}+a_{2d-1}x^{2d-1}+...+a_1x+a_0

has no real roots.

algebrapolynomial

Moving numbers in a square

Source: Iranian 3rd-Round MO 2019 ; mid-term Combinatorics Exam P3

8/23/2019

Cells of a

n*n

square are filled with positive integers in the way that in the intersection of the

i-

th column and

j-

th row, the number

i+j

is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to

n

and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the

i-

th column and

j-

row is written

2n+2-i-j

.

rectanglecombinatorics

Concurrency of BC, OI, AK

Source: own, Iran MO 3rd round 2019 mid-terms - Geometry P3

7/31/2019

Consider a triangle

ABC

with circumcenter

O

and incenter

I

. Incircle touches sides

BC,CA

and

AB

at

D, E

and

F

.

K

is a point such that

KF

is tangent to circumcircle of

BFD

and

KE

is tangent to circumcircle of

CED

. Prove that

BC,OI

and

AK

are concurrent.

geometrycircumcircleincenter

Ugly functional equation

Source: Iranian third round 2019 Finals algebra exam problem 3

8/18/2019

Let

a,b,c

be non-zero distinct real numbers so that there exist functions

f,g:\mathbb{R}^{+} \to \mathbb{R}

so that:

af(xy)+bf(\frac{x}{y})=cf(x)+g(y)

For all positive real

x

and large enough

y

.
Prove that there exists a function

h:\mathbb{R}^{+} \to \mathbb{R}

so that:

f(xy)+f(\frac{x}{y})=2f(x)+h(y)

For all positive real

x

and large enough

y

.

functionfunctional equationalgebra

Composing into number of integers with lower order

Source: Iranian third number theory finals problem 3

8/15/2019

Let

a,m

be positive integers such that

Ord_m (a)

is odd and for any integers

x,y

so that
1.

xy \equiv a \pmod m

2.

Ord_m(x) \le Ord_m(a)

3.

Ord_m(y) \le Ord_m(a)

We have either

Ord_m(x)|Ord_m(a)

or

Ord_m(y)|Ord_m(a)

.prove that

Ord_m(a)

contains at most one prime factor.

number theory

Inscribed pentagon

Source: Iran MO 3rd round 2019 finals - Geometry P3

8/14/2019

Given an inscribed pentagon

ABCDE

with circumcircle

\Gamma

. Line

\ell

passes through vertex

A

and is tangent to

\Gamma

. Points

X,Y

lie on

\ell

so that

A

lies between

X

and

Y

. Circumcircle of triangle

XED

intersects segment

AD

at

Q

and circumcircle of triangle

YBC

intersects segment

AC

at

P

. Lines

XE,YB

intersects each other at

S

and lines

XQ, Y P

at

Z

. Prove that circumcircle of triangles

XY Z

and

BES

are tangent.

geometry

3

Problems(7)