MathDB

Isosceles triangles among a group of points

Source: Iran 3rd round 2012-Special Lesson exam-Part1-P2

7/27/2012

Consider a set of

n

points in plane. Prove that the number of isosceles triangles having their vertices among these

n

points is

\mathcal O (n^{\frac{7}{3}})

. Find a configuration of

n

points in plane such that the number of equilateral triangles with vertices among these

n

points is

\Omega (n^2)

.

combinatorics proposedcombinatorics

Van der Warden Theorem!

Source: Iran 3rd round 2012-Special Lesson exam-Part 2-P2

9/15/2012

Suppose

W(k,2)

is the smallest number such that if

n\ge W(k,2)

, for each coloring of the set

\{1,2,...,n\}

with two colors there exists a monochromatic arithmetic progression of length

k

. Prove that

W(k,2)=\Omega (2^{\frac{k}{2}})

.

functionprobabilityarithmetic sequencecombinatorics proposedcombinatorics

Infinitely many pairs of rational numbers

Source: Iran 3rd round 2011-Number Theory exam-P2

9/19/2012

Prove that there exists infinitely many pairs of rational numbers

(\frac{p_1}{q},\frac{p_2}{q})

with

p_1,p_2,q\in \mathbb N

with the following condition:

|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.

Proposed by Mohammad Gharakhani

floor functionnumber theory proposednumber theory

Nagel point, two lines intersecting on circumcircle

Source: Iran 3rd round 2012-Geometry exam-P2

9/20/2012

Let the Nagel point of triangle

ABC

be

N

. We draw lines from

B

and

C

to

N

so that these lines intersect sides

AC

and

AB

in

D

and

E

respectively.

M

and

T

are midpoints of segments

BE

and

CD

respectively.

P

is the second intersection point of circumcircles of triangles

BEN

and

CDN

.

l_1

and

l_2

are perpendicular lines to

PM

and

PT

in points

M

and

T

respectively. Prove that lines

l_1

and

l_2

intersect on the circumcircle of triangle

ABC

.
Proposed by Nima Hamidi

geometrycircumcirclegeometric transformationgeometry proposed

Coloring natural numbers

Source: Iran 3rd round 2012-Combinatorics exam-P2

9/20/2012

Suppose

s,k,t\in \mathbb N

. We've colored each natural number with one of the

k

colors, such that each color is used infinitely many times. We want to choose a subset

\mathcal A

of

\mathbb N

such that it has

t

disjoint monochromatic

s

-element subsets. What is the minimum number of elements of

A

?
Proposed by Navid Adham

combinatorics proposedcombinatorics

Inequality and continued fraction

Source: Iran 3rd round 2012-Algebra exam-P2

9/20/2012

Suppose

N\in \mathbb N

is not a perfect square, hence we know that the continued fraction of

\sqrt{N}

is of the form

\sqrt{N}=[a_0,\overline{a_1,a_2,...,a_n}]

. If

a_1\neq 1

prove that

a_i\le 2a_0

.

inequalitiescontinued fractionalgebra proposedalgebra

Inequality in a convex figure

Source: Iran 3rd round 2012-Final exam-P2

9/23/2012

Suppose

S

is a convex figure in plane with area

10

. Consider a chord of length

3

in

S

and let

A

and

B

be two points on this chord which divide it into three equal parts. For a variable point

X

in

S-\{A,B\}

, let

A'

and

B'

be the intersection points of rays

AX

and

BX

with the boundary of

S

. Let

S'

be those points

X

for which

AA'>\frac{1}{3} BB'

. Prove that the area of

S'

is at least

6

.
Proposed by Ali Khezeli

inequalitiesgeometrycombinatorics proposedcombinatorics

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Problems(7)