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Equilateral triangle and a point in it ( Iran 2014)

Source: Iran 3rd round 2014-Algebra exam-P1

9/19/2014

We have an equilateral triangle with circumradius

1

. We extend its sides. Determine the point

P

inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center

P

and radius

1

, is maximum. (The total distance of the point P from the sides of an equilateral triangle is fixed )
Proposed by Erfan Salavati

geometrycircumcirclegeometry proposed

Show that for all natural number n

Source: Iranian 3rd round Number Theory exam P1

9/22/2014

Show that for every natural number

n

there are

n

natural numbers

x_1 < x_2 < ... < x_n

such that

\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}-\frac{1}{x_1x_2...x_n}\in \mathbb{N}\cup {0}

(15 points )

number theory proposednumber theory

Connected graph

Source: Iranian 3rd round Combinatorics exam P1 - 2014

9/25/2014

Denote by

g_n

the number of connected graphs of degree

n

whose vertices are labeled with numbers

1,2,...,n

. Prove that

g_n \ge (\frac{1}{2}).2^{\frac{n(n-1)}{2}}

. Note:If you prove that for

c < \frac{1}{2}

,

g_n \ge c.2^{\frac{n(n-1)}{2}}

, you will earn some point!
proposed by Seyed Reza Hosseini and Mohammad Amin Ghiasi

combinatorics proposedcombinatorics

three circle

Source: Iranian 3rd round Geometry exam P1

9/25/2014

In the circumcircle of triange

\triangle ABC,

AA'

is a diameter. We draw lines

l'

and

l

from

A'

parallel with Internal and external bisector of the vertex

A

.

l'

Cut out

AB , BC

at

B_1

and

B_2

.

l

Cut out

AC , BC

at

C_1

and

C_2

. Prove that the circumcircles of

\triangle ABC

\triangle CC_1C_2

and

\triangle BB_1B_2

have a common point.
(20 points)

geometrycircumcirclegeometry proposed

Increasing Function from What to What

Source: Iran 3rd round 2014 - final exam problem 1

9/16/2014

In each of (a) to (d) you have to find a strictly increasing surjective function from A to B or prove that there doesn't exist any. (a)

A=\{x|x\in \mathbb{Q},x\leq \sqrt{2}\}

and

B=\{x|x\in \mathbb{Q},x\leq \sqrt{3}\}

(b)

A=\mathbb{Q}

and

B=\mathbb{Q}\cup \{\pi \}

In (c) and (d) we say

(x,y)>(z,t)

where

x,y,z,t \in \mathbb{R}

, whenever

x>z

or

x=z

and

y>t

. (c)

A=\mathbb{R}

and

B=\mathbb{R}^2

(d)

X=\{2^{-x}| x\in \mathbb{N}\}

, then

A=X \times (X\cup \{0\})

and

B=(X \cup \{ 0 \}) \times X

(e) If

A,B \subset \mathbb{R}

, such that there exists a surjective non-decreasing function from

A

to

B

and a surjective non-decreasing function from

B

to

A

, does there exist a surjective strictly increasing function from

B

to

A

?
Time allowed for this problem was 2 hours.

functionalgebra unsolvedalgebra

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Problems(5)