2009 239 Open Mathematical Olympiad

Part of 239 Open Math Olympiad

Subcontests

(7)

2

Making triangles out of sticks

200

sticks are given whose lengths are

1, 2, 4, \ldots , 2^{199}

. What is the smallest number of sticks needed to be broken so that out of all the resulting sticks, several triangles could be created, if each stick could be broken only once, and each triangle can be created out of only three sticks?

Friend/Stranger graph

The company has

100

people. For any

k

, we can find a group of

k

people such that there are two (different from them) strangers, each of them knows all of these

k

people. At what maximum

k

is this possible?

2

Consecutive numbers multiplication

Alireza multiplied a billion consecutive natural numbers, and Matin multiplied two million consecutive natural numbers. Prove that these two got different results or one of them has made a mistake.

Christmas presents

11

girls wants to mail each of the other a gift for Christmas. The packages contain no more than two gifts. If they have enough time, what is the smallest possible number of packages that they have to send?

2

The Feuerbach point of ABC

The Feuerbach point (the tangent point of the inscribed circle and the nine-point circle of triangle

ABC

F

is marked in triangle

ABC

A_1

BC

AA_1

is the altitude of triangle

ABC

. Prove that the line symmetric to

FA_1

with respect to

BC

is perpendicular to

IO

O

is the center of the circumcircle of the triangle

ABC

I

is the center of its incircle.

Angle inequality

In the triangle

ABC

AA_1

BB_1

CC_1

intersect at the point

O

. It turned out that

AA_1

is the bisector, and the point

O

is closer to the straight line

AB

than to the straight lines

A_1C_1

B_1A_1

\angle{BAC} > 120^{\circ}

1

Numbers on 100-gon

Non-negative integers are placed on the vertices of a

100

-gon, the sum of the numbers is

99

. Every minute at one of the vertices that is equal to

0

will be replaced by

2

and both its neighboring numbers are subtracted by

1

. Prove that after a while a negative number will appear on the board.

1

Equal distance from the Incenter

AB, BC

CA

ABC

K, L

M

are selected, respectively, such that

AK = AM

BK = BL

\angle{MLB} = \angle{CAB}

ML = KI

I

is the incenter of triangle

CML

2

Lcm fractions

Natural numbers

a

b

are given such that the number

P = \frac{[a, b]}{a + 1} + \frac{[a, b]}{b + 1}

Is a prime. Prove that

4P + 5

is the square of a natural number.

Composite sum

The natural numbers

x, y > 1

, are such that

x^2 + xy -y

is the square of a natural number. Prove that

x + y + 1

is a composite number.

2

Prime sequence

In a sequence of natural numbers, the first number is

a

, and each subsequent number is the smallest number coprime to all the previous ones and greater than all of them. Prove that in this sequence from some place all numbers will be primes.

Interesting roads

Kostya drives a car from a village to a city, driving along three roads. Moreover, on each of these roads, he drives at a constant speed. Is it possible that a third of the distance traveled was completed earlier than a third of the time, half of the distance traveled later than half of the time, and two-thirds of the distance was earlier than two-thirds of the time?