MathDB

2

Square and Product of Lengths

ABCD

be a square. The line segment

AB

is divided internally at

H

so that

|AB|\cdot |BH|=|AH|^2

. Let

E

be the midpoints of

AD

and

X

be the midpoint of

AH

. Let

Y

be a point on

EB

such that

XY

is perpendicular to

BE

. Prove that

|XY|=|XH|

.

GCD of Products

For any positive integer

n

define

E(n)=n(n+1)(2n+1)(3n+1)\cdots (10n+1).

Find the greatest common divisor of

E(1),E(2),E(3),\dots ,E(2009).

3

2

Primes of the form n^8+n+1

Find all positive integers

n

for which

n^8+n+1

is a prime number.

Easy Diophantine equation

Find all pairs

(a,b)

of positive integers such that

(ab)^2 - 4(a+b)

is the square of an integer.

4

2

Operations on n-tuples

Given an

n

-tuple of numbers

(x_1,x_2,\dots ,x_n)

where each

x_i=+1

or

-1

, form a new

n

-tuple

(x_1x_2,x_2x_3,x_3x_4,\dots ,x_nx_1),

and continue to repeat this operation. Show that if

n=2^k

for some integer

k\ge 1

, then after a certain number of repetitions of the operation, we obtain the

n

-tuple

(1,1,1,\dots ,1).

Combinatorics problem

At a strange party, each person knew exactly

22

others. For any pair of people

X

and

Y

who knew each other, there was no other person at the party that they both knew. For any pair of people

X

and

Y

who did not know one another, there were exactly

6

other people that they both knew. How many people were at the party?

1

2

irmo

Hamilton Avenue has eight houses. On one side of the street are the houses numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses, finally returning to house 1 for a cup of tea. Throughout the entire journey he must observe the following rules. The numbers of the houses delivered to must follow an odd-even-odd-even pattern throughout, each house except house 1 is visited exactly once (house 1 is visited twice) and the postman at no time is allowed to cross the road to the house directly opposite. How many different delivery sequences are possible?

Polynomials

Let

P(x)

be a polynomial with rational coefficients. Prove that there exists a positive integer

n

such that the polynomial

Q(x)

defined by

Q(x)= P(x+n)-P(x)

has integer coefficients.

5

2

Three Variables Inequality

Hello.
Suppose

a

,

b

,

c

are real numbers such that

a+b+c = 0

and

a^{2}+b^{2}+c^{2} = 1

. Prove that

a^{2}b^{2}c^{2}\leq \frac{1}{54}

and determine the cases of equality.

Angle Bisectors in Triangle

In the triangle

ABC

we have

|AB|<|AC|

. The bisectors of the angles at

B

and

C

meet

AC

and

AB

at

D

and

E

respectively.

BD

and

CE

intersect at the incenter

I

of

\triangle ABC

.
Prove that

\angle BAC=60^\circ

if and only if

|IE|=|ID|

2009 Irish Math Olympiad

Subcontests

Square and Product of Lengths

GCD of Products

Primes of the form n^8+n+1

Easy Diophantine equation

Operations on n-tuples

Combinatorics problem

irmo

Polynomials

Three Variables Inequality

Angle Bisectors in Triangle