MathDB

2

Polish MO Final 2010, 3rd problem (parallelogram and circle)

ABCD

is a parallelogram in which angle

DAB

is acute. Points

A, P, B, D

lie on one circle in exactly this order. Lines

AP

and

CD

intersect in

Q

. Point

O

is the circumcenter of the triangle

CPQ

. Prove that if

D \neq O

then the lines

AD

and

DO

are perpendicular.

Polish MO Final 2010, 6th problem (sequence with properties)

Real number

C > 1

is given. Sequence of positive real numbers

a_1, a_2, a_3, \ldots

, in which

a_1=1

and

a_2=2

, satisfy the conditions

a_{mn}=a_ma_n,

a_{m+n} \leq C(a_m + a_n),

for

m, n = 1, 2, 3, \ldots

. Prove that

a_n = n

for

n=1, 2, 3, \ldots

.

1

2

Polish MO Final 2010, 1st problem (remainders in the set)

The integer number

n > 1

is given and a set

S \subset \{0, 1, 2, \ldots, n-1\}

with

|S| > \frac{3}{4} n

. Prove that there exist integer numbers

a, b, c

such that the remainders after the division by

n

of the numbers:

a, b, c, a+b, b+c, c+a, a+b+c

belong to

S

.

Polish MO Final 2010, 4th problem (perimeters' inequality)

On the side

BC

of the triangle

ABC

there are two points

D

and

E

such that

BD < BE

. Denote by

p_1

and

p_2

the perimeters of triangles

ABC

and

ADE

respectively. Prove that

p_1 > p_2 + 2\cdot \min\{BD, EC\}.

2

Polish MO Final 2010, 2nd problem (equation in rationals)

Positive rational number

a

and

b

satisfy the equality

a^3 + 4a^2b = 4a^2 + b^4.

Prove that the number

\sqrt{a}-1

is a square of a rational number.

Polish MO Final 2010, 5th problem (remainder of the product)

Prime number

p>3

is congruent to

2

modulo

3

. Let

a_k = k^2 + k +1

for

k=1, 2, \ldots, p-1

. Prove that product

a_1a_2\ldots a_{p-1}

is congruent to

3

modulo

p

.

2010 Polish MO Finals

Subcontests

Polish MO Final 2010, 3rd problem (parallelogram and circle)

Polish MO Final 2010, 6th problem (sequence with properties)

Polish MO Final 2010, 1st problem (remainders in the set)

Polish MO Final 2010, 4th problem (perimeters' inequality)

Polish MO Final 2010, 2nd problem (equation in rationals)

Polish MO Final 2010, 5th problem (remainder of the product)