2012 Polish MO Finals

Part of Polish MO Finals

Subcontests

(6)

1

Inequality with squares

Show that for any positive real numbers

a, b, c

true is inequality:

\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)

1

Show that circle is tangent

O

is a center of circumcircle of acute triangle

ABC

, bisector of angle

BAC

BC

D

M

be a point such that,

MC \perp BC

MA \perp AD

BM

OA

intersect in point

P

. Show that circle of center in point

P

passing through a point

A

is tangent to line

BC

1

Tournament without 4-cycles

n

n \ge 4

) took part in the tournament. Each player played exactly one match with every other player, there were no draws. There was no four players

(A, B, C, D)

A

B

B

C

C

D

D

A

. Determine, depending on

n

, maximum number of trios of players

(A, B, C)

A

B

B

C

C

A

. (Attention: Trios

(A, B, C)

(B, C, A)

(C, A, B)

are the same trio.)

1

Show that point lies on bisector

ABC

AB = AC

is inscribed in circle

o

o_1

o_2

are internally tangent to circle

o

P

Q

, respectively, and they are tangent to segments

AB

AC

, respectively, and they are disjoint with the interior of triangle

ABC

m

be a line tangent to circles

o_1

o_2

, such that points

P

Q

lie on the opposite side than point

A

m

AB

AC

K

L

, respectively. Prove, that intersection point of lines

PK

QL

lies on bisector of angle

BAC

1

Build bigger cube

Determine all pairs

(m, n)

of positive integers, for which cube

K

with edges of length

n

, can be build in with cuboids of shape

m \times 1 \times 1

to create cube with edges of length

n + 2

, which has the same center as cube

K

1

Power of rational number

Decide, whether exists positive rational number

w

, which isn't integer, such that

w^w

is a rational number.