2015 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

BMO 2015 #4: 20 consecutive integers

Prove that among

20

consecutive positive integers there is an integer

d

such that for every positive integer

n

the following inequality holds

n \sqrt{d} \left\{n \sqrt {d} \right \} > \dfrac{5}{2}

\left \{x \right \}

denotes the fractional part of the real number

x

. The fractional part of the real number

x

is defined as the difference between the largest integer that is less than or equal to

x

to the actual number

x

1

BMO 2015 #3: Oscars

3366

film critics are voting for the Oscars. Every critic voted just an actor and just one actress. After the voting, it was found that for every positive integer

n \in \left \{1, 2, \ldots, 100 \right \}

, there is some actor or some actress who was voted exactly

n

times. Prove that there are two critics who voted the same actor and the same actress. (Cyprus)

1

BMO 2015 #2: Collinearity

\triangle{ABC}

be a scalene triangle with incentre

I

and circumcircle

\omega

AI, BI, CI

\omega

for the second time at points

D, E, F

, respectively. The parallel lines from

I

BC, AC, AB

EF, DF, DE

K, L, M

, respectively. Prove that the points

K, L, M

are collinear. (Cyprus)

1

BMO 2015 #1: Inequality on a,b,c.

{a, b}

c

are positive real numbers, prove that
\begin{align*} a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}. \end{align*}
(Montenegro).