1999 Polish MO Finals

Part of Polish MO Finals

Subcontests

(3)

1

50 distinc integers with certain property

Show that one can find

50

distinct positive integers such that the sum of each number and its digits is the same.

2

Ad > be

D

be a point on the side

BC

ABC

AD > BC

E

be a point on the side

AC

\frac{AE}{EC} = \frac{BD}{AD-BC}

AD > BE

equations in integers

n

do the equations have a solution in integers: \begin{eqnarray*}x_1 ^2 + x_2 ^2 + 50 &=& 16x_1 + 12x_2 \\ x_2 ^2 + x_3 ^2 + 50 &=& 16x_2 + 12x_3 \\ \cdots \cdots \cdots & \cdots & \cdots \cdots \\ x_{n-1} ^2 + x_n ^2 + 50 &=& 16x_{n-1} + 12x_n \\ x_n ^2 + x_1 ^2 + 50 &=& 16x_n + 12x_1 \end{eqnarray*}

2

Multiple of 5050

101

distinct non-negative integers less than

5050

show that one can choose four

a, b, c, d

a + b - c - d

is a multiple of

5050

beautiful estimations with 2n integers [mod edit: reals ;) ]

Prove that for any

2n

a_{1}

a_{2}

a_{n}

b_{1}

b_{2}

b_{n}

, we have \sum_{i < j}{\left|a_{i}\minus{}a_{j}\right|}\plus{}\sum_{i < j}{\left|b_{i}\minus{}b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}\minus{}b_{j}\right|}.