2012 Silk Road

Part of Silk Road

Subcontests

(4)

1

arithmetic mean of $\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},...,\sqrt[n]{n}$ lies in

Prove that for any positive integer

n

, the arithmetic mean of

\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},\ldots ,\sqrt[n]{n}

\left[ 1,1+\frac{2\sqrt{2}}{\sqrt{n}} \right]

1

gcd of binomial (2n, 2i+1) from 0<=i<i+1

n > 1

be an integer. Determine the greatest common divisor of the set of numbers

\left\{ \left( \begin{matrix} 2n \\ 2i+1 \\ \end{matrix} \right):0 \le i \le n-1 \right\}

i.e. the largest positive integer, dividing

\left( \begin{matrix} 2n \\ 2i+1 \\ \end{matrix} \right)

without remainder for every

i = 0, 1, ..., n–1

\left( \begin{matrix} m \\ l \\ \end{matrix} \right)=\text{C}_{m}^{l}=\frac{m\text{!}}{l\text{!}\left( m-l \right)\text{!}}

is binomial coefficient.)

1

table 4x4, lines labeled 1,2,3,4, columns labeled a,b,c,d, written inside 0 or 1

In each cell of the table

4 \times 4

, in which the lines are labeled with numbers

1,2,3,4

, and columns with letters

a,b,c,d

, one number is written:

0

1

. Such a table is called valid if there are exactly two units in each of its rows and in each column. Determine the number of valid tables.

1

trapezium inscribed in circle, silk road

ABCD

BC||AD

, is inscribed in a circle,

E

is midpoint of the arc

AD

of this circle not containing point

C

F

be the foot of the perpendicular drawn from

E

on the line tangent to the circle at the point

C

BC=2CF