IZhO 2008

Points $K,L,M,N$ are repectively the midpoints of sides $AB,BC,CD,DA$ in a convex quadrliateral $ABCD$ .Line $KM$ meets dioganals $AC$ and $BD$ at points $P$ and $Q$ ,respectively.Line $LN$ meets dioganals $AC$ and $BD$ at points $R$ and $S$ ,respectively. Prove that if $AP\cdot PC=BQ\cdot QD$ ,then $AR\cdot RC=BS\cdot SD$ .

A polynomial $P(x)$ with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable $x$ ) with integer coefficients.For example,the polynomials $x^3 - 1$ and $9x^3 - 3x^2 + 3x + 7 = (x - 1)^3 + (2x)^3 + 2^3$ are good. a)Is the polynomial $P(x) = 3x + 3x^7$ good? b)Is the polynomial $P(x) = 3x + 3x^7 + 3x^{2008}$ good? Justify your answers.

Let $A = \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$ , $1\leq a_i\leq i + 1$ for each $i = 1,2\dots,8\}$ .A subset $X\subset A$ is called sparse if for each two distinct elements $(a_1,\dots,a_8)$ , $(b_1,\dots,b_8)\in X$ ,there exist at least three indices $i$ ,such that $a_i\neq b_i$ . Find the maximal possible number of elements in a sparse subset of set $A$ .

For each positive integer $n$ ,denote by $S(n)$ the sum of all digits in decimal representation of $n$ . Find all positive integers $n$ ,such that $n=2S(n)^3+8$ .

Let $A_1A_2$ be the external tangent line to the nonintersecting cirlces $\omega_1(O_1)$ and $\omega_2(O_2)$ , $A_1\in\omega_1$ , $A_2\in\omega_2$ .Points $K$ is the midpoint of $A_1A_2$ .And $KB_1$ and $KB_2$ are tangent lines to $\omega_1$ and $\omega_2$ ,respectvely( $B_1\neq A_1$ , $B_2\neq A_2$ ).Lines $A_1B_1$ and $A_2B_2$ meet in point $L$ ,and lines $KL$ and $O_1O_2$ meet in point $P$ . Prove that points $B_1,B_2,P$ and $L$ are concyclic.