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.

Let $ a, b, c$ be positive integers for which $ abc = 1$. Prove that $ \sum \frac{1}{b(a+b)} \ge \frac{3}{2}$.