Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} = x,$ $ \frac {AE}{AC} = y,$ $ \frac {DF}{DE} = z.$ Prove that (1) $ S_{\triangle BDF} = (1 - x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} = x(1 - y) (1 - z)S_{\triangle ABC};$ (2) $ \sqrt [3]{S_{\triangle BDF}} + \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$

There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$-th row and $ j$-th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say that the student is moved by $ [a, b] = [i - m, j - n]$ and define the position value of the student as $ a+b.$ Let $ S$ denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of $ S.$

As shown in the figure, quadrilateral $ ABCD$ is inscribed in a circle with $ AC$ as its diameter, $ BD \perp AC,$ and $ E$ the intersection of $ AC$ and $ BD.$ Extend line segment $ DA$ and $ BA$ through $ A$ to $ F$ and $ G$ respectively, such that $ DG || BF.$ Extend $ GF$ to $ H$ such that $ CH \perp GH.$ Prove that points $ B, E, F$ and $ H$ lie on one circle. [asy][asy] defaultpen(linewidth(0.8)+fontsize(10));size(150); real a=4, b=6.5, c=9, d=a*c/b, g=14, f=sqrt(a^2+b^2)*sqrt(a^2+d^2)/g; pair E=origin, A=(0,a), B=(-b,0), C=(0,-c), D=(d,0), G=A+g*dir(B--A), F=A+f*dir(D--A), M=midpoint(G--C); path c1=circumcircle(A,B,C), c2=Circle(M, abs(M-G)); pair Hf=F+10*dir(G--F), H=intersectionpoint(F--Hf, c2); dot(A^^B^^C^^D^^E^^F^^G^^H); draw(c1^^c2^^G--D--C--A--G--F--D--B--A^^F--H--C--B--F); draw(H--B^^F--E^^G--C, linetype("2 2")); pair point= E; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$F$", F, dir(point--F)); label("$G$", G, dir(point--G)); label("$H$", H, dir(point--H)); label("$E$", E, NE);[/asy][/asy]

(1) Prove that there exist five nonnegative real numbers $ a, b, c, d$ and $ e$ with their sum equal to 1 such that for any arrangement of these numbers around a circle, there are always two neighboring numbers with their product not less than $ \frac{1}{9}.$ (2) Prove that for any five nonnegative real numbers with their sum equal to 1 , it is always possible to arrange them around a circle such that there are two neighboring numbers with their product not greater than $ \frac{1}{9}.$

Let $ \{a_n\}^{\infty}_1$ be a sequence of real numbers such that $ a_1 = 2,$ and \[ a_{n+1} = a^2_n - a_n + 1, \forall n \in \mathbb{N}.\] Prove that \[ 1 - \frac{1}{2003^{2003}} < \sum^{2003}_{i=1} \frac{1}{a_i} < 1.\]

Let $ n \geq 2$ be an integer. Find the largest real number $ \lambda$ such that the inequality \[ a^2_n \geq \lambda \sum^{n-1}_{i=1} a_i + 2 \cdot a_n.\] holds for any positive integers $ a_1, a_2, \ldots a_n$ satisfying $ a_1 < a_2 < \ldots < a_n.$

Let the sides of a scalene triangle $ \triangle ABC$ be $ AB = c,$ $ BC = a,$ $ CA =b,$ and $ D, E , F$ be points on $ BC, CA, AB$ such that $ AD, BE, CF$ are angle bisectors of the triangle, respectively. Assume that $ DE = DF.$ Prove that (1) $ \frac{a}{b+c} = \frac{b}{c+a} + \frac{c}{a+b}$ (2) $ \angle BAC > 90^{\circ}.$

Let $ n$ be a positive integer, and $ S_n,$ be the set of all positive integer divisors of $ n$ (including 1 and itself). Prove that at most half of the elements in $ S_n$ have their last digits equal to 3.