China Girls Math Olympiad 2003

1. 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
   a) $ S_{\triangle BDF} = (1 - x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} = x(1 - y) (1 - z)S_{\triangle ABC};$
   b) $ \sqrt [3]{S_{\triangle BDF}} + \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$
2. 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.$
3. 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.
4. a) 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 $ \dfrac{1}{9}.$
   b) 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 $ \dfrac{1}{9}.$
5. Let $ \{a_n\}^{\infty}_1$ be a sequence of real numbers such that $$a_1 = 2,\quad a_{n+1} = a^2_n - a_n + 1,\, \forall n \in \mathbb{N}.$$ Prove that \[ 1 - \frac{1}{2003^{2003}} < \sum^{2003}_{i=1} \frac{1}{a_i} < 1.\]
6. 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.$
7. 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
   a) $ \dfrac{a}{b+c} = \dfrac{b}{c+a} + \dfrac{c}{a+b}$.
   b) $\angle BAC > 90^{\circ}.$
8. 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 $3$.