CGMO 2003

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 $DG \parallel{} BF.$ Extend $GF$ to $H$ such that $CH \perp GH.$ Prove that points $B, E, F$ and $H$ lie on one circle.

$A$ $B$ $C$ $D$ $F$ $G$ $H$ $E$

(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.