JBMO-SL 2002

A student is playing computer. Computer shows randomly 2002 positive numbers. Game's rules let do the following operations

• to take 2 numbers from these, to double first one, to add the second one and to save the sum.
• to take another 2 numbers from the remainder numbers, to double the first one, to add the second one, to multiply this sum with previous and to save the result.
• to repeat this procedure, until all the 2002 numbers won't be used. Student wins the game if final product is maximum possible. Find the winning strategy and prove it.

Positive real numbers are arranged in the form: $1 \ \ \ 3 \ \ \ 6 \ \ \ 10 \ \ \ 15 ...$ $2 \ \ \ 5 \ \ \ 9 \ \ \ 14 ...$ $4 \ \ \ 8 \ \ \ 13 ...$ $7 \ \ \ 12 ...$ $11 ...$ Find the number of the line and column where the number 2002 stays.

Let $a,b,c$ be positive real numbers such that $abc=\frac{9}{4}$ . Prove the inequality: $a^3 + b^3 + c^3 > a\sqrt {b + c} + b\sqrt {c + a} + c\sqrt {a + b}$

Jury's variant: Prove the same, but with $abc=2$

Let $a,b,c$ be positive real numbers. Prove the inequality: $\frac {a^3}{b^2} + \frac {b^3}{c^2} + \frac {c^3}{a^2}\ge \frac {a^2}{b} + \frac {b^2}{c} + \frac {c^2}{a}$

Let $a_1,a_2,...,a_6$ be real numbers such that: $a_1 \not = 0, a_1a_6 + a_3 + a_4 = 2a_2a_5 \ \mathrm{and}\ a_1a_3 \ge a_2^2$ Prove that $a_4a_6\le a_5^2$ . When does equality holds?

Consider integers $a_i,i=\overline{1,2002}$ such that $a_1^{ - 3} + a_2^{ - 3} + \ldots + a_{2002}^{ - 3} = \frac {1}{2}$ Prove that at least 3 of the numbers are equal.

Let $ABC$ be a triangle with centroid $G$ and $A_1,B_1,C_1$ midpoints of the sides $BC,CA,AB$ . A paralel through $A_1$ to $BB_1$ intersects $B_1C_1$ at $F$ . Prove that triangles $ABC$ and $FA_1A$ are similar if and only if quadrilateral $AB_1GC_1$ is cyclic.

In triangle $ABC,H,I,O$ are orthocenter, incenter and circumcenter, respectively. $CI$ cuts circumcircle at $L$ . If $AB=IL$ and $AH=OH$ , find angles of triangle $ABC$ .

Let $ABC$ be a triangle with area $S$ and points $D,E,F$ on the sides $BC,CA,AB$ . Perpendiculars at points $D,E,F$ to the $BC,CA,AB$ cut circumcircle of the triangle $ABC$ at points $(D_1,D_2), (E_1,E_2), (F_1,F_2)$ . Prove that: $|D_1B\cdot D_1C - D_2B\cdot D_2C| + |E_1A\cdot E_1C - E_2A\cdot E_2C| + |F_1B\cdot F_1A - F_2B\cdot F_2A| > 4S$

Let $ABC$ be an isosceles triangle with $AB=AC$ and $\angle A=20^\circ$ . On the side $AC$ consider point $D$ such that $AD=BC$ . Find $\angle BDC$ .

Let $ABCD$ be a convex quadrilateral with $AB=AD$ and $BC=CD$ . On the sides $AB,BC,CD,DA$ we consider points $K,L,L_1,K_1$ such that quadrilateral $KLL_1K_1$ is rectangle. Then consider rectangles $MNPQ$ inscribed in the triangle $BLK$ , where $M\in KB,N\in BL,P,Q\in LK$ and $M_1N_1P_1Q_1$ inscribed in triangle $DK_1L_1$ where $P_1$ and $Q_1$ are situated on the $L_1K_1$ , $M$ on the $DK_1$ and $N_1$ on the $DL_1$ . Let $S,S_1,S_2,S_3$ be the areas of the $ABCD,KLL_1K_1,MNPQ,M_1N_1P_1Q_1$ respectively. Find the maximum possible value of the expression: $\frac{S_1+S_2+S_3}{S}$

Let $A_1,A_2,...,A_{2002}$ be arbitrary points in the plane. Prove that for every circle of radius $1$ and for every rectangle inscribed in this circle, there exist $3$ vertices $M,N,P$ of the rectangle such that $MA_1 + MA_2 + \cdots + MA_{2002} +$ $NA_1 + NA_2 + \cdots + NA_{2002} +$ $PA_1 + PA_2 + \cdots + PA_{2002}\ge 6006$ .