Junior Balkan Mathematical Olympiad 2002 [Shortlists & Solutions]

1. 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.
   
2. Positive real numbers are arranged in the form
   $1361015...$
   $25914...$
   $4813...$
   $712...$
   $11...$
   Find the number of the line and column where the number 2002 stays.
3. 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 $abc=2$
4. Let $a,b,c$ be positive real numbers. Prove the inequality $$\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2}\ge \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}.$$
5. Let $a_1,a_2,...,a_6$ be real numbers such that $$a_1\ne 0,a_1{a}_6+a_3+a_4=2a_2{a}_5,a_1{a}_3\ge a_2^2.$$ Prove that $a_4{a}_6\le a_5^2$. When does equality holds?
6. Consider integers $a_i,i=\overline{1,2002}$ such that $$a_1^{-3}+a_2^{-3}+\dots +a_{2002}^{-3}=\frac{1}{2}.$$ Prove that at least 3 of the numbers are equal.
7. 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 $B{B}_1$ intersects $B_1{C}_1$ at $F$. Prove that triangles $ABC$ and $F{A}_{1}A$ are similar if and only if quadrilateral $A{B}_{1}G{C}_1$ is cyclic.
8. 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 $ABC$.
9. 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 $$\begin{gathered} |D_{1}B\cdot D_{1}C-D_{2}B\cdot D_{2}C|+|E_{1}A\cdot E_{1}C-E_{2}A\cdot E_{2}C|+ \\ +|F_{1}B\cdot F_{1}A-F_{2}B\cdot F_{2}A|>4S \end{gathered}$$
10. Let $ABC$ be an isosceles triangle with $AB=AC$ and $\mathrm{\angle }A={20}^{\circ }$. On the side $AC$ consider point $D$ such that $AD=BC$. Find $\mathrm{\angle }BDC$.
11. 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 $KL{L}_1{K}_1$ is rectangle. Then consider rectangles $MNPQ$ inscribed in the triangle $BLK$, where $M\in KB,N\in BL,P,Q\in LK$ and $M_1{N}_1{P}_1{Q}_1$ inscribed in triangle $D{K}_1{L}_1$ where $P_1$ and $Q_1$ are situated on the $L_1{K}_1$, $M$ on the $D{K}_1$ and $N_1$ on the $D{L}_1$. Let $S$, $S_1$, $S_2$, $S_3$ be the areas of the $ABCD$, $KL{L}_1{K}_1$, $MNPQ$, $M_1{N}_1{P}_1{Q}_1$ respectively. Find the maximum possible value of the expression $$\frac{S_1+S_2+S_3}{S}$$
12. 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 $$\begin{gathered} M{A}_1+M{A}_2+\cdots +M{A}_{2002}+N{A}_1+N{A}_2+\cdots +N{A}_{2002}+ \\ +P{A}_1+P{A}_2+\cdots +P{A}_{2002}\ge 6006. \end{gathered}$$

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