[Shortlists & Solutions] Junior Balkan Mathematical Olympiad 2002

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
   $ 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.
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 $$a_1 \not = 0,\, a_1a_6 + a_3 + a_4 = 2a_2a_5,\, a_1a_3 \ge a_2^2.$$ Prove that $ a_4a_6\le a_5^2$. When does equality holds?
6. Consider integers $ a_i,i=\overline{1,2002}$ such that $$a_1^{ - 3} + a_2^{ - 3} + \ldots + a_{2002}^{ - 3} = \frac {1}{2}.$$ 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 $ 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.
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 $$|D_1B\cdot D_1C - D_2B\cdot D_2C| + |E_1A\cdot E_1C - E_2A\cdot E_2C| + \\ + |F_1B\cdot F_1A - F_2B\cdot F_2A| > 4S$$
10. 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$.
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 $ 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 $$\frac{S_1+S_2+S_3}{S}$$
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 $$ MA_1 + MA_2 + \cdots + MA_{2002} + NA_1 + NA_2 + \cdots + NA_{2002} + \\ + PA_1 + PA_2 + \cdots + PA_{2002}\ge 6006.$$

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