Τετάρτη, 11 Οκτωβρίου 2017

2002 JBMO Shortlist 12

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: ${\frac{S_1+S_2+S_3}{S}}$


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