Q2 Matemática (IMO Longlists 1989)

02 | IMO Longlists | 1989 | Matemática

ABC $\overset{e}{ˊ} u m t r i \overset{a}{^} n gu l o, abi sse t r i z d o \overset{a}{^} n gu l o$ A $e n co n t r aa c i rc u n f er \overset{e}{^} n c ia d o t r i \overset{a}{^} n gu l o$ ABC $e m$ A_1 $, os p o n t os$ B_1 $e$ C_1 $s \tilde{a} o d e f ini d os d e f or ma se m e l han t e . F a \overset{c}{\c} a co m q u e$ AA_1 $e n co n t re a s l inha s q u e bi ssec t am os d o i s \overset{a}{^} n gu l ose x t er n ose m$ B $e$ C $e m$ A_0 $. De f ina$ B_0 $e$ C_0 $d e f or ma se m e l han t e . P ro v e q u e a \overset{a}{ˊ} re a d o t r i \overset{a}{^} n gu l o$ A_0B_0C_0 = 2 \cdot $\overset{a}{ˊ} re a d o h e x \overset{a}{ˊ} g o n o$ AC_1BA_1CB_1 \geq 4 \cdot $\overset{a}{ˊ} re a d o t r i \overset{a}{^} n gu l o$ ABC$.