Q2 Matemática (IMO Shortlist 1991)

02 | IMO Shortlist | 1991 | Matemática

ABC $\overset{e}{ˊ} u m t r i \overset{a}{^} n gu l o d e \overset{a}{^} n gu l o a gu d o .$ M $\overset{e}{ˊ} o p o n t o m \overset{e}{ˊ} d i o d e$ BC $e$ P $\overset{e}{ˊ} o p o n t o d e$ AM $t a lq u e$ MB = MP $.$ H $\overset{e}{ˊ} o p \overset{e}{ˊ} d a p er p e n d i c u l a r d e$ P $a$ BC $. A s l inha s q u e p a ss am p or$ H $p er p e n d i c u l a res a$ PB $,$ PC $e n co n t r am$ AB, AC $res p ec t i v am e n t ee m$ Q, R $. M os t re q u e$ BC $\overset{e}{ˊ} t an g e n t e a oc \overset{ı}{ˊ} rc u l o p or$ Q, H, R $e m$ H $. F or m u l a \overset{c}{\c} \tilde{a} o O r i g ina l : P a r a u m t r i \overset{a}{^} n gu l o a gu d o$ ABC, M $\overset{e}{ˊ} o p o n t o m \overset{e}{ˊ} d i o d ose g m e n t o$ BC, P $\overset{e}{ˊ} u m p o n t o n ose g m e n t o$ AM $t a lq u e$ PM = BM, H $\overset{e}{ˊ} o p \overset{e}{ˊ} d a l inha p er p e n d i c u l a r d e$ P $p a r a$ BC, Q $\overset{e}{ˊ} o p o n t o d e in t ersec \overset{c}{\c} \tilde{a} o d ose g m e n t o$ AB $e a re t a q u e p a ss a p or$ H $q u e \overset{e}{ˊ} p er p e n d i c u l a r a$ PB, $e p or f im,$ R $\overset{e}{ˊ} o p o n t o d e in t ersec \overset{c}{\c} \tilde{a} o d ose g m e n t o$ AC $e a re t a q u e p a ss a p or$ H $q u e \overset{e}{ˊ} p er p e n d i c u l a r a$ PC. $M os t re q u e a c i rc u n f er \overset{e}{^} n c ia d e$ QHR $\overset{e}{ˊ} t an g e n t e a o l a d o$ BC $n o p o n t o$ H.$