Concursul ,, Evaluare in matematica ,,

[alex2008]

Se considera triunghiul \( ABC \) de arie \( S \) , numerele \( x,y,z \in (0;\infty) \) si punctele \( D\in (BC)\ , E\in (AC)\ , F\in (AB)\ , \frac{BD}{DC}=x\ , \frac{CE}{EA}=y\ , \frac{AF}{FB}=z\ ; \{M\}=(AD)\cap (BE)\ , \{N\}=(BE)\cap (CF)\ , \{P\}=(AD)\cap (CF) \). Se mai considera adevarate relatiile \( \frac{MA}{MD}=\frac{x+1}{xy}\ , \frac{NB}{NE}=\frac{y+1}{yz} \) si \( \frac{PC}{PF}=\frac{z+1}{zx} \) .

a)Sa se arate ca \( \frac{S_{ADB}}{S}=\frac{BD}{BC} \) ; \( \frac{S_{AMB}}{S_{ADB}}=\frac{AM}{AD} \) si \( S_{AMB}=\frac{x}{xy+x+1}\cdot S \) .

b)Sa se arate ca \( S_{MNP}=(1-\sum_{cyc}\frac{x}{xy+x+1})\cdot S \) si ca \( \sum_{cyc}\frac{x}{xy+x+1}\le 1 \) .

c)Sa se arate ca \( \forall s \in (0;\frac{S}{4}]\ , \exists X \in (AB)\ , \exists Y\in (BC)\ , \exists Z\in (CA) \) astfel incat \( S_{XYZ}=s \) .

d)Sa se arate ca daca \( xyz=1 \) atunci \( S_{DEF} \le \frac{S}{4} \) .



Nicolae Musuroaia , Baia Mare , ,,Concursul de evaluare in matematica,,