Barem de corectare OLM Clasa a VIII-a, 2013 1. a) 2013 1 = − + + + − − − z x y xz yz xy xyz ………………………………….……...… (1p) Grupând termenii convenabil rezultă: ( ) ( ) ( ) 61 11 3 2013 1 1 1 ⋅ ⋅ = = − − − z y x ..….………... (1p) Soluţii sunt toate permutările tripletelor ( ) ( ) ( ) ( ) ( ) 2 2 201! 2 ! 6"2 2121#! 2 62 3! !12 62 ......………………….......…… (1p) b) ( ) ( ) ( ) 2 2013 3 11 61 2 1 36 2$ 36 2$ = × × = − − + .……………………………………… (2p) ( ) ( ) 2 2 2 2 2 2 2 2 1 36 2$ "2 $0 36 2$ − − = − − + ..…………………………………………….. (1p) "2 $0 36 2$ a b c d ⇒ = = = = .........................………………………………………… (1p) 2. ( ) ( ) ( ) ( ) b a b a b a G a H b a b a b a H a G + = + = 2 2 …………...……………………….. (2p) b a b a b a b a b a b a + ≤ + ⇒ + ≤ + 2 1 2 2 …………………………………….………… (2p) ( ) ( ) 0 2 2 2 2 ≥ − + ⇒ + ≤ + + ⇒ + ≤ + ab b a b a ab b a b a b a ....…………….….. (2p) ( ) ( ) ( ) ( ) ( ) 2 0 a b G a H a b H a G a b − ≥ ⇒ ≤ ..…………………………………….. (1p) 3. %ie & mi'locul se(mentului )*+,…………………………….........……..…………… (1p) -n 2 2 BC AD ME NE = = )linii mi'locii n triun(/iuri,…………......………….... (1p) 2 2 BC AD ME NE MN + = + = .......…………………………..................……………….... (2p) M E N ⇒ coliniare…………………………………...........................….....…. (1p) 0in ME BC EN AD avem BC AD deci A B C D coliniare …................…... (2p) 4. a) ABC ∆ este ec/ilateral...………................................…… (1p) DOC ∆ 1 3 10 = ⇒ ∆ OC BOA ..................................……………………………… (1p) ( ) 1 $ 3 $ 3 3 3 OP CAT AT BC OP BC AT OP AT ∆ ⊥ ⊥ ⇒ = = ⇒ = .…………………… (1p) MOP ∆ dreptun(/ic 10 MP ⇒ = ………….............................…………………………… (1p) b) 2erpendiculara din 3 pe 4+ ) n 5432, este distanţa căutată..…… (2p) 6 11 $ = ⇒ ⋅ = OS MP OP MO OS ..................................………………………………… (1p)
Barem de corectareOLMClasa a VIII-a, 20131. a) ....(1p)Grupnd
termenii convenabil rezult:......(1p)Soluii sunt toate permutrile
tripletelor .............(1p)b) .(2p)
....(1p)
.........................(1p)2. .....(2p)
.(2p)
.......(2p)
....(1p)3. Fie E mijlocul segmentului (AC)...........(1p)n
(linii mijlocii n triunghiuri)....................(1p)
.............................(2p)
coliniare...................................(1p)Din avem , deci
coliniare ......................(2p)4. a) este
echilateral...................................(1p)
~..................................(1p)
.(1p)
dreptunghic.............................(1p)b) Perpendiculara
din O pe MC (n MOP) este distana cutat..(2p)
