Page 109 - MATINF Nr. 11-12

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PROBLEME DE MATEMATICA PENTRU CONCURSURI 109

Clasa a X-a

M 230. Rezolvat , i ˆın mult , imea numerelor reale ecuat , ia

√ √
» »
3
3
2
2
2
2
2
2 (9x − 7x + 2) +6 9x − 7x + 2+18x +3 = 2 (3x − 2x + 1) +6 3x − 2x + 1+15x.
George Mihai, Slatina
M 231. Fie a 1 , a 2 , . . . , a n > 0, n ≥ 2. Demonstrat , i c˘a

1 2 3 n S n
+ √ + √ + . . . + √ ≥ √ ,
3 n S n a 1 a 2 a 3 . . . a n
a 1 a 2 a 3 a n
unde S n = 1 + 2 + . . . + n.

Cˆand are loc egalitatea?

Dorin M˘arghidanu, Corabia

M 232. Rezolvat , i ˆın mult , imea numerelor reale ecuat , ia

x
x
x
2
x
2 + 3 + 9 + 18 = x .
Marin Chirciu, Pites , ti

M 233. Rezolvat , i ˆın mult , imea numerelor reale ecuat , ia

2
3
2
x + x lg 5 + 2x lg 5 + lg 5 − 1 = 0.
Ionel Tudor, C˘alug˘areni

M 234. Fie ABC un triunghi, T ∈ Int (ABC), {M} = AT ∩ BC, {N} = BT ∩ AC s , i
{P} = CT ∩ AB. Demonstrat , i c˘a

Å ã λ Å ã λ Å ã λ
TM TN TP 3
+ + ≥ ,
TA TB TC 2 λ
pentru orice λ ∈ (−∞, 0] ∪ [1, +∞).

Mih´aly Bencze, Bras , ov