Page 110 - MATINF Nr. 11-12

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110 PROBLEME DE MATEMATICA PENTRU CONCURSURI

Clasa a XI-a

M 235. Fie A ∈ M 2 (C) astfel ˆıncˆat tr A 6= 0 s , i det A = 0. Consider˘am ecuat , ia matriceal˘a

3
X + AX(A + X) + X(A + X)A + (A + X)AX = O 2 , X ∈ M 2 (C).

a) Ar˘atat , i c˘a dac˘ X este solut , ie a ecuat , iei, atunci AX = XA.
a
b) Demonstrat , i c˘a ecuat , ia are exact trei solut , ii.
R˘amˆan valabile aceste afirmat , ii ˆın M 3 (C)?

Sorin Ulmeanu s , i Costel B˘alc˘au, Pites , ti

a
M 236. Demonstrat , i c˘a exist˘ un unic s , ir m˘arginit (a n ) n≥1 astfel ˆıncˆat
1 1 1
c n = 1 + + . . . + + − ln n, n ≥ 1,
2 n − 1 n + a n

este s , ir constant. Ar˘atat , i c˘a
1
lim a n = − .
n→∞ 6

Cristinel Mortici, Viforˆata

1 1 1
M 237. Fie x, y, z > 0 astfel ˆıncˆat + + = 2. Demonstrat , i c˘a
x + 1 y + 1 z + 1
1 1 1
√ + √ + √ ≥ 6.
xy xz yz

Mih´aly Bencze, Bras , ov

M 238. Fie a, b, c s , i d numere reale pozitive astfel ˆıncˆat a − 3 ≥ b ≥ c ≥ d.
Rezolvat , i ˆın mult , imea numerelor reale ecuat , ia

x
x
x
x
x
x
x
x
(a + 9) + b + c + d = a + (b + 3) + (c + 3) + (d + 3) .
Marin Chirciu s , i Octavian Stroe, Pites , ti
M 239. Fie a, b, c s , i d numere reale nenegative astfel ˆıncˆat
Ä √ ä
2
2
2
2
a + b + c + d − 4 ≤ 2 + 3 (a + b + c + d − 4) .
Ar˘atat , i c˘a
1 1 1 1
+ + + ≤ 2.
a + 1 b + 1 c + 1 d + 1
Leonard Mihai Giugiuc, Greci – Mehedint , i