Page 10 - RMGO 4

10 Probleme propuse

Clasa a XII-a

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MGO 156. Fie p un num˘ar prim de forma p = 4k + 3, k ∈ N . Demonstrat , i c˘a
2
2
ecuat , ia x + k + 4 = 9 nu are solut , ii ˆın Z p .
b
b
b
Stelian Corneliu Andronescu s , i Costel B˘alc˘au, Pites , ti
n
MGO 157. Fie n ∈ N, n ≥ 3. Rezolvat , i ˆın R sistemul
+

5 2
 x + 3 = x + 3x 3
1
2


5 2

x + 3 = x + 3x 4
2 3 .
 . . .


 5 2
x + 3 = x + 3x 2
1
n
Mih´aly Bencze, Bras , ov
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MGO 158. Fie n ∈ N , a ∈ R s , i b > 0. Calculat , i integrala
Z 2n−1 2n−1
x (x + a)
dx, x ∈ (0, ∞).
(x + a) 4n + bx 4n
Daniel Jinga, Pites , ti
MGO 159. Fie f : [−1, 1] → R o funct , ie de dou˘a ori derivabil˘a, cu derivata a
doua continu˘a, astfel ˆıncˆat f(−1) = f(1) = 0. Demonstrat , i c˘a are loc inegalitatea
1 Z 1 2
2
00
· f (x) dx ≥ max f (x).
6 −1 x∈[−1,1]
Florin St˘anescu, G˘aes , ti
MGO 160. Fie a, b, c, d ≥ 0 astfel ˆıncˆat ab + ac + ad + bc + bd + cd = 6.
Demonstrat , i c˘a
√
2 + 2 √
2
2
2
2
a + b + c + d + · (abc + abd + acd + bcd) ≥ 2 4 + 2 .
2
Cˆand are loc egalitatea?
Leonard Mihai Giugiuc, Drobeta Turnu Severin