Number 12 Spring 2019

Transcription

1 Number 2 Spring 209 ROMANIAN MATHEMATICAL MAGAZINE Avilble online Founding Editor DANIEL SITARU ISSN-L

2 ROMANIAN MATHEMATICAL MAGAZINE PROBLEMS FOR JUNIORS JP.66. If, b [0; + ) nd n N n 2 then: (b) n 2 n k=0 k b n k n + n + b n 2 Proposed by Nguyen Vn Nho - Nghe An - Vietnm JP.67. Let OABC be tetrhedron with AOB = BOC = COA = 90 nd let P be ny point inside the tringle ABC. Denote respectively by d, d b, d c the distnces from P to fces (OBC), (OCA), (OAB). Prove tht: () d 2 + d2 b + d2 c = OP 2. (b) d d b d c OA OB OC. 27 (c) OA d 3 + OB d3 b + OC d3 c OP 4. JP.68. Let, b, c be positive rel numbers such tht: b c 3 Prove tht: 2 + b 2 + c 2 2. JP.69. Let, b, c be positive rel numbers such tht: +b+c = 3. Prove tht: 4 b 4 + b 4 2c( 3 +) c 4 + c 4 2(b 3 +) 4 2 +b 2 +c 2 2b(c 3 +) 2 Proposed by Hong Le Nht Tung - Hnoi - Vietnm JP.70. Let x, y, z be positive rel numbers such tht: x + y + z = 3. Find the minimum vlue of: P = x 4 y 4 3 4z(x 5 + ) + y 4 z 4 3 4x(y 5 + ) + z 4 x 4 3 4y(z 5 + ) Proposed by Hong Le Nht Tung - Hnoi - Vietnm

3 Romnin Mthemticl Mgzine Spring Edition 209 JP.7. Let ABC be n cute tringle with perimeter 3. Prove tht: + m m b + 3 b m c c R + r. JP.72. Let, b, c be positive rel numbers such tht: bc =. Prove the inequlity: 4 b b 4 c 4 b c 4 4 c ( + b + c) 5 Proposed by Hong Le Nht Tung - Hnoi - Vietnm JP.73. Prove tht in ny tringle ABC, sin A + sin B + 6R sin C r JP.74. Prove tht in ny tringle ABC, h + h b b + h c c 6( + cos A cos B cos C) JP.75. Prove tht in ny cute tringle ABC, m r + m b r b + m c r c s 2 JP.76. If, b > 0, then: ( + b) sin x x + 2b + b tn x x > 4 2b (0; + b, x π ) 2 Proposed by Rovsen Pirguliyev - Sumgit - Azebijn JP.77. If, b, c 0 then: 2( + b + c) + cyc 2 + b 2 b 3( b + bc + c) Proposed by Dniel Sitru - Romni 2 c Dniel Sitru, ISSN-L

4 Romnin Mthemticl Mgzine Spring Edition 209 JP.78. If, b > 0 then: 3 + b 3 + ( 2 + b 2 ) b 2 + b b 2 > 4b 2 + b 2 Proposed by Dniel Sitru - Romni JP.79. In cute ABC the following reltionship holds: cos A b cos B + b cos B c cos C + c cos C cos A 3 8 cos A cos B cos C Proposed by Dniel Sitru - Romni JP.80. If, b 0 then: { 4b 2 + b 2 ( + b b 2 ) 4b 2 + b 2 ( 2 + b 2 )( + b b 2 ) Proposed by Dniel Sitru - Romni PROBLEMS FOR SENIORS SP.66. Let n N nd k R, k = ; n. Find: ( n ) Ω = ln (x k ) dx k= (x > mx{ k k = ; n}) Proposed by Nguyen Vn Nho - Nghe An - Vietnm SP.67. Let x, y, z be positive rel numbers such tht: xyz =. Prove tht: x 2(x 4 + y 4 ) + 4xy + y 2(y 4 + z 4 ) + 4yz + z 2(z 4 + x 4 ) + 4zx + 2(x + y + z) Proposed by Hong Le Nht Tung - Hnoi - Vietnm SP.68. Let x, y, z be positive rel numbers. Find the minimum possible vlue of: x y + z + y z + x z x + y c Dniel Sitru, ISSN-L

5 Romnin Mthemticl Mgzine Spring Edition 209 SP.69. Prove tht for ll non-negtive rel numbers, b, c b + c + + b c c b + 3 SP.70. Let, b, c, d be positive rel numbers such tht + b + c + d = 2. Prove tht: b + 3 cd + b c + 3 db + c d + 3 bc + d + 3 bcd 2 SP.7. Let, b, c be positive rel numbers such tht: bc =. Find the minimum vlue of: P = 4 b 5 5( 4 + 4) + b 4 c 5 5(b 4 + 4) + c 4 5 5(c 4 + 4) Proposed by Hong Le Nht Tung - Hnoi - Vietnm SP.72. Prove tht for ny rel numbers x, y, z: (x + y + z)(y + z x)(z + x y)(x + y z) 4y 2 z 2. SP.73. Prove tht for ny positive rel numbers x, y, z: x 2 y 2 + z 2 + y 2 z 2 + x 2 + z 2 x 2 + y 2 x 3 + y 3 + z 3 2. SP.74. Prove tht for ny positive rel numbers, b, c, x, y, z: ( 3 + 3x 3 )(b 3 + 3y 3 )(c 3 + 3z 3 ) (yz + bzx + cxy + xyz) 3 SP.75. Let x, y, z be positive rel numbers such tht: x 2 + y 2 + z 2 + 2xyz =. Find the minimum vlue of: P = x 3 + 3y 2yz + y 3 + 3z 2zx + z 3 + 3x 2xy Proposed by Hong Le Nht Tung - Hnoi - Vietnm 4 c Dniel Sitru, ISSN-L

6 Romnin Mthemticl Mgzine Spring Edition 209 SP.76. Prove tht if m [0, ), x, y, z, t (0, ), then in ny tringle ABC, with the usul nottions holds: (xm 2 + ym2 b )m+ cyc (zb 2 + twc 2 c 3 3m+ )m 2 (x + y) m+ S. (4z + 3t) m Proposed by D.M. Bătineţu-Giurgiu, Neculi Stnciu - Romni SP.77. Prove tht if m [0, ), x, y, z, t (0, ), then in ny tringle ABC, with the usul nottions holds: (x 2 + ym 2 b )m+ cyc (zh 2 c + th2 )m (4x + 3y)m+ 3 m 2 (z + t) S. m Proposed by D.M. Bătineţu-Giurgiu, Neculi Stnciu - Romni SP.78. Prove tht if m [0, ), x, y, z, t (0, ), then in ny tringle ABC, with the usul nottions holds: (x 2 + ym 2 b )m+ cyc (zm 2 c + tm2 )m (4x + 3y)m+ 3 m 2 (z + t) S m Proposed by D.M. Bătineţu-Giurgiu, Neculi Stnciu - Romni SP.79. If x [0, ) then: cos x x 3 + tn 3 x + sin x + e x Proposed by Seyrn Ibrhimov - Msilli - Azerbidin SP.80. If x, y, z R + x 2 + y 2 + z 2 = 3 n, n N then: 4 x + y + 4 x + z + 4 y + z n+ Proposed by Seyrn Ibrhimov - Msilli - Azerbidin UNDERGRADUATE PROBLEMS UP.66. Solve the eqution in R: x 3 2x 2 + 2x x 2 x x 3x 4 = x4 3x Proposed by Hong Le Nht Tung - Hnoi - Vietnm c Dniel Sitru, ISSN-L

7 Romnin Mthemticl Mgzine Spring Edition 209 UP.67. Let, b, c be positive rel numbers such tht: bc =. Find the mximum vlue of: P = b b 4 4b + 2c c 4 4c Proposed by Hong Le Nht Tung - Hnoi - Vietnm UP.68. Let be > 0 nd f : (, ) (, + ) R; f(x) = x 2 +(2+)x Find: n n 2 p k= p f (n) (k) Proposed by Mrin Ursărescu - Romni UP.69. Let be the sequence x 0 > 0 nd x p + xp xp n = p+ x n+, n N, p N. Find: n np+ x p2 +p+ n. Proposed by Mrin Ursărescu - Romni UP.70. Find: rctn(nx) ln( + x) dx n 0 + x 2 Proposed by Mrin Ursărescu - Romni UP.7. Find tht in ny cute-ngled ABC the following inequlity holds: ( sin A min sin B + sin C, sin B sin A + sin C, sin C ) cos A + cos B + cos C sin A + sin B 3 ( sin A mx sin B + sin C, sin B sin A + sin C, sin C ) sin A + sin B Proposed by Mrin Ursărescu - Romni UP.72. Let be A M 5 (R), invertible such tht: det(a 2 +I 5 ) = 0. Prove tht: Tr A = + det A Tr A Proposed by Mrin Ursărescu - Romni 6 c Dniel Sitru, ISSN-L

8 Romnin Mthemticl Mgzine Spring Edition 209 UP.73. Find: Ω = n n n 6 2 i + i=2 ( ) 2i i + 3 n i=2 ( ) 2i i Proposed by Dniel Sitru - Romni UP.74. If f : [, b] [, ); 0 < b; f integrble then: b b b 3 + f(x) + f(y) + f(z) f(x)f(y) + f(y)f(z) + f(z)f(y) dxdydz (b )3 + ( b Proposed by Dniel Sitru - Romni dx f(x) ) 3 UP.75. In cute ABC the following reltionship holds: b 2 + c 2 + 2bc b 2 + c c c c b b 2 + 2b 2 + b 2 c 2 > 9 Proposed by Dniel Sitru - Romni UP.76. Let, b be positive rel numbers such tht: + b = 2. Find the minimum vlue of: P = 3 + b b + 3 b Proposed by Hong Le Nht Tung - Hnoi - Vietnm UP.77. If x, y, z, t > then: (log xzt x)(log xyt y)(log xyz z)(log yzt t) < 6 Proposed by Dniel Sitru - Romni ( ) UP.78. Let be A = ; B = 0 (e A - exponentil mtrix) ( ) 0. Find: Ω = e A (e B ) ; Proposed by Dniel Sitru - Romni UP.79. In ABC the following reltionship holds: m 5 m b + 5 mb m c + 5 mc m 5 m m c 5 mb m 5 mc m b < Proposed by Dniel Sitru - Romni c Dniel Sitru, ISSN-L

9 Romnin Mthemticl Mgzine Spring Edition 209 UP.80. If f : (0, ) (0, ) such tht exists f(x + ) x xf(x) Ω = x ( (f(x)) x = > 0 nd exists x x ( (f(x)) 2 (f(x + )) x+ x+ (f(x)) (x + ) 2 then find: x ) x 2 Proposed by D.M. Bătineţu-Giurgiu, Neculi Stnciu - Romni Mthemtics Deprtment, Theodor Costescu Ntionl Economic, College Drobet Turnu - Severin, MEHEDINTI, ROMANIA ) 8 c Dniel Sitru, ISSN-L