Number 8 Spring 08 ROMANIAN MATHEMATICAL MAGAZINE Available online www.ssmrmh.ro Founding Editor DANIEL SITARU ISSN-L 50-0099
ROMANIAN MATHEMATICAL MAGAZINE PROBLEMS FOR JUNIORS JP.06. Let a, b, c > 0. Prove that: 8(a b c 5(ab bc ca 0(a b c (a b(b c(c a Proposed by Nguyen Ngoc Tu Ha Giang Vietnam JP.07. Prove that in any triangle ABC with incentre I the following relationship holds: AI a BI b CI c R R(R r, w a w b w c r where R is the circumradius, r is the inradius of triangle ABC and w a, w b, w c are the lengths of the internal bisectors of the angle opposite of the sides of lengths a, b, c, respectively. JP.08. If a, b > 0, then: 4 ab sin x ( tan x x b ( a > 6 ab, x 0, π x JP.09. If a, b > 0, then: (a b sin x x ab a b tan x > 6ab (0, x a b, x π JP.0. If x, y, z (0, and ABC is a triangle, then prove that: sin A x( x sin B y( y sin C z( z 4 (R r R JP.. Prove that: (i If a, b, c, d, x, y, z, t R, then: 4(a b c d x y z t 8 (a b c d (x y z t
Romanian Mathematical Magazine Spring Edition 08 (a b c d x y z t ; (ii If a, b, c, m, n, p, x, y, z R then: 5(a b c m n p x y z (a b c (m n p (x y z (a b c(m n p(x y z JP.. Prove that if a, b R, x, y, u, v R then: (a x y b u (a y v x b v (a b u JP.. Let x, y, z be real numbers such that: x y y z z x 4(xy yz zx xyz = 5 Find the minimum value of the expression: P = (x 4xy 5y (y 4yz 5z (z 4zx 5x 6x y z Proposed by Do Quoc Chinh Vietnam JP.4. Let a, b, c be positive real numbers. Prove that: (a ln ( ab ln ( bc ln ( ca 4 ln a ln b ln c = ln (abc c a b (b ln(ab ln(bc ln(ca = ln (abc ln a ln b ln c JP.5. If a, b, c are positive real numbers such that a b c = then: ( a ( b ( c 8 b c a a b c Proposed by Pham Quoc Sang - Ho Chi Minh - Vietnam JP.6. If a, b, c are positive real numbers such that abc = then: a a bc b b ca c c ab Proposed by Pham Quoc Sang - Ho Chi Minh - Vietnam JP.7. If a, b, c are positive real number then: ab c ca bc a ab ca b bc (a b c. max{(a b, (b c, (c a } Proposed by Pham Quoc Sang - Ho Chi Minh - Vietnam c Daniel Sitaru, ISSN-L 50-0099
Romanian Mathematical Magazine Spring Edition 08 JP.8. Let a, b, c be the three sides of a triangle. Prove that: a b b c c a (a b c a b c Proposed by Nguyen Ngoc Tu - Ha Giang - Vietnam JP.9. Let a, b, c be positive real numbers such that abc =. Prove that: a b c (c a b b c a (a b c c a b (b c a a b c abc Proposed by Do Quoc Chinh - Ho Chi Minh - Vietnam JP.0. Let a, b, c be positive real numbers and k [; ]. Prove that: a ab ca kbc b bc ab kca c ca bc kab 9 (k (ab bc ca Proposed by Do Quoc Chinh - Ho Chi Minh - Vietnam PROBLEMS FOR SENIORS SP.06. In ABC the following relationship holds: (a cot 0 b cot 40 c cot 80 > 9 ( a b c r a r b r c SP.07. Prove that: ( ( arctan xdx 0 0 dx arctan ( x x > 4 SP.08. If a, b, c > 0, a b c = abc then: 4(a b(a c 4(b c(b a 4(c a(c b a b c (b c (c a (a b c Daniel Sitaru, ISSN-L 50-0099
Romanian Mathematical Magazine Spring Edition 08 SP.09. If a, b, c 0; Ω(a = ( a 0 sin x dx then: x e π(bω(acω(baω(c (a b (b c (c a SP.0. Let m, x, y, z > 0 be positive real numbers and F be the area of the triangle ABC. Prove that: a m x m (y z bm y m m (z x cm z m m (x y m m F m m ( SP.. Let x, y, z > 0 be positive real numbers and F be the area of the triangle ABC. Prove that: (y z a 4 x (z x b 4 y (z x c 4 z 64F SP.. Let x, y, z > 0 be positive real numbers and F be the area of the triangle ABC with circumradius R. Prove that: x A y z sin y B z x sin z C x y sin F R SP.. If m N, x, y, z > 0; x y z = s then: m ( sx yz m ( sy zx m ( sz xy m 9(m s SP.4. Let t, u, v, x, y, z > 0 be positive real numbers, t max{u, v} and S = x y z. Prove that: ts uy vz a ts uz vx ts ux vy c 4(t u v S uy vz uz vx ux vy u v 4 c Daniel Sitaru, ISSN-L 50-0099
Romanian Mathematical Magazine Spring Edition 08 SP.5. Let a, b, c be the lengths of the sides of a triangle with inradius r and circumradius R. Let r a, r b, r c be the exradii of triangle. Prove that: 78 r 5 a6 r a b6 r b c6 r c 08R 4 (R r SP.6. A triangle with side lenghts a, b, c has perimeter equal to. Prove that: a b c a 4 b 4 c 4 (a b b c c a a. 8 cos A cos B ( r SP.7. Let ABC be a triangle with inradius r and circumradius R. Prove that: ( R r b. 9 R cos C cos A cos B cos C 9 4 SP.8. Let a, b, c > 0 such that: a b c =. Find the minimum of the expression: P = a 4 b 8 c 8 5bc b 4 c 8 a 8 4ca c 4 a 8 b 8 4ab (a b(b c(c a 6 Proposed by Hoang Le Nhat Tung Hanoi Vietnam SP.9. Let a, b, c > 0 such that: abc =. Find the minimum of the expression: a P = 4(b 6 c 6 7bc b 4(c 6 a 6 7ca c 4(a 6 b 6 7ab (a b(b c(c a 4 Proposed by Hoang Le Nhat Tung Hanoi Vietnam SP.0. In ABC the following relationship holds: a B b C c A 4πs s semiperimeter; a, b, c length s sides; A, B, C angled s measures c Daniel Sitaru, ISSN-L 50-0099 5
Romanian Mathematical Magazine Spring Edition 08 UNDERGRADUATE PROBLEMS UP.06. In ABC triangle the following relationshiph holds: cos A 4 cos B cos C 4 6 t 7 8, t = r 6 R Proposed by Vadim Mitrofanov - Kiev - Ukraine UP.07. Let ABC be a triangle with inradius r and circumradius R. Prove that: ( r cyc sin4 A ( R ( R cyc sin A r 4 r R UP.08. Let ABC be a triangle with circumradius R and inradius r. Prove that: r 6 cos4 A cos 4 B cos 4 C 6( R 8 r R 5 8 UP.09. Let a, b, c and d be positive real numbers. Prove or disprove that: (a b c d abc bcd cda dab 6 UP.0. Let m a, m b, m c be the lengths of the medians of a triangle ABC. Prove that: R m a m b m c r where R, r are the circumradius and inradius respectively of ABC. UP.. For an acute triangle ABC and a positive integer n, prove that: ( n n (sin A sin B sin C n 8 where the sum is over all cyclic permutations of (A, B, C. 6 c Daniel Sitaru, ISSN-L 50-0099
Romanian Mathematical Magazine Spring Edition 08 UP.. Solve in positive real numbers: x y y x = 8 8(x 8 y 8 4x y = xy Proposed by Hoang Le Nhat Tung Hanoi Vietnam UP.. Let x, y, z be positive real numbers. Prove that: x yz y z y zx z x z xy x y 0 6 0x y z (x y (y z (z x Proposed by Do Quoc Chinh Vietnam UP.4. Let a, b, c be the sides and R and r the circumradius and the inradius of triangle ABC respectively. Prove that: a b c 9 4r(4R r UP.5. Evaluate ln(x sin(x dx 0 (x Proposed by Arafat Rahman Akib - Dahka - Bangladesh UP.6. Prove that: (( n H4 n 6H n H( n n= (H( n 8Hn H n ( n 6H(4 n ( = 4Li 5 Proposed by Ali Shather - An Nasiriyah - Iraq, Shivam Sharma - New Delhi-India [ UP.7. Let a, b, c ; a b c =. Prove that: 4 a 4 b 4 c 0 be positive real numbers such that: 40 (ab bc ca ( a b c Proposed by Hoang Le Nhat Tung - Hanoi - Vietnam c Daniel Sitaru, ISSN-L 50-0099 7 =
Romanian Mathematical Magazine Spring Edition 08 UP.8. Show that: x p x dx = p x 0 ( p B, p ; < p < Proposed by Shivam Sharma - New Delhi - India UP.9. Prove that: ( ( H k = 9ζ(ζ(9ζ(ζ(86ζ(4ζ(74ζ(5ζ(6 7ζ( k 9 k= Proposed by Ali Shather - An Nasiriyah - Iraq, Shivam Sharma - New Delhi-India UP.0. Prove that in any acute-angled triangle ABC the following relationship holds: ( (sin A cos A > sin C cos (A B Mathematics Department, Theodor Costescu National Economic, College Drobeta Turnu - Severin, MEHEDINTI, ROMANIA 8 c Daniel Sitaru, ISSN-L 50-0099