József Wildt International Mathematical Competition

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1 József Wildt Iteratioal Mathematical Competitio The Editio XXVIII th, 28 The solutio of the problems W. - W.6 must be mailed before 26. October 28, to, str. Hărmaului 6, 556 Săcele - Négyfalu, Jud. Braşov, Romaia, beczemihaly@gmail.com; beczemihaly@yahoo.com W. We cosider a prime umber 3, two matrices A,B M Q), AB = BA, ad such that ε = cos 2 )π +isi 2 )π a). deta εb) = b). for all i {,,..., 2} c). ε ki det ε k )detε k B A) k= detb > Show that whatever m is the atural umber ad whatever the ratioal umbers b,b,...,b m, white m b k A k B m k O the k= m ) det b k A k B m k k= Flori Stăescu W2. Determie the biggest real umber α, with the propoerty that for ay fuctio f : [,] [, ) which meets the requiremets: i). f it s covex ad f ) = ii). exist ε,) such that f it s differetiable o [,ε) ad f ) ; iequality holds x 2 x dx x+α) f x) ) 2 dx f t)dt f t)dt Flori Stăescu W3. Cosider the complex umbers a,b,c,d, white module oe, which have the followig properties: a). arga < argb < argc < argd 2 Mathematics Subject Classificatio. -6. Key words ad phrases. Cotest.

2 b). Show that: 2 b+ai)c+bi)d+i)a+di) [a c)b d)] 2 abcd i+4a max{ 2 i+8ab d) ; i+4b 2 } i+4ba c) Note. If a = rcost+isit), t [,2π), the t = argz. 4 7 = 4 Flori Stăescu W4. Let F ) ad L ) be the Fiboacci ad the Lucas sequece, respectively. Compute the followig limits: a). lim )!!F + 2 ) 2 )!!F b). lim )!!L + 2 ) 2 )!!L D.M. Bătieţu-Giurgiu ad Neculai Staciu W5. Show that i ay triagle ABC with usual otatios) holds the followig iequalites: a). s 2 +r 2 +4Rr ) 2 8r4R+r) s 2 r 2 4Rr ) b). s 2 +r 2 8Rr ) 2 8 8R 2 +r 2 s 2) s 2 4RR+r) ) D.M. Bătieţu-Giurgiu ad Neculai Staciu W6. For N {},x k R+, )k =,, X t) = x t k, t R, X ) = x k = X, m [, ) show that W7. Fid the sum x k X m) x m ) ) m k X m + =! ) D.M. Bătieţu-Giurgiu ad Neculai Staciu Moubiool Omarjee W8. Cosider the real sequece a =,a =,a 2 = a,a 2+ = a +a +. Prove that N, 2 a ϕ where ϕ = Moubiool Omarjee

3 W9. Fid a example of field K, a iteger d, a ifiite subgroup G of GL d K) such that there exist N, that verify g G, g N = I d. W. Let a > be a real umber. Fid the value of the sum Here,! =! = ). 3 a )! W. Let a be a iteger. Calculate ) + +)a+ lim Γ ) ) a+ Γ, a a where Γ is the Euler gamma fuctio. W2. Let A,A 2,,A M 2 C), 2) be the solutios of the equatio ) X 2 3 =. 4 6 Prove that TrA k ) =. W3. Fid i closed form the value of ) 2 3 =3 k=+ ) k 2k 3 + = ) 2k )2+2k +) Moubiool Omarjee José Luis Díaz-Barrero José Luis Díaz-Barrero José Luis Díaz-Barrero W4. Let a,b,c ad a+b+c =. Prove that 3 4+7a2 b b 2 c+ 3 ) 4+7c a abc 5 Paolo Perfetti Paolo Perfetti W5. Evaluate ) kγk 2 +) k= Γ k+3 2 ) W6. Let N, 2 ad the umbers a i,b i R,b i > where i {,2,...,}. Prove that ) b +b b ) 2 a 2 b 2 + a2 2 b a2 2 b 2 a +a a ) 2 + Paolo Perfetti

4 + 2 ) a a 2...a ) 2 Ovidiu T. Pop W7. Prove that where a {,}. z z +a z z 2, z C Ovidiu T. Pop W8. Let a,b,c be oegative real umbers such that ab+bc+ca = 3α 2, α. Fid the miimum of +a 2 )+b 2 )+c 2 ) ad the values a,b,c) where the miimum is attaied. W9. ). Prove that the fuctio f :, + ) R, where is ijective 2). Solve the equatio fx) = e x +lx e 2x +e x l 2x 2x = e W2. Let f : [, ] [, ) be a covex ad itegrable fuctio, with f) =. Show that a). Paolo Perfetti Ioel Tudor b). x 2 fx)dx + fx)dx x 28 l+x 2 )dx > 44 Mihaela Berideau W2. Let G,m) be the -graphs with vertices ad m edges. Draw these graphs ad describe them about their plaarity ad covexity, if they have the ext characteristic polyomial: P λ) = λ 4 3λ 2 +. Fid SpecG) ad study if these graphs are cospectral. Laureţiu Moda

5 W22. Let x ) be the recurret sequece: x + = + x 2,,x = 2 If we cosider the recuret sequece y = 2 x,, study the covergece of the series: ad i the covergece case, compute its sum. Laureţiu Moda ) ) W23. Fid the smallest atural umber 2, so that ord 3 = 4 i Z, ) ad ord 3 = 5 i Z,+). Let P Z [x] be the polyomial: Fid the decompsitio of P i irreductible factors. W24. Prove that the iequality y P x) = 4x 4 x 3 + 6x 2 9 L 6 L 4 +L2 L2 2 +L4 2 ) 2L +L 2 ) + L 6 2 L 4 2 +L2 2 L2 3 +L4 3 ) 2L 2 +L 3 ) + L 6 + L 4 +L2 L2 +L 4 ) 2L +L ) + L 6 L 4 +L 2 L 2 +L4 ) 2L +L ) 2 L +2 ). 3 W25. Let a,b,c be o egative iteger umbers such that a+b = c. Prove that a ) ) b [ 2] ) i + = + c c i i= i i )) ab ) i c 2. Laureţiu Moda W26. Let x,y,z be positive real umbers, ad ad m itegers. Fid the maximal value of the expressio x+y m+)x++m)y +mz + y +z m+)y ++m)z +mx + z +x + m+)z ++m)x+my. W27. If a,a 2,...,a e, with the otatios: A := a k, G := prove that: a k,h := a k Ágel Plaza Ágel Plaza Ágel Plaza

6 A GH G AH H AG Dori Mărghidau W28. If a,a 2,...,a >, with the otatios: prove that: a). b). A [a,a 2,...,a ] := = a k, G [a,a 2,...,a ] := a k,h [a,a 2,...,a ] := a k A [a,a 2,...,a ]) A[a,a2,...,a] G [a a,aa2 2,...,aa ]; [ ] G a /a,a /a2 2,...,a /a H [a,a 2,...,a ]) H[a,a 2,...,a] Dori Mărghidau W29. Let u : R R be a cotiuos fuctio ad f :, ), ) be a solutio of differetial equatio y x) yx) ux) = for ay x, ). Fid e x ux) e x +fx)) 2 dx. W3. If x k >, k =,2,...,, the W3. For all N, the x 2 k x 4 k W x 3 k D.M. Bătieţu-Giurgiu ad Neculai Staciu x 6 k Li Yi l ) 2.) 2 where W := 2 )!! 2)!! is Wallis product. First of all, we prove the followig Lemma. Lemma. Let a >,a ad ) lim a = α 2.2) a + The W32. If x = ad lim a ) l = α 2.3) x = x +) 3 Li Yi

7 for all, the [x ] = for all, whe [ ] deote the iteger part. Tibor Jakab W33. Prove that if m, N the: π 2 si xdx π 2 cos m xdx ) +m 2 π Daiel Sitaru W34. Let be ε k = cos Prove that if x < ; y < the: 2kπ 2kπ +isi ;k,2; N. 2 i= x ε i )y ε i ) < 4+) Daiel Sitaru W35. Compute L = lim xsiπx x+ x)k 2xdx Daiel Sitaru W36. Prove that if a,b,c R;a 2 +b 2 +c 2 the: ab+bc+ca) 2 b a) 2 c a) 2 c b) 2 a 6 +b 6 +c 6 2 a 4 +b 4 +c 4 a 2 b 2 b 2 c 2 a 2 c 2) Daiel Sitaru W37. If ABCD its a iscriptible quadrilateral: AB = a, BC = b, CD = c, DA = d, the ) a+b+c+d > 2 + abcd ab+cd ad+bc Daiel Sitaru W38. Let x,y,z >, k > such that The x 2 +y 2 +z 2 +xyz = 4 x+2) k 2 + y +2) k 2 + z +2) k 2 < 2 k +2 k 2 Marius Drăga

8 W39. Compute cos 3 x+ 2kπ ), N k= Liviu Bordiau W4. Let x,x 2,y,y 2,z,z 2 >. The x +x 2 ) 2 { x 2 y +y 2 )z +z 2 ) max, y z x 2 2 y 2 z 2 } Marius Drăga ad Sori Rădulescu W4. Let A,B be two square matrices from C, a,b,c,d itegers such that a < b,q the quotiet of dividig b by a, such that cq d ad A a B c = A b B d = I. The there is a strictly iteger umber û such that B u = I. W42. Let be 2, N ad x,x 2,...,x >. The ) x +x x x x 2...x ) xi +x i2 +...x ik max i <i 2<...<i k x i x i2...x ik k ad Marius Drăga ad Marius Drăga W43. Let p,p 2,p 3 be positive real umbers ad the fuctios f,g : [,+ ) R, f x) = [ ] x], gx) = [ x [x] 2. The is true the iequality: gp ) f p 2 )f p 3 )+f p )gp 2 )f p 3 )+f p 2 )f p )gp 3 ) W44. Let x,y,z > ad α > 9 such that The f p p 2 p 3 )+gp )gp 2 )gp 3 ) x+y +z) x + y + ) = α z x 5 +y 5 +z 5) x 5 + y 5 + ) z 5 α 5 25α 4 +23α 3 95α 2 +75α 945 ) 6 ad Marius Drăga Maria Cucoaeş ad Marius Drăga W45. Let O a iterior poit of ABC triagle ad {M} = AO BC, {N} = BO AC, {P} = CO AB. We deote S = σ[bom], S 2 = σ[moc], S 3 = σ[noc], S 4 = σ[aon], S 5 = σ[aop], S 6 = σ[bop] such that S 5 S 3. The

9 2S +2S 5 3 S S 5 +S 2 +S 3 +S 4 +S 6 Maria Cucoaeş ad Marius Drăga W46. Compute cot π ) ) ) 2π 4π +cot +cot, N Maria Cucoaeş ad Marius Drăga W47. Let ABC be a acute triagle, deote A,B,C the midpoits of sides BC,CA,AB. The lies OA,OB,OC itersect the circumcircle i poits A 2,B 2,C 2. Deote x = A A 2,y = B B 2,z = C C 2. prove that ). 2). x+y +z 3r R x)r y)r z) = R 3 cosacosbcosc W48. Let be f : [, ], + ) a cotiuously fuctio. Prove that if exist α > for which where N the α is uique. x α f x)dx +)α+ W49. If x k > k =,2,...,), the x k x k W5. Compute i<j f + x)dx x i x j ) 2 Nicuşor Miculete π 2 x 3 six+λcosx)dx 5+x3 cosx+ 5+x 6 cos 2 x π 2 W5. Prove that [ where [ ] deote the iteger part. ] k4 +2k 3 +2k 2 +k + = k 4 +2k 3 k +

10 W52. Let A,A 2,...,A be a covex polygo. Prove that sia k si π+a k W53. Let a,b {2,3,4,5,6,7,8,9} ad m N. Prove that exist N such that the first digit of a are b m. W54. For all ivertable matrices A,B M 2 C) holds ). 2). 2TrAB)+TrA B)detA+TrB A)detB = 2TrA)TrB) TrABA)+TrBAB)+TrA)detB +TrB)detA = TrA)+TrB))TrAB) W55. If a N, a,p) = where p is a prime, the is divisible by p +) 2. W56. I all tetrahedro ABCD holds: ). ) a pk p ) 2) h a h b h a h c h a h d h b h c h b h d h c h d 2r r a r b r a r c r a r d r b r c r b r d r c r d r 2 W57. Prove that π 2 si 2k+ x+cos 2k+ x ) π dx 2 2 ) N 2; Daiel Sitaru W58. Let f be th Fiboacci umber defied by recurrece f + f f =, N ad iitial coditios f =,f =. Prove that ) f 6f + is divisible by 5 for ay N Arkady Alt

11 W59. Let E be a Ier Product Space with dot product ad F be proper ozero subspace. Let P : E E be orthogoal projectio E o F. a). Prove that for ay x,y E, holds iequality b). Determie all cases whe equality occurs. x y x P y) y P x) x y W6. Let x,a,h be arbitrary real umbers such that x >,a,h > ad let sequece x ) defied recursively by x = x, x + = +a +a+h x, N {}. Explore for which h the ifiite sum x coverges ad fid it i the case of covergece. = Arkady Alt Arkady Alt

József Wildt International Mathematical Competition

József Wildt International Mathematical Competition József Wildt Itertiol Mthemticl Competitio The Editio XXVII th, 17 The solutio of the problems W.1 - W.6 must be miled before 3. September 17, to, str. Hărmului 6, 556 Săcele - Négyflu, Jud. Brşov, Romi,