József Wildt International Mathematical Competition

Transcription

1 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, E-mil: 1beczemihly@gmil.com; beczemihly@yhoo.com W1. If x,y,z re positive rel umbers, d x +xy + y 3 the compute xy + yz + 3zx. W. Prove tht x x x x+1 x1 x) 4 for ll x 1. W3. Let,b,c,d,e d. = +b+c+d+e = 5; y 3 +z = 9 d z +xz +x = 16 Chg-Ji Zho Perfetti Polo b = b+c+d+e+bc+bd+be+cd+ce+de Prove tht bc. = bc+bd+be+bcd+bde+bce+cd+ce+de+cde 6 ) 3 +5 bc b Perfetti Polo W4. Let p N,p. Determie f : R R cotiuous fuctios, derivble i x =, which ferifies the reltioship: f px) = f p x), for y x R. W5. 1. Prove tht for y turl umber d for y turl umbers k, k = 1,, { 1,,..., } = {1,,...,}, there re positive itegers x 1,x,...,x 1 so tht x 1 x...x = 1 x 1 + x x. Nicole Ppcu. Determie the turl umbers, so tht for y turl umbers k, k = 1,, { 1,,..., } = {1,,...,}, there re positive itegers x 1,x,...,x 1 such tht 1 x 1 < x <... < x d x 1 x...x = 1 x 1 + x x. W6. ). Study the mootoity of the fuctio f :, + ) R where Nicole Ppcu f x) = 4xrctgx+xrctg x 3 3πx+π b). Solve i,+ ) the equtio 4xrctgx+xrctg x 3 = 3x )π.

2 Ioel Tudor W7. Let > be rel umber. Compute the vlue of the followig itegrl cost π 1/t +1 dt W8. Let p,q be iteger umbers d let A = {x R x = 3p +7 q } 3 p 7 q, p,q 1 Show tht A is either ope set or closed set i R with the usul topology. W9. Let p 5 be prime umber. Prove tht p 3 divides p p). W1. Clculte H ζ) ζ+1)), = where ζ deotes the Riem zet fuctio d H is the th hrmoic umber. W11. Clculte π six José Luis Díz-Brrero José Luis Díz-Brrero José Luis Díz-Brrero Ovidiu Furdui 1+ six)) dx. Ovidiu Furdui W1. Let p 3 be prime umber. Solve i M Z p ) the equtio ) p 1 X p =. p 1 1 Ovidiu Furdui W13. Let x,y,z) := xy +yz +xz) x +y +z ) d let,b,c be sidelegths of trigle. Prove tht,b,c) 3,b 3,c 3) 3 4,b 4,c 4) Arkdy Alt W14. Let f u,v) be cotiuous i 1,) homogeeous fuctio of order m tht is for y t > holds f tu,tv) = t m f u,v)) d let D 1 be set of strictly decresig sequeces of positive rel umbers with first term equl to 1. For y sequece x N := x 1,x,...,x,...) D 1 let S f x N ) = f x,x +1 ) if series f x,x +1 ) coverges d S f x N ) = if it diverges. Prove tht =1 =1 f 1,x) if{s f x N ) x N D 1 } = mi x [,1) 1 x m.

3 Arkdy Alt W15. For y give turl umbers m 1, k prove tht... mi m {i 1,i,...,i k } = i 1=1i =1 i k =1 or... i 1=1i =1 i k =1 = m ) 1) m i m +1) i i) s k+m i ), i i=1 where s p ) = k p,p N {}. m mi m {i 1,i,...,i k } = i=1 j=1 i 1) m i m i ) ) i i j s k+m i ) j Arkdy Alt W16. Fid umber of elemets i imge of fuctio [ ] k k : {1,,...,} N {} Arkdy Alt W17. For y turl 3 solve the system. x k+ x k+1 x k = f k,k = 1,,..., x 1 x x 1 = f 1, x x 1 x = f where f, N Fibocci umber, tht is f =,f 1 = 1 d f +1 f f 1 =, Z Arkdy Alt W18. Let p be iteger d positive rel umber. Prove tht = p p ) rct + = πp. Ágel Plz W19. Prove tht if,b,c,d R; +b )c +d ) the: c+d) bc d) 1+ d bc)c+bd) +b )c +d ) +b )c +d ) Diel Sitru W. Prove tht i y ABC the followig reltioship holds: 4 r +r) r +16rr +1r 9+ r 8r r +r Diel Sitru

4 W1. Prove tht if < < b < π the: 3 b cos4x dx > cot+cotb Diel Sitru W. Prove tht if x,y R;z [, ) the: zsix y)cosx+y)+cosx+cosy cosx+z)+cosy +z)+ z Diel Sitru W3. Prove tht if,b,c,1] the: e 1)e b 1)e c 1) e 1) 3 b c Diel Sitru W4. C p π R,R) the set of pice wise cotiuous fuctio π periodic. Let t > π, fid the optiml costt m 1 t), m t) such tht f C po π R,R), π m 1 t) f t π f m t) f. W5. A M C) such tht expa) = I. Prove A is digolisble. Moubiool Omrjee Moubiool Omrjee W6. A GL k) whe K commuity field with chrcteristic differet th. B M,1 k), C M 1, k). Suppose 1 k +CA 1 B = O k. Compute deta+bc). W7. Fid lim! 3 3!...! +1 +1)!! W8. Let γ = l+ W9. Let S = +. 1 k, lim γ = γ. Fid lim γ γ) 1)!!. 1 k, lim s = s Iochimescu costt). Fid lim s s) 1)!!. Moubiool Omrjee D.M. Bătieţu-Giurgiu d Neculi Stciu D.M. Bătieţu-Giurgiu d Neculi Stciu D.M. Bătieţu-Giurgiu d Neculi Stciu W3. Fid the probbility, so tht throwig two dice, their sum to be equl with the lst digit of the umber Lureţiu Mod W31. Let M be the set M = {f : X Y\ X =, Y = m,f surjective}. Study if there re sets X, so tht M = 1 d C 4 m+ = m 1.

5 Lureţiu Mod W3. Let G be uorieted grph, without multiedges d loops, hwig vertices. Let A be the djcecy mtrix of G, with i). A 3 = ii). P G λ) = λ 4 +1)λ +αλ,,α N is the chrcteristic polyomil of G. Fid the spectrum, SpecG) drw the grph d estblish its plrity. Fid other grph G, cospectrl d o-isomorphic with G. W33. Let K N be field with the chrcteristic p, where p is prime umber. Prove: i). 3 p,4 p,5 p determie rithmetic progressio, ii). 3 p+1 = 1+ p+1, p = 1+3 p, 3 4) p 1 = 1. Lureţiu Mod Lureţiu Mod W34. Let A,B M R) two mtrices, t lest oe oiverted so tht A +3AB +B = BA. Prove tht TrAB) = TrA)TrB). Stăescu Flori W35. Let f,g : [,1] R two covex fuctios d cotiuous o [,1] such tht f it is differetible, 1 with f cocve, d g the positive o [,1]. If f 1) f )) g) = f ) gx)dx, the i). Give exmple of such fuctios, where g is ot icresi o [,1] ii). Show tht f x)gx)dx f x)dx gx)dx. W36. Is cosidered,b,c three complex umbers, which hve tht followig properties: ). b + b c + c + b + c b + c = 1 3 b). mxrg,rgb,rgc) π If A = 3 ), b +b show tht the iequlity π rga < rccos 1 A ) W37. Prove tht i trigle ABC we hve the iequlity ) A sec + 3 ) π +A tg 8 + 4R+r. s Stăescu Flori Stăescu Flori Stăescu Flori W38. Fid sum of series l=k+1 k 3 k l+l 3 k 3 l 4 l k). Pál Péter Dályy

6 W39. Let p be positive iteger, d let m be odd positive iteger. Fid the mximl power of tht divides sum S p m) = m k p -t! Pál Péter Dályy W4. Let x, y, z be positive rel umbers such tht xyz = 1. Prove tht x 4 +y 4 +z 4 ) 3x 5 +y 5 +z 5 ). W41. Prove tht ) x b + y x + y b x +y ) for y,b,x,y R+. Pál Péter Dályy W4. Let,b,c be rel umbers, with the property < b c. 1). Prove tht 3 b+5c + 3b c+5 b + 3c +5b c 18 ). If c < 5+3b d b > 5+c 7 the 3 b+5c + b 3b c+5 + c 3c +5b 1 Ovidiu Pop Ovidiu Pop W43. Let be A,B M C) d i,m i N i {1,,...,k} such tht 1,,..., k ) = m 1,m,...,m k ) = 1 d A m1 B 1 = A m B =... = A m k B k = I, the exist r Z such tht A r = B r = I. Mrius Drăg d y W44. Let x ) 1 d y ) 1 be two sequeces of rel umbers such tht lim =, y ) lim = β R, lim x =, lim x k y = α R. Prove tht lim 1+ x 1 ) 1+ x ) x ) ) 1 ) y = α+ β. Mrius Drăg d W45. Let be p i [,1] i = 1,,...,k) such tht p 1 +p +...+p k = 1, > 1 rel umber. Prove tht: W46. If k 1 the k k p i pi) + i=1 +p 1 p +p p p k 1 p k ) +...+p 1 p k +p p p k p k 1 ). i=1 Mrius Drăg d Geerliztio. ) k ) k ) k ) k ) k Mrius Drăg W47. Compute 1 i). lim {x}{x}...{x}dx ii). b {x} dx, < b

7 Mrius Drăg W48. Let A M R) be such tht ij >, i j with the sum of elemets from every 1 lies re zero d the sum from the lie is ozero. The deta. W49. I ll trigle ABC hold W5. I ll trigle ABC hold m ) 3s mi { b) ;b c) ;c ) }. Liviu Bordiu d Mrius Drăg Mrius Drăg h r) h +r +b+c 5 b+c )h r). h +r W51. Let f : R R be cotiuous covex fucto d k,λ k > k = 1,,...,) such tht λ k = 1. Prove tht W5. Prove tht 1 i 1<...<i k λ k k k f x)dx 1 λ k k λ k k f x)dx. 4 k i 1 +3)i +3)...i k +3) = +11) ). 84 W53. If F k deote the k th Fibocci umber F = F 1 = 1, F + = F +1 +F ) the e F k+1 e F k e 1 F+4 5). F k 1 W54. If k > k = 1,,...,) the 1 rctg 1 k rctg. 1 1 k )3 W55. I ll cute trigle ABC holds sia+ cosa ) 3 3+ ) sia cos π 4 A).

8 W56. If P =, P 1 = 1 d P = P 1 +P for ll, the compute Pk+ rctg P k P k Pk+1 P k+ 1 )rctg Pk+1 P k Pk+1 P k+ +1. W57. Prove tht if,b,c,d [1, ) the: b+1 + c d+1 < < b b+c b+c+dw Diel Sitru W I y trigle ABC, we hve the followig iequlity: m +bm b +cm c +b +c b bc c+6, where,b,c re the legths of the sides: m.m b,m c,m c - the legths of the medis d the re of the trigle ABC.. I y trigle ABC, the followig iequlity holds: ) m h mb h b mc h c b c + + b c + c b + b c. W59. How my squres c you drw o fiite lttice bord defied by B = {x,y) N N : x,y 17}, i such wy tht ll vertices hve iteger coordites? Nicuşor Miculete Ovidiu Bgdsr W6. Fid the umber of segmets hvig iteger legth tht c be drw betwee poits of the fiite lttice bord defied by B = {x,y) N N : x,y 17}. W61. Let x,y,z > be rel umbers. Prove tht the followig iequlity holds: x 3 y 3 +y 3 z 3 +z 3 x 3 xyzx y +y z +z x). W6. If A >,B >,A+B π, d λ 1, the cos λa+si λb cosλa siλb siλπ cos λπ si 4 λ+1)π 4 Ovidiu Bgdsr Ovidiu Bgdsr Chg-Ji Zho d )