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# 2009b - h'qyz ` xhqnq,zihxwqic dwihnznl `ean'a cren ogand...

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Unformatted text preview: h''qyz ` xhqnq ,zihxwqic dwihnznl `ean 'a cren - ogand oexzt onqheb hxaex didi ze`vezd mekqy jk zepey zeiaew 6 lihdl zeiexyt`d xtqn edn (`) .1 ? 20 oexzt mekqy jk zepey zeiaew 6 lihdl zeiexyt`d xtqn z` a n a onqp zlhda d`viy d`vezd z` ( 1 ≤ i ≤ 6 ) u i a onqp . n didi ze`vezd-ynd ly minly mixtqna zepexztd xtqnl deey a n f` . i xtqn diaew d`ee u 1 + u 2 + u 3 + u 4 + u 5 + u 6 = n, `id { a n } ∞ n =0 dxcqd ly zxveid divwpetd . 1 ≤ u i ≤ 6 miveli`d mr F ( x ) = ∞ X n =0 a n x n = ( x + x 2 + x 3 + x 4 + x 5 + x 6 ) 6 = x 6 (1 + x + x 2 + x 3 + x 4 + x 5 ) 6 = x 6 1- x 6 1- x 6 = x 6 (1- x 6 ) 6 1 1- x 6 = x 6 6- 6 1 x 6 + 6 2 x 12- 6 3 x 18 + ... 1 1- x 6 . y mircei ep` . F ( x ) a x 20 ly mcwna xnelk , a 20 a mipiipern ep` ( * ) 1 1- x 6 = ∞ X n =0 ( n + 6- 1)! (6- 1)! n ! x n = ∞ X n =0 n + 5 5 x n . 1 okl , ( * ) iehiaa x 2 e x 8 , x 14 ly mincwna mipiiperne a 20 = 6 14 + 5 5- 6 1 8 + 5 5 + 6 2 2 + 5 5 = 6 19 5- 6 1 13 5 + 6 2 7 5 . cg` s`a miwlgzn mpi` 1000-l mieey e` miphw miirah mixtqn dnk (a) ? 4 , 5 , 6 , 14 :mixtqndn oexzt a onqp . 4 a miwlgzny 1000 e 1 oia mixtqnd zveaw z` A 1 a onqp z` A 3 a onqp . 5 a miwlgzny 1000 e 1 oia mixtqnd zveaw z` A 2 zveaw z` A 4 a onqp . 6 a miwlgzny 1000 e 1 oia mixtqnd zveaw A i ly milynd z` A i a onqpe . 14 a miwlgzny 1000 e 1 oia mixtqnd a mipiipern ep` . { 1 , 2 , 3 ,..., 1000 } dveawa | A 1 ∩ A 2 ∩ A 3 ∩ A 4 | . :lawp dgcdde dlkdd oexwr it lr | A 1 ∩ A 2 ∩ A 3 ∩ A 4 | = 1000- ( | A 1 | + | A 2 | + | A 3 | + | A 4 | ) + ( | A 1 ∩ A 2 | + | A 1 ∩ A 3 | + | A 1 ∩ A 4 | + | A 2 ∩ A 3 | + | A 2 ∩ A 4 | + | A 3 ∩ A 4 | )- ( | A 1 ∩ A 2 ∩ A 3 | + | A 1 ∩ A 2 ∩ A 4 | + | A 1 ∩ A 3 ∩ A 4 | + | A 2 ∩ A 3 ∩ A 4 | ) + | A 1 ∩ A 2 ∩ A 3 ∩ A 4 | . , x l deey e` ohwy lecb ikd mlyd xtqnd z` [ x ] a onqp iynn x xear lynl [17] = 17 , [3 . 5] = 3 , [3 . 2] = 3 , [3 . 9] = 3 . dveawa mixtqnd xtqn f` miirah mixtqn a,b,c,d m`y xirp B = { 1 , 2 , 3 ,...,n } l deey a a miwlgzny [ n a ] xear . 4-l miwlgzny mixtqn [ 20 4 ] = 5 yi , a = 4 e n = 20 xear lynl . 4-l miwlgzny mixtqn [ 19 4 ] = 4 yi , a = 4 e n = 19 l deey b-a mbe a-a mb miwlgznd B dveawa mixtqnd xtqn [ n lcm( a,b ) ] 2 xnelk , b e a ly zilnipind ztzeynd dletkd `ed...
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2009b - h'qyz ` xhqnq,zihxwqic dwihnznl `ean'a cren ogand...

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