alimov-7-algebra-gdz (542418), страница 10
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1) ( x − y ) 2 + ( x + y ) 2 = x 2 − 2 xy + y 2 + x 2 + 2 xy + y 2 = 2 x 2 + 2 y 22) ( x + y ) 2 − ( x − y ) 2 = x 2 + 2 xy + y 2 − x 2 + 2 xy − y 2 = 4 xy3) (2a + b) 2 − (2a − b) 2 = 4a 2 + 4ab + b 2 − 4a 2 + 4ab − b 2 = 8ab4) (2a + b) 2 + (2a − b) 2 = 4a 2 + 4ab + b 2 + 4a 2 − 4ab + b 2 == 8a 2 + 2b 278386. 1) (a − b) 2 = a 2 − 2ab + b 2 = b 2 − 2ab + a 2 = (b − a ) 22) (− a − b) 2 = (−1) 2 ⋅ (a + b) 2 = (a + b) 23) (−1) ⋅ ( a + b) ⋅ (a + b) = −(a + b) 24) (−1) 3 ⋅ (−a + b) 3 = −(b − a) 35) (a + b + c ) 2 = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc(a + b + c) 2 = (a + b) 2 + 2 ⋅ (a + b) ⋅ c + c 2 = a 2 + 2ab + b 2 + 2ac ++ 2bc + c 2 = a 2 + b 2 + c 2 + 2ab + 2ac + 2bca 2 + b 2 + c 2 + 2ab + 2ac + 2bc = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc ч.т.д.387. 1) 5m 2 − 10mn + 5n 2 = 5 ⋅ ( m 2 − 2mn + n 2 ) = 5 ⋅ (m − n) 2m = 142; n = 425 ⋅ (142 − 42) 2 = 5 ⋅ 10000 = 500002) 6m2 + 12mn + 6n2 = 6 ⋅ (m2 + 2mn + n2 ) = 6 ⋅ (m + n)2m = 56; n = 446 ⋅ (56 + 44) 2 = 6 ⋅ 10000 = 6000013193) − 36a 3 + 4a 2b − ab 2 = −a ⋅ (6a − ⋅ b) 2a = 4; b = 481− 4 ⋅ (6 ⋅ 4 − ⋅ 48) 2 = −4 ⋅ ( 24 − 16) 2 = −2563⎛⎝141 ⎞2 ⎠4) − 64a 3 − 8a 2b − ab 2 = −a ⋅ ⎜ 8a + b ⎟2a = −6; b = 8421⎞⎛6 ⋅ ⎜ 8 ⋅ (−6) + ⋅ 84 ⎟ = 6 ⋅ (−48 + 42) 2 = 6 ⋅ 36 = 2162⎠⎝388.
1) 1012 − 202 ⋅ 81 + 812 = (101 − 81) 2 = 4002) 37 2 + 126 ⋅ 37 + 632 = (37 + 63) 2 = 100003)4)48 2 + 2 ⋅ 48 ⋅ 18 + 182248 − 182852 − 17 2285 + 2 ⋅ 85 ⋅ 17 + 172==2(48 + 18) 248 + 18=2=15( 48 − 18) ⋅ (48 + 18) 48 − 18(85 − 17) ⋅ (85 + 17)(85 + 17)2=85 − 17 2=85 + 17 379389. 1) ( x + 2) 3 = x 3 + 6 x 2 + 12 x + 82) (3 − y ) 3 = 27 − 27 y + 9 y 2 − y 33) (2a − b) 3 = 8a 3 − 12a 2b + 6ab 2 − b 34) (3b + 2a) 3 = 27b 3 + 54b 2 a + 36ba 2 + 8a 3390. 1) 125 + 75a + 15a 2 + a 3 = (5 + a ) 32) m 3 − 12m 2 + 48m − 64 = (m − 4) 33) x 6 − 3x 4 y + 3x 2 y 2 − y 3 = ( x 2 − y ) 34) c 6 + 3c 4 d 2 + 3c 2 d 4 + d 6 = (c 2 + d 2 ) 3391. Рассмотрим двузначные числа и их квадраты (после 20 всеаналогично):a10 11 12 13 14 15 16 17 18 19 20a2 100 121 144 169 196 225 256 289 324 361 400Цифра единиц двузначного числа, квадрат которого содержитнечетное число десятков, 4 или 6.§ 23.
Применение несколькихспособов разложения многочлена на множители392. 1) 2a 2 − 2 = 2 ⋅ (a 2 − 1) = 2 ⋅ ( a − 1) ⋅ (a + 1)2) 3x 2 − 12 = 3 ⋅ ( x 2 − 4) = 3 ⋅ ( x − 2) ⋅ ( x + 2)3) 9 x 3 − 81x = 9 x ⋅ ( x 2 − 9) = 9 x ⋅ ( x − 3) ⋅ ( x + 3)4) 16 x − 4 x 3 = 4 x ⋅ (4 − x 2 ) = 4 x ⋅ (2 − x) ⋅ (2 + x )5) 8 − 72 x 6 y 2 = 8 ⋅ (1 − 9 x 6 y 2 ) = 8 ⋅ (1 − 3x 3 y ) ⋅ (1 + 3x 3 y )6) 32a 4b − 2a 2b = 2a 2b ⋅ (16a 2 − 1) = 2a 2b ⋅ ( 4a − 1) ⋅ ( 4a + 1)393.
1) 2a 2 + 4ab + 2b2 = 2 ⋅ (a 2 + 2ab + b 2 ) = 2 ⋅ (a + b)22) 2m 2 + 2n 2 − 4mn = 2 ⋅ ( m 2 + n 2 − 2mn) = 2 ⋅ (m − n) 23) 5 x 2 + 10 xy + 5 y 2 = 5 ⋅ ( x 2 + 2 xy + y 2 ) = 5 ⋅ ( x + y ) 24) 8 p 2 − 16 p + 8 = 8 ⋅ ( p 2 − 2 p + 1) = 8 ⋅ ( p − 1) 25) 27 a 2b 2 − 18ab + 3 = 3 ⋅ (9a 2b 2 − 6ab + 1) = 3 ⋅ (3ab − 1) 26) 12m 5 n + 24m 4 n + 12m 3n = 12m 3n ⋅ (m 2 + 2m + 1) = 12m 3n ⋅ (m + 1) 280394.
1) ( x 2 + 1) 2 − 4 x 2 = ( x 2 + 1 − 2 x) ⋅ ( x 2 + 1 + 2 x) = ( x − 1) 2 ⋅ ( x + 1) 22) ( x 2 + 2 x) 2 − 1 = ( x 2 + 2 x − 1) ⋅ ( x 2 + 2 x + 1) = ( x + 1) 2 ⋅ ( x 2 + 2 x − 1)3) 4 y 2 − ( y − c) 2 = (2 y − y + c ) ⋅ ( 2 y + y − c) = ( y + c ) ⋅ (3 y − c)4) 81 − ( y 2 + 6 y)2 = (9 − y 2 − 6 y) ⋅ (9 + y 2 + 6 y ) == ( y + 3)2 ⋅ (9 − y 2 − 6 y )395.
1) (a 2 + 2ab + b 2 ) − c 2 = (a + b) 2 − c 2 = ( a + b + c) ⋅ (a + b − c)2) 1 − ( x 2 − 2 xy + y 2 ) = 1 − ( x − y ) 2 = (1 − x + y ) ⋅ (1 + x − y )3) 1 − a 2 − 2ab − b 2 = 1 − (a + b) 2 = (1 − a − b) ⋅ (1 + a + b)4) 4 − x 2 − 2 xy − y 2 = 4 − ( x + y ) 2 = ( 2 − x − y ) ⋅ (2 + x + y )396. 1) a 2 − b 2 + a + b = ( a 2 − b 2 ) + (a + b) = ( a + b) ⋅ ( a − b) + ( a + b) == (a + b) ⋅ ( a − b + 1)2) a 2 − b 2 − a − b = (a + b) ⋅ ( a − b) − (a + b) = ( a + b) ⋅ ( a − b − 1)3) x − y − x 2 + y 2 = ( x − y ) − ( x − y ) ⋅ ( x + y ) = ( x − y ) ⋅ (1 − x − y )4) x 3 + x 2 − x − 1 = x 2 ⋅ ( x + 1) − ( x + 1) = ( x + 1) ⋅ ( x 2 − 1) = ( x + 1) 2 ⋅ ( x − 1)5) m 5 − m 3 + m 2 − 1 = m 3 ⋅ (m 2 − 1) + (m 2 − 1) = (m 2 − 1) ⋅ (m 3 + 1) == (m − 1) ⋅ (m + 1) ⋅ (m + 1) ⋅ (m 2 − m + 1) == (m + 1) 2 ⋅ (m − 1) ⋅ (m 2 − m + 1)6) x 4 − x3 + x − 1 = x( x3 + 1) − ( x3 + 1) = ( x3 + 1)( x − 1) == ( x + 1)( x 2 − x + 1)( x − 1)397.
1)2)3)=532 − 27 22279 − 5138 2 − 17 247 2 − 19 2=(53 − 27) ⋅ (53 + 27) 26 ⋅ 80 2 ⋅ 2 4===(79 − 51) ⋅ (79 + 51) 28 ⋅ 130 7 ⋅ 1 7=(38 − 17) ⋅ (38 + 17) 21⋅ 55 1 ⋅ 5 5===(47 − 19) ⋅ ( 47 + 19) 28 ⋅ 66 4 ⋅ 2 8(49 2 − 2 ⋅ 49 ⋅ 29 + 29) 249 2 − 19 2=(49 − 29) 2=(49 − 19) ⋅ (49 + 19)20240010==30 ⋅ 68 30 ⋅ 68 51814)=47 2 − 3 2227 + 2 ⋅ 27 ⋅13 + 132=(47 − 3) ⋅ (47 + 3)(27 + 13)2=44 ⋅ 5040 2=44 ⋅ 50 113==1160088398.
1) 19,7 2 − 8,32 + 28 ⋅ 8,6 = (19,7 − 8,3) ⋅ (19,7 + 8,3) − 28 ⋅ 8,6 == 11,4 ⋅ 28 − 28 ⋅ 8,6 = 28 ⋅ (11,4 − 8,6) = 28 ⋅ 2,8 = 78,42) 37 ⋅ 12,2 + 22,4 2 − 14,6 2 = 37 ⋅ 12,2 + ( 22,4 − 14,6) ⋅ ( 22,4 + 14,6) == 37 ⋅ 12,2 + 7,8 ⋅ 37 = 37 ⋅ (12,2 + 7,8) = 37 ⋅ 20 = 7403) 38,82 + 83 ⋅ 15,4 − 44,22 = (38,8 − 44,2) ⋅ (38,8 + 44,2) + 83 ⋅15,4 == −5,4 ⋅ 83 + 83 ⋅ 15,4 = 83 ⋅ (−5,4 + 15,4) = 83 ⋅ 10 = 8304) 97 ⋅ 2,2 − 99,6 2 + 2,6 2 = 97 ⋅ 2,2 + (2,6 − 99,6) ⋅ (2,6 + 99,6) == 97 ⋅ 2,2 − 97 ⋅ 102,2 = 97 ⋅ (2,2 − 102,2) = 97 ⋅ (−100) = −9700399. 1) x 2 + 2 x − y 2 + 2 y = ( x − y ) ⋅ ( x + y ) + 2 ⋅ ( x + y ) == ( x + y ) ⋅ ( x − y + 2), ч. т. д.2) a 2 − 2b − a − 4b 2 = (a − 2b) ⋅ (a + 2b) − (2b + a) == (a + 2b) ⋅ (a − 2b − 1), ч.
т. д.400. 1) x3 − x 2 y − xy 2 + y 3 = x 2 ⋅ ( x − y ) − y 2 ⋅ ( x − y ) = ( x − y ) ⋅ ( x 2 − y 2 ) == ( x − y) 2 ⋅ ( x + y) ; x = 12,07; y = 2,07 :(12,07 − 2,07) 2 ⋅ (12,07 + 2,07) = 10 2 ⋅ 14,14 = 14142) a3 + a 2b − ab 2 − b3 = a 2 ⋅ (a + b) − b 2 ⋅ (a + b) == (a + b) ⋅ (a 2 − b 2 ) = (a + b) 2 ⋅ (a − b) ; a = 7,37; b = 2,63 :(7,37 + 2,63) 2 ⋅ (7,37 − 2,63) = 4,74 ⋅ 10 2 = 474401. 1) 25 x 2 − 10 x − x 2 − 25 = 0 ;25 x 2 − ( x 2 + 10 x + 25) = 0 ;( x + 2)2 − (4 x)2 = 025x 2 − ( x + 5)2 = 0 ;(5 x − x − 5) ⋅ (5 x + x + 5) = 0 ;( x + 2 + 4 x) ⋅ ( x + 2 − 4 x) = 0(5 x + 2) ⋅ (2 − 3x ) = 0(4 x − 5) ⋅ (6 x + 5) = 0 ; 6 x + 5 = 0 ;2 − 3x = 0 ; x1 =x1 = −822) x 2 + 4 x + 4 − 16 x 2 = 051; 4 x − 5 = 0 ; x2 = 164235 x + 2 = 0 ; x2 = −253) x 5 − x 4 − 2 x 3 + 2 x 2 + x − 1 = 0x 4 ⋅ ( x − 1) − 2 x 2 ⋅ ( x − 1) + ( x − 1) = 0( x − 1) ⋅ ( x 4 − 2 x 2 + 1) = 0( x − 1) ⋅ ( x 2 − 1) 2 = 0( x − 1) ⋅ ( x − 1) 2 ⋅ ( x + 1) 2 = 0( x − 1) 3 ⋅ ( x + 1) 2 = 0x + 1 = 0 ; x1 = −1x − 1 = 0 ; x2 = 14) 2 x 4 − 2 x 3 − 2 x 2 + 2 x = 02 x 3 ⋅ ( x − 1) − 2 x ⋅ ( x − 1) = 0( x − 1) ⋅ (2 x 3 − 2 x) = 02 x ⋅ ( x − 1) ⋅ ( x 2 − 1) = 02 x ⋅ ( x − 1) 2 ⋅ ( x + 1) = 0x + 1 = 0 ; x1 = −1x − 1 = 0 ; x2 = 1 ; 2 x = 0x3 = 0 .
(опечатка в ответе задачника).402. 27 2 − 14 2 = ( 27 − 14) ⋅ ( 27 + 14) = (13 ⋅ 41)13 ⋅ 41 : 13 = 41, ч. т. д.403. (7 n − 2) 2 − ( 2n − 7) 2 = (7 n − 2 − 2n + 7) ⋅ (7 n − 2 + 2n − 7) == (5n + 5) ⋅ (9n − 9) = 5 ⋅ 9 ⋅ (n + 1) ⋅ (n − 1)5 ⋅ 9(n + 1)(n − 1) : 5 = 9(n + 1)(n − 1)5 ⋅ 9( n + 1)(n − 1) : 9 = 5(n + 1)(n − 1) , ч.т.д.404. 1) (a − 2) ⋅ (a 2 + 2a + 4) = a 3 − 82) (b + x) ⋅ (b 2 − bx + x 2 ) = b 3 + x 33) (2a + 3) ⋅ (4a 2 − 6a + 9) = 8a 3 + 274) (a 2 − 1) ⋅ (a 4 + a 2 + 1) = a 6 − 1405. 1) 27 a 3 − b 3 = (3a − b) ⋅ (9a 2 + 3ab + b 2 )2) x3 y 3 + 64 = ( xy + 4) ⋅ ( x 2 y 2 − 4 xy + 16)833) 8m3 + n9 = (2m + n3 ) ⋅ ( 4m 2 − 2mn3 + n6 )4) c 6 − 125d 3 = (с 2 − 5d ) ⋅ (c 4 + 5c 2d + 25d 2 )406.
Если натуральное число не делится на 3, то оно равно:m = 3p + 1 или m = 3p + 2.Возможно 3 случая:1) m = 3p+1; n =3k + 1|m2 – n2| = |9p2 + 6p + 1 – 9k2 – 6k – 1| = 3|3p2 – 3k2 + 2p – 2k|2) m = 3p + 2; n = 3k + 1;|m2 – n2| = |9p2 + 12p + 4 – 3k2 – 6k – 1| = 3|3p2 + 4p – k2 – 2k + 1|3) m = 3k + 2; n = 3p + 2;|m2 – n2| = |9k2 + 12k + 4 – 9p2 – 12p – 4| = 3|3k2 + 4k – 3p2 – 4p|Во всех трех случаях |m2 – n2| M 3.407.
Пусть n – первое натуральное число, тогда следующее числоn + 1.n3 − ( n + 1)3 = n3 − n3 − 3n 2 − 3n − 1 = − 3n 2 − 3n − 1Данное выражение не делится на 3, т. к. два слагаемых делятсяна три, а одно слагаемое, а именно 1, на 3 не делится.Упражнения к главе IV408. 1) 6 ⋅ (a + b) + (a + b) 2 = (a + b) ⋅ (6 + a + b)2) 4 ⋅ ( x − y ) + 3 ⋅ ( x − y ) 2 = ( x − y ) ⋅ (4 + 3x − 3 y )3) (a − b) + (b − a ) 2 = (a − b) ⋅ (1 + a − b)4) (a − b) 2 − (b − a) = (b − a ) ⋅ (b − a − 1)409.
1) (c − 3) 2 − (c + 3) ⋅ (3 − c) = (3 − c) ⋅ (3 − c − c − 3) = 2c ⋅ (c − 3)2) (a + 2) 2 − (a + 2) ⋅ (2 − a ) = ( a + 2) ⋅ ( a + 2 − 2 + a ) = 2a ⋅ (a + 2)3) (−b − a) ⋅ (a + b) + a 2 + b 2 = −( a 2 + 2ab + b 2 ) + a 2 + b 2 == −a 2 − 2ab − b 2 + a 2 + b 2 = −2ab4) (b − a) ⋅ (− a − b) − 3b 2 = −(b 2 − a 2 ) − 3b 2 = −b 2 + a 2 − 3b 2 == a 2 − 4b 2 = (a − 2b) ⋅ (a + 2b)410. 1) 2b ⋅ ( x − 1) − 3a ⋅ ( x − 1) + c ⋅ ( x − 1) = ( x − 1) ⋅ (2b − 3a + c)2) c ⋅ ( p − q) − a ⋅ ( p − q ) + b ⋅ ( p − q ) = ( p − q ) ⋅ (c − a + b)84411. 1) 8ax + 16ay − 3bx − 6by = 8a ⋅ ( x + 2 y ) − 3b ⋅ ( x + 2 y ) == ( x + 2 y ) ⋅ (8a − 3b)2) 14am − 7 an + 8bm − 4bn = (14am − 7an) + (8bm − 4bn) == 7 a (2m − n) + 4b(2m − n) = (7 a + 4b)(2m − n)3) 9a 2 + 6a + 1 − 4b2 = (3a + 1) 2 − 4b2 = (3a + 1 − 2b) ⋅ (3a + 1 + 2b)4) 25a 2 − 4b 2 + 4b − 1 = 25a 2 − (2b − 1) 2 = (5a − 2b + 1) ⋅ (5a + 2b − 1)412.
1) 287 2 − 287 ⋅ 48 + 239 ⋅ 713 = 287 ⋅ ( 287 − 48) + 239 ⋅ 713 == 287 ⋅ 239 + 239 ⋅ 713 = 239 ⋅ (287 + 713) = 239 ⋅ 1000 = 2390002) 73,42 + 73,4 ⋅17,2 − 90,6 ⋅ 63,4 = 73,4 ⋅ (73,4 + 17,2) − 90,6 ⋅ 63,4 == 73,4 ⋅ 90,6 − 90,6 ⋅ 63,4 = 90,6 ⋅ (73,4 − 63,4) = 90,6 ⋅ 10 = 90621 ⎞1 ⎞ ⎛1 ⎞ ⎛⎛413. 1) ⎜ 4c + x ⎟ ⋅ ⎜ 4c − x ⎟ + ⎜ 4c − x ⎟ =4 ⎠4 ⎠ ⎝4 ⎠ ⎝⎝1 ⎞1 ⎞11 ⎞ ⎛⎛⎛= ⎜ 4c − x ⎟ ⋅ ⎜ 4c + x + 4c − x ⎟ = 8c ⋅ ⎜ 4c − x ⎟4 ⎠4 ⎠44 ⎠ ⎝⎝⎝1⎞1 ⎛ 1 1 ⎞1⎛c = ; x = 2 : 8 ⋅ ⋅ ⎜ 4 ⋅ − ⋅ 2⎟ = 4 ⋅ ⎜ 2 − ⎟ = 622⎠2 ⎝ 2 4 ⎠⎝2) (0,1a − 0,2b) 2 + (0,1a − 0,2b) ⋅ (0,1a + 0,2b) == (0,1a − 0,2b) ⋅ (0,1a − 0,2b + 0,1a + 0,2b) = (0,1a − 0,2b) ⋅ 0,2a2a = −50; b = −1 :3⎛1⎞⎛⎛ 2 ⎞⎞⎜⎜ 0,1 ⋅ ( −50) − 0,2 ⋅ ⎜ − 1 ⎟ ⎟⎟ ⋅ 0,2 ⋅ ( −50) = ⎜ − 5 + ⎟ ⋅ (−10) =3⎠3⎠⎝⎝⎝⎠22= 4 ⋅ 10 = 4633ПРОВЕРЬ СЕБЯ!21.(a + 3) + (a – 3) ⋅ (a + 3) + 6a = a2 + 6a + 9 + a2 – 9 + 6a == 2a2 + 12a = 2a2 + 12a = 2a(a + 6)2.xy – 2y = y ⋅ (x – 2)3x2 – 6x3 = 3x2 ⋅ (1 – 2x); 3 ⋅ (x – 1) + y ⋅ (x – 1) = (x – 1) ⋅ (3 + y)2a2 – 4ab + 2b2 = 2 ⋅ (a – b)2; 16a2 – 81 = (4a + 9) ⋅ (4a – 9)x2 – 10x + 25 = (x – 5)2853.a2 – 3ab + 3a – 9b = a ⋅ (a – 3b) + 3 ⋅ (a – 3b) = (a – 3b) ⋅ (a + 3)a = 1; b = –1⎛1⎞: (1 – 3 ⋅ ⎜ ⎟ ) ⋅ (1 + 3) = 83⎝3⎠414.
1) (x + y) ⋅ (x2 – y2) = (x – y) ⋅ (x + y)2(x + y) ⋅ (x2 – y2) = (x + y)2 ⋅ (x – y) = (x – y)(x + y)22) (x – 2y) ⋅ (x + 2y) ⋅ (x2+4y2) = (x2 – 4y2) ⋅ (x2 + 4y2) = x4 – 16y2ч.т.д.415. 1) mn – kn – m2 + 2mk – k2 = n ⋅ (m – k) – (m – k)2 == (m – k) ⋅ (n – m – k)2) c2 – 2c + 1 – d2 – 2de – e2 = (c – 1)2 – (d + e)2 == (c – 1 – d – e) ⋅ (c – 1 + d + e)416. 1) (x2 – 1)2 – (x2 + 2)2 = (x2 – 1 – x2 – 2) ⋅ (x2 – 1 + x2 + 2) == – 3 ⋅ (2x2 + 1)2) (5 + x2)2 – (7 + x2)2 = (5 + x2 – 7 – x2) ⋅ (5 + x2 + 7 + x2) == – 4 ⋅ (x2 + 6)3) (3x – 1)2 – (5 – 2x)2 = (3x – 1 – 5 + 2x) ⋅ (3x – 1 + 5 – 2x) == (5x – 6)(x + 4);4) (7 + 5x)2 – (3x – 2)2 = (7 + 5x – 3x + 2)(7 + 5x + 3x – 2) == (2x + 9)(8x + 5).417. 1) (3x – 1)2 – (3x – 2)2 = 0(3x – 1 – 3x + 2) ⋅ (3x – 1 + 3x – 2) = 0(6x – 3) = 0; x =122) (y – 2)(y + 3) – (y – 2)2 = 5(y – 2) ⋅ (y + 3 – y + 2) = 5(y – 2) ⋅ 5 = 5y – 2 = 1; y = 33) (x + 3) ⋅ (x+ 7) – (x + 4)2 = 0x2 + 3x + 7x + 21 – x2 – 8x – 16 = 02x + 5 = 0; x = – 2,54) (y + 8)2 – (y + 9) ⋅ (y – 5) = 117y2 + 16y + 64 – y2 – 9y + 5y + 45 = 11712y = 8; y =235) (3x + 2) ⋅ (3x – 2) – (3x – 4)2 = 289x2 – 4 – 9x2 + 24x – 16 = 2824х = 48; x = 286418.
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