4. , , . , . . , , , , , , .

4.1. . ( ) , . , . . 1. + 3 5 = 18. : + 3 - 5 = 18 + 3 5 15 = 3, ( + 3)( 5) = 3. 3 , : 1) 2) 3) 4) + 3 = 1, 5 = 3 = -2, = 8 ; + 3 = -1, 5 = -3 = -4, = 2 ; + 3 = 3, 5 = 1 = 0, = 6 ; + 3 = -3, 5 = -1 = -6, = 4.

35

4. 4.1.

, . . 2. 2 + 1 . : n. 2 + 1 = n7, . n7 1 = (n 1)(n6 + n5 + n4 + n3 + n2 + n + 1) = 2. 1, 2, 2 2, : 1) n - 1 = 1, n = 2, n6 + n5 + n4 + n3 + n2 + n + 1 = 2 = 127, , 2 , 127 ; 2) n 1 = 2, n = 3, n6 + n5 + n4 + n3 + n2 + n + 1 = 1093 = . 1093 , (3, 1093) . 3) n 1 = , n = + 1 3, n6 + n5 + n4 + n3 + n2 + n + 1 n + 1 3, . 4) n 1 = 2, n6 + n5 + n4 + n3 + n2 + n + 1 = 1, n6 + n5 + n4 + n3 + n2 + n = 0. n = 0 n = -1, , 0 1 . 3. 2 + 1 . : 2 + 1 = 2, 2 - 1 = ( + 1)( - 1) = 2. 2 2. 2 = 2 2b ( b), a + b = x. + 1= 2 - 1 = 2b, 2 - 2b = 2. 2b(2-b - 1) = 2. 2b = 2 2-b - 1 = 1, b = 1 a - b = 1. = 2, b = 1 x = a + b = 3. 23 + 1 = 9 = 32, = 3. 4. n4 + 2n3 + 2n2 + 2n + 1 n. : n4 + 2n3 + 2n2 + 2n + 1 = n4 + 2n3 + n2 + n2 + 2n + 1 = n2(n2 + 2n + 1) + n2 + 2n + 1 = (n2 + 2n + 1)(n2 + 1) = (n + 1)2 (n2 + 1), n2 + 1 , n2 + 1 = 2. 2 - n2 = 1, . ( - n)( + n) = 1. - n = + n = 1, = 1 n = 0, , n = 0 .

36

4. 4.1.

5. ( + 1)( + 7)( + 8) = 2 . 8 : (2 + 8)(2 + 8 + 7) = 2. 4 (22 + 16 + 7 - 7)(22 + 16 + 7 + 7) = 42, (22 + 16 + 7)2 - 49 = 42, (22 + 16 + 7)2 (2)2 = 49. (22 + 16 + 7 2)(22 + 16 + 7 + 2) = 49. : 1) 22 + 16 + 7 2 = 1 22 + 16 + 7 + 2 = 49; 2) 22 + 16 + 7 2 = 7 22 + 16 + 7 + 2 = 7; 3) 22 + 16 + 7 2 = 49 22 + 16 + 7 + 2 = 1; 4) 22 + 16 + 7 2 = -1 22 + 16 + 7 + 2 = -49; 5) 22 + 16 + 7 2 = -7 22 + 16 + 7 + 2 = -7; 6) 22 + 16 + 7 2 = -49 22 + 16 + 7 + 2 = -1. , :

42 + 32 + 14 = 50 4 = 48 2 + 8 9 = 0 = 12; 42 + 32 + 14 = 14 4 = 0 2 + 8 = 0 = 0; 42 + 32 + 14 = 50 4 = -48 2 + 8 9 = 0 = -12; 42 + 32 + 14 = -50 4 = -48 2 + 8 + 16 = 0 = -12; 42 + 32 + 14 = -14 4 = 0 2 + 8 + 7 = 0 = 0; 42 + 32 + 14 = -50 4 = 48 2 + 8 + 16 = 0 = 12.

: (-9, 12); (1, 12); (0, 0); (-8, 0);(-9, -12); (1, -12); (-4, -12); (-7, 0); (-1, 0); (-4, 12). , .

8

1973. . : .. : , , 1987. 37

4. 4.1.

214. (,) + 2 = 7. 215. 2 2 = 12. ? 216. (,) : ) 2 7 = 2 ; b) x4 y4 = 15 ; c) 6x2 13xy + 6y2 = 4 ; d) x2 + 6y = y2 + 4x + 10 . 217. (,) + 2 = 7. 218. n = n4 + 4 . 219. n n2 + 2n + 13 . 220. 2 - 5 + 62 = 3 ? 221. n 5 + 1 = n2 . 222. n 8n2 + 10n + 3? 223. 5 . 224. = 9 cm. 225. 3 + 1 = 3 ? 226. , b bc + b = + c + 3. 227. 3 - 3 = - 2 ? 228. n m : ) 22 + 1 = n5 ; b) n4 + n2 + 1 = p ; 4 4 ) n + 4m = p . 229. , z , ( + z)2 = 2 - 2 + z2 . . 230. , + + = 232 .

38

4. 4.1.

231. m , 2 + 2 + 2 m m m = 1 m . . 232. ( + 1) = 4(+1) . ?

233. , > , 1 1 2 + = . ( - 1931.) x y p 234. 21, . . ? ( 1971.) 1 1 1 235. + = . 3, , 3. . (36. 1973.) 236. (, ) ( + 2)4 4 = 3. ( 1974.) 237. 2(2 2) = 1978, . ( 1978.) 238. ( ) . ( 1978). 239. (, ) 3 2 + 1 = 2. (41. 1978.) 240. = + + 1. ( 1987.) 241. q 2 - 2q2 = 1. ( 1990.) 242. + 3 = 10. ( 1991.) 243. 1991 . ( 1991.)39

4. 4.1.

244.

1 1 1 = (n N) x y n , n . ( 1991.)

245. n n3 n + 2n = 1992? ( 1992.) 246. 22 2 = 2 + 1994. ( 1994.) 247. n 2 7, . ( 1993). 248. 222...222 . ( 1996.) 249. (, , z) + + z = 0 3 + 3 + z3 = -18. ( 2000.) 250. n 2n + 1 3n + 1 . 5n + 3 . ( 1996.) 251. n n2 + 2n + 1997 . ( 1997.) 252. , z 1 = 0, xyz . ( 1997.) x+ y+z 253. n n2 + 2n + 2000 . ( 2000.) 254. m .

(, ) 22 + 11 + 122 + 4 + 5 + 6 = 2m. ( 1999.)

255. n , n5 + n4 + 1= p. 256. , z ( )3 + ( z)3 + (z )3 = 6.

40

4. 4.2.

257. a, b, c d ab + cd = 14

ac + bd = 11?258. , n m : ) 1! 2! ... 99! 100! = n! m2 ; b) 1! 2! ... (k-1)! k! = n! m2 .

4.2. . = , = N +

P . N , Q

P , Q . Q

, Q . 6. 2 + 2 3 = 6. : 2 + 2 3 = 6 2 + 2 6 = + 3 = ( + 3), : 1) = -3, 2 + 2 6 = -3, ( + 3) = 0, = 3 ; 2) - 3, =x 2 + 2x 6 3 = x1 . x +3 x +3

: 2.1) + 3 = 1 = -2, = - 6 ; 2.2) + 3 = 1 = -4, = -2 ; 2.3) + 3 = 3 = 0, = - 2 ; 2.4) + 3 = -3 = -6, = -6.

41

4. 4.2.

7. (, ) ( + ) = , . : ( + ) = = = ( ). :

1) = , 2 = 0, = 0, ; 2) , =px px p 2 + p 2 p2 = =p+ . x p x p xp

3) 2 - , , , : 3.1) = 1 = + 1, = 2 + ; 3.2) = = 2, = 2 ; 3.3) = 2 = 2 + , = + 1.9 , 2, 3, 5, ... . . 9 8. . : xy = 10 + . 10 + = ( + )2 9 + ( + ) = ( + )2. 0, ( + ) 0, ( + ) : 9x + 1 = + . x+y d, b 1 = d 9 0 = bd 9. 9d 9 + 1 = ( + b ) d . + 1 = ( + b)d +b ( + b)d

9 ( + b) +b

, :9

,

42

4. 4.2.

1) + b = 1, 9 + 1 = d. , + b = 1 , 0, = 1, b = 0. d = 10, , d 9. 2) + b = 3, 3 + 1 = 3d. , . 3) + b = 9, + 1 = 9d = 9d 1. 1 9, 1 = d = (9d 1)d 9, d = 1. = 8 b = 1. 81 (= (8 + 1)2). 9. . ? : . , . , : 1000 + = 3. 0, y 1000 + = 3y . x , = ( ). 1000 + = 3, (3 1) = 1000. , 100, 3 300, 3 - 1 299. 1000 299 1000 500, : 1) 3 - 1 = 1000, 3 = 1001, , 1001 3. 2) 3 - 1 = 500, 3 = 501, = 167. 3 - 1000 = 2, = 334. 3167334 = 167334.

259. 3 + 33 + 333z = 2006 . 260. a2 + 2 1. a 143

4. 4.2.

261. ( + 1)2 = 243. 262. + 7 3 = 23, . 263. 2 = 3 + 10. 264. 13 . 265. + = + 9 . 266. 3 + 11 = 2000. , . . 267. (, ) 5 + 9 = 127. 268. o ? 269. .4

270. . ( 1974.) 271. (n n ), (n + 5 m ). . ( 1991.) 272. . ( 1993.) 273. 100 , 102, 203, 304? ( 2001.) 274. , 2 , 5 . 100 100 ? ( 2000.)x 3 + 3x 2 x 10 x+2

44

4. 4.3.

275. : + 1 = 3 + . ? 276. ? 277. n . 278. , (, , z) z = ( + + z).n3 + 5 n2 + 1

,

4.3. . . , , . 10. 2 2 -1. 4 + 2006 =

: 4 + 2006 = 22 1 (2 1)2 + (1003)2 = 0 0 0, : 2 1 = 0 = 0. , : (1, 0) (-1,0). 11. : 4 + 4 = 62 + 142 53. : 4 + 4 = 62 + 142 53 (2 3)2 + (2 - 7)2 = 5. 5 1, 4, :45

4. 4.3.

1) (2 3)2 = 1 (2 - 7)2 = 4. , 2 3 = 1 2 - 7 = 2. 2 = 2 2 = 5 2 = 4 2 = 9, (, ) {(2,3); (2, -3); (-2, 3); (-2, -3)}. 2) (2 3)2 = 4 (2 - 7)2 = 1. 2 3 = 2 2 - 7= 1. 2 = 8 2 = 6 . . . 12. 2 2 + + = 1. : 2 22 + 2 + 22 = 2 2 + ( + )2 + 2 = 2. 2, 1, 0, : 1) 2 = 1, ( + )2 = 1 2 = 0. = 0, 2 = 1, (1, 0) (-1, 0). 2) 2 = 1, ( + )2 = 0 2 = 1. + = 0, | | = 1 | | = 1, (1, -1) (-1, 1). 3) 2 = 0, ( + )2 = 1 2 = 1. = 0, 2 = 1, (0, 1) (0, -1). : (1, 0); (-1, 0); (1, -1); (-1, 1); (0, 1); (0, -1). . . 13. 11 . : abc . Ka abc = 100 + 10b + c = 11(9a + b) + a b + c, 11 b + c.46

4. 4.3.

a b + c = a2 + b2 + c2, . a2 + b2 + c2 a + b c = 0. 2 2a2 + 2b2 + 2c2 2a + 2b 2c = 0 a2 + b2 + c2 + (a 1)2 + (b + 1)2 + (c 1)2 = 3. 0, 1. abc , 1, = 1. b 0, b + 1 1 b = 0 b + 1 = 1. a2 + b2 + c2 + (a 1)2 + (b + 1)2 + (c 1)2 = 1 + 0 + c2 + 0 + 1 + (c 1)2 = 3, : 1) = 0 ( 1)2 = 1. 2) = 1 ( 1)2 = 0. 100 101. , abc 11 a2 + b2 + c2. 0 a2 + b2 + c2 10, a, b c 3, 10. : 100, 101, 102, 103, 110, 111, 112, 120, 121, 122, 130, 200, 201, 202, 210, 211, 212, 220, 221, 300, 301 310. 11 100 101.

279. x y : ) x2 + y2 = 0 ; c) x2 + y2 = 2 ; ) x2 + y2 = 4 ; b) x2 + y2 = 1 ; d) x2 + y2 = 3 ; f) x2 + y2 = 10.

280. x y x2 + y4 = 2x - 1?

281. 4x2 + y2 = 12x + 4y 12. 282. x4 + y2 + 2y = 1, . 283. x, y, z x2 + 4y2 + z4 = 2x - 20y - 23?47

4. 4.3.

284. 2 + q2 = r , p, q r ? ? 285. 2 + 2 + z2 + + + z = 1 . 286. 2 + 2 + z2 + + + z = n. n : ) ; b) .

287. 2 + 2 = + .

(7. 1941.)288. , b

2 + b2 + 2 + 3 < b + 3b + 2. ( 1965.)289.

: + = zt z + t = xy. (29. 1966.)290. : 2n (n

n 3) 72 + 2, . (48. 1985.)291. 2 + 2 = 1990

. ( 1990.)292. ( > ),

+ , , : 245. ( 1992.)293. (m, n) n3 + 1 . (35. 1994.) mn 1

294. 2 + 2 + z2 = + z + z. . ( 1995.) 295. 222...222 . ( 1996.) 296. 1, ... 1998 12 + ... + 19972 = 19982 . . ( 1997.)

48

4. 4.3.

297. : + + z = 3 3 + 3 + z3 = 3. ( 2002.) 298. 2m2 + n2 = 2mn + 3n. ( 2003.) 299. , , z t 2 + 2 + z2 + t2 = ( + z + t). ( 2004.) 300. 2 6 + 132 = 100. ( 2005.)

301. 2 + 2 + z2 = 2006 ? 302. 12 + 22 +...+ 20002 = 2(1 + 2 +...+ 2000) 1999. 1, 2, ... 2000. 303. 2 + q2 = r (q > 5) , r = 10 + 3 (). . 304. 2 + 2 + z2 = , , z . 305. 12 + 22 + ... + 2 = 12 ... = 2006 ?

4.4. . , , . , , . , , , . 14. ?

49

4. 4.4.

: abcd . abcd = (a + b + c + d ) 4 . 54 < 1000 abcd < 104,

5 < + b + c + d < 10. , + b + c + d 6, 7, 8 9. 64 = 1296, 74 = 2401, 84 = 4096 94 = 6561, 2401 = 74, 9. 15. . : ab . ab = 10 + b = a3 + b2. ab = 10 + b 99 a3 + b2 99, 1 4. 10 3 = b2 b (10 2) = b(b - 1). b(b - 1), , , (10 2) .

. (10 2) 0, 10 2 0, . 3. 3 = 2. (10 2) = b(b - 1) = 12, b = 4. , 24 ( = 23 + 42). , 90 , . , . 16. . : : 1) , = 2, = 1. 2) 10 + = 2 + 2, (10 - ) = ( -1). ( - 1) , . {2, 4, 6, 8} (10 - ) 18 24, . 3) , 100 + 10 + z = 2 + 2 + z . 100 + 10 + z = 2 + 2 + z2 381 = 243, = 1 = 2.2

3.1) = 1, 100 + 10 + z = 1 + 2 + z2 99 + 10 + z = 2 + z2. 99 > z2 10 > 2, .

50

4. 4.4.

3.2) = 2, 200 + 10 + z = 4 + 2 + z2. 200, 200, . 4) 3 , 10-1, 100, 10-1 100, 10-3 , , 3. 17. , b, c, d

1 a2

+

1 b2

+

1 c2

+

1 d2

= 1?

: , , 1 < a b c d. Ta je

1 d2 3

1 c2 1

1 b2 1

1 a2 1

,

4 d2

1 a2

+

1 b2

+

1 c2

+

1 d2

=1

4 a2

.

a2 4, a = 2, 1.

3 3 . b2 4, b = 2. 2 4 b d b c d 2 1 1 2 2 , c2 4, o c = 2. O + = 2 2 2 2 4 c d c d2

2

+

2

+

2

=

d = 2, a = b = c = d = 2. 18. , , z ! + ! = z!? : ! + ! = z! ! 1 ! 1, < z < z. ! + ! = z! = z(z 1)...( + 1)( - 1)...321 = z(z 1)...( + 1)!. ! = z(z 1)...( + 1)! ! = ! ( z(z 1)...( + 1) 1). : 1) < , ! < !, ; 2) = , z(z 1)...( + 1) 1 = 1, z(z 1)...( + 1) = 2 z = 2. = = 1. 3) > , 2 ! = ( - 1)...( + 1) ! = ! (z(z 1)...( + 1) 1).

51

4. 4.4.

! ( - 1)...( + 1) = (z(z 1)...( + 1) 1. , , . = = 1 z = 2. 19. 2 + + 2 = 22 . : : 1) 1 ,

12

+

1 1 1 1 1 + = 1 . 2 2, . 2 2 2 12 +

32

1 1 3 + =1 , 2 3, 2 2

= 1. = 1, = - 1, . 2) 1 , . 20. (x, y, z) a , z, + z + z - z = 2. : , . : 1) = 1, + z + z - z = + z = 2, (1, 1, 1) . 2) = 2, 2 + z + 2z - 2z = 2 - z + 2z = 2. z - 2 - 2z + 4 = ( - 2)(z - 2) = 2, : - 2 = 1, z - 2 = 2 - 2 = 2, z - 2 = 1. (2, 3, 4) ; (2, 4, 3) ; (3, 2, 4) ; (3, 4, 2) ; (4, 2, 3) ; (4, 3, 2). 3) > 2, > 2 z > 2, : z 3, z 3z z 3z. 3z 3 + 3z + 3z, z + z + z 2 = + z + z - z 0, . 21. 2 + = 4 + 3 + 2 + .

52

4. 4.4.

: , , . 4 1 42 + 4 + 1 = 44 + 43 + 42 + 4 + 1. (2 + 1)2 = 44 + 43 + 42 + 4 + 1 = 44 + 43 + 2 + 32 + 4 + 1 = (22 + )2 + ( + 1)(3 + 1). (2 + 1)2 = 44 + 43 + 42 + 4 + 1 = 44 + 2 + 4 + 43 + 82 + 4 - 52 - 3 = (22 + + 2)2 52 3. , (22 + )2 + ( + 1)(3 + 1) = (2 + 1)2 = (22 + + 2)2 (52 + 3). : 1) = - 1, 2 + = 0, = 0 = -1. 2) - 1, ( + 1)(3 + 1) > 0. 52 + 3 > 0 , (22 + )2 < (2 + 1)2 < (22 + + 2)2. (22 + )2 (22 + + 2)2 (22 + + 1)2, (2 + 1)2 = (22 + + 1)2. (2 + 1)2 = 44 + 43 + 42 + 4 + 1 = (22 + + 1)2 = 44 + 2 + 1 + 43 + 42 + 2 + 1, 2 2 = 0, . 0 2: 2.1) = 0, 2 + = 0, = 0 = -1. 2.2) = 2, 2 + = 30, = 5 = -6. : (0, -1); (-1, -1); (0, 0); (-1, 0); (5, 2); (-6, 2).

308. . 309. . 310. ? (, ) ? 311. ? 312. , z, ? 313. p, q r , p4 + q4 = r4 . .53

4. 4.4.

314. , b n,

a + bb + ccc = n 4 . ?315. . ? 316. . 317. ? 318. , ? 319. , ? 320. (, ) - 3 = 4 - 16. 321. 4 + 23 + 2 11 + 11 = 2, . 322. n , 4 n2. 323. : 11! + ... + ! = 4 . ? 324. (, ) : 10 + 11 = + 2000. 325. 1, 2, ... 2006 12 + 22 +...+ 20062 = 2040. 1, 2, ... 2006. 326. , b, c, d, : 2 < b < c < d < . 1 1 1 1 1 2 + 2 + 2 + 2 + 2 = 1 . a b c d e

1 1 + = 1 x 2 xy y 2 . ( 1952.)

327.

1

+

328. (26. - 1963.)54

z zx + + = 3. y

4. 4.4.

329. ( ) 2 10 22. (10. 1968.) 330. (, ) - ( + 2)4 4 = 3. ( - 1974.) 331. b . 2 + b2 + b q r. (, b) q2 + r = 1977. (19. - 1977.) 332. n ( - 1)! + 1 = n. ( 1980.) 333. (, ) 3 + 2 + 2 + 3 = 8(2 + + 2 + 1). ( - 1980.) 334. 3 3 = + 61. (15. 1981.) 335. 6 + 33 + 1 = 4 . ( 1981.) 336. 1 1 1 1 + + = x y z 1983 . ( 1983.)

337. , b, c, d, 1 1 1 1 1 + + + + = 1 , a : 2 < < b b < a b c d e c < d < . ( 1995.) 338. , b, c 1 1 1 + + = 1. ( 1996.) ab bc ca 339. (5 + 1)2 = abcd , . ( 1996.) 340. 3 3 = + 25. ( 2000.) 341. (n) , (n) n. n (n) + (n) = n. ( 2002.) 342. 31 . , . ? ( 2003.)55

4. 4.4.

343. (, b) , . (44. 2003.) 2ab 2 b3 + 1 344. 2 + 1 . (22. 2005.)a2

345. 1, 2,... 1 < 2 < ... < ( 2).

1 x 12

+

1 x 22

+ ... +

1 xk2

= 1 .

346. ? * , , . , , . , ( ) .

4.5. , , () () . 22. 2 + 4 = 555...555 ( 555...555 n ).

56

4. 4.5.

: : 1) n = 1, 2 + 4 = 5, . = 1, = 1. 2) n 2, : 2.1) = 2 ( ), 2 + 4 = 42 + 4 = 4(2 + ) = 555...555, . 2.2) = 2 + 1 ( ), 2 + 4 = 42 + 4 + 1 + 4 = 555...555, 42 + 4 + 4 = 4(2 + + ) = 555...554. , 2(2 + + ) = 277...77. , , .

= = n = 1. 23. . : ab . ab = 10 + b = (a + b)2. 10 + b = + b + 9 = (a + b)2, 9 = (a + b)(a + b 1). (a + b) (a + b 1) , 9 , . :

1) ab = 25, , 25 (2 + 5)2 = 49 ; 2) ab = 49, 49 (4 + 9)2 = 169 ; 3) ab = 64, , 64 (6 + 4)2 = 100 ; 4) ab = 81, , 81 = (8 + 1)2 = 92. 24. 2 + 2 + z2 = 2007 . : 2, 2, z2 2007, , , , : 1) = 2m + 1, y = 2n + 1 z = 2k + 1 (m, n, k Z). 2 + 2 + z2 = (2m + 1)2 + (2n + 1)2 + (2k + 1)2 = 4m2 + 4m + 1 + 4n2 + 4n + 1 + 4k2 + 4k + 1 = 2007. 4m(m + 1) + 4n(n + 1) + 4k(k + 1) = 2004. 4 m(m + 1) + n(n + 1) + k(k + 1) = 501. m(m + 1), n(n + 1) k(k + 1) , , , 501 .57

4. 4.5.

2) = 2m, y = 2n z = 2k + 1. 2 + 2 + z2 = (2m) + (2n)2 + (2k + 1)2 = 4m2 + 4n2 + 4k2 + 4k + 1 = 2007. 4m2 + 4n2 + 4k(k + 1) = 2006. 2 2m2 + 2n2 + 2k(k + 1) = 999. , , . 2

25. .

2 + 2 = 21999

: 21999 , 2 2 , : 1) , . = 2m + 1 y = 2n + 1 ( m, n ). 2 + 2 = (2m + 1)2 + (2n + 1)2 = 4m2 + 4m + 1 + 4n2 + 4n + 1 = 21999. 4m(m + 1) + 4n(n + 1) = 21999 2 = 2(21998 1), 2m(m + 1) + 2n(n + 1) = 21998 1. , , . 2) , . = 2m y = 2nb, b , m n 1000. 2 + 2 = (2m)2 + (2nb)2 = 22m2 + 22nb2 = 21999, : 2.1) m > n 1, 22m2 + 22nb2 =22n(22m - 2n2 + b2) = 21999. 22m - 2n2 + b2 = 21999 - 2n. (22m - 2n2 + b2) , 21999 - 2n , . 2.2) m = n, 22m2 + 22nb2 =22n(2 + b2) = 21999, . 2 + b2 = 21999 - 2n. 21999 - 2n 2, : 2.2.1) n 998, 21999 - 2n 4. b , 2 + b2 4 2, . 2.2.2) n = 999, 2 + b2 = 21999 - 1998 = 2, = b = 1, = = 2999. 2.3) 1 m < n, 22m2 + 22nb2 =22m(2 + 22n 2m b2) = 21999 2 + 22n 2m b2 = 21999 2m. (a2 + 22n-2m b2) , 21999 2m , . , = = 2999. 58

4. 4.5.

345. 2 + 2 = 999...999 ( 1) . 346. 2 + 42 = 1000? 347. 2 22 +8z3 = 3 . 348. 22 52 = 7 349. 192 + 2 = 2? 350. 2 1 = 2. 351. 2 + = 1026? 352. ! + 2 = 123456789. 353. 1 p q p + q = 3. 354. , ! + 1 = 2.

355. 11 . (2. 1960.) 356. 2 + 2 = 1990 . ( 1990.) 357. 2 + 2 + z2 = 1980 . ( 1980.) 358. 1991 . ( 1991.) 359. (, ) 2 1992 = 2. ( 1992.) 360. ? ( 1992.)59

4. 4.5.

361. m n m2 + n n2 + m ? ( 1999.)

362. , ! + ! = 2. 363. 4 = 2 + 2 + 54321. 364. , q r . 2p + 3q = 5r. 365. 2 + 2 = 22, 3 + 5 = 23, 3 + 13 = 24, 3 + 29 = 25, 3 + 61 = 26, ... + q = 2, p q , , . 2 ?*

(), 2, , , .

4.6. = , . , 3, 4 1, . = = . 26. q + q = q2 40. : + q = q2 40 , + 40 = q2 q = q(q - 1). , 2, + 40. = 2, q(q 1) = + 40 = 2 + 40 = 42, q = 7.60

4. 4.6.

2 . 27. . : xy , 10 + = + . 10 2 = (10 ) = 3 = ( - 1)( + 1). , 6, . (10 - ) (10 - ) 6, = 6 10 - = 6.2 3

= 4, 24 = ( 1)( + 1), = 3, 43 = 42 + 33. = 6, 24 = ( 1)( + 1), = 3, 63 = 6 + 33. , .2

28. . : ab . 10 + b = 9(a + b). ab 9, 9. + b 9 18, 10 + b 99 = 81 918 = 162. 162 , 81 = 9 (8 + 1). , . . 10. 29. (, ) ! + 5 = 6666. : 6666 5 1. ! + 5 5 1. 5 5, ! 5 1.61

4. 4.6.

5 ! 5, 5, 1, 2, 3, 4. 1! = 1 3! = 6 5 1, 5 = 6665 5 = 6660. : (1, 1333) ; (3, 1332). 30. . ? ? : xy = 10 + . 10 + + 10 + = 11( + ) = 2 + =

k2 18 . + 11

2 11 2 < 198. 2 = 121, = 11. + = 11, : 29, 38, 47, 56, 65, 74, 83 92. 31. 11 .. : z . 100 + 10 + z = 11 ( + + z). 11, 11, + z 11. + z= 0 + z= 11: 1) + z = 0, = + z. , 100 + 10 + 10z + z = 11 ( + + z + z). 11 10 + z = 2 + 2z, 8 = z, = 1, z = 8, = 9. 198. 2) + z= 11, + z = 11, 0 9. + z = 11 + . z 11 (9 + 9 + 9) = 1127 = 297, 2. 0 z 9, + z 11 = 2, z = 9, = 0. , 209 , 209 121 = 1111.

366. 32 + 52 = 345. 367. 2 1 = 2 ?

62

4. 4.6.

368. ! + 2 = 2. 369. abc abc + bca + cab .

370. ! + 48 = 216. 371. ? 372. , 2 + 2 = 10 1? 373. (, ) ! + 2 = 987654. 374. . 375. 3 + 4 = 4( + 1) . 376. , z x y z , + + = m (m ). y z x

377. - , . (2. - 1936.) 378. 5 + 34 532 1523 + 44 + 125 = 33 . (9. - 1946.) 379. 1977 ? ( 1977.) 380. -1. ( 1979.) 381. n ( 1986.)63

n (n +1) 1 . 2

4. 4.6.

382. a b ab = bbb ( - 1991). 383. . ? ( - 1994)

1234567891011 = 399*6*** . ( - 1997)384. 385. (, b) ,

2b + + b b2 + b + 7. (39. 1998.)386.

, , . . ( - 2001.)387. ,

, . ( -2001.)388. p, q r, n, 1 1 1 1 + + = . ( - 2005.) p q r n

389. 6 ! = 10? 390. 11

.

4.7. . = , (mod m), = . .

64

4. 4.7.

32. , , z + + z2 = 8 1?2 2

: 0, 1 4 - 8 (). 2 + 2 + z2 0 (0+0+0), 1 (1+0+0), 2 (1+1+0),3 (1+1+1), 4 (4+0+0), 5 (4+1+0), 6 (4+1+1). 7, . 1, , . , , 57, 2007 8-1. 33. 2 3 = 7.

: : 1) = 0, 2 = 8, = 3. 2) 1, 2 = 3 + 7 10, > 3.

2 = 3 + 7 2 0 (mod 4), 3 + 7 0 (mod 4), 3 1 (mod 4), , = 2b. 3 + 7 1 (mod 3), 2 1 (mod 3), , = 2. 22 32b = 7, (2 + 3b)(2 - 3b) = 7. 2 + 3b = 7 2 - 3b = 1. 2 = 4 3b = 3, = 2 b = 1, = 2 = 4 = 2b = 2. , (3, 0) (4, 2). 34. 1 ?4

+ 2

4

+ ... + 14

4

= 1599

: 16 {0, 1, 2,..., 15}. 2 (mod 16) ( { 0, 1, 4, 9}. 4 0 (mod 16) 4 1 (mod 16). 1 4 + 2 4 + ... + 14 4 16 { 0, 1, 2,..., 13, 14}, 15. 1599 15 (mod 16), .

65

4. 4.7.

391. . 392. 3 + 4 = 4( + 1) . 393. 15 : ) 7; b) 9. 394. , z 4 + 4 = z4 + 3.m2 + n 2 = 1981 3

395. 193 173 = 50 . (30. 1967.) 396. 1! + 2! + ... + ! = z, , z z > 1. ( 1975.) 397. 193 842 = 1984. (47. 1984.) 398. q 2 - 2q2 = 1. ( 1990.) 399. 24 2 + 16 . ( 1992.) 400. b 15 + 16b 16 - 15b . . (37. 1996.) 401. 2 = 5 4 . (15. 1998.) 402. A , z 5 + 5 + z5 = 2004 . . ( 2004.) 403. 3 + 4 4 + 3 , 7. . (8. 2004.)

66

4. 4.8.

404. 4 - 7 = n (n n 100). n ? 405. 2 = 2006 + 4.

4.8. , . O , , , , . 35. 22 + 5 + 2 = 19. : , . 22 + 5 + (2 19) = 0. 1,2 =

5

25 8( 2 19) . ( ) 4

. , 25 8(2 19) = 2, , . , 25 82 + 152 = 2, 2 + 82 = 177. 82 177, 2 4, { 0, 1, 2, 3, 4}. = 1 = 4. = 13 = 7. : (2, 1) (-3, 4), (1/2, 4) (-9/2, 1) , . 36. . 2 + + 2 = 22

67

4. 4.8.

: : 1) = 0, = 0. 2) = -1, = 1. 3) = 1, = - 1. 4) {-1, 0, 1}, (1 2)2 + + 2 = 0. . 2 4(1 2)2 = 2 42 + 44 = 44 32 = 2(42 3) , . = 0 42 3 = 2. 42 2 = 3, (2 + )(2 ) = 3. = 1 = -1. {-1, 0, 1}, . , : (0, 0); (1, -1); (-1, 1). 37. - ( + 1)( + 2)( + 3) = 1? 104

: . , . , 4 - 1 = ( 1)( + 1)(2 + 1) = ( + 1)( + 2)( + 3). , . . 4 - ( + 1)( + 2)( + 3) = 1, 4 = ( + 1)( + 2)( + 3) + 1 = (2 + 3)(2 +3 + 2) + 1 = (2 + 3 + 1 - 1)(2 +3 + 1 + 1) + 1 = (2 + 3 + 1)2 1 + 1 = (2 + 3 + 1)2. , 4 - (2 + 3 + 1)2 = 0, ( 2 - (2 + 3 + 1))(2 + (2 + 3 + 1)) = 0. : 2 + 3 + 1 - 2 = 0 2 + 3 + 1 + 2 = 0. : 1) 2 = 9 4(1 - 2) = 2 2 - 42 = 5. 2 = 9, 2 = 1. 2) 2 = 9 4(1 + 2) = 2 2 + 42 = 5. 2 = 1, 2 = 1. 8 : (0, 1); (0, -1); (-1, 1); (-1, -1); (-2, 1); (-2, -1); (-3, 1); (-3, -1). 10

1981. 2. .

68

4. 4.8.

406. 2 + + 2 = 1 . 407. , z + z2 + 1 = 0 = 3. 408. 2 + + 2 = 22. 409. (, ) 4 + 2 + 2 = 2. 410. + + 1 = + 2, . 411. 6 + 33 + 1 = 4.8 4 2

412. 2 + 2 = ? (27. - 1964.) 413. 2 + 2 = + . (7. 1941. 1947.) 414. 2 + 3 + 24 . ( 1980.)

4.9. . . , . : (x, y, z, ...) F(x, y, z, ...) = 0, . , , .

69

4. 4.9.

38. 2 + 2 = z3 . : = 5, = 10 z = 5, 52 + 102 = 125 = 53. 526+ 1026 = 536 ( ), (53)2 + (103)2 = (52)3. = 53, = 103 z = 52 . 39. + = z4 + t 5 ?2 3

: = = z = t = 0 = = z = t = 1, = 230, = 220 z = 215 t = 212. , = 230 , = 220 , z = 215 t = 212. : = 30, = 20 , z = 15 t = 212 ( ), : = 5, = b4, z = b3 t = a2 ... . F(x, y, z, ...) = 0 P(m, n), Q(m, n), R(m, n)... (m, n ) F( P(m,n), Q(m,n), R(m,n), ...) = 0. m n . 40. 2 + 2 + 1 = z2 . : 4n4 + 4n2 + 1 = (2n2)2 + (2n)2 + 1 = (2n2 + 1)2, = 2n2, = 2n z = 2n2 + 1 . . F(x, y) = 0 F(P(x, y), Q(x, y)) = 0, P(x, y) Q(x, y) . (, ) , +1 = P(x, y) = Q(x, y)) . .

70

4. 4.9.

41. 22 32 = 5 . : a, b, c d 2( + by)2 - 3(cx + dy)2 = 2x2 3y2 = 5.

222 + 4b + 2b22 322 - 6cd - 3d22 = 22 32 (22 32)2 + 2(2b - 3cd) + (2b2 - 3d2)2 = 22 32. 22 32 = 2, 2b - 3cd = 0 2b2 - 3d2 = -3. = 5, = 4, 10b - 12d = 0 2b2 - 3d2 = -3. b = 6, d = 5. , 2(5 + 6)2 - 3(4 + 5)2 - 22 32 = 5. = 2, = 1 , +1 = 5 + 6 +1 = 4 + 5 . , 22 32 = 2, 2b - 3cd = 0 2b2 - 3d2 = -3 ( = 5, b = - 6, c = 4, d = -5), ( 2 1), .

415.

2 = 7 + 2.416. , 2 + 2 + = z2 . 417. ( 1)2 + ( + 1)2 = 2 + 1 -

.418. m

(,) 2 (m2 +1)2 = 1.419. 3 3 =

2 + 2 + (+)2.

71

4. 4.9.

420. m n (1 m 1981, 1 n 1981)

(n2 mn m2)2 = 1 . m2 + n2. (22. 1981.)421. 3 + = n. n 3 2 3

(, ), . n = 2891 . (23. 1982.)422. , z + z + z = 1

. . ( 1990.)

423. (, , z)

3 3 = 2 + 2 + z2.

4.10. '''' 11 , . :12 '' . , (. +). , (, , ) .

11 12

() , : , 2004, . 53 54.

72

4. 4.10. ''''

. , ( ) . , , , . , .'' 42. 2+2 =3z2.. : = = z = 0. . (, , z) = ++ z . 2 + 2 = 3z2 3 ( 3 + 1, 3 1, 2 2 3 1, 2 + 2 3 1, 2). = 31 = 31. 2 + 2 = 9 (12 + 12) = 3z2. 3(12 + 12) = z2, z 3, z = 3z1. 3(12 + 12) = 9z12, 12 + 12 = 3z12. (1, 1, z1) . =++z=31+31+3z1= 3 (1+1+ +z1) = 31, 0 < 1 =

< . 3

, . 43. 3+23= 4z3 : = = z = 0. . 13 (, , z) , . 3 + 23 = 4z3. 3 = 4z3 - 23 = 2(2z3 3). , . = 21. 813 = 4z3 - 23. 2 413 = 2z3 - 3 , = 21. = 21 413 = 2z3 - 813.13

18. - 1955.

73

4. 4.10. ''''

2 213 = z3 - 413. z , . z = 2z1. 213 = 8z13 - 413, 2 13 + 213 = 4z13. 1 =

, 2

z . 2 , 21 =,

2

z1=

2

z , , 22

+ 2 = 1 (2 + 22 ) = = 1 4z 2 = 4 z 2 2 2 2 2 22 , , , z 2, , . 44. 2 ( + 1) (2 + 1).

2

2

: (2 + 1) px = y2 + 1 (2 + 1) qy = 2 + 1. D(, ) = d, = d = d. d = 2d2 + 1 qd = 2d2 + 1. d ( - 2d) = 1 d (q - 2d) = 1, d = 1, . px qy = (y2 + 1)(2 + 1) = 22 + 2 + 2 + 1,

2 + 2 + 1 + + 1. n = 2 2

> 2, n = 2 + 2 + 1, x2 - nyx + y2 +1 = 0 (*). (, ) (*) . (*) , f(x) = 2 - nyx + y2 + 1 = 0. f(ny x) = (ny x)2 ny(ny x) + y2 +1 = n2y2 - 2ny + x2 - n2y2 + nyx + y2 +1 = x2 - nyx + y2 +1 = 0. , (, ) (*), (ny x, ) (*).

74

4. 4.10. ''''

ny x > 0 , ny x . f(x) (, ny x), . f() = 2 - ny + y2 +1 = (2 n)y2 +1 0. = = 1 , 3 - n 0, n = 3. (, ) (ny x, ), (, ) (3y x, ), (1, 1) (2, 1), (5, 2), (13, 5),... (F2n-1, F2n+1), F 1, 1, 2, 3, 5, 8, 13, 21,...( F1 = F2 = 1, F+2 = F + F+1). 14

424. 2 + 2 + z2 =

z .

425. 3 + 23 + 4z3 - 6xyz = 0

( 1948.)426. 2 + 2 + z2 = 2xyz

= = z = 0 . (12. 1949.)427. , , z u

2 + 2 + z 2 + u2 = 2xyzu. (12. 1949.)428. , z 2 + 2 + z2 = 22.

( 1976.)429. 3 + 33 + 9z3 - 9xyz = 0 -

= = z = 0. ( 1983.)430. 2 + 2 = 3(u2 + v2).

( 2003.)431. 2 + 2 + z2 = 2004xyz, ,

z . ( 2004).14

, , : , , 2004.

75

4. 4.11.

4.11. '''' . : 2 + 2 + + D + E + F = 0 (A, B, C, D, E, F ) ( , ). Ao je (, ) , (, ) = ( ) ( 0). , . . 15 :

45. 32 - 2 = 26 . : : = + t ; = + t. (, ) = (3, 1), = 3 + t; = 1 + t. 3(3 + t)2 (1 + t)2 = 26, t ( t (3 2) + 18 2) = 0. t = 0 t = , t =18 22 3

18 22 3

. t = 0

, =

3 2 2 + 92 3

=

18 2 3 2 3

.

= 1, (-5, -7), = 2 (17, 29).

15

: , , 1974.

76

4. 4.11. , =m : = n 18mn m 2 3n 2 m 2 3n 2 3m 2 2mn + 9n 2 m 2 3n 2

=

.

46. 2 - 22 + 3 + 4 5 + 3 = 0 . : (, ) = (1, 0), = 1 + t; = t. (1 + t)2 2(t)2 + 3(1 + t) t 4(1 + t) + + 5t + 3 = 0. t ( t (1 22 + 3) + 8 2) = 0. t = 0 t =8 2 2 2 3 1

. = .

2 2 + 5 3 2 2 3 1

=

8 2 2 2 2 3 1

. . 16 47. 192 + 5b2 = c2 . : : 1) = 0, = b = = 0. 2) 0, =

b =

192 + 52 = 1. 1 3 1 3 (, ) = , = + t = + t 8 8 8 8 (15 + 19) . , t = 4 (5 2 + 19)

16

: : , , 1974, . 50-51.

77

4. 4.11.

=5 2 30 19 8 (5 2 + 19)

=

15 2 38 + 57 8 (5 2 + 19)

.

= ( 5 2 30 19 ); b = ( 15 2 38 + 57 ) c = 8 (5 2 + 19) , . . = : b = 15m 2 38mn + 57 n 2 n2

m n

;

=

.

5m 2 30mn 19n 2 n2

c = 8

5m 2 + 19n 2 n2

432. 2 + 22 = 3z2, , z . 433. : 2 - 152 = z2; 2 z = z 2; 2 + 32 = 5z2.

78