Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://lnfm1.sai.msu.ru/~olympiad/2002/files/sol2002teor.pdf Äàòà èçìåíåíèÿ: Sat Mar 15 15:46:14 2008 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 23:28:29 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 5

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8 . 1. . (.. , 2002.) ­ (5°37'), , , . , 41°59' ( 42°) , (, , ) 2 47 56 ( 2 48 = 2,8 ) . ! , 8 7°23'08,6" - 7°00'42,0'' = 22'26,6" 22,4', 2,8 22,4'·2,8 / 24 = 2,62' 3'. . , : , 5°34'. 2. . (.. .. , 2002.) , ( 44 ). P, ­ O, ­ C. POC ­ , . CPO 60°, 1/6 . , C O 24 : 6 = 4 , ­ . : 8 . . (.. , 2000.) ( ) . , , , . = 16°43', , ­ , 16°43' , , 73°17'. ­ (35' ). 73°52' .. . , . , , 5°. , 68° - 69° .. . (.. , 2001.) , . 360° 23 56 04 ( ), · 23 56 04 / 360°. = 90° 5 59 01. . (.. -­.. .. , 2002.) ­ , , II , . , . , , ­ , .

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6.

. (.. , 2000.) 29,53 ( , , , , ). , , , ­ . ­ , . 360° 29,53 = 708,7 , 0,5° ­ 708,7·0,5°/360° 1 . , , (, ), ( ). , , - , ; ­ . , , , , "". , , . ­ , ­ , . . (. 377 " - IV".)

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9 . 1. . (.. , 2002.) ­ (5°37'), , , . , 41°59' ( 42°) , (, , ) 2 47 56 ( 2 48 = 2,8 ) . ! , 8 7°23'08,6" - 7°00'42,0'' = 22'26,6" 22,4', 2,8 22,4'·2,8 / 24 = 2,62' 3'. . , : , 5°34'. 2. . (.. .. , 2002.) , ( 44 ). P, ­ O, ­ C. POC ­ , . CPO 60°, 1/6 . , C O 24 : 6 = 4 , ­ . : 8 . . (.. , 2000.) ( ) . , , , . = 16°43', , ­ , 16°43' , , 73°17'. ­ (35' ). 73°52' .. . , . , , 5°. , 68° - 69° .. 22750 . (.. , 2002.) 2,512 . . , , , 2,512 . , 5- 4- 700/250 = 2,80 , 6- 5- 1900/700 = 2,71 , 7- 6- 5300/1900 = 2,79 , 8- 7- 14600/5300 = 2,75 . 2,512, ( ) . , . : . ­ 2002 .

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-. (.. , 1992, 2000.) . , R ( ­ , 0, 0), . . , R ( , 4/3··R3, 0, 0), g = G·M/L2 = G··4/3··R3/L2, L ­ . , , L R, . g = G··4/3··R3/R2 = G··4/3··R. T = 2·(l/g)1/2, g g T/T = -(g/g)/2. : 2T/T = -(G··4/3··R) / g. = 0 - 0 = -0, 2T/T = (G·0·4/3··R) / g, : R = (3/2)·(T/T)·g / (G·0). 2·10-5 %, T/T = 2·10-7, , 5,0 . : () 10 .

6.

. (.. , 2000.) 29,53 ( , , , , ). , , , ­ . ­ , . 360° 29,53 = 708,7 , 0,5° ­ 708,7·0,5°/360° 1 . , , (, ), ( ). , , - , ; ­ . , , , , "". , , . ­ , ­ , . ­ 2002 .


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. (. 377 " - IV".)

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1. 10 . . (.. .. , 2002.) ? , , (- ­ "").

, , . ( « »), , , . , , . h S : V = h·S / 3. : , . , ­ . , . r1 - ( ), r - 15m. ·r1 ·r, S1 = (r1)2 S = (r)2. : V1 = r1·( r1)2 / 3 = 2 r13 / 3, V = r·( r)2 / 3 = 2 r 3 / 3. : N1 / N = V1 / V = (r1 / r) r1 = r (N1 / N)1/3. r , , m. M = 5m , m = 15m. 5 lg r - 5 = m - M, ­ 2002 .
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r1 r :


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lg r = (m - M + 5) / 5, lg r = 3, r = 1000 . , N1 / N = 10, : r1 = r (1/10)1/3 460 . : ( ), . 2. . (.. .. , 2002.) 2 /, 24 . , , ­ , , 48 . , . . 24 361°. , 24 ( ), 23 56 ( , ­ 24 â 365,26 / 366,26 = 23 56 04). 4 24- (4 : 60) â 15° = 1°. , , . . , 23 56 04 â 2 / = 47,869 47,869 / 2 = 7,621 . . , , 24- 4 131 . . (.. .. , 2002.) , 0,5. , . , , . , 2 , «» 2000 sin(0,5) 2000 0,5 / 2105 = 510-3 = 5 . , 2 ­ , , . 200 ­ 2002 .

3.


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100 , 2 , 1/2, 1/200 . /D, ­ , 5000 å = 5·10-7 . -8 = 1/200 (1/200)·(1 /200000) = 2,5·10 , D = / 5·10-7 / 2,5·10-8 = 20 . . 1 0,1. 10 . . 4. 22750 . (.. , 2002.) 2,512 . . , , , 2,512 . , 5- 4- 700/250 = 2,80 , 6- 5- 1900/700 = 2,71 , 7- 6- 5300/1900 = 2,79 , 8- 7- 14600/5300 = 2,75 . 2,512, ( ) . , . : . 5. -. (.. , 1992, 2000.) . , R ( ­ , 0, 0), . . , R ( , 4/3··R3, 0, 0), g = G·M/L2 = G··4/3··R3/L2, L ­ . , , L R, . g = G··4/3··R3/R2 = G··4/3··R. T = 2·(l/g)1/2, g g T/T = -(g/g)/2. : 2T/T = -(G··4/3··R) / g. = 0 - 0 = -0, 2T/T = (G·0·4/3··R) / g, : ­ 2002 .


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R = (3/2)·(T/T)·g / (G·0). 2·10-5 %, T/T = 2·10-7, , 5,0 . : () 10 . 6. . (.. , 2002.) d. : M = ·d·S, d = M / ·S. , ; : 2·103 /3 2·104 /3. = 2,7·103 /3. , 10 . , M = 5·103 . 7 654 321 2 7,7 2. : d = 5·103 / (2,7·103 /3·7,7·106 2) 2,4·10-7 1/4 . , , . : 0,03 0,4 .

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1. 11 . . (.. .. , 2002.) ? , , (- ­ "").

, , . ( « »), , , . , , . h S : V = h·S / 3. : , . , ­ . , . r1 - ( ), r - 15m. ·r1 ·r, S1 = (r1)2 S = (r)2. : V1 = r1·( r1)2 / 3 = 2 r13 / 3, V = r·( r)2 / 3 = 2 r 3 / 3. : N1 / N = V1 / V = (r1 / r) r1 = r (N1 / N)1/3. r , , m. M = 5m , m = 15m. 5 lg r - 5 = m - M, ­ 2002 .
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r1 r :


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lg r = (m - M + 5) / 5, lg r = 3, r = 1000 . , N1 / N = 10, : r1 = r (1/10)1/3 460 . : ( ), . 2. . (.. .. , 2002.) 2 /, 24 . , , ­ , , 48 . , . . 24 361°. , 24 ( ), 23 56 ( , ­ 24 â 365,26 / 366,26 = 23 56 04). 4 24- (4 : 60) â 15° = 1°. , , . . , 23 56 04 â 2 / = 47,869 47,869 / 2 = 7,621 . . , , 24- 4 131 . . (.. .. , 2002.) , 0,5. , . , , . , 2 , «» 2000 sin(0,5) 2000 0,5 / 2105 = 510-3 = 5 . , 2 ­ , , . 200 ­ 2002 .

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100 , 2 , 1/2, 1/200 . /D, ­ , 5000 å = 5·10-7 . -8 = 1/200 (1/200)·(1 /200000) = 2,5·10 , D = / 5·10-7 / 2,5·10-8 = 20 . . 1 0,1. 10 . . 4. . (.., 1999, .., 2002.) : . ( ) , ( ). , , , . « » . . , . , , . ­ , ­ . , . 5. . (.. , 1999.) , , , F ~ 1/R2. , , « » . , , M0 M1. , , ( ­ R0), ( ­ R1) ­ , ( - ). , II , , , « » : M1 = M0·(1 + R0/R1)/2 0,829 M0, (GM)/GM0 (1 - R0/R1)/2 0,171. , . E = mc2, p = mc, , p = E/c. P/t ­ - , . P/t : P/t = (p/t). , , p p = 2p0 = 2E0/c. : ­ 2002 .


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F = P/t = (2·E / t) / c. , , , R : E/t = A·(R0/R)2·S, : F = 2AS(R0/R)2/c. , (1 - R0/R1)/2 , : (1 - R0/R1)·GM0m/2R2 = 2AS(R0/R)2/c, (1 - R0/R1)·GM0m = 4ASR02/c, S = (1 - R0/R1)·(GM0/R02)·(mc/4A). , , GM0 = 42R03/T02, T0 ­ . S = (1 - R0/R1)·(2R0/T02)·(mc/A). S = 0,342·( 6.
2

·1,5·1011/1015)·(104·3·108/1,4·103) 1,1·106 2 = 1,1 2.

, . . (.. , 2002.) , d. : M = ·d·S, d = M / ·S. , ; : 2·103 /3 2·104 /3. = 2,7·103 /3. , 10 . , M = 5·103 . : d = 5·103 / (2,7·103 /3·1,1·106 2) 1,7·10-6 = 1,7 . , . : 0,2 3 . , , 7,7 2 = 2,7·103 /3: d = 5·103 / (2,7·103 /3·7,7·106 2) 2,4·10-7 1/4 . , , , . : 0,03 0,4 .

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