7/17/2019 se_04-05

http://slidepdf.com/reader/full/se04-05 1/4

 

International Mathematical Olympiad

Preliminary Selection Contest 2004 — Hong Kong

°ê»Ú¼Æ¾Ç¶øª L¤Ç§J2004 - »´ ä¿ï ©ÞÁÉ

6th June, 2004

2004 ¦ ~ 6 ¤ë 6 ¤é

Time allowed: 3 hours®É ¡ G3 ¤p®É

Instructions to Candidates:

¦ ҥͶ· ª ¾¡ G

 1. Answer ALL questions.

¥»¨ ÷¦ UÃD¥þµª¡ C

2. Put your answers on the answer sheet.

½Ð±Nµª ®×¼g¦ bµª ÃD¯È¤W¡ C

3. The use of calculators is NOT allowed.

¤£¥i Ï ¥Î pº â¾÷ ¡ C

7/17/2019 se_04-05

http://slidepdf.com/reader/full/se04-05 2/4

  2

1. Let  ABCDEFGH  be a rectangular cuboid. How many acute-angled triangles are formed by joining any three vertices of the cuboid? (1 mark)

³ ]  ABCDEFGH ¬°ª ø¤èÅé¡ C Y§â¥ô· N¤T Ó³ »ÂI³ s°_¡A¥i² Õ¦ ¨ ¦ h¤Ö Ó¾U¨ ¤¤T¨ ¤§Î ¡ H  ¡ ] 1 ¤À¡ ^

2. A collector has  N  precious stones. If he takes away the three heaviest stones then the

total weight of the stones decreases by 35%. From the remaining stones if he takes awaythe three lightest stones the total weight further decreases by

5

13. Find N . (1 mark)

¤@¦ W¦ ¬Âîa¾Ö¦ ³ N ¶ôÄ_¥Û¡C Y¥L®³ ¨ «³ Ì «ª º ¤T¶ôÄ_¥Û¡A º »òÄ_¥Ûª º Á «¶q· |´ î ¤Ö

35%¡ C Y¥L±q¾l¤Uª º Ä_¥Û¤¤¦ A®³ ¨ «³ Ì »´ ª º ¤T¶ô¡ A º »òÄ_¥Ûª º Á «¶q· |¦ A´ î ¤Ö5

13¡ C

¨ D N¡C   ¡ ] 1 ¤À¡ ^

3. Denote by [a] the greatest integer less than or equal to a. Let  N  be an integer,  x and  y be

numbers satisfying the simultaneous equations [ ] 2 2[ ] 2 3 x y N  y x N 

+ = ++ = −

. Find x in terms of N . (1 mark)

§Ú Ì §â¤p©ó©Î µ¥©ó  a ª º ³ Ì ¤j¾ã¼Æ°O§@[a]¡C³ ]   N ¬°¾ã¼Æ¡ A¥B x ©M y º ¡ ¨ ¬Áp¥ß¤èµ{  [ ] 2 2

[ ] 2 3

 x y N 

 y x N 

+ = +

+ = − ϒΧ♦∆ x¡ Aµª ®×¥HN ª í ¥Ü¡C   ¡ ] 1 ¤À¡ ^

4. A  palindrome is a positive integer which is the same when reading from the right handside or from the left hand side, e.g. 2002. Find the largest five-digit palindrome which is

divisible by 101. (1 mark) Y¬Y¥¿¾ã¼Æ¤£½×±q¥ª ±©Î ¥k ±Åª ° _¬Ò¬Û¦ P¡ ] ¨ Ò¦ p¡ G2002¡ ^¡ A«h¸ Ӽƺ Ù¬°¡ u¦ ^¤å¼Æ¡ v¡ C D¥i³ Q101 ¾ã°£ª º ³ Ì ¤j¤ ¦ ì ¦ ^¤å¼Æ¡ C ¡ ] 1 ¤À¡ ^

5. Evaluate3 4 2004

1! 2! 3! 2! 3! 4! 2002! 2003! 2004!+ + +

+ + + + + +L . (1 mark)

¨ D3 4 2004

1! 2! 3! 2! 3! 4! 2002! 2003! 2004!+ + +

+ + + + + +L  ª º È¡ C ¡ ] 1 ¤À¡ ^

6. If( )( )( ) 2004

( )( )( ) 2005

a b b c c a

a b b c c a

− − −=

+ + +, find

a b c

a b b c c a+ +

+ + +. (1 mark)

Y( )( )( ) 2004

( )( )( ) 2005

a b b c c a

a b b c c a

− − −=

+ + +  ϒΑ♦∆

a b c

a b b c c a+ +

+ + +  ϒΧ ¡ ] 1 ¤À¡ ^

7. Let  D be a point inside triangle ∆ ABC  such that  AB DC = , 24 DCA∠ = ° , 31 DAC ∠ = °  and 55 ABC ∠ = ° . Find ∠ DAB. (1 mark)

³ ]   D ¬°   ∆ ABC ¤ º ª º ¤ @ ÂI ¡A¨ Ï ±o   AB DC = ¡ A 24 DCA∠ = ° ¡ A 31 DAC ∠ = ° ¡ A¥B 55 ABC ∠ = ° ¡C¨D∠ DAB¡C   ¡ ] 1 ¤À¡ ^

7/17/2019 se_04-05

http://slidepdf.com/reader/full/se04-05 3/4

  3

8. It is known that 999973 has exactly three distinct prime factors. Find the sum of these prime factors. (1 mark)

¤wª ¾999973 è¦ n¦ ³ ¤T Ó¤£¦ Pª º ½è¦ ]¼Æ¡ C D³ o¨ Ç½è¦ ]¼Æ¤§©M¡ C ¡ ] 1 ¤À¡ ^

9. A person picks n  different prime numbers less than 150 and finds that they form an

arithmetic sequence. What is the greatest possible value of n? (1 mark)¬Y¤H¿ï ¨ ú¤Fn Ó¤£¦ Pª º ½è¼Æ¡A C Ó§¡ ¤p©ó 150¡ C¥Lµo² {³ o¨ ǽè¼Æ² Õ¦ ¨ ¤@ Óµ¥®t¼Æ¦ C¡ C Dn ª º ³ Ì ¤j¥i¯à È ¡ C   ¡ ] 1 ¤À¡ ^

10. Given a positive integer n, let ( ) p n   be the product of the non-zero digits of n. Forexample, (7) 7 p   = , (204) 2 4 8 p   = × = , etc. Let (1) (2) (999)S p p p= + + +L . What is

the largest prime factor of S ? (1 mark)

¹ ï ©ó¥¿¾ã¼Æn¡A³ ] ( ) p n   ¬° n ª º ©Ò¦ ³ «D¹ s¼Æ¦ r¤§¿n¡ C  ¨ Ò¦ p¡ G (7) 7 p   =   ¡B (204) 2 4 8 p   = × =  µ¥¡ C

³ ] (1) (2) (999)S p p p= + + +L   ¡ C º »ò¡ AS ³ Ì ¤jª º ½è¦ ]¼Æ¬O¬Æ»ò¡ H  ¡ ] 1 ¤À¡ ^

11. If

1 1 1

1 2 3 4 2003 20041 1 1

1003 2004 1004 2003 2004 1003

 A

 B

= + + +   × × ×

  = + + +   × × ×

L

L

, find A

 B. (1 mark)

Y

1 1 1

1 2 3 4 2003 2004

1 1 11003 2004 1004 2003 2004 1003

 A

 B

= + + +   × × ×

  = + + +   × × ×

L

L

¡

 A

 B¡ C ¡ ] 1 ¤À¡ ^

12. Find the number of 6-digit positive integers abcdef   satisfying the following two

conditions:

(a) Each digit is non-zero.

(b) a b c d e f  × + × + ×  is even. (1 mark)

¨ D² Ŧ X¥H¤U¨ â Ó±ø¥óª º ¤»¦ ì ¥¿¾ã¼Æabcdef   ª º ¼Æ¥Ø¡ G

(a) ¨ C¦ ì ¼Æ¦ r¬Ò¤£µ¥©ó¹ s ¡ C

(b) a b c d e f  × + × + ×  ¬O°¸ ¼Æ¡ C ¡ ] 1 ¤À¡ ^

13. Find the area enclosed by the graph 2 2 | | | | x y x y+ = +  on the xy-plane. (2 marks)

¨ D xy §¤¼Ð¥ ±¤W¥Ñ¹ Ï ¹ ³   2 2 | | | | y x y+ = +  ©Ò³ ò¥Xª º ±¿n¡ C ¡ ] 2 ¤À¡ ^

14. Determine the number of ordered pairs of integers ( , )m n  for which 0mn ≥  and3 3 399 33m n mn+ + = . (2 marks)

¨ D º ¡ ¨ ¬ 0mn ≥  ¤Î 3 3 399 33m n mn+ + =  ª º ¾ã¼Æ§Ç°¸ ( , )m n  ª º ¼Æ¥Ø¡ C ¡ ] 2 ¤À¡ ^

7/17/2019 se_04-05

http://slidepdf.com/reader/full/se04-05 4/4

  4

15. A natural number n  is said to be lucky  if 2   n−  and 12   n− −   have the same number ofsignificant figures when written in decimal notation. For example, 3 is lucky since

32 0.125− =  and 42 0.0625− =  have the same number of significant figures. On the otherhand 52 0.03125− =  has one more significant figure than 0.0625, so 4 is not lucky. Giventhat log 2 0.301=   (correct to 3 significant figures), how many lucky numbers are less

than 2004? (2 marks)

¹ ï ©ó¦ ÛµM¼Æn¡AY   2   n−  ©M 12   n− −  ¨ â¼Æ¥H¤Q¶i¨ î ª í ¥Ü®É¦ ³ ®Ä¼Æ¦ rª º ¼Æ¥Ø¬Û¦ P¡ A«h  n º Ù ¬°¡ u© ¹ B¼Æ¡v¡ C Ò¦ p¡G 32 0.125− =  ©M 42 0.0625− =  ¨ â¼Æ¤¤¦ ³ ®Ä¼Æ¦ rª º ¼Æ¥Ø¬Û¦ P¡ A¦ ]

¦ ¹ 3 ¬O© ¹ B¼Æ¡ C¥t¤@¤è ±¡ A¥Ñ©ó 52 0.03125− =  ¤ñ 0.0625 ¦ h¤F¤@ Ó¦ ³ ®Ä¼Æ¦ r¡ A©Ò¥H4 ¤£¬O© ¹ B¼Æ¡ C¤wª ¾log 2 0.301= ¡ ] · ǽT¦ Ü 3 ¦ ì ¦ ³ ®Ä¼Æ¦ r ¡ ^¡ A°Ý¦ ³ ¦ h¤Ö Ó© ¹ B¼Æ¤p

©ó 2004¡ H ¡ ] 2 ¤À¡ ^

16. A positive integer n is said to be  good  if 3n is a re-ordering of the digits of n when theyare expressed in decimal notation. Find a four-digit good integer which is divisible by 11. (2 marks)

¹ ï ©ó¥¿¾ã¼Æn¡ A Y¦ b¤Q¶i¨ î ª í ¥Ü¤¤¡ A¾ã¼Æ 3n ¥i±q n ª º ¼Æ¦ r¸ g «· s±Æ¦ C¦ Ó±o¥X¡A«hn º Ù¬°¡u¦ n¼Æ¡ v¡ C D¤@ Ó¥i³ Q11 ¾ã°£ª º ¥|¦ ì ¦ n¼Æ¡ C ¡ ] 2 ¤À¡ ^

17. From any n-digit ( 1n > ) number a, we can obtain a 2n-digit number b by writing two

copies of a one after the other. If2

b

a is an integer, find the value of this integer. (2 marks)

¹ ï ©ó¥ô·N n ¦ ì ¼Æa¡]¨ 䤤   1n >   ¡^¡A§â a ³ s¼g¨ ⦠¸ ¥i±o¨ ì ¤@ Ó2n ¦ ì ¼Æb¡C Y  2

b

a ¬°

¾ã¼Æ¡ A D¦ ¹ ¾ã¼Æª º È¡ C ¡ ] 2 ¤À¡ ^

18. Find the sum of all x such that 0 360 x≤ ≤  and 2 2cos12 5sin 3 9 tan cot x x x° = ° + ° + ° . (2 marks)

¨ D©Ò¦ ³ º ¡ ¨ ¬0 360 x≤ ≤  ¤Î 2 2cos12 5sin 3 9 tan cot x x x x° = ° + ° + °  ª º   x Ȥ§©M¡C   ¡ ] 2 ¤À¡ ^

19.  ABC  and GBD are straight lines.  E  is a point on CD produced, and  AD meets EG at F . If∠CAD = ∠ EGD, EF  = FG, : 1: 2 AB BC  =  and : 3 : 2CD DE  = , find : BD DF . (3 marks)

 ABC ©MGBD §¡¬°ª ½½u¡AE ¬O CD ©µ½u¤Wª º ¤@ÂI ¡A¥B AD ¥æ EG ©ó F¡C Y   ∠CAD  =∠ EGD¡AEF  = FG¡A : 1: 2 AB BC  =   ¡A¥B  : 3 : 2CD DE  =   ¡A¨D : BD DF ¡ C  ¡ ] 3 ¤À¡ ^

20. For any positive integer n, let ( ) f n  denote the index of the highest power of 2 which

divides !n , e.g. (10) 8 f    =  since 8 4 210! 2 3 5 7= × × × . Find the value of(1) (2) (1023) f f f + + +L . (3 marks)

¹ ï ©ó¥ô¦ 󥿾ã¼Æn¡ A³ ]   ( ) f n  ¬°¥i¾ã° £ n! ª º 2 ª º ³ Ì ° ª ¼¾ ª º «ü¼Æ¡ C Ò¦ p¡ G¦ ] ¬°8 4 210! 2 3 5 7= × × × ¡ A ©Ò¥H   (10) 8 f    =   ¡C¨D (1) (2) (1023) f f f + + +L  ª º È¡ C ¡ ] 3 ¤À¡ ^

End of Paper¥þ¨ ÷§¹