1、 - 1 - 10 2015/2016 1 2015 a 3 16 a . 4 3 16 2 3 2 1 a a a 4 a . . . . a 2 4 3 a . 2. 2015 12 P AB C AB E CB . 2 PO 1 OA AOC P PA OE . 10 10 arccos - 2 - PA OE 10 10 arccos . . ( ) 3. 2016 ABCD A1B1C1D1 E F BC BB1 EF . (A) AA1 (B) A1B1 (C) A1D1 (D) B1C1 D . 4. 2016 12 2 1 6 2 6 . 1 AA1O1O OO1 AC 5 6
2、 11 AB 3 B1 C AA1O1O . 1 2 O1B1 OC . - 3 - 1 V 2 S 2 2 . . . - 4 - 1. 2014 , 8 . 24 2 (5 1) 3 24 . . 2. 2013 , 10 l r O A B BC OA BC 6 l r _. 3 3 tan 63 r l 3 l r . 3. 2012 , 5 2 _ 4. 2011 , 7 ( ) 3,3,2 _ - 5 - 5. 2010 , 6 P ABCD 6 PA ABCD PA 8 _ 96 6. (2009 , 5) , ABCD A1B1C1D1 2, 4, BD1 AD _.( ) 5
3、 arctan 1 BD1 AD . 1 , 5 2 4 2 2 2 1 CD , 1= 5 2 5 2 1 BC CD . - 6 - 1= 5 arctan . 7. (2009 , 6) O1 O2 4 2 1 S S , 2 1 R R =_. 2 4 4 4 2 2 2 1 2 2 2 1 2 1 R R R R S S , 2 2 1 R R . 8. (2009 , 8) 2, _. 3 8 9. (2009 , 16) , , 3 4, 4, , B , , B . 10. 2007 , 7 1 1 1 C B A ABC 90 ACB 2 1 AA 1 BC AC B A 1
4、 AC . - 7 - 6 6 arccos 11. 2007 , 16 12 ABCD P 2 PA PA ABCD 60 ABCD P V . 23 3 PO ABCD O. AO O ABCD PAO PA ABCD . - 8 - PAO 60 2 PA . 3 PO 1 AO 2 AB 3 3 2 2 3 3 1 3 1 ABCD S PO V . 12. 2006 , 16 . A 48 B 18 C 24 D 36 D 13. 2005 , 12 a 2 ) 0 ( 5 , 4 , 3 a a a a . a _. 15 0 3 a - 9 - a 5 24 2 a +28 a
5、4 24 2 a +32 a 3 24 2 a +36 12 2 a +48 20 12 48 12 28 24 2 2 2 a a a 3 15 0 a 1. 2014 , 19 12 2 P ABC , 1 2 3 PPP 1 2 3 PPP V . 4 22 3 - 10 - 1 2 1 3 2 3 4 PP PP PP P ABC 2 P ABC O BO AC D D AC O ABC PO ABC 2 2 3 33 BO BD 26 3 PO 1 1 3 2 6 2 2 22 3 2 2 3 3 V 2. 2013 , 19 O ABC 2 1 3 3 33 - 11 - 3. 2
6、012 , 19 P ABC PA ABC D PC 2 BAC AB 2 23 AC PA 2. - 12 - (1) P ABC (2) BC AD ( ) (1) 43 3 ; (2) 3 arccos 4 (1) 1 2 2 3 2 3 2 ABC S P ABC 1 1 4 3 2 3 2 3 3 3 ABC V S PA . (2) PB E DE AE ED BC ADE( ) BC AD ADE DE 2 2 AE AD 2 22 2 2 2 3 cos 2 2 2 4 ADE 3 arccos 4 ADE . BC AD 3 arccos 4 . 4. 2011 , 20 A
7、BCDA1B1C1D1 1 AA1 2 (1) BD AB1 ( ) (2) AB1D1C (1) 10 arccos 10 ; (2) 2 3- 13 - 5. 2010 , 20 4 9.6 8 S ( ) (1) r S ( 0.01 ) (2) 0.3 ( ) (1) r 0.4( ) Smax ) ;(2) - 14 - 6. 2008 , 16 ( 12 ) 2 1 1 1 1 ABCD ABCD E BC1 DE ABCD 5 arctan 5- 15 - 7. 2006 , 19 14 2 1 6 2 8 ABC ABC 90 , 1 ABC AB BC . 1 11 BC AC 2 1 AC ABC S 45 1 A ABC . 1 ; 2 2 6 1C1 B1C1 AC ( ) - 16 - B1C1 AC 1 ABC, 1 A1C ABC 2 , 1= 2 . A1-ABC V= 3 1 S 1= 2 6 . 8. 2005 , 17 12 1 1 1 1 D C B A ABCD M N 1 BB BC AB=4 AD=2 D B 1 ABCD 60 D B 1 MN . arctan 2 1- 17 - DB1C=arctan 2 1 . B1D MN arctan 2 1 .
