Nội dung

Tuyển tập Olympic Toán toàn quốc 1993 - 2005 DongPhD. No rights reserved. Share and share alike. TUYEN TAP OLYMPIC TOAN SINH VIEN TOAN QUOC 1993 - 2005 (de thi & loi giai) dua theo Cac de thi Olympic toan sinh vien toan quoc cua Nguyen Van Mau, Le Ngoc Lang, Pham The Long, Nguyen Minh Tuan, Hanoi, 2006. &KŒQJ  &˜F Ž² WKL 2O\PSLF FK½QK WKØF   &|X  2O\PSLF QˆP  9¿QJ  1J•\ WKØ QKªW w W ? r ) D 7°P W£W F F‘F PD WU¤Q WKÒF VDR FKR F 0 w W d ( 1 ) 3 ( d E &KR 1z VÁ QJX\ƒQ yd , 333, yz  Kd , 333, Kz WKºD PQ ‡L«X NL¯Q €ŸW C yd Kd L N N N L yz Kz ) (3 d L y d Kd y d K1 NNN h y 1 Kd d L y K 1 1 NNN \)h H N N N y z Kd y z K1 NNN 7¶QK / I‚ \/ &|X  + D &KR W w?D ) 8s- 1y FÊD W w?D WUƒQ b E 7¶QK W¶FK SK‚Q DUFWJ ?, {) 8$b1 ( I y d Kz y 1 Kz n n3 O N d L y z Kz ?  , ? ; b3 7°P PÂW QJX\ƒQ KŽP ?1 L d Q? $ 3 1 d L wWJ?D    2O\PSLF QP  &|X  D &KR KŽP VÁ W w?D [‘F ‡·QK YŽ F¼ ‡’R KŽP E¤F KDL OLƒQ WÍF YŽ NK„QJ ‡¾QJ QK£W E”QJ NK„QJ WUƒQ E£W NÓ ‡R’Q QŽR FÊD b %L®W U”QJ ‡¾ WK· FÊD KŽP VÁ r ) W w?D F—W ‡†ÃQJ WK•QJ y? L Kr L  ) W’L ED ‡L¬P SK‚Q EL¯W &KÑQJ PLQK U”QJ W¾Q W’L ?( ; b VDR FKR W II w?( D ) ( YŽ W II w?D ‡¿L G£X TXD ? ) ?( 3 E .¶ KL¯X hz w D OŽ W¤S KÇS W£W F F‘F ‡D WKÑF YÆL K¯ VÁ WKÒF F¼ E¤F NK„QJ Y†ÇW TX‘ z  &KR KDL VÁ SK‚Q EL¯W y, K ; b h1 w D [‘F ‡·QK WKHR F„QJ WKÑF ;©W ‘QK [’ W U hn w D €w?D ; hn w D U W w€D ) €w? L yD  €w? L KD3 L +ºL W F¼ SKL OŽ WRŽQ ‘QK NK„QJ" LL 7¶QK W d w(D  9¿QJ  1J•\ WKØ KDL &|X  &KR ( a a d &KÑQJ PLQK U”QJ YÆL P½L y, K ; X SK† QJ WU°QK F n  yF L K ) ( F¼ ¶W QK£W PÂW QJKL¯P WKR PQ ‡L«X NL¯Q /F  / a 1  3 $ YŽ ( a r c L  &KÑQJ PLQK U”QJ 1 0 rwDUFWJ r  ?D  OQ w{BR ? d L r 1 D3 &|X  &KR ( a ? a +ºL NKL QŽR WK° [\ UD G£X ‡•QJ WKÑF &|X  &KR €w?D W) FRQVW OŽ ‡D WKÑF YÆL K¯ VÁ WKÒF &KÑQJ PLQK U”QJ Q®X K¯ SK† QJ WU°QK T? $ € € z €w0D R$t 0Q0 ) (, ( $? € € Z €w0D {BR 0Q0 ) (3 ( F¼ QJKL¯P WKÒF WK° VÁ QJKL¯P WKÒF FK± F¼ WK¬ OŽ KÐX K’Q &|X  &KR KDL PD WU¤Q WKÒF YX„QJ ‡¾QJ F£S \ YŽ 4 *L WKL®W U”QJ I‚w\ L 4D W) ( YŽ I‚w\  4D W) ( €ŸW w W \ 4 - ) 3 4 \ &KÑQJ PLQK U”QJ I‚ - W) (    2O\PSLF QˆP  &K† QJ  &‘F ‡« WKL 2O\PSLF FK¶QK WKÑF 0‹Q WKL ‡™L VÈ &|X  &KR yd , y1 , yn , y; ; b &KÑQJ PLQK U”QJ PD WU¤Q C I d L y1 y1 y; yn h y1 d L y1d y; yn n n \)h 1 H yn y; d L y1 y1 O y1 d L y1d y; yn NK QJK·FK 7°P \d  &|X  &KR y7i w7, i ) d, 333, zD OŽ F‘F VÁ WKÒF QJX\ƒQ +\ JLL K¯ SK† QJ WU°QK Td € ?d ) ydd ?d L yd1 ?1 L N N N L ydz ?z € € € 1 € € zd? ) y1d ?d L y11 ?1 L N N N L y1z ?z 1 1 € € NNNNNNNNN € € € d € Z ? z ) yzd ?d L yz1 ?1 L N N N L yzz ?z 1 &|X  &KR PD WU¤Q 7¶QK W¿QJ &|X  7°P w {BR 1$i z \i ) R$t 1$i z  R$t 1$i z {BR 1$i z W  i ; L3 C€ ) \€( L \€d L N N N L \€zd  w€, z ; L D3 j$8 ? ( D j$8 z WURQJ ‡¼ o OŽ PD WU¤Q ‡ Q Y· YŽ C d \)H ?  z Q d z w\  oD , ? ?I zO, z ; L d   2O\PSLF QP  &|X  &KÑQJ PLQK U”QJ Q®X \ OŽ PD WU¤Q YX„QJ F£S z YÆL \1 ) o WK° K’QJ w\ L oD L K’QJ wo  \D ) z3 &|X  &KR \ OŽ PD WU¤Q YX„QJ F£S KDL WKºD PQ ‡L«X NL¯Q \1 ) \ &KÑQJ PLQK U”QJ ‡¬ \  \ ) ( OŽ PD WU¤Q YX„QJ F£S   OŽ PD WU¤Q NK„QJ  ‡L«X NL¯Q F Q YŽ ‡Ê OŽ W¾Q W’L PD WU¤Q YX„QJ F£S KDL ( VDR FKR )\  ( L (\  (3 0‹Q WKL *L–L W½FK &|X  &KR z VÁ QJX\ƒQ G† QJ y> , K> ; b w> ) (, d, 333, zD &KÑQJ PLQK U”QJ SK† QJ WU°QK z 3 ?L wy> R$t >? L K> {BR >?D ) ( >)d F¼ QJKL¯P WURQJ NKRQJ w$, $D &|X  &KR KŽP VÁ W w?D OLƒQ WÍF YŽ F¼ ‡’R KŽP F£S PÂW WUƒQ w(, D YŽ NK„QJ SKL OŽ KŽP K”QJ &KR y, K OŽ KDL VÁ WKÒF WKºD PQ ‡L«X NL¯Q ( c y c K &KÑQJ PLQK U”QJ SK† QJ WU°QK ?W I w?D  W w?D ) F¼ ¶W QK£W PÂW QJKL¯P WKXÂF wy, KD &|X  D &KR KŽP VÁ W U >y, KH yW wKD  KwyD Ky >y, KH, YÆL y c K YŽ WKºD PQ ‡L«X NL¯Q /W w?D  W wrD/ c /?  r/, ?, r ; >y, KH YŽ ? W) r3 &KÑQJ PLQK U”QJ SK† QJ WU°QK W w?D ) ? F¼ GX\ QK£W PÂW QJKL¯P WKXÂF >y, KH E &KR KŽP VÁ W w?D NK YL WUƒQ >y, KH YŽ ? ; >y, KH WK° /W I w?D/ c /W w?D/ &KÑQJ PLQK U”QJ W w?D # ( YÆL P½L ? ; >y, KH  &K† QJ  &‘F ‡« WKL 2O\PSLF FK¶QK WKÑF &|X  ;©W W¶FK SK‚Q {z ) D 7¶QK {z  E &KÑQJ PLQK U”QJ 8; ( $ ?z ;  ?Q? wz ; L D3 {z c 11zLn w1zD 1 3 d &|X  7°P ‡’R KŽP F£S z FÊD KŽP VÁ r &|X  D &KÑQJ Wº U”QJ W¶FK SK‚Q 8 NK„QJ SKÍ WKXÂF YŽR  E 7¶QK   Q? wd L ?1 Dwd L ? D ( {) DUFWJ ? W’L ? ) ( 8 2O\PSLF QˆP  Q? $ 3 ? L ?1 L ? L d 0‹Q WKL ‡™L VÈ &|X  &KR \ OŽ PD WU¤Q YX„QJ F£S z VDR FKR \d ) n\ 7¶QK GHWw\dx \D3 &|X  &KR F‘F VÁ y7i L yi7 ) (, 7, i ) SK† QJ WU°QK T € ydd ?d L yd1 ?1 L N N N L ydz ?z € € € zy ? L y ? L N N N L y ? 1d d 11 1 1z z € € € € Zy ? L y ? L N N N L y ? zd d z1 1 F¼ QJKL¯P NK„QJ W P WK†ÃQJ zz z d, 1, 333, z &KÑQJ PLQK U”QJ K¯ )( )( )( NNNNNNNNN   2O\PSLF QP  w W 1 d  7¶QK - z ÑQJ YÆL P½L z QJX\ƒQ G† QJ d 1 &|X  &KR PD WU¤Q - ) FKR WU†ÆF &|X  &KR KDL VÁ WKÒF SK‚Q  ‡†ÇF [‘F ‡·QK QK† VDX T € z? K7i ) y € Z K *L VÏ 7¶QK EL¯W y, K YŽ FKR 4 ) wK7i D OŽ PD WU¤Q YX„QJ F£S NKL 7 ) i, W i, 7 L i ) 1z, NKL 7 ) NKL 7 ) i, 7 ) i ) 1z L d3 I‚ 4 ) ; E 3 >)( > w?  yD> 3 &|X  &KR PD WU¤Q YX„QJ \ ) wy7i D F£S z z k dD F¼ K’QJ „ ;©W PD WU¤Q \ ) w\7i D WURQJ ‡¼ \7i OŽ SK Q EÈ ‡’L VÁ FÊD y7i WURQJ \ 7°P K’QJ FÊD \ &|X  &KR 4 OŽ PD WU¤Q YX„QJ F£S z YŽ OŽ PÂW VÁ WKÒF WKºD PQ ‡L«X NL¯Q GHWw4  oD ) ( YÆL o OŽ PD WU¤Q ‡ Q Y· &KÑQJ PLQK U”QJ YÆL P½L y( , yd , 333, yz ; b z ; L WD ‡«X F¼ z z p3 Q 3 > y> 4  y> > o ) ( GHW >(  >( 0‹Q WKL *L–L W½FK &|X  &KR KŽP VÁ W w?D OLƒQ WÍF YŽ QJK·FK EL®Q WUƒQ >(, KH YŽ FKR y ; >(, KH3 &KÑQJ PLQK U”QJ 8y 8K K W w?DQ? y W w?DQ?3 &|X  ;©W ‡D WKÑF ( ( d Qz 1 >?  dDz H 1z z Q?z &KÑQJ PLQK U”QJ Q®X W w?D OŽ ‡D WKÑF E¤F E wE c zD WK° hz w?D ) 8d d W w?Dhz w?DQ? ) (3  &K† QJ  &‘F ‡« WKL 2O\PSLF FK¶QK WKÑF &|X  &KÑQJ PLQK U”QJ KŽP VÁ l 1 db? W w?D ) ( NK YL Y„ K’Q W’L ? ) ( NKL ? W) ( ?)( &|X  &KR KŽP VÁ W w?D NK YL Y„ K’Q WUƒQ b YŽ WKºD PQ F‘F ‡L«X NL¯Q  - k ( VDR FKR W wzD w?D/ a - ? ; b, z ; L, pdQ ) ( z ; L 3  W z &KÑQJ PLQK U”QJ NKL ‡¼ W w?D # (, ? ; b3 &|X  &KR KŽP VÁ W w?D OLƒQ WÍF WUƒQ >(, dH &KÑQJ PLQK U”QJ 8s- /W w?D/ ) j$8 € ?;>(,dH L p 8d ( /W w?D/€ Q? Qd € 3 &|X  7°P W U b b WKºD PQ F‘F ‡L«X NL¯Q D W w? L rD a W w?D L W wrD w?, r ; bD W w?D ) d3 E j$8 ? ( ? &|X  7¶QK   {z ) 8$ ( R$t z? Q?, z ; L3 R$t ? 2O\PSLF QˆP  0‹Q WKL ‡™L VÈ &|X  &KR hz w?D OŽ ‡D WKÑF E¤F z YŽ FKR E ; L  &KÑQJ PLQK U”QJ L 1®X hz w?E D FKLD K®W FKR ?  d WK° Q¼ FKLD K®W FKR ?E  d LL 1®X hz w?E D FKLD K®W FKR ? yD> WK° Q¼ FKLD K®W FKR w?E yE D> wy W) (D   2O\PSLF QP  &|X  &KR PD WU¤Q w W y K \) , y, K,  ; b3 (  L &KÑQJ PLQK U”QJ Q®X \dE ) ( WK° \1 ) ( LL ;‘F ‡·QK y, K,  VDR FKR W¾Q W’L z ; L ‡¬ &|X  &KR 7¶QK \z w W d ( )  ( d C Iz C I 1 ( ( ydd wzD yd1 wzD ydn wzD H( n (O ) Hy1d wzD y11 wzD y1n wzDO 3 ( d 1 ynd wzD yn1 wzD ynn wzD j$8 z y11 wzD 3 yn1 wzD &|X  &KR \ OŽ PÂW PD WU¤Q YX„QJ F£S z &KÑQJ PLQK U”QJ Q®X \ NK„QJ NK QJK·FK WK° F¼ WK¬ WKD\ WK® F‘F SK Q WÏ y7i FÊD \ EÄL VÁ  KRŸF VÁ  F¸Q F‘F SK Q WÏ NK‘F Y¢Q JLÑ QJX\ƒQ ‡¬ QK¤Q ‡†ÇF PD WU¤Q PÆL C NK QJK·FK &|X  &KÑQJ PLQK U”QJ Q®X y W) ( WK° K¯ T € y? L wd  KDr L F L wd  QD0 ) y € € € z wK  dD? L yr L wQ  dDF L 0 ) K € ? L wd  QDr L yF L wK  dD0 )  € € € ZwQ  dD? ) r L wd  KDF L y0 ) Q OX„Q OX„Q F¼ QJKL¯P K, , Q ; b  0‹Q WKL *L–L W½FK &|X  .KR V‘W W¶QK NK YL FÊD KŽP VÁ W w?D ) /?  d/ /?  1/ N N N /?  dE/3 &|X  &KR K ; b 7¶QK y j$8 L 8y y wd L Q? ?1 Dwd L K ?D 3