2014全国高中数学联赛B卷试题及答案.pdf

2014 c?IpmK (B ?) ?Y ?W?K? 0 ? 8 ?=?OKJ?o?IO. ?K? ?g?Yg1?. AO?5?=?YXJ)S?) ?p?)?g?n!?(?S? gIO. 1? ?!W?K? 1!e f(x) ?d?g?: (0,0),(1,1) ? (2,3) ?K f1(2) = 3 2. )? y = 2 ?:? ?3 2,2 ? f?3 2 ? = 2l? f1(2) = 3 2. 2!3Xe?N ABCD A0B0C0D0? A0 BD C0?u arccos 1 3 ( ?n?L). )? AC ? BD ?:? O. w, AC BD?nR?nA0O ? C0O ?R? u BD? A0OC0. ?b?N?c? 1. ? A0C0= 2,A0O = C0O = 6 2 . dun? cosA0OC0= 1 3? A 0 BD C0?u arccos 1 3. AOI?5?n?L?X 2arctan 2 2 ?. 3!u8 R ?f8 U?3 R ? fU(x) = ( 1x U 0x 6 U . XJ A,B 8 R ?f8K fA(x) + fB(x) 1 ?7? A,B p?8. )?u? x R,fA(x) + fB(x) = 1 fA(x),fB(x) ?k? 1= x A B. A B = R. ,?fA(x),fB(x) =k? 1= A B = . l? A,B p?8. p?8?kN?X A B = R,A B = . 1 4!XJn?/ ABC ?n?S? A,B,C ? cotA,cotB,cotC ?g?K ? B ? 3. )?P x = cotA,y = cotB,z = cotC?= 2y = x + z. dun?S? ?u x,y,z ?k?u?u“ y 0. ,?A + B = C % ? xy 1 x + y = z, = xy + yz + zx = 1. ? z ? x2 2xy + (1 2y2) = 0. ?u x ?k ?O“ = 12y2 4 0. cotB = y 3 3 = B 6 3. ? A = B = C = 3 ? ? B ?. 5!? an ?v?a1= 2 2 + 1,a2= 2 + 1,a n+1+ an1= n anan1 + 2(n 2) K?“ an= n 2+ 1 (n 1). )?O?A?“ an= n 2+ 1?e58By?(J. w , n = 1,2 ?(?. b n 6 k ?“o ak+1= ak1+ k ak ak1 + 2 = ?k 1 2+ 1 ? + k 1 2 + 2 = k + 1 2+ 1. = n = k + 1 ?“?. ?y?(?. 6!F1,F2? x2 a2 + y2 b2 = 1 (a b 0) ?:P ?:. XJ PF1F2 ? 1tanPF1F2= 1 2,tanPF2F1 = 2. K a = 15 2 . )?b F1= F1(c,0),F2= F2(c,0),(c 0)? P = P(x0,y0). ? ? F1,P ? k1= tanPF1F2= 1 2? F2,P ? k2= tanPF2F1= 2. d? y0= 1 2(x0 + c) y0= 2(x0 c) . l?)? x0= 5 3c,y0 = 4 3c. qdu 1 = SPF1F2= 1 2|F1F2|y0 = 4 3c 2? c = 3 2 . P :?I? ? 53 6 , 23 3 ? . l? 2a = |PF1| + |PF2| = q (x0+ c)2+ y2 0 + q (x0 c)2+ y2 0 = 215 3 + 15 3 = 15. = a = 15 2 . 7!?B?K3?e? 52 ?/? 5 ?k? ?i (?i1 J, Q, K, A) ?V 0.49 (?O?V?(? 0.01). )?Pfl ? AK A ?fl ? A ?“? 5 ?i?”? 5O? A ?V. fl ? A ?)?1?3 13 ?i? 5 ?i?k C5 13 ?1? ?z?i?s?z?i?k 4 s?5 ?i?k 45 ?J. fl ? A ? 45C5 13 ?fl ?. du“3 52 ? 5 ?”?k?fl ? C5 52u A u)?V P(A) 0.5071l? P(A) = 1 P(A) 0.4929. 2 8!? g(x) = px(1 x) 3m 0,1 ?K y = xg(x) ? x ?/? 16. )?w, g(x) ? x?/? A L xg(x) ? x? ?/. ? 2x = 1 ? A ?m?. ? xg(x) + (1 x)g(1 x) = xg(x) + (1 x)g(x) = g(x),0 6 x 6 1 2. ?“?A?Xe? ?y3?n?/“A ?m?To? =? 1 4 ?1 2 ?2 = 16. ?!)?K? 9!? an ?c n ? Sn|?v Sn+ Sn+1+ Sn+2= 6n2+ 9n + 7 (n 1). fi ? a1= 1,a2= 5? an?“. )?k S1= 1,S2= 6,S3= 6 + a3? S1+ S2+ S3= 22 %? a3= 9,S3= 15. ? ? a4= 13. ?d Sn?4?“?k Sn+1+ Sn+2+ Sn+3= 6n2+ 21n + 22 an+1+ an+2+ an+3= Sn+3 Sn= 12n + 15= an+ an+1+ an+2= 12n + 3,n 2. ?d“? an+1+ an+2+ an+3= 12n + 15 ? an+3 an= 12. a3k1,a3k? a3k+1O? 12 ?O a2= 5,a3= 9 ? a4= 13. 5? a1,a2,a3,a4? 4 ?. an? 4 ? ?“? an= 4n 3 (n 1). )? a3= 9,a4= 13,.? an? “ an= 4n 3?e5B8By?“ w, n = 1,2 ?“b n 6 k,(k 2) ?“o? n 6 k ?k Sn= 2n(n + 1) 3n = 2n2 n. ?4?“?k ak+1= Sk+1 Sk= (Sk+1+ Sk+ Sk1) 2Sk Sk1 = 6(k 1)2+ 9(k 1) + 7 4k2+ 2k 2(k 1)2+ k 1 = 4k + 1 = 4(k + 1) 3 ?y“ 3 10!? x1,x2,x3?“? x3 10x + 11 = 0 ?n?. a) fi ? x1,x2,x3?3m (5,5) ?n? b) y arctanx1+ arctanx2+ arctanx3= 4. (?IO?p? a) ?S?J? 5 b) ? 15 ) )?a) ? P(x) = x3 10x + 11?kn?. du P(5) = 64,P(4) = 13,P(3) = 14,P(2) = 23,P(1) = 20,P(0) = 11,P(1) = 2,P(2) = 1,P(3) = 8,.? w?3n?m (4,3),(1,2) ? (2,3) ?: P(x) ?UC?n?O3 n?mS? 4,1,2 B?v?n?Cq?. 5?u?3m (4,3) ?XJ? 3 ?( ?. ?vk(?o?. b) b x1 0,y0 0) ?:“3L: P ? ? ?C,D,E O? 1? 22? 31? 3?:“yn? AD,BE ? CP ? :“ )?3? x0x 4 + y0y = 1 C,D,E n:?IO? (2,1), ?4(y 0+ 1) x0 ,1 ? , ? 2, x0+ 2 2y0 ? . 4 ?5AD ? x0(x + 2) 2(2 + x0+ 2y0) + y = 0, BE ? x 2 + 2y0(y + 1) 2 + x0+ 2y0 = 0. ?e5k?Xe? ? S: x0(x + 2) 2(2 + x0+ 2y0) + y + ?x 2 + 2y0(y + 1) 2 + x0+ 2y0 ? = 0. w, SL AD ? BE ?:. XJ?L C? C(2,1) ? )? = 1. ? S1?L AD ? BE ?:? C ?. ?y P 3? S1?= x0(x0+ 2) 2(2 + x0+ 2y0) + y0 ?x 0 2 + 2y0(y0+ 1) 2 + x0+ 2y0 ? = 0. ?yXe?“? x0(x0+ 2) 4y0(y0+ 1) 2(2 + x0+ 2y0) = x0 2 y0. ? 2 + x0+ 2y0? x2 0 2 + x0 2y2 0 2y0= ?x0 2 y0 ? (2 + x0+ 2y0) = x0 2y0+ x2 0 2 2y2 0. CP ?(L AD ? BE ?:. ?I?(?. ?,?K?5?. )?|c? AD ? BE ?O?X?: Q ? I(J ? 8y0(y0+ 1) (2 + x0+ 2y0)2 2x0y0 , x0(x0+ 2) (2 + x0+ 2y0)2 2x0y0 ? ? CQ ? kCQ= 1 x0(x0+ 2) (2 + x0+ 2y0)2 2x0y0 2 8y0(y0+ 1) (2 + x0+ 2y0)2 2x0y0 = 2 + 2y2 0+ x0+ 4y0+ x0y0 4 + x2 0+ 4x0+ 4y0+ 2x0y0 = (2 + x0+ 2y0)(y0+ 1) (2 + x0+ 2y0)(x0+ 2) = y0+ 1 x0+ 2, ?u? CP ?. Q 3 CP ?I?(?. )?n?| Ceva n. du EA AC CB BD DP PE = x0+ 2 2y0 2x0 4(y0+ 1) y0+ 1 x0+ 2 2y0 y0 = x0(x0+ 2) 2(x0+ 2) 4y2 0 , ? P(x0,y0) 3? ? x2 0 4 + y2 0 = 1? x0(x0+ 2) 2(x0+ 2) 4y2 0 = x0(x0+ 2) 2(x0+ 2) + x2 0 4 = 1. 5 = EA AC CB BD DP PE = 1. ? Ceva nAD,BE ? CP n?:. 1? ?!Xe?H n? R ?:A,B,C n:K,?n?:. a) y H n?/ ABC ?R% b) yn?/ ABC ?u R. (?IO?a) ? b) ? 20 ) )? O1,O2,O3On?%Xe? o/ AO2HO1? O2BO3H ?u R?!/. ?5AO1k O2H k BO3?. o/ ABO3O1?1o/= AB k O1O3?. a) du CH ? O1O3!/ CO1HO3?pR? CH ? AB R?. ?nAH,BH ?OR?u ABC ? BC,AC. ?y? H ABC ?R%. b) du AB = O1O3?n BC,AC O?u O1O2? O2O3.n?/ ABC ? O3O1O2?. ? H ?n?%?l? R O3O1O2? R. ?y ? ABC ?u R. ?!3?IX f(x) = ax + 4,(a 6= 0) ? y = f1(x) ?T kn?: a ?y?(?. 6 )? f(x) = ax + 4 ? f1(x) = 1 a(x 2 4),x 0 ?:?I? (u,v). ?Jw?:?I?K?. 5? u,v ?v?|? u2= av + 4 v2= au + 4 ?|?“? (u v)(u + v + a) = 0. ?)?O ? u = v ? u + v + a = 0 ? u2= av + 4 . ? u,v ?|? ?3?“?|?UkO?. XJ a 0o?31? u + v + a 0.? u = v ? u2= av + 4 ?)?: (,?)K?). ?K8?b?kn?: ?Uu). ?k a 0? a 3a2 + 16 0. = |a| 2. q? a 2a,b ?“? a+b ?. b? akxbky = ab k?) x,y? 0 ? 23 ?n?/. 3n?/ ABC ?S?.? 11 ?: a) y?3?:?m?l?u?u 1 b) y?3?:?m?l?u 1. (?IO?a) ? b) ?O? 20 ? 30 ) )?a) Xe? ABC 16 ? 3 2 ?n?/u m?8?n?/?n?n?/?5? w? ABC ?m? 10 ?8/CX“ 7 ?Jw?p? 10 ?8/? 1. ?w? 10 ?“T”. un?/ ABC S?.? 11 ?:?T?n?k?8/?:. ?3 ?8/?:?m?l?m?l?L 1. ?By?I?(?. ?5?T?E?. XJe? 10 ? 1 2 ?CX ABC?(?. b) ?y