CÁC DẠNG TOÁN THI HỌC SINH GIỎI -...

CC DNG TON THI HC SINH GII GII TON TRN MY TNH IN T KHOA HC

(Tp I: Trung hc C s) Bin son: PGS TS T Duy Phng

LI NI U

Ti liu ny c bin son cho lp tp hun gio vin Gii ton trn my tnh in t nm hc 2011-2012 ca S Gio dc v o to H Ni. Ti liu c bin son da theo bn tho cun sch Cc dng ton thi hc sinh gii Gii ton trn my tnh in t khoa hc, Tp I: Trung hc C s.

Ti liu tp hp cc thi hc sinh gii Gii ton trn my tnh in t v c chia lm tm Chng: S nguyn, S hc, i s, Thng k, Dy s, Lng gic, Hnh hc, v Cc bi ton khc. Tc gi c gng phn loi tng i y v t m cc dng ton trong mi Chng. Cc thi trong mi dng c sp xp theo theo tiu ch: T d n kh, u tin cc thi nhng nm gn y. Tuy nhin, sp xp ny c tnh cht ch quan v tng i. Bn c c th sp xp li theo quan im c nhn. Do khun kh ca Ti liu, cc thi khng c li gii. Bn c c th t gii hoc xem li gii chi tit ca phn ln cc thi trong Ti liu tham kho [1]-[10].

Ti liu (v bn tho cun sch) c bin son da trn cc bi ging ti cc lp Bi dng gio vin t nm 2000 n nay. Xin chn thnh cm n B Gio dc o to, cc S Gio dc o to cc tnh, thnh ph, to iu kin tc gi thc hin cc bi ging v hon thin bn tho cun sch ny.

Do hn ch v khun kh ca Ti liu cng nh hn ch v thi gian, thng tin v kin thc ca tc gi, Ti liu cha th c gi l hon chnh. Xin chn thnh cm n nhng kin ng gp. Th t trao i xin c gi v a ch:

T Duy Phng, Vin Ton hc, 18 Hong Quc Vit, H Ni.

in thoi: 0983605756; E-mail: [email protected]

H Ni, thng 9 nm 2012

Tc gi

2

CC DNG TON THI HC SINH GII

GII TON TRN MY TNH INT KHOA HC

(Tp 1: TRUNG HC C S)

Chng 1 S NGUYN TRN MY TNH IN T

Dng ton 1 Hc m chi Chi m hc

Ngay c hc sinh lp 4, lp 5 cng c th s dng my tnh nghin cu ton (pht hin cc qui lut n tng trong cc s t nhin). Di y l mt s v d.

Bi 1.1 Hy tnh trn my:

1) 9 + 9 = (18) v 9 9 = (81);

2) 24 + 3 = (27) v 24 3 = (72);

3) 47 + 2 = (49) v 47 2 = (94);

4) 263 + 2 = (265) v 263 2 = (526);

5) 497 + 2 = (499) v 497 2 = (994).

iu th v y l: Tng v tch cc s trong mi cp ch khc nhau v v tr cc ch s. Cn cc s nh vy khng?- Cha c cu tr li.

Bi 1.2 Dng my tnh kim tra:

20129999=20117988; 19099999=19088091.

Qui lut: Nhn mt s c bn ch s vi 9999 c mt s c 8 ch s, bn ch s u chnh l s bt i 1 n v, bn ch s sau ph vi bn ch s cho c 9999. Hy kim tra trn my tnh v chng minh qui lut trn.

Bi 1.3 Dng my tnh pht hin iu th v sau: 34=(12); 3334=(1122); 333334=(111222); 33333334=(11112222); 3333333334=(1111122222)

Chng minh rng (mi nhm c k ch s 3):

3 3 3 34 1 12 2 .k k k k

=

Bi 1.4 Dng my tnh kim tra bng cu chng sau v chng minh:

1) 1234567899=111111111; 2) 12345678918=222222222;

3

3) 12345678927=333333333; 4) 12345678936=444444444; 5) 12345678945=555555555; 6) 12345678954=666666666; 7) 12345678963=777777777; 8) 1234567899=888888888; 9) 1234567899=999999999.

Bi 1.5 Hy kim tra trn my tnh: Nhng cp s sau y c tch khng i khi ta i ch cc cc ch s trong mi tha s. C hay khng nhng cp s tng t?- Cha c cu tr li.

1) 12 42 = 21 24 = 504. 2) 1263 = 2136 = 756. 3) 12 84 = 21 48 = 1008. 4) 13 62 = 31 26 = 806. 5) 13 93 = 31 39 = 1209. 6) 14 82 = 41 28 = 1148. 7) 23 64 = 32 46 = 1472. 8) 2396 =32 69 = 2208. 9) 24 63 = 42 36 = 1512. 10) 24 84= 42 48=2016. 11) 26 93 = 62 39=2418. 12) 34 86 = 43 68 =2924. 13) 36 84 = 6348 = 3024. 14) 46 96=64 69= 4416.

Bi 1.6 Vit chn s 1, 2, 3,..., 8, 9 thnh bn s: mt s c ba ch s v ba s c hai ch s sao cho c th chia bn s thnh hai cp s c tch bng nhau. Ngi ta mi ch bit 10 trng hp sau y:

15823 = 7946=3634; 13827=6954=3726; 13429=6758=3886; 17423=6958=4002; 14629=5873=4234; 18627=5493=5022; 17432 = 9658=5568; 15832=6479=5056; 58412=7396=7008; 53214=9876=7448.

chng minh c rng tch 53214=9876=7448 l ln nht c th c, nhng vn cn cha bit tch 15823 = 7946=3634 c phi l nh nht khng. V cng cha bit cn s no na c tnh cht trn khng. Hy dng my tnh kim tra tnh ng n ca cc tch s trn v tm thm (hoc chng t khng cn) nhng s nh vy.

Bi 1.7 1) Tch 9801 hoc 3025 chia thnh s c hai ch s, cng li v em bnh phng, ta li c chnh s :

2 29801 (98 01) 99 .= + = 2) Tng t: 2 23025 (30 25) 55 .= + = Hy kim tra trn my tnh. Cn cc s nh vy khng?

Bi 1.8 Ly s 37 nhn vi tng cc ch s ca n, mt khc tm tng cc lp phng ca cc ch s , ta s c hai kt qu nh nhau:

3 337.(3 7) 3 7 .+ = + Tng t,

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3 348.(4 8) 4 8 ;+ = + 3 3147.(14 7) 14 7 ;+ = + 3 3111.(11 1) 11 1 ;+ = + 3 3 31.2.3.(1 2 3) 1 2 3 .+ + = + +

Hy kim tra trn my tnh. Cn c cc s khc c tnh cht ny khng?

Bi 1.9 Hy kim tra trn my tnh: Nhng s sau y c tng (tng bnh phng, tng lp phng) cc ch s ca chng bng tng (tng bnh phng, tng lp phng) ca cc ch s vit theo th t ngc li (cc s i xng vi cc s cho):

1)12 32 43 56 67 87 18211 78 76 65 34 23 21.+ + + + + = = + + + + +2 2 2 2 2 2 2 2 2 2 2 212 32 43 56 67 87 18211 78 76 65 34 23 21 .+ + + + + = = + + + + + 3 3 3 3 3 3 3 3 3 3 3 312 32 43 56 67 87 1248885 78 76 65 34 23 21 .+ + + + + = = + + + + +

2) 13 42 53 57 68 97 330 79 86 75 35 24 31.+ + + + + = = + + + + + 2 2 2 2 2 2 2 2 2 2 2 213 42 53 57 68 97 22024 79 86 75 35 24 31 .+ + + + + = = + + + + +3 3 3 3 3 3 3 3 3 3 3 313 42 53 57 68 97 1637460 79 86 75 35 24 31 .+ + + + + = = + + + + +

Bi ton: Tm tt c cc b 6 s c hai ch s tha mn iu kin trn.

Bi 1.10 Kim tra trn my tnh cc tng sau v rt ra qui lut:

1+2=3; 4+5+6=7+8; 9+10+11+12=13+14+15; 16+17+18+19+20=21+22+23+24+25.

Qui lut: Ly s u tin l 2n , cng lin tip vi n s tip theo ( 2 1n + ,, 2n n+ ), kt qu bng tng ca n s tip theo: ( 2 1n n+ + ,, 2 2n n+ ). Ngha l, vi mi n ta c:

2n +( 2 1n + )++( 2n n+ )=( 2 1n n+ + )++( 2 2n n+ ). (*)

Bi 1.11 Trong cc ng thc sau, v tri v v phi u c nhng ch s ging nhau:

42: 3 = 4.3+2; 63: 3=6.3+3; 95: 5=9+5+5; (2+7).2.16=272+16 102 2 1022 = ; 2(8 9) 289+ = ; 82 1 128 = ; 3 34.2 4 : 2 34 : 2.= =

Hy dng my tnh kim tra. Cn c cc s khc c tnh cht ny khng?

Bi 1.12 Tng ca cc s l u tin Hy tnh trn my:

1 + 3 = = (2); 1 + 3 + 5 = = (3);

1 + 3 + 5 + 7 = = (4); 1 + 3 + 5 + 7 + 9 = = (5).

Chng minh cng thc tng cc s l u tin sau bng qui np ton hc: 21 3 5 ... (2 1) .n n+ + + + =

5

Dng ton 2 Tnh ng kt qu vt qu kh nng hin th trn mn hnh ca my tnh khoa hc (kt qu vt qu 10 ch s)

Mt hn ch ca my tnh in t khoa hc l kh nng hin th trn mn hnh thng ch l 10 hoc 12 ch s. V vy, nhiu bi ton khng th tnh c trn my tnh in t khoa hc, nu khng c sng to trong s dng my tnh. Th d sau y minh ha iu .

Bi 2.1 (B Gio dc v o to, Trung hc C s, 2007-2008) Tnh gi tr ca biu thc:

2 2A 135791 246824 .= +

Bi 2.2 (B Gio dc v o to, lp 9, 2004) Tnh kt qu ng ca cc tch sau:

20032003 20042004;M = 2222255555 2222266666.N =

Bi 2.3 (S GD v T Tha Thin Hu, Trung hc C s, 01.02. 2007) Tnh kt qu ng (khng sai s) ca cc tch sau:

1) P = 1123200611232007; 2) Q = 77777555557777799999

Bi 2.4 (S Gio dc v o to Tha Thin-Hu, Lp 8-9, 2005) Tnh kt qu ng ca cc tch sau:

3344355664 3333377777;M = 3123456 .N =

Bi 2.5 (Thi chn i tuyn Ph Th, Lp 12 THBT, 2005) Tm s cc ch s ca: 3659893456789325678 342973489379256.P =

Bi 2.6 (Chn i tuyn thi Khu vc, S GD T Lm ng, 2004) Tnh tng cc ch s ca (999 995)2 .

Bi 2.7 (Phng Gio dc v o to ng Triu, 2011-2012) Tnh A = 999 999 9993 .

Bi 2.8 (S Gio dc v o to Tha Thin-Hu, THCS, 01.02.2007) Tnh chnh xc gi tr ca biu thc sau (nu quy trnh bm phm) :

P 3 33 333 ... 33...33,= + + + + trong ch s cui cng gm 13 ch s 3.

Bi 2.9 (S Gio dc v o to Qung Nam, lp 8, 2009-2010) Tnh ng tng: S = 5 5 5 51 2 3 ... 75 .+ + + + Tnh ng tch: M = 1.2.319.20 (M=20!).

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Dng ton 3 Tm thng v s d khi chia mt s t nhin a cho mt s t nhin b

Dng ton 3.1 Tm thng v s d khi chia mt s t nhin a c khng qu 10 ch s cho mt s t nhin b trn my tnh in t

Bi 3.1 (B Gio dc v o to, lp 6-7-8, 2001)

1) Vit mt qui trnh bm phm tm s d khi chia 18901969 cho 2382001. 2) Tm s d khi chia 18901969 cho 2382001. 3) Vit qui trnh bm phm tm s d khi chia 3523127 cho 2047.

Bi 3.2 (S Gio dc v o to Thi Nguyn, 2002-2003) Tm thng v s d ca php chia: (Lp 12 Trung hc ph thng) s 123456789 cho 23456. (Lp 12 Trung hc B tc) s 3456789 cho 23456.

Bi 3.3 (S Gio dc v o to Cn Th, 2001-2002)

(Lp 6) Chia 6032002 cho 1905 c s d l 1r . Chia 1r cho 209 c s d l

2r . Tm 2.r (Lp 8) Chia 19082002 cho 2707 c s d l 1r . Chia 1r cho 209 c s d l

2r . Tm 2.r

(Lp 8) Vit qui trnh bm phm tm phn d ca php chia 19052002 cho 20969. (Lp 9) Vit qui trnh bm phm tm phn d ca php chia 26031931 cho 280202.

Bi 3.4 (S Gio dc v o to Cn Th, lp 9, 2002-2003) Vit qui trnh bm phm tm phn d ca php chia 21021961 cho 1781989.

Bi 3.5 (Chn i tuyn. S Gio dc v o to Thp H Ch Minh, 3) Tm s nh nht c 10 ch s bit rng s khi chia cho 5 d 3 v khi chia cho 619 d 237.

Dng ton 3.2 Tm thng v s d khi chia mt s t nhin a vt qu 10 ch s (vt qu kh nng hin th ca my) cho mt s t nhin b

Bi 3.6 (B Gio dc v o to. Trung hc C s, 2006) Tm s d trong mi php chia sau: a) 103103103: 2006; b) 30419753041975: 151975; c) 103200610320061032006: 2010.

Bi 3.7 (S Gio dc v o to Thp H Ch Minh, THCS, 2000-2001)

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Tm s d trong php chia: a) 1234567890987654321:123456; b) 157 : 2001.

Bi 3.8 (S Gio dc v o to Ha Bnh, Trung hc C s, 2007-2008) Tm cc s a v b bit 686430 8a b chia ht cho 2008.

Bi 3.9 (Trng Ph thng Trung hc Trn i Ngha, lp 6, 2001) Tm s d trong php chia 175 : 2001.

Bi 3.10 (B Gio dc v o to, lp 12, 2003) Tm s d khi chia s 20102001 cho s 2003.

Bi 3.11 (S Gio dc v o to Qung Nam, lp 8, 2009-2010) Tm s d khi chia 20102009 cho 2008.

Bi 3.12 (S Gio dc v o to Sc Trng, 28.11.2010) Tm s d trong php chia 2811 cho 2010.

Bi 3.13 (B Gio dc v o to, Trung hc C s, 2009-2010) Tm s d (trnh by cch gii) trong cc php chia sau: 1) 20092010: 2011; 2) 22009201020112012: 2020; 3) 1234567890987654321: 2010.

Bi 3.14 (S Gio dc v o to Qung Nam, lp 9, 2009-2010) Tm s d khi chia S = 25 + 210 + 215+ + 245 + 250 cho 30.

Bi 3.15 (S Gio dc v o to Qung Ngi, lp 9, 2009-2010) 1) Tm s d trong php chia 22010 cho 49. 2) Tm s t nhin nh nht c 10 ch s bit rng: S chia cho 17 d 2, chia cho 29 d 5.

Dng ton 4 Phn tch mt s ra tha s nguyn t

Bi 4.1 (S Gio dc v o to Ph Yn, lp 9 Trung hc C s, 10.2.2009) Phn tch ra tha s nguyn t s P = 2450250.

Bi 4.2 (S Gio dc v o to Sc Trng, Trung hc C s, 2003-2004) Phn tch cc s 20387 v 139231 ra tha s nguyn t.

Bi 4.3 (S Gio dc v o to Tha Thin-Hu, THCS, 01.02.2007) Phn tch s 9405342019 ra tha s nguyn t.

Bi 4.4 (S Gio dc v o to Tha Thin Hu, lp 9, 01.12. 2006) Phn tch thnh tha s nguyn t cc s sau:

252633033 v 8863701824.

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Bi 4.5 (S Gio dc v o to Qung Nam, lp 8, 2009-2010) Phn tch s 311875250 thnh tch cc tha s nguyn t.

Bi 4.6 (S Gio dc v o to Sc Trng, Trung hc C s, 2003-2004) Phn tch cc s 20387 v 139231 ra tha s nguyn t.

Bi 4.7 (S Gio dc v o to Tha Thin-Hu, THCS, 2005-2006) Phn tch cc s 252633033 v 8863701824 ra tha s nguyn t.

Bi 4.8 (S Gio dc v o to Ha Bnh, Trung hc C s, 2007-2008) Phn tch cc s 8563513664 v 244290303 ra tha s nguyn t.

Bi 4.9 (S Gio dc v o to Thi Nguyn, Lp 9, 2003) Hy tm tt c cc c s ca 2005.

Bi 4.10 (B Gio dc v o to, 2001. chnh thc) (Lp 10-11) Tm cc c nguyn t nh nht v ln nht ca s 2 2215 314+ . (Lp 10) Tm s ln nht v s nh nht trong cc s t nhin c dng 1 2 3 4x y z m chia ht cho 7. (Lp 11) Tm s ln nht v s nh nht trong cc s t nhin c dng 1 2 3 4x y z m chia ht cho 13.

Bi 4.11 (S Gio dc v o to Thp H Ch Minh, Lp 9, 11.1.2009) Hy tm tt c cc s t nhin l bi ca 2009 c dng 7 *13*1.

Bi 4.12 (S Gio dc v o to Thi Nguyn, Lp 12, 2002-2003) S 112 1 l hp s hay nguyn t?

Dng ton 5 Tm c s chung ln nht (USCLN) v bi s chung nh nht ca hai hay nhiu s

Bi 5.1 ( chn i tuyn thi hc sinh gii Gii ton trn my tnh. S Gio dc o to Thi Nguyn, lp 9, 2002- 2003) Tm c s chung ln nht ca 7729 v 11659.

Bi 5.2 ( chn i tuyn thi hc sinh gii Gii ton trn my tnh, S Gio dc v o to Thi Nguyn, lp 12, 2004) Tm USCLN ca hai s 1754298000a = v 75125232.b =

Bi 5.3 (S Gio dc o to Cn Th, 2002- 2003) (Lp 9) Tm c s chung ln nht ca hai s 11264845 v 33790075. (Lp 9) Tm c s chung ln nht ca hai s sau: 1582370a = v

1099647.b =

Bi 5.4 (B Gio dc o to, lp 12, 2002)

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Tm c s chung ln nht ca hai s sau y: a = 24614205, b = 10719433.

Bi 5.5 (S Gio dc o to Cn Th, lp 6, lp 8, 2001- 2002) Tm cc c chung ca bn s sau: 222222; 506506; 714714; 999999.

Bi 5.6 (S Gio dc o to Thp H Ch Minh, Trung hc c s, 2000-2001) Tm USCLN v BSCNN ca: a) 9148 v 16632; b) 75125232 v 175429800.

Bi 5.7 (Trung hc C s, S Gio dc v o to Ha Bnh, 2005-2006) Tm USCLN v BSCNN ca hai s a =457410, b =831615.

Bi 5.8 (Trung hc C s, S Gio dc v o to Sc Trng, 2004-2005) Tm USCLN v BSCNN ca hai s 1) a =9148, b =16632; 2) a =75125232, b =175429800.

Bi 5.9 (Tp ch Ton Tui th 2, s 25 v 27, thng 3 v thng 5, 2005) Tm USCLN v BSCNN ca hai s a =3022005, b =7503021930.

Bi 5.10 (Tp ch Ton hc v Tui tr, thng 11, 2004 v thng 1, 2005) Tm USCLN v BSCNN ca hai s a =1234566, b =9876546.

Bi 5.11 (S Gio dc v o to Thnh ph H Ch Minh, Lp 9, 2001) Tm USCLN v BSCNN ca hai s: 1) 4047 v 8316. 2) 150250464 v 350859600.

Bi 5.12 (Chn i tuyn. S Gio dc v o to thp H Ch Minh, 2003) Tm UCLN v BCNN ca hai s 2419580247 v 3802197531.

Bi 5.13 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008) Tm USCLN ca 40096920, 9474372 v 51135438.

Bi 5.14 (S Gio dc o to Tha Thin-Hu, lp 8, 9, 11, 2005) Cho ba s A =1193984; B =157993; C =38743. Tm UCLN ca ba s , ,A B C ; Tm BCNN ca ba s , ,A B C vi kt qu ng. Bi 5.15 (S Gio dc v o to Qung Nam, lp 8, 2009-2010) Tm c chung ln nht v bi chung nh nht ca ba s a = 9200191; b=2729727; c = 13244321.

Dng ton 6 Bi tp v cc ch s trong mt s

Bi 6.1 (Chn i tuyn. S Gio dc v o to thp H Ch Minh, 2003) 1) Tm ch s hng n v ca s : 172002

2) Tm ch s thp phn th 456456 sau du phy trong php chia 13 cho 23.

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Bi 6.2 (S Gio dc v o to Qung Ngi, lp 9, 2009-2010) 1) Vit quy trnh bm phm lin tc tm chu k ca phn thp phn trong kt qu php chia 85 cho 47. 2) Ch s thp phn th 2010 sau du phy ca php chia cu 2) l s no?

Bi 6.3 (S Gio dc v o to Tha Thin Hu, lp 9, 01.12. 2006) Tm ch s l thp phn th 112007 k t du phy ca s thp phn v hn

tun hon ca s hu t 10000

29.

Bi 6.4 (Chn i tuyn thi Khu vc, S GD T Lm ng, 2004)

Tnh tng ca 12 ch s thp phn u tin sau du phy ca 121

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Bi 6.5 (S Gio dc o to Thp H Ch Minh, Trung hc c s, 2000-2001) Ch s thp phn th 2001 sau du phy l ch s no khi ta: a) chia 1 cho 49; b) chia 10 cho 23.

Bi 6.6 (S Gio dc v o to Qung Nam, lp 9, 2009-2010) Tm ch s thp phn th 242010 sau du phy trong php chia 1 cho 49.

Bi 6.7 (S Gio dc o to thnh ph H Ch Minh, THPT, 2000-2001) C bao nhiu ch s khi vit s 300300 .

Bi 6.8 (B Gio dc v o to, lp 12 Trung hc Ph thng, 11.3.2011) Tm s cc ch s khi vit trong h thp phn ca s 20109 .

Bi 6.9 (S Gio dc v o to ng Nai, lp 9, 2004-2005) Tm ba ch s tn cng ca s

999 .

Bi 6.10 (B Gio dc o to, Lp 9, 2002. d b) 1) Tm hai ch s cui cng ca s 1999 2000 20012 2 2 .+ + 2) Chng minh ton hc (kt hp vi my tnh) cho khng nh trn.

Bi 6.11 (Phng Gio dc v o to huyn ng Triu, Lp 9, 2011-2012) Tm hai ch s tn cng ca tng D = 22001 + 22002 + 22003.

Bi 6.12 (S Gio dc v o to Qung Nam, lp 8, 2009-2010) Tm ch s thp phn th 252010 sau du phy trong php chia 17 cho 19.

Bi 6.13 (S Gio dc v o to Ph Yn, lp 9, 2009-2010) Trong h thp phn, s A c vit bng 100 ch s 3, s B c vit bng 100 ch s 6. 1. Tch AB c bao nhiu ch s ? 2. Tm 8 ch s tn cng ca hiu 20092010.C AB=

11

Dng ton 7 Phng trnh nghim nguyn

Bi 7.1 (S Gio dc v o to Ty Ninh, lp 9, 2008-2009) Cho phng trnh

1657 367 23x y = ( vi ,x y ) (1) 1) Vit cng thc tng qut nghim nguyn ca phng trnh (1) 2) Tm 8 nghim nguyn ca phng trnh (1).

Bi 7.2 (S Gio dc v o to Qung Nam, lp 8, 2009-2010) Tm x sao cho 15 + 25 + 35 + ...+ x5 = 10923365376.

Bi 7.3 (S Gio dc v o to Ty Ninh, lp 9, 2008-2009) Tm tt c cc nghim nguyn dng ca phng trnh:

2 6 5 3041994.xy x y =

Bi 7.4 (S Gio dc v o to Thp H Ch Minh, Lp 9, 11.1.2009; S Gio dc v o to Qung Nam, lp 9, 2009-2010) Tm tt c cc cp s nguyn dng (x; y) tho mn phng trnh: x4 x2y + y2 = 81001.

Bi 7.5 (S Gio dc v o to Sc Trng, lp 9, 2008-2009) Tm cc cp s ( x, y) nguyn dng nghim gn ng ca phng trnh:

5 25 20(72 ) 16277165.x x y =

Dng ton 8 My tnh in t tr gip gii ton

Bi 8.1 (B Gio dc v o to, Trung hc C s, 2008-2009) 1) S chnh phng P c dng P 17712ab81.= Tm cc ch s a, b bit rng a b 13.+ = 2) S chnh phng Q c dng Q 15cd26849.= Tm cc ch s c,d bit rng 2 2c d 58.+ = 3) S chnh phng M c dng M 1mn399025= chia ht cho 9. Tm cc ch s m, n. Bi 8.2 (S Gio dc v o to Tha Thin Hu, lp 9, 01.12. 2006) Tm cc ch s sao cho s 567abcda l s chnh phng

Bi 8.3 (S GD v T Tha Thin Hu, Lp 8, 01.02. 2007) 1) Tm s t nhin b nht m lp phng s c 4 ch s cui bn phi u l ch s 3. Nu quy trnh bm phm. 2) Tm s nguyn dng nh nht c ba ch s abc sao cho

3 3 3abc a b c .= + +

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C cn s nguyn dng no tha mn iu kin trn na khng? Nu s lc cch tm.

Bi 8.4 (S Gio dc v o to Thp H Ch Minh, Lp 9, 11.1.2009) Tm s t nhin c 3 ch s abc bit rng 5 .abc c ab= +

Bi 8.5 (Phng Gio dc v o to huyn ng Triu, Lp 9, 2011-2012) Tm s nguyn dng nh nht c ba ch s abc sao cho 3 3 3abc a b c= + + . C cn s nguyn dng no tha mn iu kin trn na khng? Nu s lc cch tm.

Bi 8.6 (B Gio dc v o to, Trung hc C s, 11.3.2011) Mt s t nhin c bn ch s, bit rng, nu vit thm ch s 1 vo bn tri ca s v vit thm ch s 8 vo bn phi ca s ny th c mt s mi c su ch s ng thi s ny bng 34 ln s ban u. Tm s . Trnh by tm tt cch gii.

Bi 8.7 (S Gio dc v o to Tha Thin Hu, lp 9, 01.12. 2006) Tm cc s t nhin n (2000 < n < 60000) sao cho vi mi s th

3na 54756 15n= + cng l s t nhin. Nu quy trnh bm phm.

Bi 8.8 (S Gio dc v o to Tha Thin-Hu, Lp 9, 01.02.2007) Tm hai s nguyn dng x b nht sao cho khi lp phng mi s ta c mt s c 2 ch s u (bn phi) v 2 ch s cui (bn tri) u bng 4, ngha l 3x 44......44.= Nu quy trnh bm phm.

Bi 8.9 (Phng Gio dc o to huyn Bo Lm, Lm ng, 2004) Cho

2 2 2 2 2 2M=12 +25 +37 +54 +67 +89 ; 2 2 2 2 2 2N=21 +78 +34 +76 +23 +Z Tm Z 3M=2N.

Bi 8.10 (S Gio dc v o to Qung Ngi, lp 9, 2009-2010)

Tnh .1 + 2 + 3 +...+ 23 + 24A =

Bi 8.11 (S Gio dc o to Cn Th, Lp 6, 2001-2002) C bao nhiu s chia ht cho 9 gm nm ch s c vit bi cc ch s 1,2,3.

Bi 8.12 (S Gio dc o to Cn Th, Lp 6, 2001-2002) C bao nhiu s chia ht cho 9 gm 6 ch s c vit bi cc ch s 2, 3, 5?

Bi 8.13 (S Gio dc o to Ph Yn, Trung hc C s, 10.2.2009)

13

Tm cc s x, y sao cho khi chia xxxxx cho yyyy c thng l 16 d l r, cn khi chia xxxx cho yyy cng c thng l 16 nhng c s d l r 2000. Nu cch gii.

Bi 8.14 (Thi chn i tuyn Trng THCS ng Nai-Ct Tin, 2004) 1) Tm s t nhin a nh nht m a2 bt u bi ch s 15 v kt thc bi 56? 2) Cho s t nhin n (5050 n 8040) sao cho an = 80788 7n+ cng l s t nhin. 2a) an phi nm trong khong no? 2b) Chng minh rng an ch c th l mt trong cc dng sau: an = 7k + 1 hoc an= 7k 1 (vi kN).

Bi 8.15 (S Gio dc o to Cn Th, Lp 6, 2002-2003) S 123 1 chia ht cho hai s t nhin nm trong khong 70 n 79. Tm hai s .

Bi 8.16 (B Gio dc v o to, lp 12 Trung hc Ph thng, 11.3.2011) Php tnh nng ln ly tha ri ly modulo ca cc s nguyn theo s nguyn

,p tc l tnh ( )mod ,kC N p l khng kh khn, ngay c vi nhng s cc ln. Nhng php tnh ngc li, tc l tm ra N khi bit , , ,C k p thng c gi l php khai cn bc k modulo ,p li l vic v cng kh khn. Trong trng hp tng qut, vi cc s nguyn ln, bi ton ny l khng th gii c ngay c vi cc siu my tnh mnh nht hin nay. Tuy nhin, khi p l s nguyn t v k khng c c chung vi 1p th nh nh l Fermat nh, ngi ta pht hin ra rng c th thc hin c php khai cn ny bng cch tm s d sao cho

( )1mod 1dk p v tnh ra N bng cng thc ( )mod .dN C p kim nghim iu ni trn, em hy: 1) Tm s ( )230512345 mod 54321 ;C 2) Tm s N sao cho ( )52209 mod89897 56331.N =

Chng 2 S HC TRN MY TNH IN T

Dng ton 1 Tnh ton vi cc phn s

Bi 1.1 (S Gio dc o to ng Nai, Trung hc C s, 1998; Phng Gio dc o to huyn Bo Lm, Lm ng, 2004) Tnh (kt qu c ghi bng phn s v s thp phn):

14

123 581 5213 2 4 .52 7 28

A = +

Bi 1.2 (S Gio dc o to Cn Th, Trung hc C s, 1998) 1) Tnh:

1 3 5 7 9 11 13 152 4 8 16 32 64 128 256

A = + + + + + + + .

2) Tnh 1994 1993 2 1993 19941994 212121

1992 1992 1994 19931993 1994 434343B = +

+ .

3) So snh cc phn s sau:

1927

; 19192727

;191919272727

;1919191927272727

.

Bi 1.3 (S Gio dc v o to Ph Yn, lp 9, 2009-2010)

Cho biu thc: A = 1 1 3 2 5 7 1 13 .3 4 8 9 12 18 24 36

A = + + + + + + +

B s no trong tng trn A = 2?

Bi 1.4 (S Gio dc v o to Ph Yn, lp 9 Trung hc C s, 10.2.2009) 1) Tnh biu thc:

2 2 22

2 3 2 3

3

2 3 2 3

2 2 2 1 1 12 11 2008200820087 7 7 3 3 3: : .1 1 1 2 2 22 2009200920091 27 7 7 3 3 3

A

+ + + +

= + + + +

2) Tm s hu t x bit: 5 5 5 10 10 105 1012345679 43434317 89 113 23 243 611: .11 11 11 3 3 3333333333 51515111 317 89 113 23 243 611

x

+ + + + = + + + +

Dng ton 2 Tnh ton vi s thp phn

Bi 2.1 (Thi thi th vng Tnh, Trng THCS ng Nai, 2004) Thc hin php tnh

A = 6712,53211 : 5,3112 + 166143,478 : 8,993

Bi 2.2 (Thi chn i tuyn Trng THCS ng Nai-Ct Tin, 2004)

Thc hin php tnh (kt qu vit di dng hn s)

A = 5322,666744:5,333332+17443,478:0,993.

Bi 2.3 (Phng Gio dc o to huyn Bo Lm, Lm ng, 2004)

15

1) Tm h bit: 3 3 3 31 1 1 1= + + .h 3,218 5,673 4,815

2) Tnh

3 2 41,6: 1 .1,25 1,08- :25 25 7C= + +0,6.0,5: .1 5 1 2 50,64- 5 -2 .2

25 9 4 17

Bi 2.4 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008) Tnh gi tr ca x t phng trnh sau

3 4 4 10,5 1 1, 25 1,8 37 5 7 2 35, 2 2,5 .

3 1 3 415, 2 3,15 2 4 1,5 0,84 2 4

::

:

x

+ = +

Bi 2.5 (S Gio dc v o to, Cn Th, lp 6, 2001-2002) 1) Tnh v lm trn n 6 ch s thp phn:

3 : 0,4 0,09 : (0,15 : 2,5) (2,1 1,965) : (1,2 0,045) .0,32 6 0,03 (5,3 3,88) 0,67 0,00325 : 0,013

C = + + +

.

2) Tnh v lm trn n 5 ch s thp phn:

13 7 7 1 1( 1,4 2,5 ) : 2 4 0,1 : (70,5 528 : 7 )84 180 18 2 2

D = + .

3) Tm x v lm trn n 4 ch s thp phn:

[ ]1 1 1 1 1( ... ) 140 1,08: 0,3 ( 1) 11.21 22 22 23 23 24 28 29 29 30

x+ + + + + + =

Dng ton 3 Lin phn s (phn s lin tc)

Bi 3.1 (S Gio dc v o to Tha Thin Hu, lp 9, 01.12. 2006) Cho dy s (biu thc c cha n tng phn s) :

11u 22

= + ; 21u 2 12

2

= ++

;

31u 2 12 12

2

= ++

+

;; n1u 2 12... 12 12

2

= +

++

.

Tnh gi tr chnh xc ca u5, u9, u10 v gi tr gn ng ca u15, u20.

Bi 3.2 (S GD v T Tha Thin Hu, Lp 8, 01.02. 2007)

16

Cho lin phn s: 12 12 ... 12

1

nu

x

= ++

++

(biu thc c cha n tng phn s).

Tm x bit 201687u1696

= (kt qu ly vi 4 ch s phn thp phn).

Nu quy trnh bm phm.

Bi 3.3 (B Gio dc v o to, Lp 6, 7, 2002) Tnh gi tr ca biu thc v vit kt qu di dng phn s:

1) 53 42 52 42 52

3

A = ++

++

+

. 2) 17 13 13 13

4

B = ++

++

.


Tnh 4D=5+ 46+ 47+ 48+ 49+10

Bi 3.5 (S Gio dc v o to Hi Phng, 2003-2004)

Cho 1230 .5102003

A = ++

Hy vit A di dng lin phn s [ ]0 1, ,..., .nA a a a=

Bi 3.6 (S Gio dc v o to Cn Th, 2001-2002) Tnh:

1) (Lp 6) 11 .11 11 11 11 11

1 1

++

++

++

+

17

2) (Lp 7) 12 12 12 12 12 12 12 12

2

++

++

++

++

.

3) (Lp 8) 13 13 13 13 13 13

3

+

+

+

.

4) (Lp 9) 11 12 13 14 15 16 17 18

9

++

++

++

++

Bi 3.7 (S Gio dc v o to Cn Th, Lp 9, 2002-2003) Tnh:

19 .28 37 46 55 64 73 829

C = ++

++

++

++

Bi 3.8 (V ch New York, 1985. Cu hi tip sc)

Bit: 15 1 ,117 1 1a

b

=+

+

trong a v b l cc s dng. Hy tnh .b

18

Bi 3.9 (Phng Gio dc v o to ng Triu, 2011-2012) Tm x trong ng thc sau, kt qu vit di dng phn s.

1 1 1 . 43 2 12 3 15 3 14 5 17 4 26 78 9

x

= + + + + +

+ + + + +

Bi 3.10 (S Gio dc v o to Thi Nguyn, Lp 9, 2002-2003) 1) Vit quy trnh bm phm tnh:

3 117 .12 51 231 11 312 117 72002 2003

A = + ++ +

+ ++ +

2) Gi tr tm c ca A l bao nhiu?

Bi 3.11 (S Gio dc v o to Sc Trng, lp 9, 2008-2009)

Tm nghim ca phng trnh:

5 1 5( 3 3 .7 43 21 2 1 6 5(7 ) 4 24 1 5 43 1 3 5 21 4 55 3 6 21 3 33 74 7

x

+ + = + +

+ + + + + + + + + +

+

Bi 3.12 (S Gio dc v o to ng Nai, lp 9, 2004-2005) Tm s dng x tha mn phng trnh:

12005 .12005 12005 12005 12005

x

x

= ++

++

+

Bi 3.13 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008) Tm cc s t nhin a, b, c, d, e bit (ch ghi kt qu):

19

5584 1 .110511

1

ab

cd

e

= ++

++

Bi 3.14 (S Gio dc v o to Tha Thin Hu, lp 9, 01.12. 2006) Tm cc s t nhin a, b, c, d, e, f, g bit

200620072008

1a 1b 1c 1d 1e 1fg

= ++

++

++

Bi 3.15 (B Gio dc v o to, Trung hc C s, 11.3.2011)

Tm x tha mn ng thc sau y (2,5 im)

4 .2011 61993 632010 31994 112009 20111995 20081996 20071997 20061998 20051999 20042000 200320012002

x=

++

+

+

+

Bi 3.16 (B Gio dc v o to, Lp 6-7, 2001. d b)

1) Lp quy trnh bm phm tnh gi tr ca lin phn s: 11 11 12 11 12 11 12

1

M = ++

++

++

+

.

2) Tnh 3 .M

20

Bi 3.17 (B Gio dc v o to, Lp 6-7, 2001) 1) Lp quy trnh bm phm tnh gi tr ca lin phn s:

13 17 115 11292

M = ++

++

2) Tnh M .

Dng ton 4 Ton phn trm

Bi 4.1 (Phng Gio dc v o to huyn ng Triu, Lp 9, 2011-2012)

Tm 2,5% ca:

3 3 56 3 .55 14 6 .

(21 1, 25) : 2,5M

=

Dng ton 5 T l thc

Bi 5.1 (S Gio dc v o to Ph Yn, Trung hc C s, 11.3.2011)

Bit ,x y l hai i lng t l nghch. Hy in s thch hp vo trng:

x 4 0,25 23

11

y 13 813

15 16


V mt tm ba ln mt ng h hnh vung v dng cc v tr ch gi lm cc ng bin (xem hnh). Nu t l din tch ca 1 trong 8 min tam gic (nh min gia 12 gi v 1 gi) v T l din tch ca 1 trong 4 t gic (nh t gic gia 1

gi v 2 gi). Tnh t s T .t

XI

II

VII I

XI I

V

I

VII

I I I

IV

IX

X

VI

Bi 5.3 (B Gio dc v o to, Trung hc C s, 11.3.2011) Mt hn hp gm 5 cht v nng 5327256605 gam. Bit rng t l v khi lng gia cc cht nh sau: t l gia cht th nht vi cht th hai l 2:3, t l gia cht hai vi cht th ba l 4:5, t l gia cht th ba v cht th t l 7:6, t l gia cht th t v cht th nm l 11:7. Hy tm v cho bit mi cht c trong hn hp ny nng bao nhiu gam. Trnh by tm tt cch gii.

21

Dng ton 6 Ton thi gian v Ton chuyn ng


Tnh +

=

h ph gi h ph gi h ph gi

h ph gi h ph gi h ph gi

(8 47 57 7 8 51 ).3 5 7E .18 47 32 : 2 5 9 4 7 27


Tnh +

=h ph gi h ph gi h ph gi

h ph gi h ph gi

(8 45 23 12 56 23 ).3 5 7E .16 47 32 : 2 5 9

Bi 6.3 (S Gio dc o to ng Nai, Trung hc C s, 15.2.1998) Tnh

6 47 29 2 58 38 .1 31 42 3

g ph gi g ph gi

g ph giC

=

Bi 6.4 (S Gio dc o to H Ni, Lp 10, vng Trng, 1996) Tnh

3 47 55 3 5 11 456 52 17

h ph gi g ph gi

h ph giB +

=

Bi 6.5 (S GD v T H Ni, vng Chung kt, THCS v THPT, 18.12.1996) Tnh A bit

22 25 18 2,6 7 47 359 28 16

g ph gi g ph gi

g ph giA +

=

Bi 6.6 (B Gio dc v o to, Trung hc C s, 2008-2009)

1) Mt chic thuyn khi hnh t bn sng A. Sau 5 gi 10 pht, mt chic can chy t A ui theo v gp chic thuyn cch bn A 20,5 km. Hi vn tc ca thuyn, bit rng can chy nhanh hn thuyn 12,5 km/h (kt qu tnh chnh xc ti 2 ch s thp phn). 2) Lc 8 gi sng, mt t i t A n B, bit qung ng AB di 157 km. i c 102 km th xe b hng my phi dng li sa cha mt 12 pht ri i tip n B vi vn tc t hn lc u l 10,5 km/h. Hi t b hng lc my gi, bit rng t n B lc 11 gi 30 pht. (kt qu thi gian lm trn n pht).

Bi 6.7 (S Gio dc v o to Ph Yn, lp 9, 2009-2010) Hai ng t A v B cng chuyn ng trn mt ng trn c ng knh 20m xut pht cng mt im. Nu chng chuyn ng cng chiu th c 20 giy li gp li nhau, nu chng chuyn ng ngc chiu th sau 4 giy li gp nhau. Tm vn tc ca mi ng t.

Bi 6.8 (S Gio dc o to ng Nai, Trung hc C s, 15.2.1998)

22

Tm thi gian mt ng t di chuyn ht on ng ABC di 127,3 km bit 75,5AB = km v c di chuyn vi vn tc 26,3 km/gi v on BC c di chuyn bng vn tc 19,8 km/gi.

Bi 6.9 (S GD v T H Ni, vng Chung kt, Trung hc C s, 18.12.1996) a) Tnh thi gian bng gi, pht, giy mt ngi i ht qung ng ABC di 435 km bit rng on AB di 147 km c i vi vn tc 37,6 km/gi v on BC c i vi vn tc 29,7 km/gi. b) Nu ngi y di chuyn vi vn tc u (37,6 km/gi) th n C sm hn khong thi gian l bao nhiu?

Dng ton 7 Tnh li sut v tng trng

Bi 7.1 (B Gio dc v o to, Trung hc Ph thng, 19.3.2010) Mt ngi mua xe my tr gp vi gi tin l 20.000.000 , mc li sut 1,2%/thng vi qui c mt thng tr 800.000 c gc v li. Hi sau 12 thng k t ngy ngi y mua xe s tin cn n l bao nhiu ng? Sau mt nm li sut li tng ln l 1,5%/thng v ngi li tr mt thng 1.000.000 ng c gc v li (tr thng cui cng). Hi sau bao nhiu thng ngi y tr ht n? (thng cui tr khng qu 500.000 ).

Bi 7.2 (B Gio dc v o to, Trung hc C s, 2007-2008) Dn s ca mt nc l 80 triu ngi, mc tng dn s l 1,1% mi nm. Tnh dn s ca nc sau n nm, p dng vi n 20= .

Bi 7.3 (S GD v T Tha Thin Hu, Trung hc C s, 01.02. 2007) Dn s ca mt thnh ph nm 2007 l 330.000 ngi. 1) Hi nm hc 2007-2008, d bo c bao nhiu hc sinh lp 1 n trng, bit trong 10 nm tr li y t l tng dn s mi nm ca thnh ph l 1,5% v thnh ph thc hin tt ch trng 100% tr em ng tui u vo lp 1? 2) Nu n nm hc 2015-2016, thnh ph ch p ng c 120 phng hc cho hc sinh lp 1, mi phng dnh cho 35 hc sinh th phi kim ch t l tng dn s mi nm l bao nhiu, bt u t nm 2007? (Kt qu ly vi 2 ch s phn thp phn).

Bi 7.4 (S Gio dc v o to Tha Thin Hu, lp 9, 01.12. 2006) Li sut tit kim ca mt s ngn hng hin nay l 8,4% nm i vi tin gi c k hn mt nm. khuyn mi, mt ngn hng thng mi A a ra dch v mi : Nu khch hng gi tit kim nm u th li sut 8,4% nm, sau li sut nm sau tng so vi nm trc l 1%.

23

Hi nu gi 1.000.000 ng theo dch v th s tin s nhn c l bao nhiu sau : 10 nm, 15 nm ? Nu s lc cch gii.

Bi 7.5 (S Gio dc v o to Qung Ngi, lp 9, 2009-2010) Dn s tnh A hin nay (u nm 2010) l 1.700.000 ngi. 1) Hi nu t l tng dn s trung bnh hng nm l 1,2% th sau 10 nm na dn s ca tnh A l bao nhiu ngi ? 2) Mun dn s ca tnh A c khong 2.000.000 ngi vo u nm 2020 th phi c t l tng dn s hng nm l bao nhiu ? (lm trn n 2 ch s thp phn)?

Bi 7.6 (Phng Gio dc v o to huyn ng Triu, Lp 9, 2011-2012) 1) Dn s huyn A nm 2011 khong 93300 ngi. Hi nm hc 2011-2012 c bao nhiu hc sinh lp 1 n trng. Bit rng trong khong 10 nm tr li y t l tng dn s trung bnh mi nm l 1,5% v vic huy ng hc sinh ng tui vo lp 1 l 100% ( nu tm tt cch gii v ly kt qu lm trn n hng n v). 2) Nu n nm hc 2016-2017, huyn ch giao ch tiu lp 1 l 35 lp, mi lp 35 hc sinh th phi hn ch t l tng dn s trung bnh mi nm l bao nhiu, bt u tnh t nm 2011 tr i (nu tm tt cch gii v ly kt qu lm trn n hai ch s phn thp phn).

Bi 7.7 (S Gio dc v o to Sc Trng, lp 9, 2008-2009) Sau 3 nm, mt ngi ra ngn hng nhn li s tin c vn ln li l 37337889,31 ng. Bit rng ngi gi mc k hn 3 thng theo li kp, vi li sut 1,78% mt thng. Hi s tin ngi y gi vo ngn hng lc u l bao nhiu? ( li kp: l li nhp vn v s tin c c li tip tnh li theo quy nh).

Bi 7.8 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008) Mt ngi vay vn mt ngn hng vi s vn l 50 triu ng, thi hn 48 thng, li sut 1,15% trn thng, tnh theo d n, tr ng ngy qui nh. Hi hng thng, ngi phi u n tr vo ngn hng mt khon tin c gc ln li l bao nhiu n thng th 48 th ngi tr ht c gc ln li cho ngn hng? b) Nu ngi vay 50 triu ng tin vn mt ngn hng khc vi thi hn 48 thng, li sut 0,75% trn thng, trn tng s tin vay th so vi vic vay vn ngn hng trn, vic vay vn ngn hng ny c li g cho ngi vay khng?

Bi 7.9 (Thi th vng tnh, Trng THCS ng Nai-Ct Tin, 2004)

24

Mt ngi hng thng gi vo ngn hng s tin l 10 000 000 vi li sut 0,55% mt thng. Hi sau hai nm ngi y nhn c bao nhiu tin li? (lm trn n hng n v).


1) Dn s ca mt nc trong hai nm tng t 30.000.000 ngi ln n 30.048.288 ngi. Tnh t l tng dn s hng nm trong 2 nm (kt qu lm trn n ch s hng n v).

2) Mt ngi hng thng gi vo ngn hng s tin l 1 000 000 vi li sut tit kim 0,45% mt thng. Hi sau hai nm ngi y nhn c bao nhiu tin (kt qu lm trn n ch s hng n v).

Bi 7.11 (S Gio dc o to thnh ph H Ch Minh, vng 1, 24.11.1996) S tin 58.000 c gi tit kim theo li kp (sau mi thng tin li c nhp thnh vn). Sau 25 thng th c c vn ln li l 84.155 ng. Tnh li sut /thng (tin li ca 100 trong 1 thng).

Bi 7.12 (S Gio dc o to H Ni, vng 1, 18.12.1996) Dn s mt nc l 65 triu, mc tng dn s 1 nm l 1,2 %. Tnh dn s nc y sau 15 nm.

Chng 3 I S TRN MY TNH IN T

Dng ton 1 Ly tha

Bi 1.1 (B Gio dc v o to, Trung hc C s, 11.3.2011)

Tnh gi tr ca biu thc 2 3 4

75 6

9,87 6,54 : 3,21 .1 3 5 7 9

11 13 17 19 23

A = +


Tnh gi tr ca biu thc 2 3 4

42 3

1, 25 15,37 3,75A .1 3 2 5 24 7 5 7 3

=

+

Bi 1.3 (S Gio dc v o to Tha Thin-Hu, lp 9, 2006) Tnh gi tr ca cc biu thc:

25

3 2 2 2

3x 2y x 16yBx 4y 9x 6xy 4y

+= + + +

4 4

2 2

x 16yx 4y

+

.

khi 1) x 5 ; y 16 ;= = 2) x 1,245 ; y 3,456.= = Bi 1.4 (Chn i tuyn thi Khu vc, S GD T Lm ng, 2004) Tnh

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

4 4 3 2

2 3

1, 23456789 0,76543211 1,123456789 . 0,76543211

1, 23456789 . 0,76543211 16. 1,123456789 . 0,76543211 .

+

+


Rt gn biu thc (kt qu vit di dng phn s) 4 4 4 4 4 4 4

4 4 4 4 4 4 4

(1 4)(5 4)(9 4)(13 4)(17 4)(21 4)(25 4)(3 4)(7 4)(11 4)(15 4)(19 4)(23 4)(27 4)

+ + + + + + +=

+ + + + + + +C

Bi 1.6 (Thi thi th vng Tnh, Trng THCS ng Nai, 2004) Rt gn biu thc (kt qu vit di dng phn s)

4 4 4 4 4 4 4

4 4 4 4 4 4 4

(1 6)(7 6)(13 6)(19 6)(25 6)(31 6)(37 6) .(3 6)(9 6)(15 6)(21 6)(27 6)(33 6)(39 6)

+ + + + + + +=

+ + + + + + +C

Bi 1.7 (S Gio dc v o to Sc Trng, lp 9, 2008-2009) Tnh gi tr ca cc biu thc:

4 4 4 4

4 4 4 4

1 1 1 12 4 6 .... 20084 4 4 4 .1 1 1 11 3 5 .... 20074 4 4 4

N

+ + + + = + + + +

Dng ton 2 A THC

Dng ton 2.1 Tnh gi tr ca a thc v phn thc i s

Bi 2.1 (Thi chn i tuyn. S Gio dc v o to Thp H Ch Minh, 2003) Tm gi tr ln nht ca hm s ( ) 21, 2 4,9 5,37f x x x= + (ghi kt qu gn ng chnh xc ti 6 ch s thp phn).

Bi 2.2 (S Gio dc v o to Thanh Ha, lp 10, 18.4.2000) Cho hm s 4 3 25 3 1y x x x x= + + . Tnh y khi 1,35627x = .

Bi 2.3 (S GD v T H Ni, vng 1, cp Trung hc C s, 18.12.1996) Tm 5 4 3 2( ) 17 5 8 13 11 357P x x x x x x= + + khi 2,18567x = .

26

Bi 2.4 (S Gio dc v o to thnh ph H Ch Minh, vng 1, 24.11.1996)

Tnh 5 4 2

3 2

3 2 3 14 3 5

x x x xAx x x + +

= + +

khi 1,8165x = .

Bi 2.5 (S Gio dc v o to ng Thp, lp 9, 2008-2009) Cho 7 6 5 4 3 2( ) 7 35 5 9 39 1.P x x x x x x x x= + + Tnh (2) 2 (5) (3).P P P+

Bi 2.6 (Chn i tuyn. S Gio dc v o to Thp H Ch Minh, 2003) Tm gi tr ca tham s m bit ti 2,5x = gi tr ca a thc ( ) ( )4 3 22 5 3 2 5f x x x x m x m= + + + l 0,49.

Bi 2.7 (S Gio dc v o to ng Nai, lp 9, 2004-2005) Cho 4,35467; 5,64753; 7.a b n= = = So snh cc s sau:

2 1

2

1 ...1 ...

n

n

a a aAa a a

+ + + +=

+ + + + v

2 1

2

1 ... .1 ...

n

n

b b bBb b b

+ + + +=

+ + + +

Bi 2.8 (S GD v T H Ni, vng chung kt, Trung hc C s, 18.12.1996)

Tnh 2 3 4

2 3 4

11

x x x xAy y y y

+ + + +=

+ + + + khi 1,8597;x = 1,5123.y =


1) Tnh 6 4 3 24 3 2 7 6 11x x x x x+ + + vi 3,1226.x =

2) Tnh 6 4 3 24 3 2 7 6 11x x x x x+ + + vi 23 .513

x = ++

3) Tnh 2 2 2

2 2 2

22

x y z xyx z y xz+ ++ +

vi 3 ; 1,5; 13, 4.4

x y z= = =


1) Tnh 5 4 37 12 3 5 7,17E x x x x= + vi 7,1254.x = 2) Cho 2,1835x = v 7,0216.y = Tnh

5 4 3 3 4

3 2 2 3

7 3 10 9 .5 8

x y x y x y xyFx x y y

+ + =

+

Bi 2.11 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008) Tnh gi tr ca biu thc(ch ghi kt qu):

2 2 2 2

( 5 )( 5 ) 5 55 5

x y x y x y x yBx y x xy x xy

+ += + + +

vi 0,987654321;x = 0,123456789.y =

27

Dng ton 2.2 Tm s d trong php chia a thc ( )P x cho n thc ax b+ vi ,a b l cc s bit

Bi 2.12 (Phng Gio dc o to huyn Bo Lm, Lm ng, 2004) Tm s d r ca php chia:

5 4 26,723 1,658 9,134 .3, 281

x x xx

+

Bi 2.13 (S Gio dc v o to H Ni, Trung hc C s, vng chung kt, 18.12.1996; Ph Yn, lp 9 Trung hc C s, 10.2.2009) Tm s d ca php chia:

1) 3 29 35 7 ;

12x x x

x +

2)

3 3, 256 7,321.1,617

x xx

+

Bi 2.14 (S GD v T ng Nai, vng Tnh, Trung hc C s, 15.12.1998) Tm s d ca php chia:

5 3 26,723 1,857 6, 458 4,319 .2,318

x x x xx

+ ++

Bi 2.15 (S GD v T Thp H Ch Minh, vng 1, PTTH, 15.3.1998) Tm s d trong php chia (kt qu ly 3 s l):

14 9 5 4 2 723 .1,624

x x x x x xx

+ + +

Dng ton 2.3 iu kin a thc ( )P x m+ chia ht cho x a


Cho 7 6 5 4 3 2( ) 5 2 4 9 2 10 .P x x x x x x x x m= + + + + Tm m ( )P x chia ht cho a thc 2.x +

Bi 2.17 (S GD v T H Ni, vng Chung kt, 1996; S GD v T Thanh Ha, lp 10- 11, 12, 2000)

Tnh a 4 3 27 2 13x x x x a+ + + + chia ht cho 6x + .


Bit ( ) 5 4 3 24 12 3 2 5 7.P x x x x x x m= + + + + Tm m ( )P x chia ht cho 13.x


Cho a thc ( ) 7 6 4 35 8 7,589 3,58 65 .P x x x x x x m= + + + + 1) Tm m ( )P x c nghim l 0,1394.

28

2) Vi m va tm c, tm s d khi chia ( )P x cho nh thc 2,312.x + 3) Vi m tm c, hy in vo bng sau (lm trn n ch s hng n v).

x 2,53 4,72149 15

34 3 6,15 5 76 7+

( )P x


Cho a thc ( ) 10 8 4 3 7,589 3,58 65 .P x x x x x x m= + + + + . 1) Tm m ( )P x c nghim l 0,3648. 2) Vi m va tm c, tm s d khi chia ( )P x cho nh thc 23,55.x 3) Vi m tm c, hy in vo bng sau (lm trn n ch s hng n v).

x 2,53 4,72149 1534

3 6,15 5 76 7+

( )P x

Dng ton 2.4 Phn tch a thc thnh nhn t

Bi 2.21 (S Gio dc v o to Ph Yn, lp 9, 10.2.2009)

Phn tch a thc sau ra tha s: 3 2( ) 4 16 9 9.P x x x x= + +

Bi 2.22 (S Gio dc v o to Ty Ninh, lp 9, 2008-2009) 1) Phn tch a thc sau thnh nhn t:

5 4 3 2( ) 43 1829 68423 736948 23029900.Q x x x x x x= + + 2) Cho a thc:

( )( ) ( )( )( )2 2 2 2 2( ) 36 11 30 11 31 11 12 9 20 13 12 .R x x x x x x x x x x x= + + + + + + + + + +a) Phn tch ( )R x thnh nhn t. b) Tm nghim ca a thc ( ).R x

Dng ton 2.5 Ton tng hp v a thc


a thc 6 5 4 3 2P(x) x ax bx cx dx ex f= + + + + + + c gi tr l 3 ; 0 ; 3 ; 12 ; 27 ; 48 khi x ln lt nhn cc gi tr tng ng l 1 ; 2 ; 3 ; 4 ; 5 ; 6.

1) Xc nh cc h s a, b, c, d, e, f ca P(x) . 2) Tnh gi tr ca a thc P(x) vi x = 11; 12; 13; 14; 15; 16; 17; 18; 19; 20.

Bi 2.24 (B Gio dc v o to, Trung hc C s, 2007-2008) Cho a thc 4 3 2P(x) x ax bx cx d= + + + + tha mn:

P(0) 12, P(1) 12, P(2) 0, P(4) 60.= = = =

29

1) Xc nh cc h s a, b, c ca P(x) . 2) Tnh P(2006) . 3) Tm s d trong php chia a thc P(x) cho 5x 6 .

Bi 2.25 (S Gio dc v o to Tha Thin Hu, lp 9, 01.12. 2006) Cho a thc ( ) 3 2P x ax bx cx d.= + + + Bit :

( ) ( ) ( ) ( )P 1 27 ; P 2 125 ; P 3 343 ; P 4 735.= = = = 1) Tnh ( ) ( ) ( ) ( )P 1 ; P 6 ; P 15 ; P 2006 . (Ly kt qu chnh xc) 2) Tm s d ca php chia P(x) cho 3x 5.

Bi 2.26 (S Gio dc v o to Tha Thin-Hu, THCS, 01.02.2007)

Tm cc h s a, b, c ca a thc 3 2P(x) ax bx cx 2007= + + bit:

Khi chia P(x) cho (x 16) th c d l 29938 v chia cho 2x 10x 21 +

th c a thc d l 10873 x 3750

16 (kt qu chnh xc).

Bi 2.27 (S Gio dc v o to Sc Trng, 28.11.2010)

1) Cho a thc bc ba ( ).f x Bit ( ) ( ) ( ) ( )0 10, 1 12, 2 4, 3 1.f f f f= = = = Tnh (10).f 2) Cho a thc bc bn ( )f x c h s ca bc cao nht l 1 v

( ) ( ) ( )1 3; 3 11; 5 27.f f f= = = Tnh gi tr ( ) ( )2 7 6 .A f f= + Bi 2.28 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008)

Cho a thc ( ) 3 2 .P x x ax bx c= + + + 1) Tm , ,a b c bit rng khi x ln lt nhn cc gi tr 1,2; 2,5; 3,7 th ( )P x nhn cc gi tr tng ng l 1994,728; 2060,625; 2173,653.

2) Tm s d r ca php chia a thc ( )P x cho 12 1.x 3) Tm gi tr ca x khi ( )P x c gi tr l 1989.


Cho ( ) 5 4 3 2 132005.P x x ax bx cx dx= + + + + + Bit ( ) ( ) ( ) ( )1 8, 2 11, 3 14, 4 17.P P P P= = = = Tnh (15).P


Cho 4 3 2( ) .P x x ax bx cx d= + + + + Tnh (20) (10)2

P P bit

(1) 5; (2) 7; (3) 9; (4) 11.P P P P= = = =

30

Bi 2.31 (S Gio dc v o to Qung Nam, lp 9, 2009-2010)

1) Cho bit ( ) 5 4 3 24 3 2 7f x x x x x ax= + + + + khi chia cho 5x + c d l 2009. Tm .a 2) Cho a thc ( ) 5 4 3 215 85 223 274 119P x x x x x x= + + v ( ) ( )( )( )1 2 3 .Q x x x x=

Gi ( )R x l a thc d khi chia ( )P x cho ( ).Q x 2a) Xc nh ( ).R x

2b) Tnh ( ) 22010 .R

Bi 2.32 (S Gio dc o to Lm ng, 2005)

1) Bit ( ) 5 4 3 24 12 3 2 5 7.P x x x x x x m= + + + + Tm m ( )P x chia ht cho 13.x 2) Cho ( ) 5 4 3 2 .P x ax bx cx dx ex f= + + + + + Tnh (12)P bit

( ) ( ) ( ) ( ) ( )1 1 11; 2 2 47; 3 107.P P P P P= = = = =


Cho a thc ( ) 4 3 2 .f x x ax bx cx d= + + + + Bit ( ) ( ) ( ) ( )1 0; 2 4; 3 18; 4 42.f f f f= = = =

1) Tm cc h s , , ,a b c d ca ( ).f x 2) Bit f(2010) l s nguyn, tnh tng cc ch s ca f(2010) (vit qui trnh bm phm).

Bi 2.34 (S GD v T Qung Nam, vng 1, PTTH, 15.3.1998)

Cho a thc ( ) 5 4 3 2 .P x x ax bx cx dx e= + + + + + Bit ( ) ( ) ( ) ( ) ( )1 3; 2 9; 3 19; 4 33; 5 51.P P P P P= = = = = 1) Tnh cc h s , , , , .a b c d e 2) Tnh chnh xc (2010).P Bi 2.35 (Phng Gio dc o to huyn Bo Lm, Lm ng, 2004)

1) Tnh 5 4 37 12 3 5 7,17E x x x x= + vi 7,1254.x =

2) Cho 2,1835x = v 7,0216.y = Tnh 5 4 3 3 4

3 2 2 3

7 3 10 95 8

x yx y x y xyFx x y y

+ + =

+

3) Tm s d r ca php chia 5 4 26,723 1,658 9,134 .

3, 281x x x

x +

31

4) Cho 7 6 5 4 3 2( ) 5 2 4 9 2 10 .P x x x x x x x x m= + + + + Tm m ( )P x chia ht cho a thc 2.x +

Dng ton 3 CN THC

Bi 3.1 (B Gio dc v o to, Trung hc C s, 11.3.2011) Tnh gi tr ca biu thc

3 5 7 9 112 3 4 5 6 7 8 9 10 11 12 .B = + + + + +


3 5 3 5 2009 13,3B

3 2 5 3 7 2 3 5 4 7

+ + =

+ + +


x 1 2 xC 1 :x 1 x 1 x x x x 1

= + + +

vi 143,08x = .

Bi 3.4 (S GD v T Tha Thin Hu, Trung hc C s, 01.02. 2007) Tnh gi tr ca biu thc, ly kt qu vi 2 ch s thp phn :

N 521973 491965 1371954 6041975 1122007 .= + + + +

Bi 3.5 (S GD v T H Ni, vng Trng, 1996; Thanh Ha, Lp 11, 2000)

Tnh: 56

47

1,815 2,732

4,621A =

Bi 3.6 (S Gio dc v o to thnh ph H Ch Minh, vng 1, 24.11.1996)

Tnh: 4 2,3

57

(1,345) .(3,143)(189,3)

x =

Bi 3.7 (S Gio dc v o to H Ni, vng Trng, 18.12.1996)

Tm x vi 43 5

74

2,3144

3,785x = .

Bi 3.8 (S Gio dc v o to ng Nai, vng Tnh, 15.2.1998)

32

Tnh 3 7

175

816,13712,35

B =


Tnh ( 3 1) 6 2 2 12 18 128 .B = + + +

Bi 3.10 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008) Tnh gi tr ca biu thc(ch ghi kt qu):

321930 291945 2171954 3041975 .A = + + +

Bi 3.11 (Thi thi th vng Tnh, Trng THCS ng Nai, 2004) Tnh gi tr biu thc (lm trn vi 5 ch s thp phn)

3 73

295

18,9 91,526 : 4 6113 .51 6635,4677 3,5 : 5 : 3,9 7183 11513

B+

= + ++ +


Tnh gi tr ca biu thc (lm trn n 5 ch s thp phn)

3 53

3 4 5 6 72

25

18,9543 981,635 : 4 7113 : 3 4 5 6 78151 6589,43111 3,5 :1 : 3,9814 7173 9513

B+

= + + + + + ++ +

Bi 3.13 (S Gio dc v o to ng Nai, lp 9, 2004-2005)

Tnh biu thc 3 22 2 1A x x= + + vi

3 31 23 513 23 513( 1).3 4 4

x + = + +


1) Tnh 2 21 999999999 0,999999999I = + +

2) Tnh 6 6 61 999999999 0,999999999

999999999+ +

Bi 3.15 (S Gio dc v o to Ty Ninh, lp 9, 2008-2009)

33

Cho biu thc

( 2)( 1002) ( 2)( 1002) ( 2)( 1002) .

( )( ) ( )( ) ( )( )a a b b c cAa a b a c b b a b c c c a c b

= + +

a) Rt gn biu thc .A b) Tnh gi tr ca biu thc A khi 2008; 2009; 2010.a b c= = =

Bi 3.16 (Phng Gio dc v o to ng Triu, 2011-2012)

Tnh tng

2 2 2 2 2 21 1 1 1 1 1C= 1 1 ... 1 .1 2 2 3 2009 2010

+ + + + + + + + +

Bi 3.17 (S Gio dc v o to Sc Trng, lp 9, 2008-2009) Cho biu thc

( )1 1 8 3 2 1 21 1 .9 1 13 1 3 1 3 1 1 1x x x x xA x

x xx x x x x

= + + + + +

Tnh gi tr ca biu thc A khi 2 2 2 2 .2 2 2 2

x += ++


Cho 23,1 2 5( ) 1,32 7,8 3 2.6, 4 7, 2

f x x x= + +

1) Tnh (5 3 2).f 2) Vi gi tr no ca x th ( )f x t gi tr nh nht? Tm gi tr nh nht ca

( ).f x


Cho a thc 4 3 22 5( ) 2 3 2 ;7 11

f x x x x x= + + 2( ) 3 1.g x x x= +

1) Tnh 54( 3 ) ( 5 13);7

f g + +

2) Tnh 543 ( 5 13 .7

g f g +

Dng ton 4 GII PHNG TRNH V H PHNG TRNH

Dng ton 4.1 Gii h hai phng trnh bc nht hai n

Bi 4.1 (S Gio dc v o to ng Nai, 1998; Phng Gio dc o to huyn Bo Lm, Lm ng, 2004)

34

Gii h phng trnh (ghi kt qu 9 s l thp phn):

1,372 4,915 3,1238,368 5,214 7,318

x yx y =

+ =

Bi 4.2 (S GD v T Thp H Ch Minh, vng Chung kt, THPT, 15.3.1998) Gii h phng trnh (ghi kt qu 9 s l thp phn):

3,6518 7,3249 4,68211,4926 6,3571 2,9843

x yx y+ =

+ =

Bi 4.3 (S GD v T H Ni, vng 1, Trung hc C s, 18.12.1996)

Gii h phng trnh 13,241 17,436 25,16823,897 19,372 103,618

x yx y+ =

=

Bi 4.4 (S Gio dc v o to ng Thp, lp 9, 2008-2009)

Gii h phng trnh o

15 12 2008sin120 os15o

x yx y c

+ =

+ = +

Dng ton 4.2 Gii h ba phng trnh bc nht ba n

Bi 4.5 (S Gio dc v o to Ph Yn, lp 12 B tc THPT, 2009-2010)

Gii h phng trnh 4 5 2 5;

3 2 4 8;3 5 10.

x y zx y z

x y z

+ = + = + + =

Bi 4.6 (S Gio dc v o to Thp H Ch Minh, Lp 9, 11.1.2009)

a thc bc ba ( )P x c (1) 5, (2) 7, (3) 9P P P= = = v (4) 12.P = Tnh (2009)P .

Dng ton 4.3 Gii h bn phng trnh bc nht bn n

Bi 4.7 (S Gio dc v o to Ph Yn, lp 9 Trung hc C s, 10.2.2009)

Gii h phng trnh bc nht bn n:

407 ;276

23 12 46 12 21; 11 23 63 11 823 3 23 4 277 ;33 14 105 11 56022 24 22 24 14 .

207 23 21 55 45

x y z t

x y z t

x y z t

x y z t

+ + + = + = = + + + =

.

35

Dng ton 4.3 Gii phng trnh bc hai

Bi 4.8 (S Gio dc v o to ng Nai, Trung hc C s, 1998) Gii phng trnh (ghi kt qu 9 s l thp phn):

22,354 1,542 3,141 0.x x = Bi 4.9 (S Gio dc v o to thnh ph H Ch Minh, vng 1, 24.11.1996)

Gii phng trnh 21,85432 3,21458 2,45971 0x x = Bi 4.10 (S Gio dc v o to Thanh Ha, lp 10, 18.4.2000)

Gii phng trnh: 21, 23785 4,35816 6,98753 0x x+ = Bi 4.11 (S Gio dc v o to thnh ph H Ch Minh, vng 1, cp Trung hc Ph thng v Ph thng Chuyn ban, 15.3.1998)

Gii phng trnh (ghi kt qu 7 s l): 21,9815 6,8321 1,0581 0x x+ + = . Bi 4.12 (S GD v T Thp H Ch Minh, Chung kt, THPT, 15.3.1998) Gii phng trnh (ghi kt qu 9 s l thp phn):

22.3541 7.3249 4.2157 0x x+ + = Bi 4.13 (S Gio dc v o to Thanh Ha, lp 10, 18.4.2000)

Cho parabol ( )P c phng trnh: 24,7 3,4 4,6.y x x= Tm ta

0 0( , )x y ca nh S ca parabol. Tm giao im ca Parabol ( )P vi trc

honh.

Bi 4.14 (B Gio dc v o to, Trung hc C s, 11.3.2011) (5 im)

Bit rng x l mt s thc khc 0, tm gi tr nh nht ca biu thc 2

2

2010, 2011 2 2012, 2013 .2014, 2015

x xQx

+=

Trnh by tm tt cch gii.


1) Cho 2 2

112 32 ; ; sin 60 cos30

2 3 2 3o oA B C

++

= = = ++ +

Gii phng trnh 2 0.Ax Bx C+ =

2) Gi 1x l nghim m ca phng trnh 2 1 0.x x+ = Tnh gi tr ca biu

thc 81 1 110 13 .P x x x= + + +

3) Cho a thc 10 5( ) 1P x x x= + + . Gii phng trnh 2(2) ( 12) 2008 0.P x P x+ =

36

Bi 4.16 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008)

Cho x1000 + y1000 = 6,912; x2000 + y2000 = 33,76244. Tnh A = x3000 + y3000.

Bi 4.17 (S Gio dc v o to Ty Ninh, lp 9, 2008-2009)

Gii phng trnh 2 2 2(2 3 1) 6 9 1 0.x x x x + =

Bi 4.18 (S GD v T ng Nai, Trung hc C s, vng Tnh, 15.2.1998)

Cho ,x y l hai s dng, gii h phng trnh 2 2

2,317;

1,654.

xy

x y

= =


Gii h phng trnh

2 2 66,789

5,78

x yxy

= =


Gii h phng trnh

2 2 55,789

6,86

x yxy

+ = =

Bi 4.21 (S Gio dc v o to H Ni, Trung hc Ph thng, vng chung kt, 1996; S Gio dc v o to Thanh Ha, lp 10, 2000)

Gii h phng trnh 2 2

0,681

19,32

xy

x y

= + =

0, 0x y> > .

Dng ton 4.4 Gii phng trnh bc ba

Bi 4.22 (Phng Gio dc v o to huyn ng Triu, lp 9, 2011-2012)

Phng trnh 3 22 10 0x ax x b + = c hai nghim 1 22; 3.x x= = Tm a, b

v nghim x3 cn li.

Dng ton 4.5 Mt s bi ton khc v phng trnh v h phng trnh

Bi 4.23 (S Gio dc v o to Qung Nam, vng 1, PTTH, 15.3.1998) Gii h phng trnh

12 3 5;3 5

2 3 6.3 5

x yx y

x yx y

+ + = + =

37

Bi 4.24 (S Gio dc v o to Qung Nam, lp 9, 2009-2010) Cho phng trnh x2 ax + 1 = 0 (aZ) c 2 nghim l x1, x2 . Tm a nh nht sao cho x15 + x25 chia ht cho 250.

Bi 4.25 (B Gio dc v o to, lp 12 Trung hc Ph thng, 11.3.2011) Gii phng trnh [ ]2 2010 2011 0.x x + = Bi 4.26 (S Gio dc v o to Tuyn Quang, lp 12 THPT, 19.10.2011) Gii cc h phng trnh:

1) 3 2

3 2

1 2( ); ( , )

1 2( ).x x x y

x yy y y x

+ = + + = +

2)

12 5;2

2 6.2

x yx y

x yx y

+ + = + =


Tnh gn ng nghim h phng trnh:

2

2

2 6;

2 6.

xy

yx

+ = + =


Tm mt nghim ca phng trnh 5 33 20 75 85 0.x x x + + =

Bi 4.29 (S GD v T Tha Thin Hu, Lp 9, 01.02. 2007) Gii phng trnh (ly kt qu vi cc ch s tnh c trn my) :

2 22007 2008 x x 0,1 = 20+ 2008 2007 x x 0,1+ + + + + .

Bi 4.30 (B Gio dc v o to, Trung hc C s, 2007-2008) Gii h phng trnh

3 2

2 2

13x 26102x 2009x 4030056 0;

(x x 4017)(y y 1) 4017 3.

=

+ + + + =

Chng 4 TON THNG K TRN MY TNH IN T


Cho bng s liu sau. Hy tnh tng s trng ( x ); s trng trung bnh ca mi con g ( x ); phng sai ( 2x ) v lch tiu chun ( x )?

S trng 12 13 14 15 16 17 18 19 20 21

S g m 6 10 14 25 28 20 14 12 9 7

38

Bi 1.2 (S Gio dc v o to thnh ph H Ch Minh, vng 1, 24.11.1996) Cho s liu:

Bin lng 135 642 498 576 637 Tn s 7 12 23 14 11

Tnh tng s liu, s trung bnh v phng sai 2n (2n ly 4 s l).

Bi 1.3 (S Gio dc v o to H Ni, lp 10, vng Trng, 1996) Cho s liu:

Bin lng 7 4 15 17 63 Tn s 2 1 5 9 14

a) Tnh s trung bnh ;X b) Tnh phng sai 2.n

Bi 1.4 (S GD v T ng Nai, vng Tnh, cp Trung hc C s, 15.2.1998) Cho s liu:

S liu 173 52 81 37 Tn s 3 7 4 5

Tm s trung bnh X , phng sai 2X (2n ) (kt qu ly 6 s l).

Bi 1.5 (S Gio dc v o to Thp H Ch Minh, vng 1, THPT, 15.3.1998) Qua k thi, 2105 hc sinh xp theo im s nh sau. Hy tnh t l phn trm (ly 1 s l) hc sinh theo tng loi im. Phi bm t nht my ln phm chia in xong bng ny vi my Casio?

im 0 1 2 3 4 5 6 7 8 9 10 S h/s 27 48 71 293 308 482 326 284 179 52 35 T l

Bi 1.6 (Thi vo 10 Ph thng Trung hc Ngh An, 1996) Mt hc sinh lp 9 ca trng ph thng c s TT c kt qu kim tra v mn ton vi 10 ln im nh sau: 7, 8, 6,7, 7, 8, 9, 6, 10, 7. a) Lp bng phn phi thc nghim, tnh im trung bnh ca hc sinh . b) Tnh phng sai, lch tiu chun v cho bit ngha lch ny.

Bi 1.7 (B Gio dc v o to, lp 12 Trung hc Ph thng, 11.3.2011) Theo kt qu iu tra dn s, dn s trung bnh nc Vit Nam qua mt s mc thi gian (n v: 1000 ngi):

Nm 1976 1980 1990 2000 2011 S dn 49160 53722 66016,7 77635 88434,6

39

1) Tnh t l % tng dn s trung bnh mi nm trong cc giai on 1976-1980, 1980-1990, 1990-2000, 2000-2011. 2) Nu c duy tr t l tng dn s nh giai on 2000-2011 th n nm 2015 v 2020 dn s ca Vit Nam l bao nhiu? 3) km hm tng dn s, ngi ta ra phng n: K t nm 2011, mi nm phn u gim bt %x ( x khng i) so vi t l % tng dn s nm trc (ngha l nu nm nay t l tng dn s l %a th nm sau l ( )%.a x Tnh x s dn nm 2015 l 92,744 triu ngi. Kt qu chnh xc ti 4 ch s thp phn sau du phy. Nu s lc qui trnh bm phm trn my tnh gii.

Chng 5 DY S, TNG CA DY S V DY TRUY HI

Dng ton 1 Tnh tng

Bi 1.1 (S GD v T Thp H Ch Minh, vng chung kt, 24.11.1996)

Mt hnh vung c chia thnh 16 (mi cnh 4 ). th nht c t mt ht thc, th nh c t 2 ht, th ba c t 4 ht,... v t lin tip nh vy n cui cng. Tnh tng ht thc c t vo 16 ca hnh vung.

Bi 1.2 (S GD v T Thp H Ch Minh, vng 1, cp THPT, 15.3.1996)

Mt cp s nhn c s hng u 1 1,678u = , cng bi 98

q = . Tnh tng 17S

ca 17 s hng u tin (kt qu ly 4 s l).


Mt ng trn ni tip trong mt hnh vung c cnh bng 2,3358909 , sau ni tip trong hnh trn mt hnh vung v qu trnh c tip din nh th mi. Nu gi nS l tng cc din tch ca n hnh trn u tin ni tip nh th. Tnh 20S .

Bi 1.4 (S Gio dc v o to Qung Ngi, lp 9, 2009-2010) Cho tam gic u th nht cnh a c din tch l S1, ni trung im cc cnh ca tam gic u th nht ta c tam gic u th hai c din tch l S2, ni trung im cc cnh ca tam gic u th hai ta c tam gic u th ba c din tch l S3. Lm tng t ta c tam gic u th n c din tch l Sn. 1) Lp cng thc tnh S = S1+S2+ +Sn theo a. 2) p dng: Tnh S vi n = 20; a = 301cm.

Bi 1.5 (S Gio dc v o to Tha Thin Hu, lp 9, 01.12. 2006)

40

Cho dy s un n1 1 1 11 1 1 12 4 8 2

=

.

Tnh u5 (chnh xc) v u10, u15, u20 (gn ng).

Bi 1.6 (S GD v T Tha Thin Hu, Trung hc C s, 01.02. 2007)

Tnh tng: 1 2 99 100 .S ...

2.3 3.4 100.101 101.102= + +

Ly nguyn kt qu hin trn mn hnh.

Bi 1.7 (S Gio dc v o to ng Nai, lp 9, 2004-2005) Tnh gi tr ca biu thc

24 20 16 4

26 24 22 2

(8,18012004) (8,18012004) (8,18012004) ... (8,18012004) 1.(8,18012004) (8,18012004) (8,18012004) ... (8,18012004) 1

A + + + + +=+ + + + +

Bi 1.8 (S Gio dc v o to ng Nai, lp 9, 2004-2005) Tnh tng

2 21!.3 2!.7 3!.13 ... !( 1) ... 12!(12 12 1).S k k k= + + + + + + + + + +


Cho 1 1 1 1... .

1.3.5 3.5.7 5.7.9 2003.2005.2007S = + + + +

1) Tnh gn ng .S 2) Tnh ng S (biu din di dng phn s).

Bi 1.10 (S Gio dc v o to Qung Nam, lp 8, 2009-2010) Cho dy ( )nu xc nh bi:

1 2 31 1 1 1 1 1; ;

1.3.5 1.3.5 3.5.7 1.3.5 3.5.7 5.7.91 1 1... ( 1, 2,3..)

1.3.5 3.5.7 (2 1)(2 1)(2 3)n

u u u

u nn n n

= = + = + +

= + + + = + +

1) Lp qui trnh bm phm tnh s hng tng qut .nu 2) Tnh ng gi tr 50 60, .u u 3) Tnh ng 1002.u

Bi 1.11 (B Gio dc v o to, Trung hc C s, 11.3.2011) Tnh gi tr ca biu thc sau

1 1 1 1... .1.2.3.4 2.3.4.5 3.4.5.6 2011.2012.2013.2014

C = + + + +

Bi 1.12 (S Gio dc v o to Tuyn Quang, lp 12 THPT 19.10.2011)

41

Gi Sn = 1 1 11 ...2 3 n

+ + + + vi n N.

Chng minh: 2 2 2 21 2 3

1 1 1 1... 2.2 3 nS S S nS

+ + + + <

Bi 1.13 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008) Tnh

1 1 1 1 1 1 1 1 1 11 1 1 ... 1 ...2 2 3 2 3 4 2 3 4 10

S = + + + + + + + + + + +

chnh xc n 4 ch s thp phn.

Bi 1.14 (Chn i tuyn thi khu vc, S GD v T Lm ng, 2004)

Cho 1 2 3 100k a a a a= + + ++ v 2 22 1 .

( )kka

k k+

=+

Tnh .k

Dng ton 2 Tnh theo dy truy hi


Cho dy s xc nh bi cng thc : 2n

n 1 2n

3 13xx1 x++

=+

vi 1x 0,09= , n 1,2,...=

1) Vit quy trnh bm phm lin tc tnh n 1x + theo nx . 2) Tnh 2 3 4 5 6x , x , x , x , x (vi 10 ch s trn mn hnh). 3) Tnh 100 200x , x (vi 10 ch s trn mn hnh).

Bi 2.2 (B Gio dc v o to, Trung hc C s, 2007-2008; Phng Gio dc v o to huyn ng Triu, Lp 9, 2011-2012)

Cho dy s 0 1a = ;2

1

1 1n nn

n

a aa

a++ +

= vi 0; 1; 2; 3;n =

1) Lp quy trnh bm phm tnh 1na + trn my tnh cm tay;

2) Tnh 1a , 2a , 3 4 5 10, , ,a a a a v 15.a

Bi 2.3 (S Gio dc v o to Ph Yn, Trung hc Ph thng, 10.2.2009)

Cho tp hp cc s v hn sau: 1 2 3 4, , , ...,4 9 16 25

P =

.

1) Vit cng thc s hng tng qut . 2) Tnh s hng th 35. 3) Vit quy trnh bm phm lin tc tnh tng 30 s hng u tin.


42

Cho dy s c s hng tng qut 211 .n

nU in

= +

( 1i = nu n l, 1i = nu n chn, n l s nguyn 1n ). Tnh tng 20 s hng u tin ca dy s.

Bi 2.5 (B Gio dc v o to, Trung hc Ph thng, 11.3.2011)

Tnh gn ng gii hn ca dy 3 3 3 35 5 5 ... 5nU = + + + + ( n du cn). Tm 0n vi mi 0n n th nu gn nh khng thay i (ch xt n chn ch s thp phn), cho bit gi tr 2010.u Nu qui trnh bm phm tnh .nu Tm 0n vi mi 0n n th nu c phn nguyn v chn ch s thp phn ngay sau du phy l khng i. Tnh gi tr 2011.u Vit qui trnh gii.


Cho dy s: 1 1 1

... ,1 2 2 3 1

vnu u u u u unn

= + + ++ + +

trong :

1 11; 2 ( 1).n nu u u n= = + >

1) Tm cng thc tnh vn theo n ( 1n > ). 2) Tnh gi tr 2010.v

Dng ton 3 Dy Fibonacci v dy Lucas

Bi 3.1 (B Gio dc v o to, lp 12 B tc THPT, 11.3.2011)

Dy s { }na c xc nh nh sau: 1 2 2 15, 3, 4 5n n na a a a a+ += = = + vi mi n nguyn dng. Tnh tng 12 s hng u ca dy s .


Cho dy s : 1U 2= ; 2U 3= ; n 1 n n-1U 3U 2U 3+ = + + vi n 2. 1) Lp quy trnh bm phm tnh n 1U + trn my tnh cm tay.

2) Tnh 3 4 5 10U , U , U , U v 19U .

Bi 3.3 (Chn i tuyn. S Gio dc v o to thp H Ch Minh, 2003)

Cho u1 = 17, u2 = 29 v un+2 = 3un+1 + 2un (n 1). Tnh u15.


1) Cho 1 11,1234; 1,0123 ( ; 1)n nu u u n N n+= = . Tnh 50.u

2) Cho 2

1 1 2

3 135; ( ; 1)5

nn

n

uu u n N nu+

+= =

+. Tnh 15.u

3) Cho 0 1 1 23; 4; 3 5 ( 2).n n nu u u u u n = = = + Tnh 12.u

43


Cho dy s un c xc nh nh sau

1 2

2 1

3, 23 2 , 3n n n

u uu u u n

= = =

.

1) Vit 7 s hng u. 2) Vit quy trnh bm phm lin tc tnh tch 7 s hng u tin.


Cho dy s: 1 2 2 12, 3,...., 3 2 ; 1,2,3,...n n nu u u u u n+ += = = + = Tnh gi tr ca 20 21,u u v 22.u

Bi 3.7 (Thi th vng tnh, Trng THCS ng Nai-Ct Tin, 2004)

1) Cho dy 1 2 1 13; 11; 8 5 ( 2).n n nu u u u u n+ = = = 1a) Lp quy trnh bm phm tm s hng th 11u ca dy. 1b) Tm s hng 1u n 12u ca dy.

2) Cho dy 1 2 311; 15;u u u= = =2

11

1

53 2

n nn

n n

u uuu u

+

= + +

vi 3.n

2a) Lp quy trnh bm phm tm s hng th nu ca dy. 2b) Tm s hng 8u ca dy.

Bi 3.8 (Phng Gio dc v o to Bo Lm, Lm ng, 2005)

1) Cho 1 11,1234; 1,0123. ( ; 1).n nu u u n N n+= = Tnh 50.u

2) Cho 2

1 1 2

3 135; ( ; 1).5

nn

n

uu u n N nu+

+= =

+ Tnh 15.u

3) Cho 0 1 1 23; 4; 3 5 ( 2).n n nu u u u u n = = = + Tnh 12.u


Cho dy 1 2 1 15; 9; 5 4 ( 2).n n nu u u u u n+ = = = +

1) Lp quy trnh bm phm tm s hng th nu ca dy.

2) Tm s hng 14u ca dy.


Cho dy s (1 2) (1 2) ,

2 2nn n

U + + = 1, 2,..., .n k=

1) Chng minh 2 12 .n n nU U U+ += +

2) Vit qui trnh bm phm tm s hng th n


44

Cho dy s

( ) ( )n nn

1 2 1 2U

2 2

+ = vi n 1,2,...=

1) Chng minh rng: n 1 n n 1U 2U U+ = + vi n 1. 2) Lp quy trnh bm phm lin tc tnh n 1U + theo nU v n 1U vi

1 2U 1, U 2.= = 3) Tnh cc gi tr t 11U n 20U .

Bi 3.12 (B Gio dc v o to, Trung hc c s, 2005)

Cho dy s ( ) ( )3 2 3 2

.2 2

n n

nu+

=

Lp cng thc truy hi tnh 2nu + theo 1,nu + .nu

Bi 3.13 (S GD v T Tha Thin Hu, Trung hc C s, 01.02. 2007)

1) Cho dy s vi s hng tng qut c cho bi cng thc :

( ) ( )n n

n6 2 7 6 2 7u

4 7+

= vi n = 1, 2, 3,

1a) Tnh u1, u2, u3, u4, u5, u6, u7, u8. 1b) Lp cng thc truy hi tnh un+1 theo un v un-1.

2) Hai dy s vi cc s hng tng qut c cho bi cng thc :

1 1

n 1 n n

n 1 n n

u 1; v 2u 22v 15uv 17v 12u

+

+

= = = =

vi n = 1, 2, 3,

2a) Tnh 5 10 15 18 19 5 10 15 18 19u , u , u , u , u ; v , v , v , v , v . 2b) Vit quy trnh bm phm lin tc tnh n 1u + v n 1v + theo nu v nv .


1) Tnh chnh xc gi tr biu thc:

14 14(5 2 6) (5 2 6) .A = + +

2) Cho ( ) 11 2 3 4 1 .nnS n+

= + + +

Tnh tng 2005 2006 2010.S S S S= + ++

Bi 3.15 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008)

Cho dy s sp xp theo th t 1 2 3 1, , ,..., , ,...n nU U U U U + Bit 5 6 1 1588; 1084; 3 2 .n n nU U U U U+ = = = Tnh 1 2; .U U


45

Cho cc s 1 2 1, ,..., , ,...n nu u u u + tha mn 1 2 , 1n n nu u u n+ ++ = v

2 503; 30.u u= =

Tnh gi tr ca 1 2 3 48... .S u u u u= + + + +

Bi 3.17 (Chn i tuyn thi khu vc, S GD v T Lm ng, 2004)

Cho 2 21 2 1 17; .n n nu u u u u+ = = = + Tnh 7.u

Chng 6 LNG GIC TRN MY TNH IN T

Dng ton 1 Tnh gi tr ca biu thc lng gic


0 0 0 0

0 0 0 0 0

3sin15 25 4cos12 12 sin 42 20 cos36 15 .B2cos15 25 3cos 65 13 sin15 12 cos31 33 sin18 20

+ +=

+ +

Bi 1.2 (S Gio dc v o to Tha Thin Hu, lp 9, 01.12. 2006) Tnh gi tr ca cc biu thc:

5 0 4 03

3 7 0 3 0

235,68 cot 23 35 cos 69 43A62,06 tan 69 55 sin 77 27

=

.


Tnh gi tr biu thc 3 0 3 0 2 0 0

0 0 0

sin 90 cot 30 cos 45 tan 20 .2 7 sin108 cos32 tan 64

P +=+


Tnh gn ng gi tri biu thc (vi 4 ch s thp phn) gi tr biu thc:

3 0 0

2 0 3 00 5 0

sin 42 tan 80 cot 17 sin 10 .sin1 cos 22

P += +

Bi 1.5 (Phng Gio dc v o to ng Triu, 2011-2012)

Tnh gi tr ca biu thc ( kt qu ly vi 4 ch s thp phn): 3 0 3 0 2 0 0

0 0 0

sin 90 cot 30 cos 45 tan 20 .2 7 sin108 cos32 tan 64

B +=+

Bi 1.6 (S Gio dc v o to Ph Yn, lp 12 B tc THPT, 2009-2010) Cho:

0 0 2 4 8cos 75 cos15 ; cos cos cos ;9 9 9

A B = =

46

0 0 0 00 0

1 1 tan 9 tan 27 tan 63 tan81 .sin18 sin 54

C = + +

Tnh gi tr biu thc 2 .D A B C= + +

Bi 1.7 (S Gio dc v o to Ph Yn, lp 9 Trung hc C s, 10.2.2009) Cho

3 0 3 0 2 0

4 0 2 0 3 0

2 3 3 sin 90 cot 30 cos 45 ;tan 60 sin 30 cos 60

A + +=+4 0 2 0

03 0

1 sin 40 cos 20cot 55 .3 tan 108

B = +

Tnh .C A B= +

Bi 1.8 (S GD v T Tha Thin Hu, Lp 9, 01.02. 2007)

Tnh gi tr ca biu thc M vi 025 30 , = 057 30 = (kt qu ly vi 4 ch s thp phn):

( )( ) ( )( )2 2 2 2 3 3M 1 tan sin 1 cot cos 1 sin 1 cos = + + + ( )( )2 21 sin 1 cos . + +


Tnh gi tr ca biu thc: 3 2 2 3 2 3

3 2 2 3 2 3

(1 sin 17 34 ) (1 tan 25 30 ) (1 cos 50 13 )C(1 cos 35 25 ) (1 cot 25 30 ) (1 sin 50 13 )

+ + =

+ +


1) Tnh P=o o o

o o

sin25 12'28''+2cos45 -7tg27cos36 +sin37 13'26''

2) Cho cosx = 0,81735 (gc x nhn). Tnh : sin3x v cos7x

3) Cho sina = 0,4578 (gc a nhn). Tnh: Q=2 3cos a-sin atga

4) Cho cotgx = 1,96567 (x l gc nhn). Tnh 2 3 2 3

3 3

tg x(1+cos x)+cotg x(1+sin x)S= .(sin x+cos x)(1+sinx+cosx)


Cho cotg=0,05849 (00 < < 900). Tnh: 2 3 6 8

3 3

(sin cos ) cotsin

tg gtg

+ +

=+

D


47

Cho cotg = 0,06993 (00 < < 900). Tnh:

4 5 7 3

3 3 5

(1 cos ) cot (1 )(sin )(1 3sin )

tg g tgtg

+ + =

+ +D

Bi 1.13 (Thi thi th vng Tnh, Trng THCS ng Nai, 2004) Cho cotg = 0,05849 (00 < < 900). Tnh:

4 3 5 7 3 3

3 3 5

(sin cos ) cot (sin )(sin )(1 3sin )

tg g tgtg

+ +

=+ +

D

Bi 1.14 (S Gio dc v o to thp H Ch Minh, THPT, 11.1.2009)

Cho tan 2(2 3 )x x = < < v 2sin 3cos 1(0 )y y y + = < < . Tnh gn ng vi nm ch s thp phn:

a) 2 3

2 4

sin cos2 tan 3cot

x xAx x

=+

b) 2 2 2 2

2 2 4 2

tan ( ) cot ( )sin ( ) cos ( )

x y x yBx y x y

+ =

+ + +

Bi 1.15 (S Gio dc v o to ng Thp, B tc THPT, 2008-2009)

Cho 3 35 3 29 12 5 ; 20 4901 20 4901A B= = + + Tm mt nghim thuc (0;2 ) ca phng trnh tan 0A x B+ =

Bi 1.16 (S GD v T Ty Ninh, THCS v B tc THPT, 2008-2009) Tnh gi tr ln nht gi tr nh nht ca hm s:

( ) cos 2 2 cosf x x x=

Chng 7 HNH HC TRN MY TNH IN T

Dng ton 1 Phng php ta

Bi 1. 1 (B Gio dc v o to, Trung hc C s, 2007-2008) Cho hai ng thng:

(d1): 3 1 3y x2 2+

= + ; (d2): 5 1 5y x2 2

= .

1) Tnh gc to bi cc ng thng trn vi chiu dng trc Ox (chnh xc n giy). 2) Tm giao im ca hai ng thng trn (tnh ta giao im chnh xc n 2 ch s sau du phy). 3) Tnh gc nhn to bi hai ng thng trn (chnh xc n giy).


48

Cho 3 ng thng: (d1): ( ) ( )2 33x 2y 6 ; d : 2x 3y 15 ; d : x 3y 6. = + = + =

Hai ng thng (d1) v (d2) ct nhau ti A; hai ng thng (d1) v (d3) ct nhau ti B; hai ng thng (d2) v (d3) ct nhau ti C.

1) Tm ta ca cc im A, B, C (vit di dng s thp phn). Tam gic ABC l tam gic g? Gii thch. 2) Tnh din tch tam gic ABC (vit di dng phn s) theo on thng n v trn mi trc ta l 1 cm. Tnh s o ca mi gc ca tam gic ABC (chnh xc n pht).

Bi 1.3 (Thi chn i tuyn Trng THCS ng Nai-Ct Tin, 2004) Tm gc hp bi trc Ox vi ng thng y=ax+b i qua hai im A(0;-4) v B(2;0).

Bi 1.4 (S GD v T Tha Thin Hu, Trung hc C s, 01.02. 2007) 1) (Lp 9) Cho ba hm s sau y :

8y x 27

= (1), 3y x 38

= (2) v 18y x 629

= + (3)

1a) V th ca ba hm s trn mt phng ta ca Oxy. 1b) Tm ta giao im A(xA ; yA) ca hai th hm s (1) v (2), giao im B(xB ; yB) ca hai th hm s (2) v (3) ; giao im C(xC ; yC) ca hai th hm s (1) v (3) (kt qu di dng phn s hoc hn s). 1c) Tnh cc gc ca tam gic ABC (ly kt qu trn my). 1d) Vit phng trnh ng thng l phn gic ca gc BAC (h s gc ly kt qu vi hai ch s phn thp phn).

2) (Lp 8) Cho ba im ( ) ( ) ( )A 2;5 , B 4;2 , C 7; 1 . T nh A v ng cao AH, ng phn gic AD v ng trung tuyn AM (cc im H, D, M thuc cnh BC). Cho bit tnh cht ca ng phn gic

trong tam gic :DB AB .DC AC

=

2a) Tnh din tch tam gic ABC. Nu s lc cch gii. 2b) Tnh di cc on thng AH, AD, AM v din tch tam gic ADM (kt qu ly vi 2 ch s phn thp phn). n v o trn cc trc ta l cm.

Bi 1.5 (S Gio dc v o to Thp H Ch Minh, Lp 9, 11.1.2009)

Trong h trc Oxy, tm ta cc giao im A, B ca ng thng 0,5 2 2 3( ) :1 3 0,6 2

d y x = + +

v parabol (P) : 20,2y x= (chnh xc n 4

ch s thp phn).

49


Trong mt phng ta Oxy, hai ng thng 1( )y x d= v 25 3( )2

y x d= + ct nhau

ti C. ng thng 31( )y d= ct 2( )d ti B v ct 1( )d ti A.

1) Tnh s o gc B ca tam gic ABC. 2) Tnh chu vi v din tch tam gic ABC. 3) Tnh bn knh ng trn ngoi tip tam gic ABC.


Cho ng thng (d): y = 5x+3 v (d): y = -5x + 4. 1) V ng thng (d) v (d) trn cng mt h trc to . Tm giao im A gia hai ng thng (d) v (d), giao im B, C ln lt ca (d) v (d) vi trc Ox. 2) Tnh gc .BAC


Trong h ta vung gc Oxy cho ng thng c phng trnh: 2 4y x= + . A l im trn ng thng c 11Ax = . H AH vung gc Ox ( H thuc Ox). Tnh din

tch tam gic OAH.

Dng ton 2 Hnh hc phng

Bi 2. 1 (B Gio dc v o to, Trung hc C s, 2010-2011)

T gic ABCD c mt ng cho l 21 ,AC cm= cc gc 0 0 025 , 37 , 35DAC DCA BAC= = = v 032 .BCA = Tnh chu vi P v din

tch S ca t gic . Trnh by tm tt cch gii.


1) Hnh ch nht ABCD c di cc cnh AB m= , BC n= . T A k AH vung gc vi ng cho BD . 1a) Tnh din tch tam gic ABH theo m v n .

1b) Cho bit m 3,15 cm= v n 2,43 cm= . Tnh din tch tam gic ABH (chnh xc n 4 ch s thp phn). 2) Hnh thang vung ABCD (AB // CD) c gc nhn BCD ,= di cc cnh BC m, CD n.= = 2a) Tnh din tch, chu vi v cc ng cho ca hnh thang ABCD theo m,n v . 2b) Tnh (chnh xc n 4 ch s thp phn) din tch, chu vi v cc ng cho ca hnh thang ABCD vi

50

m 4,25 cm,= n 7,56 cm,= o54 30 . = 3) Cho tam gic ABC vung ti A . T A k AH vung gc vi BC ( H thuc BC ). Tnh di cnh AB (chnh xc n 2 ch s thp phn), bit rng din tch tam gic AHC l 2S 4, 25 cm ,= di cnh AC l m 5,75 cm.=


1) Tam gic ABC c cc cnh AB 31,48= (cm), BC 25,43= (cm), AC 16,25= (cm). Vit quy trnh bm phm lin tc trn my tnh cm tay v tnh chnh xc n 02 ch s sau du phy gi tr din tch tam gic, bn knh ng trn ngoi tip v din tch phn hnh trn nm pha ngoi tam gic ABC. (Cho bit cc cng thc tnh din tch tam gic l

S p(p a)(p b)(p c)= ; abcS4R

= ).

2) T im M nm ngoi ng trn (O; R) k hai tip tuyn MA, MB vi ng trn. Cho bit MO 2R= v R 4,23= (cm). Tnh chnh xc n 2 ch s sau du phy: 2a) Phn din tch ca phn t gic MAOB nm pha ngoi ng trn (O ; R). 2b) Din tch phn chung ca hnh trn ng knh MO v hnh trn (O ; R).

3) Cho ng trn ng knh AB 2R.= M v N l hai im nm trn ng trn sao cho cc cung AM ; MN v NB bng nhau. Gi H l hnh chiu ca N trn AB v P l giao im ca AM vi HN. Cho R 6,25= (cm). 3a) Tnh gc MBP (chnh xc n giy). 3b) Cho hnh v quay mt vng xung quanh trc BM. Tnh din tch xung quanh v th tch hnh do tam gic BMP to thnh (chnh xc n 2 ch s sau du phy).

Bi 2.4 (S Gio dc v o to Tha Thin-Hu, Lp 9, 01.02.2007) 1)Cho tam gic ABC c cnh BC = 9,95 cm, gc 0ABC 114 43 12 , = gc

0BCA 20 46 48 . = T A v ng cao AH, ng phn gic trong AD, ng phn gic ngoi AE v ng trung tuyn AM. 1a) Tnh di ca cc cnh cn li ca tam gic ABC v cc on thng AH, AD, AE, AM. 1b) Tnh din tch tam gic AED (kt qu ly vi 2 ch s phn thp phn). 2) Cho t gic ABCD ni tip trong ng trn (O) bn knh R = 4,20 cm, AB =7,69cm, BC = 6,94cm, CD = 3,85cm. Tm di cnh cn li v tnh din tch ca t gic ABCD (kt qu ly vi 2 ch s phn thp phn).

51

Bi 2.5 (Thi th vng tnh, Trng THCS ng Nai-Ct Tin, 2004) 1) Cho tam gic ABC vung ti A c ng cao l AH . Cho bit AB=0,5, BC=1,3. Tnh AC, AH, BH, CH gn ng vi 4 ch s thp phn. 2) Cho tam gic ABC c AB = 1,05 ; BC = 2,08 ; AC = 2,33 . 2a) Tnh di ng cao AH. 2b) Tnh di trung tuyn AM. 2c) Tnh s o gc C. 2d) Tnh din tch tam gic ABC.

Bi 2.6 (Chn i tuyn thi khu vc, S Gio dc v o to Lm ng, 2004) 1) Cho tam gic ABC vi 3 cnh BC = 5,1123; AB = 3,2573; AC = 4,7428. Tnh ng phn gic trong AD.

2) Tia phn gic chia cnh huyn thnh hai on 135

7v

2227

. Tnh hai cnh

gc vung. 3) Cho hnh thang vung ABCD, ng cao AB. Cho gc BDC = 900;Tm AB, CD, AC vi AD=3,9672; BC=5,2896. 4) Cho tam gic ABC vi ba cnh BC = 14; AB = 13; AC = 15. Gi D, E, F l trung im ca BC, AC, AB v { } { }; ;Q BE FD R DF FC= = { } .P AD EF= Tnh:

2 2 2 2 2 2

2 2 2 .AQ AR BP BR CP CQm

AB BC AC+ + + + +

=+ +


Cho ABC vung ti A, cnh AB = c, AC = b.

1) Tnh khong cch d t chn ng phn gic trong ca gc vung n mi cnh gc vung. Vi b = 5,78914 cm; c = 8,911456 cm. Tnh khong cch .

2) Cho ABC c ba cnh a = 17,894 cm; b = 15,154 cm; c = 14,981 cm. K ba ng phn gic trong ca ABC ct ba cnh ln lt ti A1, B1, C1.

Tnh phn din tch c gii hn bi ABC v A1B1C1?

3) Cho t gic li ABCD ni tip trong mt ng trn bn knh R, c cc snh

a = 3,657 cm; b = 4,155 cm; c = 5,651 cm; d = 2,765 cm. Tnh phn din tch c gii hn bi ng trn v t gic ABCD.

Bi 2.8 (S Gio dc v o to Ty Ninh, lp 9, 2008-2009) 1) Cho na ng trn tm O ng knh AB = 2008. T mt im C trn bn knh OB k mt tia vung gc vi AB ct na ng trn D. Tip tuyn vi na ng trn

52

(O) ti D ct ng thng AB ti E. Tm v tr im C trn OB sao cho DE = CD + CB. 2) Cho hnh bnh hnh ABCD ( AD > AB) c chu vi bng 26; 120oABC = . Bn knh ng trn ni tip ca tam gic BCD bng 3 . a) Tnh di cc cnh AB, BC ca hnh bnh hnh ABCD. b) Vit cng thc v tnh din tch hnh bnh hnh ABCD. 3) Cho na ng trn ng knh AB = 2008. Trn tia i ca tia AB, ly im P sao cho AP = 1004. Qua P v ct ct tuyn PCD ( C nm gia P v D ) sao cho

1004 2.CD = a) Tnh di on PC v PD b) Tnh di cc on CA, AD, BD

Bi 2.9 (Phng Gio dc o to huyn Bo Lm, Lm ng, 2004) 1) Cho tam gic ABC vung A vi AB=4,6892 cm ; BC=5,8516 cm. Tnh gc ABC (bng n v o ), tnh di ng cao AH v phn gic trong CI. 2) Cho tam gic ABC vung cn A. Trn ng cao AH, ly cc im D, E

sao cho AE=HD= 14

AH. Cc ng thng BE v BD ln lt ct cnh AC F

v G. Bit BC=7,8931 cm. 2a) Tnh din tch tam gic ABE 2b) Tnh din tch t gic EFGD.

3) Cho ngi sao 5 cnh nh hnh bn. Cc khong cch gia hai nh khng lin tip ca ngi sao AC=BD=CE= = 7,516 cm. Tm bn knh R ca ng trn i qua 5 nh ca ngi sao.

Bi 2.10 (Phng Gio dc v o to huyn ng Triu, lp 9, 2011-2012) 1) Vit cng thc tnh din tch S ca hnh thang bit di hai ng cho l

1l v 2l v on thng ni trung im ca hai y l d .

p dng bng s: (Ly kt qu n 5 ch s thp phn) 1 302,1930 ;l m= 2 503,2005 ;l m= 304,1975 .d m=

2) Mt hnh vung v mt hnh tam gic u cng ni tip mt hnh trn c bn knh bng 1cm, sao cho mt cnh ca tam gic song song vi mt cnh ca hnh vung. Gi S l din tch phn chung ca tam gic v hnh vung. Hy lp cng thc v tnh gn ng .S


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Cho tam gic ABC vung ti A. Bit 5,2538AB m= , gc 40 25'oC = . T A v ng

phn gic AI v trung tuyn AM ( I v M thuc BC) a) Tnh di ca cc on thng AI, AM. b) Tnh t s din tch tam gic AIM v din tch tam gic ABC.

Bi 2.12 (S Gio dc v o to ng Thp, lp 9, 2008-2009) 1) Cho tam gic ABC c phn gic trong AD ( D thuc on BC). M l trung im AB. AD ct CM ti I. Tnh din tch tam gic ACI khi AB = 5; AC = 5; BC = 7. 2) Hnh vung ABCD c cnh bng 1 ni tip trong ng trn tm O. M l im trn cung BC sao cho gc MAB 30o= . a) Tnh gc AOM. b) Tnh din tch t gic AOMB

Bi 2.13 (S Gio dc v o to Thp H Ch Minh, Lp 9, 11.1.2009) Cho hnh ch nht ABCD c AD = 4,9. Trn cnh AD ly im M sao cho AM= 1,5. Gi I l giao im ca BM v AC. Gc IDC c s o bng 070 . 1) Tnh ID (chnh xc n 2 ch s thp phn). 2) Tnh AB ( chnh xc n 2 ch s thp phn). 3) Tnh gc BIC ( , pht, giy) 4) Tnh bn knh ng trn ni tip tam gic BIC (chnh xc n 2 ch s thp phn).

Bi 2.14 (S Gio dc v o to Sc Trng, 28.11.2010) Tnh din tch hnh trn ni tip tam gic ABC. Bit AB=4, BC=5, AC=6.

Bi 2.15 (S Gio dc v o to Qung Ngi, lp 9, 2009-2010) 1) Cho ng gic li ABCDE. Tnh s o cc gc trong ca ng gic, bit

13A = 17B = 19C = 23D = 29E. 2) Cho hnh ch nht ABCD c BC = a ; AB = b .K CK vung gc vi BD 2a) Tnh din tch tam gic AKD theo a v b 2b) Tnh din tch tam gic AKD vi a = 5,67cm ; b = 3,45cm ( kt qu ly 4 ch s thp phn).

Bi 2.16 (Chn i tuyn. S Gio dc v o to thp H Ch Minh, 2003) Cho ng gic u ABCDE c di cnh bng 1.Gi I l giao im ca 2 ng cho AD v BE. Tnh (chnh xc n 4 ch s thp phn): 1) di ng cho AD. 2) Din tch ca ng gic ABCDE. 3) di on IB. 4) di on IC.

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Bi 2.17 (S Gio dc v o to Ph Yn, lp 9, 2009-2010) 1) Tnh chu vi hnh trn ni tip tam gic vung ABC, bit cnh AB=1,7321 cm v cnh huyn BC = 2,4495 cm. 2) Cho tam gic ABC vung ti A . Gi D, E l hai im trn cnh huyn BC sao cho BD =DE = EC. Bit di on AD = sin x , AE = cos x, vi

02

x <

55


Cho tam gic ABC, trn cnh AB, AC, BC ln lt ly cc im M, L, K sao cho t gic KLMB l hnh bnh hnh. Bit SAML= 42,7283 cm2, SKLC = 51,4231 cm2 . Hy tnh din tch tam gic ABC (gn ng vi 4 ch s thp phn).

Bi 2.22 (S Gio dc v o to Ph Yn, lp 9 Trung hc C s, 10.2.2009)

Trong hnh di y, dy PQ v MN song song vi bn knh OR = 1. Cc dy MP, PQ v NR u c di bng a, dy MN c di bng b. Tnh a2 b2.

a

a

b 1

a

O

N

RP

M

Q


Mt mnh ba c dng mt tam gic cn ABC vi 25AB AC cm= = v 14 .BC cm= Lm th no ct t mnh ba ra c mt hnh ch nht

MNPQ c din tch bng 117

din tch ca tam gic ,ABC trong ,M N

thuc cnh ,BC cn ,P Q tng ng thuc cc cnh , .AC AB Trnh by tm tt cch gii.

Dng ton 3 Hnh hc khng gian


1) Hnh chp t gic u O.ABCD c di cnh y BC a= , di cnh bn OA = l . 1a) Tnh din tch xung quanh, din tch ton phn v th tch ca hnh chp O.ABCD theo a v l . 1b) Tnh (chnh xc n 2 ch s thp phn) din tch xung quanh v th tch ca hnh chp O.ABCD bit a 5,75 cm, 6,15 cm= =l . 2) Ngi ta ct hnh chp O.ABCD cho trong Cu 4.1 bng mt phng song song vi y ABCD sao cho din tch xung quanh ca hnh chp O.MNPQ c ct ra bng din tch xung quanh ca hnh chp ct u MNPQ.ABCD c ct ra. Tnh th tch hnh chp ct c ct ra (chnh xc n 2 ch s thp phn).

Chng 8 CC BI TON KHC


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Mt mnh sn hnh ch nht c chiu rng v chiu di tng ng l 7,6m v 11,2m c lt kn bi cc vin gch hnh vung c cnh 20 .cm Cho rng din tch phn tip gip nhau gia cc vin gch l khng ng k. Ngi ta nh s cc vin gch lt t 1 cho n ht. Gi s trn vin gch th nht ngi ta t ln 1 ht u, trn vin th hai ngi ta t ln 7 ht u, trn vin th ba ngi ta t ln 49 ht u, trn vin th t ngi ta t ln 343 ht u,...v c t theo cch cho n vin gch cui cng trn sn ny. Gi S l tng s ht u t ln cc vin gch ca sn . Tm ba ch s tn cng bn phi ca s 6 5.S + Trnh by tm tt cch gii.

Bi 8.2 (B Gio dc v o to, Trung hc C s, 2010-2011) Mt ci sn hnh ch nht c lt kn bng cc vin gch hnh vung c cnh 5 ,cm xen k mt vin mu en vi mt vin mu trng v khng c hai vin cng mu c ghp cnh nhau. Cho bit din tch phn tip gip nhau gia cc vin gch l khng ng k. Nu hng th nht theo chiu rng ca sn ny c 2011 vin mu en v tt c c 22210983 vin gch c lt th sn ny c chiu di v chiu rng l bao nhiu mt? Trnh by tm tt cch gii.

Bi 8.3 (B Gio dc v o to, Trung hc C s, 2010-2011) (5 im)

Mt qu bng r theo tiu chun quc t c dng hnh cu vi bn knh

12,09R cm= (hnh bn). Ngi ta mun to ra cc ti dng hnh hp ng c np bng ba (cng v nhn) ng c 12 qu bng r ni trn.

Nu cha tnh din tch cn c cho cc mp dn th din tch ba t nht to ra c mt ti nh th l bao nhiu

2 ?cm Trnh by tm tt cch gii.

Bi 8.4 (B Gio dc v o to, Trung hc Ph thng, 11.3.2011) (5 im)

Tm h s ln nht trong khai trin nh thc ( )115 7 .x + Bi 8.5 (Phng GD v T huyn B Trch, Qung Bnh, lp 9, 4.7.2008) Cho a thc Q(x) = ( 3x2 + 2x 7 )64. Tnh tng cc h s ca a thc chnh xc n n v.


Khai trin biu thc (1 + 2x + 3x2)15 ta c a thc :

57

0a + a1x + a2x + + a30x30.

Tnh gi tr chnh xc ca biu thc :

0 1 2 3E a 2a 4a 8a ...= + + 536870912a29 + 10723741824a30

Bi 8.7 (Gio dc v o to Tuyn Quang, THPT, 19.10.2011)

Gii phng trnh 2 4 233 1 1 0.

3x x x x + + + =


Cho hm s ( )y f x= , bit (1) 0,73579

( )( 1)1 . ( )

ff nf nn f n

= + = +

, vi n l s nguyn

dng. Tnh 1 .

(2005)f


Cho: (1) 1; ( ) ( ) ( )f f m n f m f n mm= + = + + (m, n nguyn dng) Tnh (10)f v (2008)f .

Bi 8.10 (S Gio dc v o to Ty Ninh, lp 9, 2008-2009) Cho a thc ( )f x bc bn v tha mn hai iu kin:

( 1) 0( ) ( 1) ( 1)(2 1)

ff x f x x x x =

= + +

1) Tm a thc ( )f x ni trn. 2) Tnh tng 1.2.3 2.3.5 3.4.7 .... 2008.2009.4017.S = + + + +

TI LIU

1. T Duy Phng, Gii ton trn my tnh in t. Nh xut bn Gio dc, 2003, 2005.

2. T Duy Phng, Nguyn Th Thch, Cc thi hc sinh gii Gii ton trn my tnh Casio 1996-2004. Nh xut bn Gio dc, 2004, 2005.

3. T Duy Phng, Phm Th Hng L, Mt s dng ton thi hc sinh gii Gii ton trn my tnh-Phng trnh sai phn. Nh xut bn Gio dc, 2005, 2006, 2008.

4. T Duy Phng, Hng dn s dng, thc hnh gii ton v mt s thi THCS, THPT trn my tnh in t Sharp. Nh xut bn Gio dc, 2006.

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5. T Duy Phng, Chuyn bi dng hc sinh gii Gii ton trn my tnh in t-H m v ng dng. Nh xut bn Gio dc, 2007.

6. T Duy Phng, Chuyn bi dng hc sinh gii Gii ton trn my tnh in t-Ton thng k. Nh xut bn Gio dc, 2007.

7. T Duy Phng, Cc dng ton thi hc sinh gii Gii ton trn my tnh in t, Tp I: Trung hc C s, Tp II: Trung hc Ph thng. (Bn tho), 2012.

8. Trn Minh Chu, T Duy Phng, Nguyn Khc Ton, Tuyn tp cc thi Gii ton trn my tnh (Trung hc C s 2003-2010), Nh xut bn Gio dc (S ra).

9. Phm Th Nhn, T Duy Phng, Trn D Sinh, Tuyn tp cc thi Gii ton trn my tnh (Trung hc Ph thng, 2003-2012) (Bn tho), 2012.

10. Cc sch, tp ch, cc trang WEB v ton v my tnh.

CC DNG TON THI HC SINH GII GII TON TRN MY TNH IN T KHOA HC LI NI U Bi 1.1 Hy tnh trn my: 1) 12 42 = 21 24 = 504. 2) 12 63 = 21 36 = 756. 3) 12 84 = 21 48 = 1008. 4) 13 62 = 31 26 = 806. 5) 13 93 = 31 39 = 1209. 6) 14 82 = 41 28 = 1148. 7) 23 64 = 32 46 = 1472. 8) 23 96 =32 69 = 2208. 9) 24 63 = 42 36 = 1512. 10) 24 84= 42 48=2016. 11) 26 93 = 62 39=2418. 12) 34 86 = 43 68 =2924. Bi 1.7 1) Tch 9801 hoc 3025 chia thnh s c hai ch s, cng li v em bnh phng, ta li c chnh s : 2) Tng t: 1) Bi ton: Tm tt c cc b 6 s c hai ch s tha mn iu kin trn. Dng ton 1 Tnh ton vi cc phn s Dng ton 2 Tnh ton vi s thp phn Dng ton 6 Ton thi gian v Ton chuyn ng Dng ton 1 Ly tha Dng ton 2.1 Tnh gi tr ca a thc v phn thc i s

Cho hnh ch nht ABCD c AD = 4,9. Trn cnh AD ly im M sao cho AM= 1,5. Gi I l giao im ca BM v AC. Gc IDC c s o bng

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