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Cac Dang Toan Thi HSG THCS

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  • 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

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  • 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):

    N N NN3 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;

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  • 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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  • 4

    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+ + + + =

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  • 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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  • 6

    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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  • 7

    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