"o"2nk +irir2nk ,
nn
k:0,t,2, ...,fl-I.
2. CHUNG MrNr{ MENH nn CUA GAUSS
2.L DA drmg dugc da gi6c d6u 17 canh nOi
ti6p trong mQt hinh trdn b6n kinh 1, chi cin
dlmg dusc dopn thEng d9 dai
"or] 1*"rn
't7
hinh vE). C6ng viQc titip theo sE ld hoin todn kh6ng kh6 khdn.
2.2. Chnng ta ki hi6u e ld mQt gi6 tri phrlc cua
cdn bfc 17 cua i10n vi 11t; 2n 2n t=-!l=cosl7+lslnlT DAt 4=t+t-l (1-e2a5-z 22: €4 + t4 6z: e8 + e-8 ))1 =21 *22 yz:4+62 y3=ztz2 Yc=A€z \:h-tlz xz=lt*lq
NhQn x6t ring t6t ch chc sti ndy d6u h c6c s5 thgc. Ngodi ra, ek + e-k
( zm 2krl\ ( -ztm -zkn\-l cos_+ l+l cos_+lsm_ | -l cos_+ l+l cos_+lsm_ | \ lz 17) \. rz t7 ) ^ 2l$c : ZCOS-. t7
Boi vi zr :2cos? 't7 n6n ta chi cAn dlmg tlopn
thing c6 d0 ddi ld zr.
2.3. ek - at7-k, v6i mgi k.
Thft vfy znIl + t<) zn(n + t) znIl + t<) zn(n + t) tt t+K - cos r lsln I7 t7 ="o"( ,n*24) +i"in( zn*?4) \ tz,/ \ t7) 2nk 2nk :cos-+lsln--6'r. L7 t7 16 2.4. Zek =0. k=0
Thflt vfly, theo c6ng thric tinh tdng c6c s5^i ^i hpng cria cdp s6 c6ng. ta c6 16 ^17 r \-^k_o -r_n Lb ft=O A-l 16 2.5. \ek = -1. k=1 t6 t6
Thptvpy Z"o =Z"o -60 =0-1:-1.
k=l k=0
8
2.6. lGk + e-k): -1.
k--t
C6ng thric ndy tlugc chimg minh nhd vio
viQc sir dung c6c k6t qu6 2.6vd2.3.
2.7. yryr. : lzlq: -1. Th6t vdy
yty2 =(et + e-t + ea + ea)(e2 + e1 +e8 +e-8)
: €3 + e-l + e9 + e-7 + el + e-3 + e7 + e-9
+e6 +€2 +et2 +e-4 +e-2 +e-6 +* +€-12
8
:\(ek + e-k): -1.
k--1
Tuong tU d6i voitich yya.
2.8. x1 * x2 : -1, x1x2 = -{.
2.9. Nh6 ring n6u cho tru6c c6c dopn thing c6 d0 ddi t, lpl vt Iq I tni
"ni bing dpng cp
thu6c vd compa c6 th6 dlmg dugc c6c dopnthing c6 d0 ddi bing gi6 tri tuy€t diSi cria c6c thing c6 d0 ddi bing gi6 tri tuy€t diSi cria c6c nghiQm cria phuong trinh bpc hai
x2+Pxiq=0
,A
(n6u c5c nghiQm ndy ld c6c s6 thgc).
Boi vi \ * x2 : -1, xtx2 = -4 n6n x1, x2 1d
hai nghiQm (th1rc) ctra phuong trinh
x2 - x - 4: O, c6 nghia ld c6 th6 dlmg dugc
c6c dopn thing c6 <I0 ddi lxll vd lx2l.
Vi y1 + !2 = xt vit yryz = -1 n6n c6 th6 dpg
du-o. c c5c ttopn thlng c6 dO dai
ly,l "a lyrl. ruitr .y, * lq = xz vit yty+ = -1 n6n c6 th6 dlmg tr .y, * lq = xz vit yty+ = -1 n6n c6 th6 dlmg
dusc c6c dopn thing c6 dO dai ly3l
"d lyal.
Cu5i ctng, tu c6c ding thirc \ * 22 = h vd,
zrz2 = y3 chirng ta c6 th6 dlmg Clugc do4nthing c6 dO dei 21, yddo tl6 d1mg cluo. c rla gi6c thing c6 dO dei 21, yddo tl6 d1mg cluo. c rla gi6c
ddu 17 canh (chi bene dung cg thudc vd compa).
3. r\{QT SO NHaN XET
3.1. S'ir dqng c6c ktit q"a tr6n chring ta c6 bii5u
thric sau day d6i v6i cos]:
I7
"or'":-l(-t t7 16' +Jfi +J:+ -zJfi +
3.2. YC sau ngudi ta dd chimg minh ctugc
ring mQt da gi6c ddu n cpnh c6 th6 dpg
tluqc blng thu6c vd compa khi vd chi khi
n = 2k Fr.Fz...Fr,, trong d6 F; ld chc sO
nguy6n t6 dpng 22' +l @6 Fermat).
*{<.*,k*********
(Nhdn d6y cfing n6i th6m ring vi5n Fermat dd
nghi v6n littdt cit c6c s6 d4ng 22'+1 d6u li
si5 nguyOn t6. Tuy nhi6n, Euler ddbilc b6 diAu ndy khi dua ra sd 2zt +l=4294967297 c6
u6c si5 ld 641. Trong ki nguy6n chring ta dang s5ng, vdi vigc su dung m6y tinh hi€n ilai viQc
ki6m tra mQt s6 c6 ph6i ld s6 nguy6n t6 hay kh6ng tr6n c6c sO .6t l6n de tr0 n6n kha thi
hcm, dir vfly cho drin nay v6n chua tim ra mQt
si5 nguydn t6 Fermat niro kh6c ngodi c6c s5 5,
17,257 vd 65537).
anzln -ta[v*zJ-v ^t d-r7) W
ruQT PriugNG pl-lAP ...
(TiAp theo trang 40)
e sinNMH = HN.sinNHMMN MN Tt (4), (5) vd (7) suy ra . :-::->-- HN.sinEIe HN.2^,ltS2 > sinNMH : MN ^ M6t Kh6c: NMH = NAH ctng chdn cung N11 ) n€n > sinNAH : HN.sinBAC e h2 -r, -( wvz!)' = q -72e .tnp I zJtsz ) 728 729 2404q :9-'" .' '" '' (theo (6)) 728 2916 - 2404s 2t5g Jztsg. o hz -g_-;.".'. :7 o h - -:!(cm). 2912 29t2 41182 YQy: vs.1sg= 1.rru..,s, r =!.z.Jla2 3 P44152
:0?'"m3)'D6 6
pC t6t thric bdi b5o, xin mdi c5c b4n gi6i th6m bdi to6n tu luyQn sau:
Cho hinh ch6p S.ABCD cri AB : Scm, AI) :7cnt,
BC : 6cn, Etrb: l35o ,FEa:45", got: tao
,:
hiti hai m\t plfing (SAD) vd (ABCD) ltdng3A", gac: tqo b6'i hai m\t phang (SBC,') r'a 3A", gac: tqo b6'i hai m\t phang (SBC,') r'a
(ABCD) hting 6ao. Tinh thd fich kh6i chdpS,ABC:D. S,ABC:D.
(7)
MN.27
(hai g6c nQi ti6p
=H*.2@ tg.
MN.27
Xet LNHAvuOng tai Nc6: , fu:ffi {S)
Tri (8) va (9) suy ra ru = Y,?l=
2Jt82
Xdt LHSAvu6ng tqi H c6: SA2 : SH2 + HAz
(dinh li Pythagore)
a3z - h, *( MN'27\' o t, :r, -( tt't't:l1'
I zJrsz / \ 21t82 )
TOflN HOC ea6*&8.$.os*as?c66***i$4'&..*..m,..ac{o.rG..?6ras'o.*6sdo!6 .r@tYc ryan' /6 - 6IUd'iE6 wl
Euii [,ip 6di:
t,
/m(r/l/#(/4/,94/ ?
(Dd ilang ffan Ddc san Tpdn hoc vd fufi trd Sd 12, trang 44)
,A""6A..6A."6A."6A".6a""6A".6,a""64..6,4."6
,f
mAT ToAN CIIIA TO
C6 bei to6n chia tO v6i nQi dung:
Trong t6 16 ho, sinh co 3 hoc sinh gidi,5 hpc
sinh khd, 8 hoc sinh trung binh. Hdi cd bao nhihu cdch chia sd hpc sinh do thdnh 2 to, mdi
td 8 ngwdi sao cho 6 mdi O aiu cd hoc sinh
, :..; , ,
gioi va moi to co it nhat hai hpc sinh kha? Trong mQt cui5n s6ch tham kh6o c6 loi giii
bdi torin niy nhu sau:
Bdi to6n tucrng tlucrng vdi bdi toin sau: Cd
bao nhiAu cdch chgn 8 h7c sinh trong d6 c6 x
hqc sinh gi6i, y hpc sinh khd vd 8 - (x + y)
hpc sinh trung binh, d dd 1, 1 x I 2, 2 3 y < 3. Vfly s6 c6ch chia li:
q c? cA + c!.c].c{ + cl .cl .c{ + cl.cl.cl
=7560.
Bpn c6 t6n thdnh loi gi6i d6 kh6ng? D6p 6n
cira bpn ld bao nhi6u cdchchia?
INAN TRQNG TIIUYET(561 NgO Gia nt, TP. Bdc Ninh, Tinh Biic Ninh) (561 NgO Gia nt, TP. Bdc Ninh, Tinh Biic Ninh)
TO6N HOC .rtt.t...t....rrr.i*oetcsrooe*t* *
E--
A+ xu*A?' *Aru tr-' t Prt;
rJiic sair *o 7f
Tlila soan I tS?8, Shd fiiiir:$ 1,{r, l{il l{oi
ST BiAlt {aill: 8t+.3St810t7
nI - trail pSri&t h*nh, Tvi sli : 0s.S51 fl1 ss6
E!neii: t$aiih0Glusit$sviBl*em@sma!l.esLr
Yryp uki Y$effi ffiWf; wffi YW&fr Tffiffi
&lmth*mmtErs umd Ymu$k ffimgmxilte
,iEan co vixr xwoa rtac
GS. T sKH.NcuvEN ceNs roaN
GS.TSKH.rnANveNNm;Nc
rs.Ncr-rvsNvANvo.Nc
cs. ooaN quirrtu PGs.rs. TRANVANHAo
Tdng bi6n hQp : TS.rnAl.r HUU NA,N{
nrtr ndu{} nrnx r'4t'
Ariilr rttA C U n g-#r* r Xf] a :r' l ta:'t
Chri tich Hoi d6ng Thanh viOn
N)G Gi6o duc Viet Nam
NGr/".NC,oTRANAI
Tdng Gidm deic kiem Tdng bien t{P
NXB Gi6o duc Vi6t Nam
cszs.WveNnuNc
Thu ki Tda soqn : ThS.ffd QUINC vfNft
rs. rnAN oixs csAu, rirs. NcurEN elrr ofilqc, rs. rnAN Nev o(rNc, rs. NcwEN vrNn otI, rs. Ncwer{
;^" ilJr. ;ffi-6,l ;rE,
"1i, "cs,ri r-E qu6c sAN, ras. PHAM vAN HUNG, PGs. rs. vU rruxs ruryrGs. rsrli. NcrryfiN vAN rraAu, ong NGUyEN rsic ltmu, zs. PHAM THI BACH NGoC, PGs. rs. NGIJYEN DANG Gs. rsrli. NcrryfiN vAN rraAu, ong NGUyEN rsic ltmu, zs. PHAM THI BACH NGoC, PGs. rs. NGIJYEN DANG pHAr, rcs. rs. T4 DUy pnril. Nc, ri?s. NGLTvEN rnd rn4.cH, Gs. TsKrl. DANG HirNG nraNc, PGs. 7s. pneN ooAN THOAI, rnS.vt rlrrl rnUy, PGS. TS. VU DUONG THUY, GS.TSKH. NCo vffir TRUNG.
Nguydn Minh Tudn - Girip em giii nhanh bdi tflp ho6 hoc hrlu cd bing phttong ph6p
bAo todn nguy6n td .................................. "3 l Kh*i ti':ric lriri Loirn ti:i nlu?
Trudng Son - C6c doan th&ng ddng quY
trong tam gtdc .........".........'.. -3{r
Fli6n riin dzrv hoc tofn
Hodng ThdTA - MQt Phirong Ph6P tim
drrdng cao trong khdi da di6n................. li:l
lilridu cdch gi*i cliu mot hai to6n
Vi€t Hdi - Cdc drrdng th&ng giao nhau trong trl di6n...... .....'......'...... 4i
Kd chu-n-,en lich su todn h*c
Bili Drtc Hi€n - Gauss vd bdi to6n drrng da gi6c d6:u 17 canh bing thrr6c vh compa. '{i
Don vuiln'fo5n............ ................ 4T
wri q441 ; qa 6c,HIEf, , rr{+ti$ LoNG
Cha'bdn; xgAtul lfun
TfrRffiN{fi $G ruAV
tiifp ban 6n ttp
Nguydn AnhVil - Ut g dung cta mQt m6nh
dd ............. ............... t
Luong Cang Hdo - Tinh tdng c6c C! cini
dtng khai tridn nhi thrlc Newton thudn
tuy..........'.
Phuong pluip gii! torin
Pham Bdc Phrt - Bei toen tlldng giao gita d6 thi hdm sd vi dudng th&ng .... 1 I ilhuiin bi cho hi ttri tot nglriep 'flFltr)'f, thi riltr
rlui hoc. cao damg
Trdn Qudc Ludt - Binh IuAn vd dd thi tuydn
sinh dai hoc khdi A, A1 n[m 2014 .........17
Nguy€n Vdn Cudng - MOt sd ky n5.ng stt
drpg lrrgng li6n hop dd giAi PT-BPT v6 tj'
)'t
Nguy1nVdnThudn - MQt sd dang bdi tAp
dao dQng trong c6c dd thi tuydn sinh dai
NHUNG TU TUONG CO BAN
.?..,P
AN CHUA TRONG TOAN HOC PHO THONG
/1f u6n sdch ndry gi6i thi6u tdng quan mOt s6 tu tuong co bAn
V-LxuyOn su6t todn hgc phd thOng, h6 trq viOc giAng day vd hoc tflp b0 mOn To6n cho gido vi6n, sinh viOn, nhdt ld sinh vi6n
cdc truong Su pham - nhung d6itugng tham gia giAng dqy toan hoc sau ndy. Nhung vdn dd cudn s6ch dd cflp lii nh0ng vdn dd m0, m6ivdn d6 ddu co thd lam phong phri vdr sAu sdc th6m, tuy
thuQc vdo d6i tuong su dqng,
NQi dung cudn sdch bao gdm '12 ch0 dd: Tiein trinh mo rQng cdc tQp sd; Tu tuong vecto trong hinh hOc so cdp; Tu tu1ng biSn thi6n; DOng lac c(ta dai sd; Dung hinh; Ti4p tuydn c(ta duong cong; Ta tuilng tdi uu; Khdo siit hdm so; Tu tuilng li)n tuc; Ta
tuhng tap hW; M1t viti tu tuong tu hinh hoc Euclid; M)t vai
nguy1n lf phd qudt trong todn hQc. Th0ng qua 1 2 ch0 dd, tdc gid mu6n ldm ro hdu h6t nh0ng tu tudng c6t ydu trong toan hqc phd
th0ng. Tdt cA ddu duot xem xet tu tiSn trinh llch s[r phdt tridn,
c0ng nhu sU soi sdng qua cdc tu tuong todn hqc hiQn dai. Ben
cqnh do lii nh0ng nguy6n li phd quiit, nh0ng vdn dd con khiSm
khuy6t vd cdch khdc phuc.
Tac gi6 PGS. TS Duong Qudc Vi€t lii nguoi di c6 nhi6u
nam giAng day o c6c truong dqi hgc, nhdt ld cdc truong Dai
hgc Su pham.
NHAXUAT BAN GIAO DUC VIET NAM
%rdru tr;;ruV Varlt thiiu or)ruV [,q,n dot
Sd trang: 248
Khdsdch:16x24cm Gi6 bia: 40 00S ddng
Bqn dpc co thd mua sach tgi cdc Cdng Q Sdch - ThiAt bi Tradng h7c d cac dia phaong hodc cac C*a hdng
sdch cita Nhd xudt bdn Gido dtlc ViQt l{am;
T9i TP. Hi NQi: 45 PhO Vqng ; 187 ,187C Gi6ng Yd ;232 Tdy Son ;25 Hi,r;, Thuydn ;
51 Ld Dric ; 45 Hdng Chu6i ;Ngd 385 Hodng Qu6c Viqt ;
17T2 - 17T3 Trung Hod - NhAn Chinh ; Tod nhd HESCO V[n QuSn - Hd DOng,
T4i TP. Di Ning: 78 Pasteur ; 145 L€Lqi ;223 LO Dinh L1i.
T4i TP. HA Chi Minh: 2A Dinh Ti6n Hodng, Qufln I ;231Nguy6n Vdn Cu, quQn 5 ;
16 Einh Ti6n Hodng, phudng 1, qufln Binh Thpnh.
Tpi TP. CAn Thtr: 162D Eudng 3/2, phudng XuAn Kh6nh, qu4n Ninh Ki€u. T4i Website b6n hirng trqc tuy6n: www.sach24.vn
heo thOng 16, hdng ndm LSnh dao NXBGD tai Ha Noid6u d6n dq vd ch0c mtrng t6 fnai
gidng nem hec mdi & c6c trudng k6t nghTa tr6n
dia bdn TP. Ha NOi. NhCrng lcyi ch0c mung t6t
dqp nh6t cung v0i mon qud c6 y nghTa, nhCrng
cu6n sSch tham khdo mcyi, ch6t luEng b6 sung
cho thu vien c0a nha trucrng ld tinh cim cia c6n b0 c6ng nh0n vi6n NXBGD tai Ha NOi d6i vdi c6c thAy co gi6o, c5c em hoc sinh nhdn dip
ndm hoc moi2014-2015.
, *{r:d*;
)igi;.2 gitii p*i:tg litu {ti ,:: t 1fi11 $g*7,1iti1*\
;**f**