&
ffiE
r- ;:i-:Er
t.
I
I
I
I
I
[=".
I t!-,
[ ''
I
ilr
fl
t
$Ae tEA..HAI ftralru_ M$T
Tbng biitt tdp : I{gui-Sn Cfri:h To*u ,:
Ptrd I0n9 bitr,t lrjp: l!'96 Set Tir r ru so :
x,*
70 -i'r'drr ;l,rrrg Dao * Le NOi
Eiu ntli t 8281f,
-,/ *#'iffire
.i,nt ellr pFroNG
It} ngAnf vIEN KHOA HOC *,I*T *;; t "l
5i
43
1S75 *"rss5
kt
I
i
t
rff6i, chuydn tJri }6n er$ vS* {,*stktt
I
"l,F
thge a:
++++
(.r, Y +- zi - {r:,4) *. {e" r)
\r * z, gi : ("r, U) + (:, y)
toi,it : [", o]l * "ti ii
(r)
ir\
i3)
e-ar7l) {u,.;'i -i
rt$'a i r,. ,' I
' I/lnl I
fo
I \'r
i ^,
!i'
i .,'
| -?'-
i-A r
nh*'ra---I*-*''
"*
*{irrl t
{
t
1'' :, 6ri5Cr .1; J ri:.) - t t
li,
t,* ?;!1.1 !i+('J51 .? I
,
tr.oiig rlii a ia 96,: e rig hai vei;ta i..i":* 'int'
ci8 ria.,rg vi iir{-t ring neu "2: 1 v' := !' (tt' u} = 8'
ihi ta c[ag r:o:
,i.2 - t, ;iZ .;- l, irr', p'i = U
Qri v6r, rt:gi i:ii[i YFtli] kli{ri' rr' p i:t} lhl 1r!Aii
++
lich t irro E, e :
'i:J'l U ' '?P
y=A*t+yap
v& kbi dd:
+*+'*
(*, y) t* lrru l- ,e, f le * Ezul
4+
i",yl = rrtrr (u u) + {., iYt "'l'
ll're:
(3: ry, ,, r1f;1o * istgt { rtyl)& "{ }}U?i
t
lt*r gi,i, t* hl1 'lait htti ve ctu ifrv -t i ii f'
ehr:o vetto UE, *i a, cr)ng hu*ng v':ri t;' idir
vr*to dryt ", ,rj$ vudnq gde v0l u' Gao cho gSe
tfis b*i v.{'lo $, r bBnS * r/?
f,
r!i'
rlryq p{6#&s {*e $f;&t sg{Tff Tfi*tre p4&s sps#6
trGrYf,n* T'flil(i IIi(}
1. 'Irong nhilrr s6 cia br{o Tr:{5 iruc vA 'i rrbi
tr6, c{c tran. <19c dfl drrnc gitli thi{tr vE tfu:h v&
hrt6ng r'ir* hai ve*tt:r. lli'r il\ lnsl hArn sE thrrr''
',:fr4 hii vtttcr, tuydn ttrr:1i rA dldi rri'ng' cirn goi
In m$t deng rong tuydn ttntr dl6i xr'rng' Ctr 1ti"'
i{ r'i,ti }rai veet{t tty ! r. 9, ngtrrrilr ri;t; tirrrnil
ung m$t sd thr'" kf "hifu l:'r (r,g) r":'; dttrre gQi
li fir}1 rA hw&ng, vivi cle iinh chflt sal dtr''
iluq* th6a mfrn, vg! moi vreirl {. S, , vi) nroi sl'
Vr'ri ti::ii-iC huuu$ "i r'rr: g';' li'iit.iii)' rr6lr*i
i,,, ,ti"i-' u*r.1ia dtl '!r:,, 6d' ' ;t xii-r'" dirng- irintr
f'r,r,'' o.i,t lilria t'-iin t' i:i 1*li1 pir'li'r ir'rili dl.ri i:ii
1'4et o.
,' \,i, r:!.,urrrr* ditr, 'r-x;1, ll:r 'l:'' ';'i' ti'i r;'"'i
,,u*,"g .uui ru5 *i 'fii .rriflg tlu'rt' ri': riirrh nirr l'
,1i,,",,oi* git tt; titr ni' rril*l';rt "i" iir i'iia
klr0ng r-i tili0r:g ild'ng pht''ring si' l.r' {":u llA tt ndu
ta clel tru6c c{c gili iri :
+++$
i, - ti, oi = o)>r), tu, r.,) * (p,,i) '- t',
'Nirtr vi.r'iA nltt6n lii'lti:iir iir'ii vo hlr6ng'
ph*i clro ii -s4 1511{r lA <r}tl. c)0 vA lr.
flrr diBu ki.t:n tl) vi hiEu thr?n satr di-r su1' ra
LI {,{ }
-q 4'+ ?+'llrirr+rtile, i5)
r - - i.1.. r.: rE d y + l].f lt? + tf .: ), {i,
Chii n&n !n iriiiri r:ri iliBu ki(n
ar ,-. .-a: ) tt. (CI)
:, !]? i:i dlrrrre ri ngirie lrinir hoc cria tich vd
hrrrlng, l* r:l l quv rilic rang ndtr.u. r' lA hri vec-
to ilrr;.: vi ttr?c i* I * lu J = 1) vr) t'u6ng ,6c ut)t
nltari tiri t *
i.- riit = r, G,*r) = s,lt- ti,it=r
SB chung minir r&ng quv rrdc nhu trBn co
hiQu lrlt vri mt;i rr$pt*v1cto clon r i vii6ng g'ir,
ta triy lfil' th&m lhli ft' ,tr'i'li-rp 'r'i ';a vudng gdc
vdri nii$.n, '1iir di diing th'i1' r':1ng ihintr l):
{{ )
t."
tlr
f4
I
la co: ;,, l; ,;i
FnF ?-B €
g '= ig leosa ' u f- lr7 isina ' u
Ilo do +:* 4 _--
. (r" g),= [r I . ly l*oae
$frn6 th{re {r0n clro ta .{' nghin Linh hoc cira
tinh v0 huilng Vi ctng chinh tl&ng ttr*c fl1",
nsuoc lai, cd tht ISy lilrn tIlnh nghil cho g6c
hinh hge ct cr'ra hli veeto (t{rri ciin hei lia).
Ciing theo (7), diBu liiQn r.in -.a dri cho lrai
4F&
i'ecto llh6c htr0ng x, v lrrong gric r.iyi nhau ia:
. *>+
- (r, Y)=ii
{. Rr}}'gi$ niu giCri ttritri vri h,,n,1t.rr mdt harn
+,i tu_rtn tintr Llrie riia..l vct.to-lri.r i, l.lrOng tldi
x*ng nhrr tich r'0 hudng, rni lai ln phiin rri,nq, .
tric l[ gid tli r:iia nr_i tlBi ridu (kh0ng dbi gi:l rri
tu-r€t d0i) khi ta hoin l.! hri tecto. H.)m sd n*r,
ilrroc goi lA nr0t dqnq nga,ii ci'r.,r hrli,vect<y. Ta
sF gqi nd li iielr nyorir cira lrai recto, hg hieu
.'*4 --& ..*
t.., gl,;S :r:. i, la nh&ng veclo tiir y. \.r{r"[ s1,.11
vectir *, g, ; \,a vdi ingi sri thuc a. ta s€ i6
niru vsi tich vd htr6ng:
(e)
rd (g), ir{i), {11} ta suy ra:
4
lr.. Ul - l:ria 1 .rzu, ri lE + glol
lr.ql= (.r'taz - r,ryr) [u. r]
l;:;l = ittyz - rssl) 5il2)
"l'l:ti ta dtii yriyi ticlr ngoAi e tia hai vect,: trong
miit phing, chi eln cho giA tri Df 0 bfia n,lr
t'oi hai verlr: chon tri1, y, kh6c khOng v& l<hdng
diing phtrong, ltl dri dE xic dinh lii.
l,!,it 11n,, chit crr lrAn eria lleh rr,1r,ii ti:
_*I)ibu kien eiin vi rfti cho.hai rr,lo r:,Li
g $$ Anng phuong v6i nhrrr l,i":
lr,9J=c
llpnh dB tr€n suy la d€i d*rig tri (l:t).
6. Bi.v gid d& co tlrr*r' -i, nghi* hinh hqrr:, i:r
qui *ac rsng I = *l l<hi li:ri lerlu-I,*i th Aou
r i vi. tq.o l.hanh m,)t g;dre vrrdng theo chiEu thudn,
trie gdc efra u vit r,l = *'ri?. Qui urlc* niry ed
hi$u iqc vdi mgi c{p vecto nhu vr|1...
'['hgc thd, mgi e{p uo.ioi'ri'don r.!*v:i vusng
gr.rc th.ugn cd thh phin tfch theo c{p ol', nf,"
dE ldm & trEs (?r. Vi f a ilriv r.irrg
t:; - r* I -+ 11117': :,: + I
Ycri hai vecto b,it ki ;:;r tn chon ul cnng
huorrg vdi .rr {xern l-rinh l), thi ta cd
fi---
.r. -_ I,xi .r
li = 11; ji:,,r?. u i- lyl 3in&" r,
\'f.v: i;''; = l'liiii uiu "
Di: ln j nghia hinh llf Ef,f [r, gl. ]',duclii Jri]-
giii trr tuyQt ddi, thi [:r:" yl ]rfrL&1firr tirh cira
hiah biuh hinh i'€ y/ri 2 \.tei:aij ;.r", q idinr caah.
i{
/j
)t
J
' *..,
ls.**-"
o *,--^']F A
ttlnh'3
Ndrr lr iluy tr-O-c r ir'rg di$n trir:h mring ti*u *
n€u grlc " =-({-l:}
> ti vs rnang ri:l u*. n6rr (r'j-u
thi iihi dr) [* qi li didri ii,:h c,i liasr,] cria liini.i
bintr hirnh.
IL;iI-- rL-i:'ii-,t J
tfuf ?'is Eit-ha i
Ir.... l/l = l*, egl = " tr" yl I
Nhulg, klr{"e vfii tich v0 hrrong :
'>'+ + .+
ir. Yl= * l,l, .,:l
ei tdt rrhi€n li F>,t+
i.r', 3l =t g
'l-ich ngoiil dinh ughia nbu tr€n, cr)n goi lt ti'elt
lfcfi, holc tith tharl ddu.
5. Cfing n$15*i tich r u huong, ta h[1. ehon
hai vecto u! p nilo ttu. khAc khring vit khdng
d$ng phucrng. T5t nhi*n:
ilil=0, lilfi=,r,
Cdn [u, ul, ta d{t
lf'*'l = 6#rt
3 Id rndlt r0 thuc khde kh6rrg, chon tu.r,i,
{:*)
Ling vOi cip lu, af. Ktri d6, hirn lr, y] li hoal
toin r{e tljnh. Qiii v&y, lri i
.r:.rln -l- rzu
y a17p * 11zu
{ 10)
(u)
I
l. f6nr l*i. trong'irinh lror: prhing, v#i hai
telto tirv ]' r. I/, ta l'cAc t1 !nh tioh r0 hrrdng r'd
tich r;4oni lA:
G}= l.lJA r "r , J
1... ,;1.- lr I ir;t sin s l
vS lr: hsi biiltr llrt'tc 1r&ri, Itl stir
7'*^',., *t -*t
,1. l,l)" -tl lr" g l' =. u' . !t'
l)irrX tl'rri. ni-i'r:hr: ih.{r'qLl a$
nguiri "ri Iirlr Id lrut';rig.
Fiai tich v6 tru6ng.vi tieh rgorli i[Eu-co'ung
dr,rng rQug rei trong hinh hoc vccto ptriing' N6i
ctrung, lilri ngrroi ta chi sei 116n,,nhitng-tinh chrit
*h,r,iibrn thirlg lrnlg, direing thing"dDng ilui'
rlu*ng thing siing so;rg' doarr ilring rhia tleo lj-
i0. r-i's"{ di{rt tir:lr" tr'tc lir nhiiug tinh ch{t gr.ri Ii
ilnh chit a fin. tiri ngudi ta cl:i chn dling ddn
tit'h ngoAi rua tlr0i' t.eir l. lri rrgtirii lu xCt ci ddrr
nhirngli,ntr cL{t r:tg li€n clriarr den dd eldl vir sd''
thi ngudi ta r:itt dring ddn tich vd hrrdngr mi
dei khi cfr n'g ndn tlrirrg.,:i tich lgorii nita'
il{) "
t:r :.
( l5)
.lr* giire 1i, hf
ilii Hs{# #iA
wffi srroffi
*nr"r
\-rJ1\-'
ffi&
':["R&?' Yff ffifr
qAc wffiru
BAI 'i*A}I
ffiemG
.",
[1 lli L,urn iL)li rr. ' i:ri. cli,' iran g[;; iil;3rru !l triritrtg
flL i.,ai tt:, n 11::| r.r': r r i'i: iu th.rt', Sir, it''rrt; lir;
binir llirig vrl,i tlhalt, nghla iA n0ri l-ir t! ao
d&i e {ac r du tri il,l, ciio nl::iu t}ri irir6ng larl
tlra.v dbi irili tual. i'.hiing hqn ':ia'; bni to{l:
('l'im t5t c& . ci.ic sli ngrr.r,,0n t6 a, i, c srio clto
obc.{o}*[r:''i- rdD. ital li aT'inr irietd ngrr-f€n
rluonS Jr .{, : s;lt, clio .}S: = 1} i :: *U * : D'"
lir nhirng btli irtxn ('o linii t'hi: v*a trBu.
DE girii nlrti'ng l.rl}i lnin dang n.hrr tlrd cd thh
cd nl'lil:rr cAch Llrfrr.: nhau, tt\y ihuQc'vAo trlng
bii ct-i thb."l'rong biii bAo *hi1 rr&-r't6i nrudh
gi,ri thi$rr r 6i e&r: brrn ni{rt phurrr; phAp giii
cAu lr&i tol{li !l'&i!,
'It bist r*ng, n*rl ,:lt'] tr'sfr .llrr.re 6i.. "2,:. (ia'
lhi bao girS i* i:i.rng ro tii& srip clrung lheo !!rr'r
tu i,5r:g d.dr hog,e gi6.rrr ditrri. Clir 'rrflrr e$tt * 1.
u2,..., #o ti,r* gis lirr.-r nrc! lr*i lr-rirn iriuir r.lfing
116i uhnu, thi l;at: gic' ta'eirng cri tlrh gi6ri bAi
to6ri vdi gi& tili{r i,; t( a: { I < an }io[r: a1>'
a?pt..Pan
'Trtr6r: i'rdt ts'gili i,li ira; lrlfln I'ttn niu ri tlan'
Bii toirl 1. f inr irli , ri ctlc sd rrgogdn {r} a.
i;. c ,lgo tho abr, {. ib.;- l,t + r t.
Giti: Yi eAc sG a, b, 'e binh ili'rng r'6i nhau
nen khOug rn{t tinh t"bng'qu5t ta Si{ tbi6t
e(e(c.
Suy ra cb r' iri' * c,r { 3&c.
NEu ap$*$br(ale '|cb i lle -ica(oeb,
mAu tbu$.n v6i lidi ra. V*y a = ? ( ri e ngulBn
t'1). Do dd :
vi qiras{; S*il
Zht. 12b -i- 6r f 2c -) l/c{ iib > t/2Eb<5
o Ii = ? => c ngu-r,0n i6 bf;t lii',
e 6:li *ic=:3 ho{c e l=5.
11 6rri lai nghiQrn ciil bAi t6{t} iA:
s * 2, b*2, c E p u$u)'en t6 vA cAe ho{n vi'
trofle o =?, S=ii, c =3 r*A c{c ho{n v!.
BAi SoAn *. ?'irrr ld! cri cdc r$ tguy€n duotig
r, ll. : \iio cl?o 19: =9 -f ,j + u + :,
Giii : llo :r', lJ, -- tharn gia vlo I::ii IoAn binh
tling vrli nhau n0n. lil:0rrg nirt tini.r tbng qirAt
t:i gi; thidr r'> g ) , ) ] l'a ed cic .kiri rrflng
snlt :
i. Ci ba sd brlng I : kh0ng lhrir irti r': pl:u,rt-lg
lltuh.
?. rli. g -. - :: Iei1,; i- l1 * a: rt) lr
:', ,)> S.)'1, : -: l'"*ry '' iU * .r - I
ii=(.r'- t)(!, -. 11. rri r - lb y- l. l6n e' -
= 11, g - I *- I vir-:r; :"::12. g = 2, .'= l.
1a>y),)>j.
Eet u =a,-'j, u = ll ^3. a. . : * 2
+ a> u) ,u ) 0 "r'e ta c6:
li'i rr f o * rr' = (rr + 2)(u +2)(u; * !1 =
J oru, * 2(,ap ,i- u,y .t- g,u) * 4(rr a u *_ru) f I
Suy ra : 7:u.utli-2(uo*'ulu*uru)+ 3(u-1-u*rrr) (*)
- N€u ta >0 thi v6 phli cria (*) ldn hsn hoic
b&ag 16, n&n ding th&c {') khdng xA1' ra. !r{6u
rr,=0tr'7-2uu*3(a+p) r,i ia ls.i xtl 2
tludng hqp:
0 = 0 =* '7 ='$u kh0ng th} xi.v ra,
u } 0 + u-? *bI + ?r{ol-l(u*p) }E : vd
f a-v
1-
lri
.'1
il
ri
T6rn lai nghi0nl ciia pirrrorrg irinh la:
'J', -" tz, u -- z, : - I \ lt eac fto;r l) \ i.
Bii toAn 3. Iim t2 s$ ngtulin Liw,tr;,1 . ,i,j,)
cho tbng crie e/rrin.g &dng llrlr cria clrin17.
Gi&i: Gt:i cAc s6 phiri tim li xl. ir,z,...
Rd ring :rt, a'?....,.tta trinh d&ng r.6i rrh.au
la gi6 thi{t c-t} rz...2 t,p} t
Ta gii i phuong llinh
.r1J2... rl2:.1'l *::z -'f .. +.1112
'I'rI (ll w" *1r2... ;12 ( 12r1
,?ri 2'
n0n
(r)
(?)
hay t2... r:tz --( tZ (3)
Vi 2a-= 16 > 12 nen lu i:)) ra rhdy trolrg il
s6 ca, .ls!...,.r1a iitrOng thA cei lron 3 s6 lorr
hcln 1. YSv trong t2 st1..r'1,.r'2..... .1rt* khOng thb
cd hon 4 sd l6n hrrl i. Do d<l 'rhi c6 thB xfiv
r.a 5 [rrrirrrg hop:
l. 11 = tZ = ... = tl2 :-:: L izhOng ihdra rnin
phuvng trinh (2). t
2' *i )sz:.1rS: ... = r.l2: I, t& i2) rilrt' r,t:
c1 : l1 * ct."| 1l = 0 ngn ohriDg ktrong tht th6a
rnf; n (2).
i. nr> EA> {ts: rurt: ".. : .,,l1 = i, suy ra:
Elaz: 10 { ;e1 * 12 €} (tr -* t) {*2 - l) : 11 : 1l.l
\ri 13-. l) :.e2- I +.r! * I : ll,. :l:t*i : i'
Tlt d6 su' r!
zt:12. :7."Z:2 r'A rS: 'f4 - ... =..zlZ = l.
4. :Ub tz) tl ) ri = ...-rlz: l. irt (2]
suJi ra &ieLx-t: I * .rt * rr * ct.
'ltrel bAi loAn 2. la c(r .t 1 = 12, ra = 2. .r'r : r :
kh6ng th6a mln vi rs ) L
5. t,tb.L-7>.rB2. u2 rs:...-= r'r* * l. 'I'r'r
(2) su.v ra rlrzns:rae 8 * -rt * 12 f rS * rl (4)
D{t tlt: .r 1 * }, !12: *z - 2, r7g = rB -. ?.
!4as4* 2, su1' .r^ yrb 0 Vi = I. X.3. '[ r:,r tar
e15 {1) ttrong ri rir7ng 1'{.-i
l6*yr*ilz rus+ lli: {!.lt.r2) lrJz*2i (Us- 2} t g++2t
=, tri * g(gr * yz +"43 * ya)*...
h"r
f (rt '- a2] { .r'l t- r'2 - .1 3 !' ,r'tr) = it
\
),u, - o;1ii--.", *.JZ 1 .!tl ji4; = tl
I (r:a *'id:r - r'i *.r': - :r.:+ L,l ti : lr
[ {rrt - d4}.r I -f (,rz '- *4i 13 * ia3 - i:4i ,t 3 : I
( ,r*r2tfvrrr+:0(tt)
I - ct -'- 1'? + rg 'i' ;r4 = 0 ($')
* i -iol -12-'.t't+er4 : s (c')
I (or*uu).ri'i-(uz*er) re+(nr*aa) .rs = I (d')
L{y (a'} + {c') d,}qc e,1 = rr; (a') * (6') auqe
.rl s ss * .t4 sur ra rca = 0 ([') * (c'] adrpc
,l + .tS = r4 sttt rtr x2 E f,A
lha.v .rg::: rr3 = 0 v;r: pl:rrong tri.h {u'-) .r}' "n
xl=#4..!{at-e4)
tr'Ar vr'vi o1 )> ,12 ): a3 ) aa thi nghi{lr tira h6 id :
.r? =.Tlr'= ii" r'I =.r4 = |l{at *,i*')
Cliir r' riing n$r. (ti. ltz. r]S, tl.1 c{r rtr,it lltri ltr
khAt' 1hi nglrignr ring cd rntit tht'r iu trrrrpg 1i11g,
Ching han ,*,, a2 ) a4 ) nt )ns
thi: .rl = 1:4 ,::0 r,A rA - rS = llteZ - uS.)
'f r'€n day, la ile rlt nrdt sd LrAi to.{.r vE giii
ohLrong lrini;. Ii*v gio ta xet m0t sd .bii loi n
iogi khite.
Bei to$rn 5. {ldc sd dtong a, b. c /hdo mdn
di'6u kiQtr gi, d? odi mqi sd ttt rthii:t n. cde sd
oE, &4, c: ld dd dr)i ed,c cqnh cfia m$t la'n gltic?
6lii:' Khlrng indt tl*h thfrl lhng quA:. ta gi&
llri6t rAng olc)>
t) Fi6r"r t)>b)2 c -)> 0 llii I -)$ia )tla Yr c|t
s0 hta. citr d0u nhir- hsu I nln l6rr lai sd trt
nhiirn n rli lOn sart eiro- (bio)o +'(c,'n)o'{ t khi'
at6 ,n + co {.i no r*n \'ui n i!qi, rru. Iro, .ro khf;ng
phLi i;i 3 raoh ciia ir;di i*;rr gil,.c.
2i 5sri ,.r : b>r-')r0 thi oh = b*>',")-t1
ltii nroi n st1 {r nhiEn. vir r5 rilng rhrirg ia tl6
di i 3 canh cria rn$t t$rn giic.
\. i vg1' di&u ui4n chn tirn c&a a, i, c l{ hai
e6 l,in nhst phii b&ng nlrarr.
Giii: lt thAl rfing { ed rltli' ni,il lihrrt ;rlrau
at, oz, dg, q4 thanr gi.a vao bii loArr i'inh iliing
v6i nhau, nGn':khCng niSl tinl: tbng rpril, tri gili
il,i6!
at)ar)rg>"ue
li h i rlo li6 ( li trrcrr-rg i!rong vl!i :
i (ol *d2).r2 .{n1 -os).rr3-t (cr1 -44}.rr= } (u)
J lrr- o2t.tt+ {oz'-os)r3f (u2 - o4).r'+ = I (b) 1r1
) (at ,r;l),r'l-r(/rZ -ol).r'z'rt,tc-u4) x4: I (,') '-'
rl iot- ua).ri'i(n2-o-l).l.e*(ar -era),rs* I (tl)
\\
Tu hii ('l). tl,Lre hii'i, 0ic"r,lr(,p tinh (ai -- (b) I
(l;) -'(,'): (c) - 1,ir ',, illyua i:0 irr,rrrg rittr"nq:
0=711il I llz *,t3 * u+) * ...
Suv ra
!j1= !]2 = y3 : y4 = 0 -> :rl = t2 = t3 =;rtE2
T6m iqi cAe sd etrn t im lA
fil = I3. 12:2, .rB =.r4 =.tS =... = X"1g ,. I
ho{o
,1'I:.1 2=.l'B :- f* = 2: 15 E .r'6 : ... =..r,ll - I
8Ai toan 4. C}o a1, a!, a$, d4 ld, 4 sc) t/rgc
rldi nt.ot khac nhau. Gttii t* phunng trinh tctr:
I
\ lrr - aalrg*-lor - oelrs +lor - arl re = I
llrr. a1l 11 -5 !o, - nrlrs .i- loa - d.lr{ = I ,,,
) Iaa - a1l.r1 * log - ael rz + iaa * otlcl = l "'
i lal - oil.rr * lnt azl..z + la+* asl13 = !