f (x)> x,Yx elR hoflc f (x)< x,Vx e JR.
Xdt hdm si5 g:lR. + IR. sao cho
8(x) = f txl - x -L,Vx e lR..
Tradng hqp 1: f (x)> x,V.r e IR. SUY ra,s(x) > -2, rr e IR. Khi d6 g(x) ld hdm sO liC, ,s(x) > -2, rr e IR. Khi d6 g(x) ld hdm sO liC,
J
Qrc vd grfl)t = fU'tx))-f@+ r\'-/ 3 =2x - + b -2f ln -2
= -Zg(x). Voi -r e IR., xdt ddy (u,)(n e N) sao
cho zo = x vd u,*r= f (un),Vn e N.
Khi d6 ta c6 g(u,) = (-2)" g(x), Vn e N.
N6u s(ir) > 0, khi d6
ffista 2k+t)= -co. D6 Iddiduv6li.Ni5u s@)<0, khi db diduv6li.Ni5u s@)<0, khi db
)*dq)---co.
D6 ld di0u vd li. VQY 8(x) = 0.
Trudng hqp 2: f (x) < x,Yx e IR. Tucrng tu nhu trucrng hqp 1 ta cflng c6 g(x) = 0, Vx e lR..
Trong mQi truorlg hqp ta d6u c6g(x) = 0, Vx e R. Suy ra f(*)=*+!,vx e IR.. g(x) = 0, Vx e R. Suy ra f(*)=*+!,vx e IR..
Thir lai tathdy hnm /:lR. -+ R. sao cho
.f (x) =, ++,Vx e IR' th6a mdn J
f (f (x))+ f (x)=2x + b,V; e IR'
Vfly cap s6 thUc khdng dm (a,b) th6a m6n cl6
bdi khi vd chi L,hi a =2,b > 0.
BAI TAP
1. Tim tdt ch c6c him s6 1i6n tgc / :1R. -+ IR sao cho f (x2 *f,> =/(.r), Vx e IR.
^ -. TOHN HOC
2. Tim tdt ch. cdc hdm li6n fi+c f : IR. -+ R. sao
{
cho /tx2 + j.\= /1x;.Vx e IR.
JO
3. Tim tit cd cdc hdm 1i6n fic f : lR -+ lR saocho f (x2 -6) = /(x),Vx e IR.. cho f (x2 -6) = /(x),Vx e IR..
4. Tim tdt cL cic hdm li6n fic .f : lR. -+ lR. saocho f (f (f (x)))+ f (x) = 2x,Vr e IR. cho f (f (f (x)))+ f (x) = 2x,Vr e IR.
5. Tim tet cd c6c hdm s6 li6n \rc f :lR-+R. sao cho f (f (f (x+y + xy))) = f (x)+ f (y)+ f (xy),Vx,y e IR.
6. Tim tdt ci cdc hdm s6 li6n ruc / :lR -+ IR sao
cho/(x+/(y)) =2y+ f (x), Vx,y e IR..
7. Tim t6t cd cdc hdm sd li€n ruc / :lR. -+ lR sao cho/("r+y) f (.-y)=(f {r)f {i)', Vx,y e JR.
8. Tim tdt cd cic hdm liOn Wc f :[-t;t]-+ R
sao cho f (2x'-l)=Z{(x),Vx e [-t;t].
9. Tim tht cit, circ hdm 1i0n \tc f :R + lR. sao cho
f (x + f (y)) = y + f (x+ 1), V.r,y e R.
I0. Tim tit cd chc hdm li6n Lrc f :R -+ lR sao
cho/(x-y) f (y - z)J'Q -x) + 8 = 0, Vx,y,z e IR
(vMo 2006).
I 1. Tim tdt ch cdc him 1i6n fic f :lR -+ R sao cho xf (y) + f (x) - ry -r = f (xy - f (x) -yl(-r) + 1), Vr,y e R.
12. Tim tlt ci cilc hdm li6n fic .f :lR. -> R sao
cho f (f Q))= 71x|+6x, Vx e lR.
ilt{t{c s s rsilN +rge vil rudr rnrt
'' i*lmilnxufrn
/-{oy, fau, fg4,yrt,it, ,ono 1Anxa.
B,a*, ry *rA Arn arg unn s/if,i,ic snl sry fgc nt nghn'rtoa. ic snl sry fgc nt nghn'rtoa.
THONG BAO
Moi c6c b4n cl{t mua D6NG TAP TAP crri roAN IISC vA TU6I TRE NAM 2016.
Gii bia: 199.000 tl6ng t4i c6c co so Buu tliQn tr6n ci nu6c ho{c t4i rda soan.
Moi chi ti6t xin li6n h€:
TAp cHi T0AN Hgc vA ruor rRf
Tdng 12, Tda nhd Diamond Flower, s6 l'Hodng Eqo Thtiy, Thanh Hd Ni,i.
DT - Fax Ph6t hanh, Tri su: (04) 35121606Email: toanhoctuoitrevietnam@gmai l.com Email: toanhoctuoitrevietnam@gmai l.com
NGUYEN TTIIU OtI
(X6m 3, Qu)nh LiAn, TX. Hodng Mai, Ngh€ An)
t.nrro-ror, T?!I#t[
TTENG ANH OUA GAG BAT TOAN
eAr s6 ro
Problem. How man,v wqys can we arrange the nttmber,v 1,2,3,4, and 5 lo get a 5 - cligil
nurnber w'hich i.s divisible by l2'/
Solution. A number is divisible by 12 if andonly if it is divisible by 3 and 4. A number with only if it is divisible by 3 and 4. A number with
the 5-digits I,2,3,4, and 5 is always divisible by 3 since l+2+3+4+5 is divisible by 3. On the other hand, a numberu brd, is divisible by 4 if
-----
and only if de is divisible by 4. Hence the
choices for de are 12,24,32,or 52. For each
Bdi totin. (1998 APMO) Cho tam g.iac ABC vd D lit c'hdn dLrintg t'trokd ti' A. Gpi E v,d F ld c'qc' di€m ndnt tr1n drdng thang di qtro D sao kd ti' A. Gpi E v,d F ld c'qc' di€m ndnt tr1n drdng thang di qtro D sao
r:ho AE t'u6ng g6c vit'i BE. AF t'tt6ng g6c vrti CF vd E. F kltong u'img vo'i D. Gio su' M vd ll lt)n lurtr lu tung di€m c'rrq tot' tlotur thing BC vit
EF. Chimg minh ring AN tttong g6c vo'i Ml{.
Ldi gidi. Chgn rli6m,4 lirm g6c tga tlQ Yd hrrc hodnh song song voi EF.
Gia;ir tqa dO cua cilc di6m D, E, F ldn luqt ld (d; *), ("; *), (f; *).
Trong truong hW *: 0, thi D = E, mdu thu6n vcri gi6 thi6t.
Viv4ytac6thiigi6sir m*O.Do BELAE n€nducrng thingBEdiquaEc6hes6g6cld -9. Oo
m d6 phucrng trinh cluong thtng BE 1d'. ex + my - e2 - m2 = 0. Tucrng tg phucmg trinh cira CF vd BC
lAnluqtlir fx+my-f2-m2 =0 vh dx+my-d2-m2=0.
M{t kh6c, c6c du}ng thbng BE vit BC lAn luqt cit cir" dudng thhng BC vit CF tai B vit C. Tri d6 ta
timduoc: g( d*r.*- d'\ -( ' " 4)' \ -; )' c ' \ -; )' c
[d+ I:m--m
)Bdi vi MvdNl6'nluqt ld trung ditim cita Bdi vi MvdNl6'nluqt ld trung ditim cita
c6c doan thing BC vit EF. n€n M(, . !,* -#)
B r (+, -) "e u r,r (-a,5#). * (+,-)
Trid6 MN.AN =0<> MN LAN.
Lwu !,. Ta c6 th6 chgn g5c tqa d0 t4i D vit tryc hodnh chria canh pC. Tuy nhiOn viQc tinh to6n toadO cic di6m E vd F sE pirr" tpp hcyn. Tdt hon h6t ld chgn dudng thing qua D, E, F ldm tryc hodnh. dO cic di6m E vd F sE pirr" tpp hcyn. Tdt hon h6t ld chgn dudng thing qua D, E, F ldm tryc hodnh.
NhQn xdt. Cilcbqnsau c6 bdi dich t6t hcrn c6: Hn NQi: Nguydn Huong Giang, 10A13, THPT Ngqc T6o, Phric Thg, Vil Thanh Thdo, 10 To6n, THPT Scm Tdy; Hii Duong: Nguydn Ddng Son, 11A, THPT Nam S6ch,
Nam S6ch; NghQ An: LA Thi Thily Dung, Mai Thi Kim Chi,11Al, THPT Ctra Lo, TX. Cira Ld; Thira Thi6n
Hudz phan CZ"t ruint, Phwtc, ti roan 2, THPT Quiic Hgc Huti, tp.Hui5; Vinh Long: Hu)nh L!' Vdn Anh,
I lTl. THPT chuy6n Nguy6n Binh Khi6m.
no nAr @dNai)
choice, there are 3l : 6 ways to fill in the first 3
positions. Therefore there are 4x6:24 ways to
alrange the numbers l, 2, 3, 4, and 5 to get a
5-digit number which is divisible by 12.
rrj'vIING
arrange : sbP xCP 5 - tligit nttntltr'r : s6 c6 5 cht so
ctivisibte bv : chia h6t cho
NGUYEN PHV HOANG LAN
(Trudng EHKHTN, DHQG, Hd N|i)
^,- TOAN HOC
*rr*+
M
craroAp: 1 :0 !
1Oi adng fiAn TH&TT sO 47l,thdng9 ndm20t6)
PhAn fich sai ldm. Ban hoc sinh tt6 rld 6p dpng
sai dinh l!,: N€u limu,=a,limv,=b(a, b hfru
han), thi lim(u, + v,) =limu, + limv, : a + b.
CAn nhd ring dinh llyi tr6n chi thing trong trucrng
hqp t6ng un + y, c6 hiru han s6 hang.
Do limfl*1*. .1.) td gioi han cua tong vo
\te n n)
han n6n kh6ng th6 thlrc hi6n bit5n ddi:
.. (t r IrmJ t\
-+-+... +- |
\r n n)