[0 1<n<3
) NhQn xdt. Chi c6 3 bpn tham gia gi6i vd c6 duy
nhet 1 ldi gi6i dring tr6n d6y ohabqn Sang.
DANG HUNG THANG
Bii Trc1446. Cho p ld m6t s6 nguy€n 16. Tim
tat cti cdc da tlr.rc .f(x) voi hQ s6 nguyAn sao
clto t'oi moi s6 nguy1n duong n, J(n) lit uo'c
t'tict p" - 7.
Ldi gi,rti. Ki higu A ld t|tp t qp t?t cit
.cdc uoc
s6 nguy6n cin p - 1. Nhpn x6t rdng t?t cit cdc
da thric hing sO,7(x):- b, b e A thodmdn bdi to6n (vi Vn e N*, p" -li (p-I) ib). Ta sE
chimg minh do lit tdt cd c6c da thric thoi mdn bdi to6n. PhAn chimg tdn tai da thricl(x) tho6 mdn bdi to6n, da thfic flx) c6 bAc ducrng. Gi6 su n e N* vlr q lduoc s5 nguy6n t6Aatki cua
fl"). Ta c6 .f(n+q)-f(n)t (r+q-n)=q.K6t hqp vli f (n) K6t hqp vli f (n) '. q > .f (n + q) i q.Nhu vQy P' -l t .f(n) i q, p"*, -l i f(n + q) i q ) pn+4 -l-(p" -l) = p'(pt *l) i q.
Tt c6c sg ki6n trOn ta suy ru p * e, pq -1i q.
Mpt kh6c theo Dinh liFermatpq : p (mod q)
), pq -l-(p, - p): p-1:.q (1)
X6tneN., n -1 i(p*l). 56c6csOn nhuth6ld vO han, trong khi d6 AldtQp hqp hiru han vd ld vO han, trong khi d6 AldtQp hqp hiru han vd
da thtrc flx) c6 bAc duong, suy ra c6 sd n nhu
vQ,y mdfln) G A.}l4dtl$6c p' - l: (p - l).8,
frong d6 B: l-t + f *...1- p * 1=n (modp - l)
: 1 (modp- 1)
= (p - l,B) : 1.
Dofln) e A > c6 u6c s6 nguy6n td q cnaftn) md B .i q
= p -l I q. Mduthudn v6i (1). trF Nhfn x6t. Ddy ld bdi toiln dathric - s6 hoc hay, kh6. F Nhfn x6t. Ddy ld bdi toiln dathric - s6 hoc hay, kh6.
Trong s6 circ bqn grri loi gihi ror Tod so4n c6 ba bqn
gi6i dung: Hn NQi: Nguy€n ViQt Anh, 12T1, THPT
chuy6n Nguy6n HuQ; Hi Tinh: Trd:n Hdu Mqnh
Cadng,12T1, THPT chuy6n Hd TTnh; Binh Dinh: Mai
Ti€n Ludt,12T, TIIPT chuy6n L6 Quj'D6n.
[k]=t tmc ;y1:1
*] Irfrl] =1y12 +[xl-l <+f trut :gt2 +ty)-r f trut :gt2 +ty)-r
NGUYEN MINH DTIC
BiitiT111446. Ki hi)u lal ld so nguy€n lcrn nhdt kh6ngwrcrt quii a. Cho x,y ld cac s6 thtrc dtong
thoa md,n txl Lyl.: 301. Tim gid tri lcsn nhat vd
nho nhdt cua bi€u thwc P: [xlx]] + [yly]].
Ldi gi,rti Ta c6: lxlfyl: 304
=[x] ,ly7+ 0. Md
x,y duongn6n: x > [x] > 0, y 2bl > 0. Ki hiQu
{;r} Upnan l€ cinx,vfly: 0 < {r} = r- [x] < 1 .
Khi ti6: r = [(rxr +1,])r,t]*[(Ut +{y})Dt]
= txl2 +[{x}t ]]+Dl' *[{y}pt] >21.xltyt:2s0a.
^ l[x] = 1Y1= 3g2
P=230* e - { -I[ro'{,}] = [:o'py1] : o
161:91=39z r , \
lo={,},{r}.# L 3
Su dpng tinh ch6t :Yor n eZ, n) I, a elR thi nla)<fnal! nfal+ n -I (1)
ta c6:
P = [x[r]] +lylyll< [x]2 +[x] -1+lyl2 +[y]-1
=lxl2 +lyl'+lxl+lyl-2 (2).Do [r] > 1, [.y] > 1 n6n { t t'l- txr:'l -.1) > 0 Do [r] > 1, [.y] > 1 n6n { t t'l- txr:'l -.1) > 0
[ ([xJ' - 1X[yl' - l) > o
I rrt*tyt < [x][yl+ l=3oa + I
<>< I t 12 +l-ll2 <l*l'lyl'+ I : 308 + I +P(308+304; P:308+30a kl=l hmc [y]=1 kl*1-*<x<[x]+1 M+r-fr <y<Lvl+l t, nuo,rr-rorr, '?!I#EE 28
I x el;2), y. [so' + r -*;roa + r)
I " L goo' )
<)t
L, = [:o- *r-4;3oa *r),r ett;2)