R o3b+b3c+13o 452 (5) suy ra af +bc3 +cn3 r Tn (2), (4) ( ,R_ l, - rI2lTl[X1, )vd Z;) 452 ,.*[, (dpcm). tr
)Nhfn x6t M 4 > 2 n6n nong kr5t h4n cua bdi tortur co
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th6 thay sO 7 boi s6 2. B4n Tnmng Hodng W,llT,
THPT chuy&r Ngoy&, Dinh Chiiiu, D6ng Thdp vi b3n D6 Nguydn Wnh Huy,l0 To6n, PTNK - DHQG TP. Hd Chi Minh c6 loi giii ihing cho bdi to6n ndy.
HO QUANGVINH
BlitiT9l439. Tim tdt ctt cac so nguy€n drong
le n sact cho 15n + I chia h€t c,ho n.
Ldi gi,rtL (Theo da sO c6c bpn).
Yor n: 1 thi hi6n nhi6n 15" + I chia htit cho
n, n€n n: I ld mQt nghiQm cria bdi torin.
X6t n > 1 ld nghiQm cira bii toin. Ggi p ld udc nguy6n tO nh6 nh6t ctia n vd k li s6 nguy€n duong nh6 nh6t sao cho tsk - |chia hi5t chop.
Khi d6 l52n - 1 : (15" + 1X15n - 1) chia h6t
cho n n6n l52n - I chia htit cho p. Nhfn th6y
I
Bni T10/439. Tim th ca cdc c(tp hdm s6
/. R -+ R.;g: R -> IR sao cho vrti mgi x,.)' € IR
ta co f(x + g(v)) : x.J(il - y.s@) + g(r).
Ldt gidl GiA str f (x), g(x) ld c6c hdm s5
thoi mdn bdi to6n.
Choy:ltac6 f (x+s(1)) =xf(l).
Suy ra f (x) = (, - s(t))./(1), Vx e IR.
Cho x = I thi /(1) =(1-s(1))/(t) =g(t)./(l) =0.
C6 hai trudng hgrp xAy ra:
Trudng hqp 1: "f(1) = 0.
Khi d6 .f (x) : (* - s(].)) "f(1) = 0, Vx e R . Do d6 -yS@)+ g(x) = 0, Vr, "y € IR. Cho y = g
tac6 g(x) = 0, Vx e lR'.
Nhu vpy .f (x) = g(x) = 0, Vx e IR. . Thir lai ta ..a ,( .
thity chc hirm s6 ndy tho6 mdn bdi toan.
Trudng hqp 2: g(1) = 0,
"F(1) * 0. D{t a = f (L).
Ta c6 f (x) = (r - s(l)) f (1) = ax.
Do d6 a(x + g(y)) - x.ay - ys@) + g(x).
Cho x = 1ta nh0n du-o. c
a(l + g(y)) - a! > g(Y) = Y -r.
Suy ra a(x + y- 1) = x.ay - y(x -l)+ (x - 1).
X6t h9 s5 ctra ry ta c6 a = l.
Nhu vpy .f (x): x, g(x) = x -1. Ding thrictr6n vcri a =1 xhc nh4n c6c hdm sO ndy cfrng tr6n vcri a =1 xhc nh4n c6c hdm sO ndy cfrng thod mdn bdi toin.
t<tit luan: c6 hai cSp cdc him s6 thoi m6n bdi
-^Ul:l: :j?;': yiit.
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