Nhung, Nguydn Quang Thdng (THPT chuy6n LC Qry Ddn, Bi Ria - Vffng'firu); Thdn Minh Duy, Nqul€.n Thi Huy€n, LA Trung Thdnh (THPT chuy6n Bdc Giang, Bfc Giang);.Nguydn D*c Tudn (THPT chuy6n Bic Ninh, Bic Ninh); Phqm Luong Hdo, Dqng H6 Trung (THPT chuy6n LC Qulf D6n, Binh
Dinh); Biti Vdn Binh, Trwrng Vdn Hodng (THPTchuydn Quang Trung, Binh Phudc); Hu)nh Bdch chuydn Quang Trung, Binh Phudc); Hu)nh Bdch Khoa (THPT chuyCn Tr6n Hrmg Dpo, Binh Thu$n); Nsu,-in Kinh Luan (THPT chuy6n NguySn Quang OIeu, Oiing Thnp); Dinh Nguyln Hodng Kim.(THPT chuydn Hirng Vuong, Gia Lai); Phqm TiAn Khoa (THPT chuy6n Bi6n Hda, Hi Nam); Vil Tudn HiAn (THPT chuydn Hd Ndi-Amsterdam,-Hh NflD, LA Bd Trudng Giang (THPT chuy6n Nguy6n HuQ, Hi NQi); Phan Nhett iuy, Nguydn Vdn Nghia, Nguydn Thanh Nhd, Nguydn Vdn ThA, Nguydn Duy.Tuan (THPT chuyen Hd Tinh, Hn fnD; Xh6nS Ti€n Dgt, Nguydn Tuan HiQp, Ngryi, ViQt Hodng (THPT chuy6n
Nguy6n riai, nai Duong); oa rii1Drtng, Duong Minh Hi€u, NCuy.en Huy Hodng, Vfr H6ng Qudn (THPT chuy6n TrAn Phir, Hii Phdng); Nguy€n Thily Trang (THPT chuy6n Hoirng Vdn Thu, Hda Binh); Nguydn Mqnh Cadng, Trdn Vdn Hi(p, Tran fhi Tlu
ruud, Ct'," Minh Huy, Trin Huy Phan, LA ViQt Tudn,Ddo C6ng Tu (THPT chuy6n Hrmg YCn, Hung YOn); Ddo C6ng Tu (THPT chuy6n Hrmg YCn, Hung YOn); LA Thi Thdo (TIIPT chuydn Nguy6n Tat Thanh, Kon Tum); Ddo Sctn Trd (THPT chuy6n Thdng Long, Lim Diing); Phqm Quang Huy (THPT chuy6n LO
H6ng Phong, Nam Dinh); Trdn Nguy1n Ldn, NguyOn Trung Phuc (THPT chuy6n Phan Bdi ChAu, NghQ
An); Nguyiin Duc Minh (THPT chuy6n Luong VIn
Tqy, Ninh Binh); NgtryAn Thdnh Dqt, NguyAn Hdi Ddng, Hua Hd Phurtng, Phqm Qui Vaqng (THPT chuy6n Htng Vuong, Phri Thg); Phan Xudn Thdnh
Lam, Tran Ngoc Tin (THPT chuyCn Luong VdnCh6nh, Phf YOn); Bili Quang Dilng, Trdn Khdnh Ch6nh, Phf YOn); Bili Quang Dilng, Trdn Khdnh Linh, Nguy€n Minh Nggc, Hodng Thanh Vi€t (TIIPT chuy6n Vd Nguy6n Gi6p, Quing Blnh); Nguy€n
^ - TOHN HOC
C6ng Thdo (THPT chuy6n LC ThAnh T6ng, Qu6ng Nam); LA Kha Tr{ (THPT chuy6n LC KhiCt, Quing
Ngn\; Dodn Thi Hi€n, Trinh Minh Nhqt (THPTchuy6n Ha Long, Quing Ninh); LA Thdnh Dqt chuy6n Ha Long, Quing Ninh); LA Thdnh Dqt
(THPT chuy6n Hoang LO Kha, Tiy Ninh); Phqm Thdnh Nam (THPT chuy6n Th6i Binh, Thfi Binh);
LA Thuy DiQu, Nguy€n Dinh Lwong (THP] chuyCn
Lam Son, Thanh H6r); Trdn Minh HiAn GHPT
chuyCn Ti€n Giang, Ti6n Giang); Trin Cao Nhi€m (THPT chuy6n Nguy6n Binh_Khi6m, Vinh Long); Nqtydn Khdc Vi€t Anh, Nguy€n ViQt Hodng, Nguy€n
H6ng Nggc, Nguy€n Qu6c Trung (THPT chuyCn Vinh
Phirc, Vinh Phric); Vti H6ng Qudn (THPT chuy6n NquVdn T6t I'hanh, YGn B6i); Hodng lrung Diing,
Trdn Quang Huy, Hodng Duc KiAn, Trdn Minh T'i€n, Trlnh Huy Ip GHPT chuy6n KHTN - qHaG fl_u
NQi); Nguy€n Eac Duy, Hodng Dinh Hi€u, Nguy€n Phan Qu6c Khang, Dodn Minh Kh6i, Hu)nh Pham
Minh NguyAn, Nguy€n Vb Thanh ThiQn (PTNK -
DHQG TP. H6 Chi Minh); Tq Khdnh Hd, Trin
Qu6c Nam, Hodng Anh Qudn (THPT chuyCn DHSP, DHSP Hn NOi); Trdn Quang Htry,, Phan NguyAn KhOi (TIIPT Chuy6n, DH Vinh).
o Gidi Khuydn khich: Ngydn Bd NguyAn, Nerry)" Pha6c Thdng (THPT chuy6n LC Quy D6n, Bi Ria - Vflng Tiu); Nguy€n Thanh Ldm,. Thdn Thi Qajnh (TIIPT Chuy6n, Bic Giang); Trdn Minh Danh, Li, Hodng Th,iAn (TIPT Chuy6n, Bgc Li6u); Nguy€n
Thanh HiAu, Ngqt€n Vdn Tdm, Mdn Bd Tudn (THPT Chuy€n, Bnc Ninh); lqrryA" Xudn Quang (TlfPT Chuy6n, BGn Tre); Trdn HiQp NguyAn (TTIPT chuy€n L6 Quf D6n, Binh Dinh); Nguy€n Gia Bdo (TTIPT chuydn Htng Vucmg, Binh Duomg); Biti C6ng Minh, LA Si Quan (TI{PT chuy6n Quang Trung, Binh Phu6c); ldm Philng Phwbc Vinh (TEIPT chuy6n Phan Nggc Hiti:n, Ci Mau); L€ Vil Nam Anh, Phgm NSuyAn
Thi(n. Huy, Ngry€, Minh Qudn, H")*h NgA lghi Truyin (TTIPT chuydn L6 Quj D6n, Dir Ning);
Nguy€n Thdnh Phdt (T.[PT chuydn Nguy6n Du, DIk Lifr.); Nguydn Phi HiAu, Lmt Hodng,Hry, Ng4tdn C6ng Philc (THPT chuy6n Luong Th6 Vinh, DOng l{;ai); Hujmh Trung Tin GryT chuydn Nguy6n Dinh Chi6u, D6ng Thfp); NguyAn Qu6c Vii Hodng, Dinh
Thdnh Nam (TIIPT chuy6n Htng Vucmg, Gia Lai); Dodn Tudn Anh, Ng.qt4n Dac Huy GHPT chuy6n Bi6n Hda, Hi Nam); Khudt Dtqt H6ng, Dinh Xudn Tilng
GHPT Son Tdy, Hn NQi); Trtrcrng Minh Quang, Ddo Minh Qudn (TIIPT chuy6n Nguy6n Trii, Hii Duong); Hodng HiQp, Nguy€n Trung Ki€n, Dacrng Minh Trung (TIIPT chuy6n Trdn Phri, Hii Phdng); Khuong Anh Til (THPT chuydn Hodng Vdn Thp, Hda Binh); Truiz Hodng Thinh (TTIPT chuy6n LC Qui Ddn, Khfnh Hda); Ddng fha nilng (THPT chuydn Hulnh Man D4t, Ki6n Giang); Ddo Huy Hdi, N6ng Ngpc.Qudn (TIIPT chuy6n Chu Vdn An, L4ng Son); Trdn ViQt
Huy, Phqm NhQt Huy, NSuyA" Ddc Thdnh (TIIPT chuy6n Ld Cai, Lio Cai); Hodng Lir E*c Chinh (THPT chuydn Th5ng Long, L0m EOng); Nguy€n Giip Phaong Dq (TIIPT Hfu Nghia, Long An); Bll Tudn Anh, Ngtqt€n H6ng Edng, DqnS fhi H6ng Vdn
(TIIPT chuyCn LO H6ng Phong, Nam Dinh); D{rg Ngoc Anh, Hd 8w! Anh (TI{PT chuy6n Luong VAn
T1ry, Ninh Binh); NguyAn Dwc Thudn (THPT3huy6.n
Htng Vuong, Phri Thg); LA Bdo Dqi, Ngay€n Trdn HQu, Nguy€n Thi Tnic Nha (TIIPT chuyCn Luong Vdn Chanh, Ph[ Y0n); Biti Vdn Tudn (TIIPT chuydn Vd Nguy6n Gi6p, Quing Binh); Truong Trdn Tdn Phudc (THPT chuy6n Nguy6n Binh Khi6m, Quing Nam); NSuyA, LA Duy, Dodn Qao Khd (TIDT chuy6n L6 Khi6t, Quing \en[; Trdn Duc Hi€u, D6 Gia KhiAm,
LA Quang Thdng Q}IPT chuy6n Hp Long, Quing Ninh); Trucrng Nhdt NguyAn GIIPT chuydn LC Quf D6n, Quing 'fa); Nguy4n Thdnh Chung (TIIPT
chuy6n Th6i Binh, Th6i Binh); Ngd Thu Thdo (TIIPT
chuyCn ThSi Nguy6n, Thii Nguy6n); Vd Thdnh Eqt, N4,ryen H6 Minh Phwdc, Vd Bd,Minh Thdng, Le ViAt Minh Th6ng_GHPT chuy6n Qu6c hgc, Thita Thi6n -
HuG); NguyAn Minh Th6ng (THPT chuyCn Ti6n Giang, Ti6n Giang); Biti Manh Khang, Npyen Dodn Hodng Ldm, Ddng Nh|, DSng VQQuanS Thinh (TIIPT chuyCn L6 H6ng Phong, TP. HO Chi Minh), Vd Thanh Dqt (TIIPT cfruyCn TrAn Dai Nghia, TP._HO Chi Minh); Pham Tdn Kha (TIIPT chuy6n Nguydn ThiQn Thanh,
Tri Vinh); Nguy€n Hd Duy (TTIPT chuyCn Tuy6n Quang, Tuy0n Quang); Trdn Nguy4n Tudn V! (TTIPT chuy6n Ng"yen Binh Khi6m, V-rnh Long); Biti Ti€n Diing, Nguy€n Trudng Giang (TIIPT chuy6n VTnh Phric, Vinh Phfc); NS"yA" Xudn Trudng (TT{PT
chuy€n Nguy6n T6t Thanh, YGn B6i); Nguym Thp Khang GryT chuy6n KHTN - DHQG Hn NQi);
Nguy€n Khdc Duy, Vd Hi€n Th6ng @TNK - DHQG TP. Hii Chi Minh); Nsrryi" Viilt Hodng, Phqn Ngpc Khanh, Vfr D*c Vdn (TIIPT chuydn DHSP, EHSP HA NQi); Phan Bd Hodng, Hodng Nhqt Minh, Qudch Thi Phucrng Thdo (TIIPT Chuy6n, DH Vinh); NghiAm
Ductng Ludn (Truimg Tnmg hgc ThUc hanh - DHSP TP. Hd chi Minh).
3. Tuy6n chgn hgc sinh tham du k) thi chgn hgc sinh dq thi Olympic Tofn hgc Qu6c t6 n5m 2016
(rMo 2016)
Cdn cir Quy ch6 thi chgn hgc sinh gi6i c6p qu6c gia
hiQn hdnh vd k6t qu6 kj, thi chqn hoc sinh gi6i qu6c gia THPT m6n Toiin, B0 GD&DT quy6t dinh triQu t4p hgc sinJl. Vii Xudn Trung (lop 12 trudng THPT chuy6n Th6i Binh, Tinh Th6i Binh, thdnh vi6n tlQi
tuy6n hgc sinh ViQt Nary dp thi IMO 2015) vd 49 hgc sinh dqt ti 27,50 di6m trd 16n trong kj. thi v'ua n6u tham dg kj,thi chgn hgc sinh dg thi IMO 2016, tlugc t6 chirc trong cdc ttgiry 24 vd251312016 tai Tht d6 He N6i.
te n.r,r-ror., T?EI#S
Trfilc ntuH OUA GAc BrI ToAtu
sAlProblem: Let .t''(r) = a,,-f," + a,,,r.\"-t + "'+ a,t be Problem: Let .t''(r) = a,,-f," + a,,,r.\"-t + "'+ a,t be
a polynomial with real coe/ficients satisfyingan2a,,-r> "'> ao>0. Prove that evety solution an2a,,-r> "'> ao>0. Prove that evety solution ze C o.f /'satis./ies lzl<1.
Remark. The above result is called the
Enestrom-Kakeya theorem.
Solution. Suppose that there exists a solution
z e C off such that I zl>l . Since /(z) = 0, we
have f(z)(z-1) = 0 . Hence(orr"t + an-rzn * "' * aoz) (orr"t + an-rzn * "' * aoz)
-(onr' + an-rzn-r + "' + ao) = 0 or equivalently,
a,l*1 =(a,-a,-r)l +(a*r-au)la +" +(q-q)z+%.
Therefore
I a,z'*r l<l (a, - a,-r)z' I + I (a,,, - an-r)2"-t I + ...+l(ar-ar)z l+lao I
Han gri'i bii dich: MuQn nhdt lit hai thang sau khi ddng bdi.
Biii rodn. Cho n ld m6t sd U' nhi€n lon hon l. Chirng minh rdng ing circ cdn bdc n cua don vi bdng 0.
Ldi giiii 1. Goi zt,...,zn ldtfit ch cilc cdnbQc n cua dcrn v!, nghia ld zt,...,znld c6c nghiCm phric cira phucrng trlnh z" - 1 = 0. Vi vf,y, ta c6 th6 phAn tich thinh nh6n ti z" -l = (z - zr)...(z - z,) Q). Khai tri6n vi5 ph6i ctra (1) tath6y hQ s6 cira z"-' b vC phii cira (1) bing (-l)(zr+...* z,). M[t kh6c,
hQ s5 cria zn-' b v|tr6i cira (1) bing 0. Do ct6 z1+ 22...+2, = 0 (di6u phii chimg minh).
Ldi siai 2. Giit srr zo =.or[24])* tr,"r+). k = 1,...,n.
\n) \r)
Ta bi6t ring chirng ld t6t ca cdc cdn bQc n cia don vi, lpi c6 Zt.zt:z2,zt.z2=z3...,zt.zn-t=zn,
zt.zn =2,. Tri do zr(2, + 22...+ zn): (2, + 22...+ z,). Ti z, * I ta c6 z, + 22"'+ zn: 0 (di6u phni chimg minh).
NhQn xet. C6c ban sau c6 kri dich t6t, grii bei sorn vA Tda so4n: Hd NQi: Nguydn Hdi YAyl2A1, TIIPT D6ng Quan, Phri Xuy6n, NSuyAn Vdn Dfing,l0Al4, TIIPT Ngqc T6o, Phric Thg; Lio Cai: Phgm Thi Hd,11 To6n,
THPT bhuy6n Lao Cri; ff,*g YGn: Vfi Minh \dnh,l2Toin, THPT chuy6n Hrmg Y6n; NghQ An: Hd Gia Bdo,
1041, TIIpT Cira Ld, TX. Cira Ld, Gia Lai: Trdn D*c Vinh,llAl,THPT Mac Dhh Chi, Chu P1h.
HO HAr (HdNii)
s66
< (l (a, - an-t)l +l (a, 1- an-z)l +
"'+ I (a, -ao) I + I aol)l zl" =l anzl'
(for the second inequality we use thecontradiction assumption that I zl>l and for the contradiction assumption that I zl>l and for the
last equality we use the fact that
inequality I a,,z'*t l'<l anz' I cannot hold if I z l> I
and a, > 0. This is a contradiction. (q.e.d.)TIJ VUNG TIJ VUNG
: m6t c6ch tuong ducrng