hio thi (o' * r' + p+t)(q+l):2p'q, suy ra
(o' - o' - p -r)(q-t)=2(t +p+1) . Nhms
UcrN (r3 - p' - p -l,p' + p+ t) = t n€n ho[c
p'-p'-p-l=l hogc p3 - p' - p-l:2 , nglfra
lit p'-p'-p-2=0 ho[c p'-p'-p-3:0.
Phuong t irh p' - p' - p-2:0 (1) c6 nghiOmnguy6n p th\ pl2. Do p ld s6 nguyOn t6 n6n ta nguy6n p th\ pl2. Do p ld s6 nguyOn t6 n6n ta
c6 p =2. Vat p =2 (ld nghiQm cta (1)) suy raq : 15.Khi d6 n=23.15:I20, kh6ng li si5 q : 15.Khi d6 n=23.15:I20, kh6ng li si5
hodn h6o. DC th6,y phuong ftinhp'-p'-p-3=O cflng kh6ng c6 nghiQm p'-p'-p-3=O cflng kh6ng c6 nghiQm nguy0n t5. oo d6 khdng th6 c6 c5c sl5 hoin h6o
d4ng n= P3q .
Thi dU 2.7. Chung rninh ,dng n€tt
n:2"' t(r' -t) n mot s6 hoiin ltao chAn, thi
r/=l*f3'+...-( ,"t' -ll .
lr,
Ldi gidi. Bing quy n?p ta c6 th6 chimg minhitusc t6ng ir:fr(ry*t)'. Gii str k:zT , itusc t6ng ir:fr(ry*t)'. Gii str k:zT ,
th.5 thi ta c6 13 +33 +-..+(zk*l)' =(r' +z' +...+(zt )' )-(r' ++' +..-+(z\') _(z*)' (zt +t)' _.,, t' (k +t)' - 4 -' 4 : k'(2k+t)' -zt' (*+t)' = k' (qk' + 4k +|-zk2 - 4k -2\ = t' (zt' -r). ^ . TORN HOC 34'cfudifi@
Thay gi6 tri cria ft vio ding thric tr€n ta dugc
di6u phii chimg minh.
3. Chc biri toSn ddnh cho b4n dgc
l. MOt s5 n du-o. c ggi ld s6 tam gidc nilu tdn t4i
sii fteN sao cho n=fr(k=*') . Chimg minhz z
ring mgi s5 hoen hio chin li sl5 tam gi6c.2. Tim tdt cb cfuc si5 t.u nhi6n n c6 tinh chdt n 2. Tim tdt cb cfuc si5 t.u nhi6n n c6 tinh chdt n va o(o(n)) ldchc s5 hod, h6o, ho{c ngusc l4i chimg to kh6ng t6n tai sd z nhu th6.
3. Chrmg t6 reng mQt s5 hodn hio c6 ft nh6n tu nguy6n ti5 ph6n biet thi c6 it nh6t mQt nhdn tu
nguy€n t6 kh6ng l'ugt qu6 ft.
4. Gie s'u qr,...,Qn li c5c si5 nguy6n ti5 phdnbiQt. Chimg minh rAng biQt. Chimg minh rAng
(4,+lxqr+l)"'(qo+l)
..2.-,,1fl2...,Q*,
Qflz...Q* __ _ (q,_t)(q,_t). .(qr_t)
k
ld <Ii€u kien can A6 n=llof ld mQt s6 hoan hao.
5. a) Chimg -irn ,arrglrr 6tt p lirsii nguy6n t6
16, thi lty thia pk kh6ng ld sri hodn hio.
b) Chtmg minh rdng s6 c6 dpng 3'.5i kh6ng ld
sO hoin hao.
6. Chimg minh r[ng n6u n=q'alal"'al,vot
,^
Q, ai ld cilc s6 nguyOn tO phdn biet,
Q,e=l (moda) vd ni5u 3 chia h€t n, thl nkh6ng ld s5 hoin h6o 16. kh6ng ld s5 hoin h6o 16.
Tii liQu tham kh:io [1]. Nguy6n Cry H_qf
9015), Li thuydt chia hh vd
d6ng du: ChW.en de b6i dudng hpc sinh gi6i todn Trung hgc phiS th6ng, Nhd xu6t b6n Gi6o duc ViQt Nam.
[2]. Malik Talbi (2014), Saudi Arabia Mathematical Competitions, Riyadl.
[3]. Michael Th. Rassias (2011), Problem - Solving and Selected Topics in Number Theory, Springer. [4]. Jean-Marie De Koninck, Armel Mercier (2007),
1001 Problems in Classical Number Theory,American Mathematical Society. American Mathematical Society.
[5]. Jonathan Pearlaman (2005), Necessary Conditions For the Non-existence Old Perfect.
nY rnr cHeN Hec srNH cl6r qudc GrA THpr
xAna goro - naOu :roeN
.l/ i lhi chon hoc sinh gioi Qu6c gia THPT ntun
't , 20 16 dd duo. c tri chirc trong c6c ngdy 6, 7 vd 8l I I
20 1 6. cho I 2 m6n hoc (ngdy OSZO 1 ta ngdy thi thuc hdnhcho ciic m6n Vdt ly, H6a hqc vd Sinh hoc). Tham du kj, cho ciic m6n Vdt ly, H6a hqc vd Sinh hoc). Tham du kj, thi, o'rn6n To6n c6 464 h1c sinh (nhidu hon 4 hoc sinh
so voi ndm 20 I 5) thuQc 69 don vi du thi; trong d6, c6 4
hgc sinh dang hgc lop l0 (em Phom lllam Khdnh, truong THPT chuyCn Hd N6i - Amsterdam, TP. Hd N6i; ern Ta Anh Ddrrg. truong THPT chuy6n Hing Vuong, tinh Phrir Tho: cm Hu.i'nh Bac:h Khoa yd em Ngul,dn Tnxntg Hai. truimg THPT chuydn TrAn
Hur.rg Dao. tinh Binh ThuAn).
l EC thi
.\'gril' thi thti'nhfu $ll12016) Bni 1 (5 di€m). Gini hQ phucrng trinh
(-
16r-y+z'=3
lz 2 ^
1*'- y" -2z = -7 (x, y, z e IR).
16*'-3y2-v -222 =o
Bni 2 (5 die@. $ Cho ddy sO 1a,; x6c dinh boi
a, =ln(znz +l)-ln(n2 +n+l), v6i n : 1,2, ...
Chtmg minh chi c6 hiru hqn s6 n sao cho
/\l
\o,j <r.
b) Cho day s6 (b,) xicdinh b6d
b,=ln(2nz +1)+1n(n2 +n+l), vdi n = 7,2, ...Chimg minh t6n tai v6 han sii n sao cho Chimg minh t6n tai v6 han sii n sao cho
{t,}.;" rrong d6 {x} ra, hf hiQu phan te criasdth,,rc x' {x}=r-[*]. sdth,,rc x' {x}=r-[*].
Bni 3 (5 aA4. Ctto tam gi6c ABC c6 B, C c6 ditlh, A thay t16i sao cho tam gi6c ABC nhgn. Gqi D ld trung <litim cua BC vd E, F tuong img ld hinh chi6u vu6ng g6c c:0;aDl6nAB, AC.
a) Ggi O ldtdm cria dudng trdn ngopi tirSp tam gi6c
ABC. EF cdt AO vd BC l6n lugt t4i M vd //. Chtmg minh ttuong trdn ngo4i tii5p tam gi6c AMN di qua mOt di6m c6 ttinh.
b) C6c tii5p my6n cta iluong trdn ngo4i 1ir6p ram gi6c AEF t4i E, F cdt nhau t4i Z Chrmg minh 7 thu6c
mQt tludng thlng c6 dinh.
Bni 4 (5 diem,).NWdta tr6ng hai loqi cdy kh6c nhau tr6n mdt mi6ng tl6t hinh chfr nhdt kich thu6c mxn
NGUYfN xTTAC MINH(Ap Khdo thi vd Kiiim dinh CLGD - BO GDE DI) (Ap Khdo thi vd Kiiim dinh CLGD - BO GDE DI)
- i. ^
6 vuong (m6i 6 h6ng mQt cdy). MQt c6ch trdng c6y dugcggi ldiin tugngni5u $u:
i) Sd luqng c6y dugc tr6ng cria hai lopi cdy blng nhau;
ii) 56 luqng ch6nh lQch cua hai lo4i c6y ffin m6i hang
kh6ng nh6 hon mQt nria s6 6 cria hang tl6 vd sli lu-o-ng
ch6nh lQch cria hai lo4i cAy tr€n m5i cQt kh6ng nh6 hon
" i^
mot nua so o cua cot do.
a) H5y chi ra mQt c6ch trdng cdy 6n tuqng khi
m = n =2016.
b) Chrmg minh ni5u c6 mQt c6ch tr6ng c6y lin tuqng th\ ci m vd r d6u ld bQi cta 4.
Ngdy thi thil'hai (7ll12016)
Bni 5 (6 di€m). Tim tdt ci cdc s6 thgc a 116 t6n tai hdm s6 /: IR. -+ IR th6a mAn:
i) f (r)=zorc;
ii) f(*+y*/(y)) =f(x)+ay v6i mqi
x,y e IR. .
Bni 6 (7 di€m). Cho tam gi6c ABC nQi ti6p tlucrng trdn (O) (vcri tdm O) c6 cdc g6c d tlinh B vd C tl6u nhgn. Ldy di6m M t€n"urrg 6D kh6ng chira I sao cho AMtJtdng vu6ng g6c vdi BC. AM chttrung truc
c;ura BC tai 7. Duong trdn ngopi tir5p tam gi6c AOT
cit(o)tqiN (N # A).
a) Chimg minh BAM =CAN.
b) Gqi I ld rdm duong trdn nQi ti€p vd G ld chdn tludng ph6n gi6c trong g6c A cila tam gi6c ABC. AI, MI, NI cdt (O) len lugt t4i D, E, F. Gqi P vi Qtuang
img ld giao di6m crta DF vor AM vd DE vor AN. Duong tron di qua P vir ti6p xric vm AD tqi I cit DF
tai H (H + D ), dulng trdn tli qua Qvitti6p xirc voi AD t4i t c1t nn @i K (K * D ). Chtmgminh iludng trdn ngopi ti6p tam gi6c GHKtrdp xirc vcri BC. Bdi 7 Q diefi. 36 {Buy6n ducmg n dugc goi li s6
hodn chinh n6u r bdng t6ng c6c u6c s6 duong cria n6 (kh6ng kC chinh n6).
a) Chrmg minh ring niSu n h sti hoen chinh 16 thi n c6 d4ng n = psnt2, trong d6 p ld s6 nguy6n t5 c6 dqng 4k * 1 , s ld sii nguy6n duong c6 d4ng 4h + I
;^
vd m ld s6 nguydn ducrng kh6ng chia h6t cho p. b) Tim tdt cb cilc sr5 nguyEn duong n ) 1 sao cho
-(- t 1\ r \ ,.\r. r r,r ,l
n - I vd: deu ld cric s6 hodn chinh. Z
2. K6t qun ch6m thi vi Gi:ii thucrng
Di0m s5 cao nh6t cria ki thi ld37140; c6 I thi sinh dat di6m s6 ndy (em Vil D*c Tdi, trudng THPT chuy€n Ld Hdng Phong. tinh Nam Dinlr).
Chieu theo Quy ch6 thi chgn HSG QG THPT hiQn
hdnh vd cdn cir k6t qu6 chAm thi, Ldnh dao B0GD&DT dd quytit dinh mic di6m cho m5i lopi gi6i GD&DT dd quytit dinh mic di6m cho m5i lopi gi6i nhu sau:
. Gidi Nhfu: tu 32 di6m di5n 40 di6m; . Gidi Nhiz tu 27 di6m diin 31,75 di6m; . Gidi Ba: ti 22 di6m ddn 26,7 5 di€m;