THI VMEO IV THNG 10

Din n Ton hc

Ngy 2 thng 10 nm 2015

1

1. CP TRUNG HC C S 2

1 CP TRUNG HC C S

Bi ton 1.1. Cho l s thc tha mn 3 = +1. Hy xc nh tt c cc b t hu t (a, b, c, d)tha mn

a2 + b + c =d

Bi ton 1.2. Cho tam gic ABC c gc A t v ng cao AH vi H thuc BC. Trn CA,ABly cc im E,F sao cho BEH = C v CFH = B. Gi BE ct CF ti D. Chng minhrng DE = DF . 4

Bi ton 1.3. Tm tt c cc s nguyn a, b, c tha mn a2 + b2 + c2 = 3(ab+ bc+ ca). 4

Bi ton 1.4. Trong Vt ngc 4, Michael Scofield phi t nhp vo T Chc ly mnh cui

cng ca Scylla. y, anh bt gp mt loi m ho bo v Scylla khi b nh cp. Loi m nyyu cu chn ra tt c cc s thuc bng m ho 2015 2015 sao cho s ny tho mn cc iu kinsau:

S nm lin trn s ny trong bng m ho phi chia 2 d 1. S nm bn phi s ny trong bng m ho phi chia 3 d 2. S nm lin di s ny trong bng m ho phi chia 4 d 3. S nm bn tri s ny trong bng m ho phi chia 5 d 4.

Hi Scofield phi chn ra bao nhiu s nh th ?

y, bng m ho n n l bng gm n2 vung, mi c nh s t 1 n n2. S 1, 2, , nc nh vo hng th nht theo th t tng dn. S n+1, , 2n c nh vo hng th hai theoth t gim dn, ...V d, bng m ho 3 3 c dng:

4

Ht cp THCS

VMEO IV Din n Ton hc

2. CP TRUNG HC PH THNG 3

2 CP TRUNG HC PH THNG

Bi ton 2.1. Vi n l mt s nguyn dng, dy s an c xc nh bi cng thca1 = 1an+1 = 1a1 + a2 + + an

2

Tm gii hn ca dy s Sn xc nh bi Sn = a1 + a2 + + an 4

Bi ton 2.2. Cho tam gic ABC c P,Q l hai im ng gic nm bn trong. Gi AP,AQ lnlt ct ng trn ngoi tip tam gic QBC v PBC ti M,N (M,N nm trong tam gic ABC).

a) Chng minh rng bn im M,N,P,Q cng thuc mt ng trn, gi ng trn ny l (I).

b) Gi MN ct PQ ti J . Chng minh rng ng thng IJ lun i qua mt im c nh khiP,Q thay i.

Ch thch: P,Q c gi l hai im ng gic ca tam gic ABC nu cc ng thng AP,BP,CPln lt i xng vi ng thng AQ,BQ,CQ qua cc ng phn gic trong gc A,B,C ca tamgic ABC. 4

Bi ton 2.3. Cho trc s nguyn dng k. Tm iu kin ca s nguyn dng m theo k sao chotn ti duy nht mt s nguyn dng n tho mn nm | 5nk + 1. 4

Bi ton 2.4. Cho n l mt s nguyn dng. Xp n bn hc sinh A1, A2, .., An cch u nhau trnmt vng trn. Pht cho h mi ngi mt s ko vi tng l m n cy ko. Ta gi mt cu hnhl "cn bng" nu vi mi bn hc sinh Ai bt k, lun tn ti t nht mt a gic u nhn Ai lmnh, v tt c cc bn hc sinh l nh ca a gic ny u c s ko bng nhau.

a) Vi n cho trc tm iu kin ti thiu ca m c th pht ko cho cc hc sinh to ra cuhnh "cn bng".

b) Gi s cc bn hc sinh c th thc hin cc bc chuyn ko, mi bc mt bn c th chuynmt vin ko cho mt bn k bn (tri hoc phi ty ) vi iu kin ngi nhn ko phi c tko hn ngi cho. Chng minh rng nu n l tch ca nhiu nht 2 s nguyn t v m tngng tha mn cu a, khi vi bt k cch pht ko ban u nh th no, ta lun c th av cu hnh "cn bng" sau hu hn ln chuyn ko. 4

Ht cp THPT

VMEO IV Din n Ton hc

CP TRUNG HC C SCP TRUNG HC PH THNG