7/31/2019 thi tuyn sinh ton chuyn PTNK 2012-2013

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I HC QUC GIA THNH PH H CH MINHTRNG PH THNG NNG KHIU

THI TUYN SINH LP 10 NM 2012Mn thi: TON (Chuyn)

Thi gian lm bi: 150 pht

Cu I .1) Gii h phng trnh:

(x y)2 = 2z z2(y z)2 = 2x x2(z x)2 = 2y y2

2) Cho hnh vung ABCD cnh a. M v N l hai im ln lt nm trn cnh AB v BC sao choAM

AB=

CN

CB= x vi 0 < x < 1. Cc ng thng qua M,N song song vi BD ln lt ct AD

ti Q v CD ti P. Tnh din tch t gic M N P Q theo a v x v tm x sao cho din tch ny lnnht.

Cu II .S nguyn dng n c gi l s iu ha nu nh tng cc bnh phng ca cc c dng can (k c 1 v n) ng bng (n + 3)2.a) Chng minh rng s 287 l s iu ha.b) Chng minh rng s n = p3 (p nguyn t) khng phi l s iu ha.c) Chng minh rng nu s n = pq (p,q l cc s nguyn t khc nhau) l s iu ha th n + 2l s chnh phng.

Cu III .a) Tm gi tr x R tha mn x2 5x + 4 + 2x 1 0

b) Chng minh rng vi cc s khng m a,b,c tha mn a + b + c = 3, ta c bt ng thca +

b +

c ab + bc + ac

Cu IV .Cho tam gic ABC vung ti A. Trn ng thng vung gc vi AB ti B ta ly im D di ngcng pha vi C i vi ng thng AB.a) Chng minh rng nu AC + B D < C D th trn cnh AB tn ti hai im M, N sao choCM D = CN D = 900

b) Gi s iu kin trn c tha mn. ng thng qua A song song vi MD ct ng thng

qua B song song vi MC ti E. Chng minh rng ng thng DE lun i qua mt im c nh.Cu V .

Cho a gic u n cnh. Dng 3 mu xanh,, vng t mu cc nh a gic mt cch ty (minh c t bi mt mu v tt c cc nh u c t mu). Cho php thc hin thao tc sauy: chn hai nh k nhau bt k (ngha l hai nh lin tip) khc mu v thay mu ca hainh bng mu cn li.a) Chng minh rng bng cch thc hin thao tc trn mt s ln ta lun lun lm cho cc nha gic ch cn c t bi hai mu.b) Chng minh rng vi n = 4 v n = 8, bng cch thc hin thao tc trn mt s ln ta c thlm cho cc nh ca a gic ch cn c t bi mt mu.

Ht

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