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1 Nguyn Tng V| Trng Ph Thng Nng Khiu

N THI VO LP 10 CHUYN TON

NM HC 2011 2012

Bi 1.

a) Cho cc s a, b, c, x, y tha

Chng minh rng nu

phn bit v khc 0 th

b) Gii h phng trnh: 22

1 2

1 2 1 4

z xy

x yz xy

!

!

Bi 2.

a) Chng minh rng trong cc phng trnh sau

c t nht mt phng trnh c nghim.b) Cho ba s thc sao cho tng ca hai s bt k khc 1. Chng minh rng trong ba

s c hai s tha 1xyx y khng thuc khong (0; 1).

Bi 3.Cho l cc s t nhin sao cho 20112011

x y

y z

l s nguyn dng. Chng

minh rng:

a)xz =y2b) Khi 1y { th 2 2 2 2 2v 4 x y z x z l hp s.

Bi 4.Cho ng trn (O) v ng thng (d) khng ct ng trn (O). P l mt imthay i trn i trn (d), t P v hai tip tuyn PM, PN n (O) (M, N l hai tip im).

a) Tm v tr ca P sao cho MN t gi tr nh nht.b) Chng minh trung im ca MN thuc mt ng trn c nh.c) Gi Q l hnh chiu ca O trn (d). V A, B l hnh chiu ca H trn PM, PN.

Chng minh rng AB lun qua i qua mt im c nh.

Bi 5. Mt bo tng gm 16 phng hnh tam gic nh hnh v. Hai phng c chung cnhth c th i t phng ny qua phng kia.

a) Mt ngi mun i tham quan ht tt c cc phng v khng mun quay li phngmnh xem. Ngi c th thc hin c khng? Ti sao?

b) Ngi c th i qua nhiu nht bao nhiu phng nu khng i li phng qua.

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8/7/2019 De on Thi Vao Lop 10 Chuyen Toan - De So 2

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2 Nguyn Tng V| Trng Ph Thng Nng Khiu

Li gii.

Bi 1.

a) Cho cc s a, b, c, x, y tha

Chng minh rng nu

phn bit v khc 0 th

b) Gii h phng trnh: 22

1 2

1 2 1 4

z xy

x yz xy

!

!

Li gii.

a) Ta c Suy ra (1)Tng t (2)T (1) v (2) ta c Suy ra

b) T phng trnh (1) ta c T phng trnh (2) ta c , vy

T (2) suy ra Vy h phng trnh c nghim v

Bi 2.

a) Chng minh rng trong cc phng trnh sau

c t nht mt phng trnh c nghim.b) Cho ba s thc sao cho tng ca hai s bt k khc 1. Chng minh rng trong bas c hai s tha

1

xy

x y khng thuc khong (0; 1).

Li gii.

a) t Ta c

Do mt trong ba s c t nht mt s khng m. Vy trong ba phngtrnh trn c t nht mt phng trnh c nghim.

b) Ta chng minh bng phng php phn chng .Gi ba s l a, b, c. Gi s ngc li khng c ba s

Nhn ba s li ta c

8/7/2019 De on Thi Vao Lop 10 Chuyen Toan - De So 2

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3 Nguyn Tng V| Trng Ph Thng Nng Khiu

Suy ra (*)Ta c

Tng t ta c

Nhn 3 bt ng thc ta c (mu thunvi (*).Vy tn ti hai s x, y tha

khng thuc khong (0,1)

Bi 3.Cho l cc s t nhin sao cho 20112011x yy z

l s nguyn dng. Chng

minh rng:

a)xz =y2b) Khi 1y { th 2 2 2 2 2v 4 x y z x z l hp s.

Li gii.

a) Ta c 2011 2011 2011 20112011

x yn x y ny nz x ny y nz

y z

! ! !

V 2011 l s v t v nn ta c Suy ra Khi ta c

b) Theo cu a ta c suy ra hoc Khi lhp s.V l hp s.

Bi 4.Cho ng trn (O) v ng thng (d) khng ct ng trn (O). P l mt im

thay i trn i trn (d), t P v hai tip tuyn PM, PN n (O) (M, N l hai tip im).a) Tm v tr ca P sao cho MN t gi tr nh nht.

b) Chng minh trung im ca MN thuc mt ng trn c nh.c) Gi Q l hnh chiu ca O trn (d). V A, B l hnh chiu ca H trn PM, PN.

Chng minh rng AB lun qua i qua mt im c nh.

Li gii.

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4 Nguyn Tng V| Trng Ph Thng Nng Khiu

a) Gi H l giao im ca OP v MN, ta c OH B MN v HM = HN.Theo nh l Pitagore th (1)V

(2)

V OM khng i nn t (1) v (2) ta c P lhnh chiu ca O trn (d) hay P | Q

b) Gi E l giao im ca OQ v MN, ta c OE.OQ = OH.OP = OM2, suy ra

, v O, Q c nh nn E c nh. Hn na OH B HE, nn H thuc ng

trn ng knh OE c nh.c) Gi C l hnh chiu ca Q trn MN, v t gic QMNP ni tip ng trn ng

knh OP nn ta chng minh c A, B, C thng hng (ng thng Simson)Gi I l giao im ca AB v OQ.V AQCM ni tip v OMPQ ni tip nn ACQ = AMQ =POQ

Hn na OP // QC suy ra POQ =OQC

T ACQ =EQC, suy ra I l trung im ca EQ.

Vy AB lun i qua im I c nh.

Bi 5. Mt bo tng gm 16 phng hnh tam gic nh hnh v. Hai phng c chung cnhth c th i t phng ny qua phng kia.

a) Mt ngi mun i tham quan ht tt c cc phng v khng mun quay li phngmnh xem. Ngi c th thc hin c khng? Ti sao?

b)Ngi c th i qua nhiu nht bao nhiu phng nu khng i li phng qua .

Li gii.

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5 Nguyn Tng V| Trng Ph Thng Nng Khiu

a) Ta c 3 phng gc ch c th i vo hoc i ra vi ch mt phng . Do 3phng gc ch c th l phng xut pht hoc phng kt thc. M xut pht vkt thc ch c hai phng, vy ngi khng th i qua tt c cc phng tha

bi.b) Ta t cc phnh nh sau:

T mt phng mu trng ch c th i vo phng t mu en, c tt c 6 phngc t mu en. Do s phng ngi c th i qua ti a l 6.2 + 1 = 13

phng.

Ta d dng c th ch ra mt cc i nh th.

Vy s phng nhiu nht ngi c th i tha bi l 13 phng.