de thi vao chuyen toan truong ptnk - 2010
TRANSCRIPT
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8/9/2019 De Thi Vao Chuyen Toan Truong PTNK - 2010
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I HC QUC GIA TP. HCM THI TUYN SINH LP 10 NM 2010
TRNG PHTHNG NNG KHIU Mn thi: TON (Chuyn)
Thi gian lm bi: 150 pht khng kthi gian pht
Cu 1. a) Cho a, b, c l cc s thc tha mn iu kin 3 3 3 0a b c a b c .
Chng minh rng trong ba s a, b, c c t nht mt s bng 0
b) Gii hphng trnh:
3 3 3 2 2 2
3
1
6 3
x y z
xy yz xz
x y z x y z
Cu 2. a) Gii phng trnh 2 2
2 1 12 2 1x x x
b) Cho tam gic ABC vung ti A v c din tch bng 1. Chng minh rng ta c btng thc 2 2 2BC AB AC
Cu 3. a) Hy ch ra mt b 4 snguyn dng phn bit m tng ba s bt k trong chng lmt s nguyn t.
b) Chng minh rng khng tn ti 5 snguyn dng phn bit sao cho tng ba s btk trong chng l mt s nguyn t.
Cu 4. Cho ng trn tm O, bn knh R v dy cung BC cnh c di 3BC R . A l
mt im thay i trn cung ln BC. GiE l im i xng ca B qua AC v F l im ixng ca C qua AB. Cc ng trn ngoi tip cc tam gic ABE v ACF ct nhau ti K ( KA).
a) Chng minh K lun thuc mt ng trn cnhb) Xc nh vtr im K tam gic KBC c din tch ln nht v tm gi tr ln nht
theo R.
c) Gi H l giao im ca BE v CF. Chng minh rng tam gic ABH ng dng vi tamgic AKC v ng thng AK lun i qua mt im cnh.
Cu 5. Trong mt gii bng c 12 i tham d, thi u vng trn mt lt (hai i bt kthi u vi nhau ng mt trn).
a) Chng minh rng sau 4 vng u (mi i thi u ng 4 trn) lun tm c ba ibng i mt cha thi u vi nhau.
b) Khng nh trn cn ng khng nu mi i thi u ng 5 trn ?Ht