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# 由一道美国数学竞赛题谈起

2006 年第 9 期

21

( 合肥工业大学数学与信息科学系 ,230009)

B F 与 CG 交于点 S , CG 与 A E 交于点 T . AT BR CS = = = TE RF SG

λ λ +1 λ λ( z + 1) x +1 故 x= λ = = . λ z +λ + 1 +1 +λ + 1 z +1 x +1 整理得 (λ + 1) x2 + (λ + 1) x - (λ + 1) λ= 0. 由于 λ+ 1 > 0 ,故有 2 x + x - λ = 0. ⑦ 1 解得 x = ( 4λ+ 1 - 1) ( 负根舍去) . 2 由式 ⑦ 得 x ( x + 1) = λ. 结合式 ⑤, 则有 ⑥ 1 x = y = z = ( 4λ + 1 - 1) . 2 由三角形面积比性质知
S △CAT AT = = λ. S △CTE TE

λ,则有
S △RS T =

1 ( 2λ+ 5 - 3 2 (λ+ 1)

4λ+ 1 ) .

S △CAT λ = ,即 S △CTE + S △CAT λ+ 1

CF AB GS · · =1 , FA BG SC AG BC ET · · =1 , GB CE TA B E CA FR · · = 1. EC A F RB

S △CAT =

λ λ+ 1 S △CA E .

② ③ ④

S △CA E CE CE = = ,于是 , S △CAB BC CE + EB

S △CA E =

1
z +1

.

CF AG BE =x, =y, = z. FA GB EC

y ( z + 1) = λ , z ( x + 1) = λ.

⑤ ⑥

=1 3λ 1 λ+ 1 · + 1 x ( 下转第 39 页)

λ
y +1

,y =

λ
z +1

,z =

λ
x +1

.

2006 年第 9 期

39
a+ b+ b- a b +4 a b 4a
2 2 2

(2) △EB 1 F

△CFB 1 . △EBB 1 或 △EB 1 F △CFB 1 . △B 1 CF.

a+ b+ c≥ b- a b a
2

=

4 a + 4 ab + b 4 a ( b - a)
t +4t +4 4 ( t - 1)
2

BB 1 B 1 E = . CF FB 1

+4 × 4
b - 1 a

=

=

B1 C B1 E 因为 BB 1 = B 1 C ,所以 , = . CF FB 1

= = =

( t 2 - 1) + 4 ( t - 1) + 9 4 ( t - 1) 1 4 1 4 ( t + 1) + 4 +
t- 1+

b 2a
2 2

9
t- 1

9
t- 1

+6

b
2

+ c-

b ≥ 0 恒成立 . 4a

≥ 1 (6 + 6) = 3. 4 当且仅当 t = 4 ,即
b 2 = 4 , b = 4 ac 时 ,上式等号 a

a+ b+ c 恒成立 ,得 m < b- a b ,则 t > 1. 从而 , a

a+ b+ c b- a

.
min

a+ b+ c b- a

= 3.
min

( 鲁有专 江苏省南京市南大附中 ,210008)

( 上接第 21 页) = 1 ( 2λ + 5 - 3 2 (λ+ 1) 4λ + 1 ) .

x ( y + 1) = p , y ( z + 1) = m , z ( x + 1) = n .

CF AG B E = = . FA GB EC AT BR CS = = 时, TE RF SG

AT BR CS = = . TE RF SG

CF AG B E = = = μ, FA GB EC

(μ - 1) 2 . 2 μ + μ+ 1

AT BR CS = = TE RF SG

AT BR CS = = = λ. TE RF SG

2

CF AG B E = = . FA GB EC

S △RS T =

[ 1 ] Sixty2second William Lowell Putnam Mathematical Com2 The petition[J ] . Amer Math Monthly 2002 ,109 (9) :827 — 839.

(μ - 1) 2 . 2 μ + μ+ 1

AT BR CS =m, = n, = p. TE RF SG

75人参加一次数学竞赛,其中第一道题做错的有39人,第二...
75人参加一次数学竞赛,其中第一道题做错的有39人,第二题做错的有26人,第一和第二题都做对的有16人,那么这两道题都做错的有( )。...

3.今年前 5 个月,小明每月平均存钱 4.2 元, 6 月起他每月储蓄 6 元,...2.某小学举行一次数学竞赛,共 15 道题,每做对一题得 8 分,每做错一题倒...