高中数学什么是函值域-2016浙江高中数学省赛试题
???3$$?fifiK??yc N??K@)fi
psg¤
?—$$
1. ??,flb56ABC?,AB ?=
AC,K???AD??8,DE⊥AB ?E,DF ⊥AC ?F,A?KE,KF
z?flA?BC???8M,N,6DEM, 6DFN
?fi%z?fiO
1
,
O
2
o???O
1
O
2
BCo
??:??
fl?,?ADfi A? fl?K,‰ 8AflAttBC,fl?K??
8G,?(ttE,
ttF ,K∠ttAE = 180
?
? ∠ABC, ∠ttAF = ∠ACB,?
DE · ttF = DB sin ∠ABC · AD sin ∠ttAF = DB ·
AD sin ∠ABC · sin ∠ACB
DF · ttE = DC sin ∠ACB
· AD sin ∠ttAE = DC · AD sin ∠ACB · sin ∠ABC
hDB = DC,??,DE · ttF = DF · ttEo
‰8E,Fz?fl?K?K?EP, FP , EFflDG,EDflFP,FDflEPz?
?? 8R,Q,H, ??∠FEH = ∠QFE, ∠QFD = ∠FED,
∠DEH =
∠DFE,fl6DEF ?,$$
ER FH DQ ED sin
∠EDtt EF sin ∠FEH FD sin ∠QFD
· ·
=
·
·
RF HD QE FD sin ∠FDtt ED sin ∠DEH EF sin
∠QFE
sin ∠EDtt sin ∠FED ttE DF
·
= 1 = · =
sin ∠FDtt sin ∠DFE ttF · DE
?1?%??%?,EH, FQ, RDu??8,?G,D,Pu8??o
?fiDtt⊥Att, BCAtt,??PD⊥BCo
hME⊥PE, ?
?P,D,E,Mfl ?PMfi A ? ? 1, ?6DEM ? fi
%O
1
fi??PM ??8,aq“,6DFN
?fi%O
2
fi??PN??8o
?O
1
O
2
MN,?O
1
O
2
BCo
2. x
i
∈
{0, 1}(i = 1, 2, · · · , n),9?8f = f (x
1
,
x
2
, · · · , x
n
)????0 ?1,K‰f
?~Knú$$??
8,flfi
D
n
(f ) =
{(x
1
, x
2
, · · · , x
n
)|f
(x
1
, x
2
, · · · , x
n
) = 0}
C1P?nú$$??8?K8;
C2P
g?10ú$$??8,???g(x
1
, x
2
, · · · ,
x
n
) ≡ 1 + x
1
+ x
1
x
2
+ x
1
x
2
x
3
+ · · · +
x
1
x
2
· · · x
n
(mod2)o
Σ
(x
1
+ x
2
+ · · · , x
n
)
< 2017o
?8?D
n
(g)?úfiK8,fl???f‰8n,?8
(x
1
,x
2
,···
,x
n
)∈D
n
(g)
):C1Px
1
, x
2
, · · · ,
x
n
??$$?U????$$2K,zKfi ??8?‰$$0?1RK,??$$???nú
n
$$??8?K8fi2
2
;
C2Pfi|D
n<
br>(g)|??8?D
n
(g)?úfi?K8,?3|D
1
(g)|
= 1, |D
2
(g)| = 1,h
.
n i
.
n i
Σ Y
Σ Y
ΣΣ
(mod2)
g(x
1
, x
2
, · · · , x
n
)
≡ 1 + x
j
=
1 + x
1
1 +
i=1 j=1 i=2 j=2
x
j
n
K|D
n
(g)| =
2
n?1
|D
n?1
(g)|(n = 3, 4, · · · ),?
n?1
n
Σ
2 + (?1)
(g) =
k n?1?k
=
n?1
? 2
n?2
+ |D
| · · ·
= ( 1)
? ·
2
|D
n
(g)| = 2
n?2
3
k=0
n+1
169
fic
n
=
Σ
(x
1
+ x
2
+ · · · +
x
n
), ??
(x
1
,x
2
,···
,x
n
)∈D
n
(g)
Σ Y
g(x
1
, x
2
, · · · ,
x
n?1
, 0) ≡ 1 + x
j
(mod2)
Σ Y
g(x
1
, x
2
, · · · ,
x
n?1
, 1) ≡ 1 + x
j
+
x
1
x
2
· · · x
n?1
(mod2)
i=1 j=1
i=1 j=1
n?1 i
n?1 i
?
?
0, nfi?8
, 1) ≡ n + 1(mod2)
=
9x
1
x
2
· · · x
n?1
= 1,Kg(x
1
, x
2
, · · · ,
x
n?1
?
1, nfi?8
9x
1
x
2
· · · x
n?1
= 0,K
n?1 i
Σ Y Σ Y
x
j
(mod2) = 0 ?? g(x
1
, x
2
, · · ·
, x
n?1
) ≡ 1 + x
j
(mod2) = 0
i=1 j=1 i=1 j=1
n?1 i
g(x
1
, x
2
, · · · ,
x
n?1
, 1) ≡ 1 +
?
?
,?3,c
?ün ≥
2?,c
n
=
n + 2c
n?1
+
|D
n?1
(g)|, nfi?8;
1
= 1, c
2
= 1o
?
2c
nfi?8.
n?1
+
|D
n?1
(g)| ? n,
ün = 2m?,
c
2m
= 2c
2m?1
+
? 2m
3
1 4
2m?2
+ (2m ? 2) + = ? + × 2 4c
2m?2
3 3
m
(3m + 1)4
?
(6m + 1)
2
=
§
9
aq“,ün = 2m + 1?,
2
2m
+ (?1)
2m+1
?
2m
2
2m?1
+ ( 1)
c
2m+1
= 2m + 1 + 2c
2m
+
3
1 4
2m?1
? (2m ? 1) + 4c = + × 2
2m?1
3 3
(6m + 5)4
m
+ 6m + 4)
3
=
§
9
2
,§
3
?c
9
= 828 < 3986 = c
11
,
c
10
= 1817 < 8643 = c
12
?,?
8fi§
1
???
??fi§
8n = 10o
3.
a
1
, a
2
, · · · ,
a
n+1
fif?8,???
?f‰
Σ Σ Σ
a
i
a
i+1
Σ
a · a ≥ ·
(a
+ a
i+1
)
i i+1 i
a + a
i+1
i=1 i=1 i=1 i=1
i
n n n n
170
??:
??
.
Σ
n
a
i
i=1
Σ .
Σ
a
i+1
n
Σ
?
?
2
2
n
n
?
n
?
n
Σ
Σ
a
i+1
?
a
i
a
i
+
a
i+1
? ?
??
Σ
i=1
i=1
?
?
?
Σ
?
=
?
?
? ?
?
i=1
i=1
2
i=1
2
.
Σ
2
n+1
? a
1
.
,
=
a
2
4
?
.
n
n
ΣΣ
(a
a a
i i+1
i
? a
i+1
2
Σ
)
) ?
4 =
(a + a
i i+1
a
i
+ a a
i
+ a
i+1 i+1
i=1 i=1
n
2
Σ
n
Σ
(a
i
? a
i+1
)
= (a +
a
) ?
i i+1
a
i
+ a
i+1
i=1
i=1
i=1
n
Σ
Σ
2
(a
i
+ a
i+1
)
??, ? fi d?
Σ
2
.
n
. Σ . Σ
Σ
ΣΣ
(a
i
+ a
i+1
)
n n
n
Σ
(a + a (a
i
?
a
i+1
)
2
(a + a )
(a
n+1
? a
1
2
)
1
i=1
) ?
? ≥
i+1 i i+1
4 4
aa
i
+
i+1
4
i=1
i
i=1
i=1
Σ
a
i+1
Σ .
n
.
n
(a
i
?
ΣΣ
2
)
2
??
(a
i
+ a
i+1
))
≥ (a
n+1
? a
1
)
a
i
+ a
i+1
i=1 i=1
???? a? fi??oK1,?K??o
4. fi?K?f‰8n,fiD
n
fin?f
8?Kof
i
(n)(i = 0, 1, 2, 3)fi8?
F
i
(n) = {a ∈ D
n
|a ≡ i(mod4)}
?úfiK8,??i = 0, 1, 2, ??f‰8,?8
f
0
(m) + f
1
(m) ? f
2
(m)
? f
3
(m) = 2017
):?f‰8m???IOz?fi
Σ
l
k
. Σ .
γ
j
β
Y
q
m = 2
α
Y
p
i
i j
i=1 j=1
??,α??fi0,p
1
, p
2
, · · · ,
p
k
fim??$$?4{1?fi?$$;q
1
, q
2
,
· · · , q
l
fim?$$?4 {3?fi?$$,fi
k l
Q Q
?k,l ?$$0 ??,??fi??“fi1o÷P =
(1 + β
i
), Q =
(1 + γ
j
)o
i=1 j=1
h???$$(??
C1Püα ≥
1?,f
0
(m) = (α ? 1)PQ, f
2
(m) = PQ;
C2Pf
1
(m) + f
3
(m) = PQ;
, ,
Q
C3Pf (m) = P o
1
2
Σ
t
k
l
γ
j
t
.
.
Σ
q
$$?m
?K?f 8
a = 2
Q
p
i
Q
j
,
β
α
t
i=1
i
j=1
r
??α
r
∈ {0, 1, · · · , α}, β
i
∈
{0, 1, · · · , β
i
}(1 ≤ i ≤ k), γ
r
∈ {0, 1, · · · , γ
j
}(1 ≤ j ≤ l)o
r r
üα ≥ 1?,a ∈ F
0
(m)üRKüα
r
≥ 2,a ∈ F
2
(m)üRKüα
r
= 1,
β
i
, γ
j
(1 ≤ i ≤ k, 1 ≤ j ≤ l),????
‰?$$PQ,?1P??o
171
??,f
1
(m) + f
3
(m)??m??
8?K8,?3,f
1
(m) + f
3
(m) =
PQ,?(?C2P??o
(?C3P????p~#?gK??fl%o
f‰8m??f
0
(m) + f
1
(m) ?
f
2
(m) ? f
3
(m) = 2017;
., ,
Σ
Q
2017 = f (m) ? f (m) = 2f (m) ? PQ =
P 2
? Q
1
3 1
2
, ,
k
Q Q
fi?8oh
??,??U
,K
P
=
(1 +
β
i
) = 2017
o
Q
2 ? Q = 1
2
i=1
l
Q
γ
j
2016
2016
??,
fifi8,??U
,K1,
2017
k
= 1, β
1
= 2016
q
j
≥ 5
;
m = p
1
9mfi?8,Kf
0
(m) = f
2
(m) =
0,?)(?C2PC3P8
j=1
9mfi?8,K?(?C1PC2PC3P8
2017 =
f
0
(m) ? f
2
(m) ? (f
1
(m) +
f
3
(m)) + 2f
1
(m)
, ,
Q
=
(α ? 1)PQ ? PQ ? PQ + 2P
2
., ,Σ
Q
1
= P (α ? 3)Q + 2
§
2
2016 2016
üP = 2017?,C$$m ≥ 2;
,
p
1
,
> 5
Q
üP =
1?,(α ? 3)Q + 2 = 2017,?3,(α ? 3)Qfi?8,?Qfi?8o
2
?1,K{8(α ? 2)Q = 2016 = 2
5
×
3
2
× 7,??2
5
|(α ? 2)o
9α ? 2 >
2
5
,Kα ? 2 ≥ 2
5
× 3,?,C$$m ≥
2
98
;
9α ? 2 = 2
5
, ??Q = 7 × 3
× 3 = 9 × 7 = 21 × 3 = 63,z?$$¤ ?;
CiPl = 3, 1
+ γ
1
, 1 + γ
2
, 1 + γ
3
fi7,
3, 3?~K??, ??,q
1
, q
2
, q
3
fiRR????4{3 ?fi8,
?m ≥ 2
34
×
3
6
× 7
2
× 11
2
;
CiiPl =
2, {1 + γ
1
, 1 + γ
2
} = {9, 7},?,m ≥
2
34
× 3
8
× 7
6
> 2
34
× 3
6
× 7
2
× 11
2
;
CiiiPl = 2, {1 + γ
1
, 1 + γ
2
} =
{21, 3},?,m ≥ 2
34
× 3
20
× 7
2
> 2
34
× 3
6
× 7
2
×
11
2
;
CivPl = 1, 1 + γ
1
=
63,?,m ≥ 2
34
× 3
62
> 2
34
×
3
6
× 7
2
× 11
2
;
??1??,fl ??,2
34
× 3
6
× 7
2
× 11
2
> 2
34
× 16
10
<
2
98
,?
2 34
× 3
6 2
m ≥
min{5
2016
, 2
34
× 3
6
×
7
2
× 11 } = 2 × 7
2
× 11
1
,K1,??? ?f‰ hüm = 2
34
× 3
6
× 7
2
× 11
2
?,mfi?8,Rα = 34, P
= 1, Q = 63,??fi§
8mfi2
34
× 3
6
× 7
2
× 11
2
o
psg¤
?s$$
5. a,b,cfi?8,a ?= 0o9~úsgS?2ax
2
+ bx + c = 0fl[?1, 1]1$$??o???
min{c, a + c
+ 1} ≤ max{|b ? a + 1|, |b + a + ?1|}
fl(?1??
fi ????,a,b,c???S?^flo
??: ??,max{|b ? a + 1|,
|b + a ? 1|} = |b| + |a ? 1|,K???? fi d?
min{c, a + c + 1} ≤ |b| + |a ? 1|
%^fl?,?flx
0
∈ [?1, 1],??2ax
2
+
bx
0
+ c = 0o
0
?$$za > 0fla < ???;
1
§
172
C1Püa > 0?,??
min{c, a + c + 1}
= c = ?2ax
2
0
? bx
0
≤ ?bx
0
≤ |b| ≤ |b| + |a ? 1|
2
1
??,Rfi§
1
?? ?üRKü|a ? 1| = 0,
?2ax
0
??,fi§
= 0, c = ?bx
0
=
|b|,?a = 1, x
0
= 0, c =
b = 0;
C2Püa < 0?,??
2
? bx
0
+ 1
min{c, a + c + 1} ≤ a + c + 1 = a ?
2ax
0
≤ a + (?2a) + |b| + 1 = |b| + 1 ? a
= |b| + |a ? 1|
1
K??,Rfi§
1
?? ?üRKüa + c + 1 ≤ c,
?2ax
2
??,fi§
= ?2a, ?bx
0
=
|b|,Ka ≤ ?1,?
0
1x
2
=
1CRx
0
flb??P,?2a ? |b| + c = 2ax
2
+
bx
0
+ c = 0o
0 0
1
??C? ?
fi8?P?C1PC2P?,fi§,R ????,a,b,c???Sz‰?^flfi(a, b,
c) =
(1, 0, 0)
?a ≤ ?1, 2a ? |b| + c = 0o
?
DC ??8fiMo‰M,O,Du8 6. ??,fl O?$$
¤fiABCD?,fi5?AC,BD^??A,KA
? flDA,DC
z???E,Fo???BE = BFo
??:??fl?,?(ME, MF, MD, MA,
MB, MC, EF ,
?M,E,O,F,D68? ?
∠MFE =
∠MDE = ∠MDA = ∠MCA,
∠MEF = 180
?
? ∠MDF
= 180
?
? ∠MDC = ∠MAC
h?MfiK
?
ADC??8?,
∠MCA = ∠MAC =?
∠MFE = ∠MEF =? ME = MFo
A?MOfl?O???~8N,?(OE,
OF, NA, NC,K
∠DEO = ∠DMO = ∠DMN = ∠DAN =? OEAN
aq“,OFNCo
?(DOflKM,flAN??8G,‰8GflttKNC,flDC??8K,K
6AttK fl6EOF ?q,?q?%fiD,K1,EFAKo
?ttKNC?A,N,C,D¤8? ?
∠AttK = ∠ANC =
180
?
? ∠ADC
=? A, tt, K, D¤8?
=?
∠AKtt = ∠ADtt
hDB⊥AC,K∠BDC = 90
?
?
∠ACD = ∠ADtt,
?∠BNC = ∠BDC = ∠ADtt =
∠AKtt,(?ttKNC,?NBAK,
h?MNfi?O ?A?,8BM ⊥NBo
K1BM ⊥AK =? BM ⊥EFo
?BMfi??EF??A?z?,?BE
= BFo
173
18
7. ?
??f‰8n,?8?flnK^????‰8x
1
, x
2
, · · ·
, x
n
,???x
2
+ x
2
+ · · · +
x
2
= 2017o
Σ
1 2 n
):?? ??
K^?f‰8??Sefi
2
,KK? K^?f‰8??Se?
18
i
= 2109 > 2017 18
?2017,K1,n ≤
17o
i=1
17
Σ
b
17K^?f‰8x
1
, x
2
, · · · ,
x
17
,??
x
2
i
= 2017
i=1
1
, §
?fi2017?3N + 8,R?9x
j
?fi3? 8?,??x
2
· · , x
17
???‰
j
≡ 1(mod3),??,x
1
, x
2
, ·
??8?K8 $$3a + 1K,?{3b + 1Kfi3? 8,??(3a + 1)+(3b
+ 1) = 17(a, b ∈ N ) =? a + b = 5,
17
Σ
??,
2
= 1785,1 2017 ? 1785 = 232o
i
i=1
1
$$?,IK8 fl8?M = {1, 2, · ·
· , 17}?,3? 8$$5K,?{12K8pfl3^fi,??§
?M?VKkK8,%?
kK??17?8,?8??Se?Oe232;
ük = 1?,üK8?M?VK~K3?
8C?88?M??$$?3 8?K8fi3b + 1K,b = 1P,??
I???(3x)
2
+ 232 = y
2
(x ∈ {1,
2, 3, 4, 5}, y > 17) ??U;
ük = ?;
C1PK8?M?VKRK3 8,fiSRK??17?3 8,h$$?VKM?RK ??3
8,fiS
RK??17??3 8,??
(18
2
+
21
2
) ? (16
2
+ 17
2
) = 220
?= 232
1üKM?VK??8?s 8?s $$~K%fi???8
?,1?
?“?;
C2PK8?M?VK~K3 8,~K3 8,fiSMfiRK3 8oaq“,$$?M
??$$?1??
??8flM ?8,??
??“?;
(19
2
+ 20
2
) ? (15
2
+
17
2
) = 247 > 232
C3PK8?M?VKRK3
8,fiSMfi~K3 8,~K3 8o? ,$$?M??$$?1???
?8flMfi??8,??
(18
2
+
19
2
) ? (12
2
+ 15
2
) = 316 >
232
?
?h“?o
ük ≥ 3?,
8?M?VKkK8a
1
, a
2
, · · · ,
a
k
,MfifiSkK8b
1
, b
2
, · · ·
, b
k
,?3,$$
k
Σ
2 2 22 2 2
(b
? i
2
? a
2
i
) ≥ (18+ 19+ 20) ?
(15+ 16+ 17) = 315 > 232
“?o
i=1
?1??,n ≤ 16o
16
Σ
2
2 2 2 2 2
2
ü ?,$$?
,?fi
,1
,
2017 ?
1496 = 521
n = 16 i
= 1496 (17 + 18
+ 23 ) ? (13
+ 14 + 16 ) = 521
?
?,?13, 14, 16?%?17, 18, 23,??1, 2, · · · , 12, 15,
17, 18, 23 ?16K^ ???? f‰ 8 ?? Se
fi2017o
????f‰8nfi16o
8. ??‰8m ≥ 2, n ≥ 3, 8?
S = {(a, b)|a ∈ {1, 2, · · · , m}, b ∈ {1, 2,
· · · , n}}
i=1
174
AfiS?$$8,9??flf‰8x
1
, x
2
,
x
3
, y
1
, y
2
,
y
3
,?8x
1
< x
2
< x
3
,
y
1
< y
2
< y
3
R
(x
1
, y
2
), (x
2
,
y
1
), (x
2
, y
2
),
(x
2
, y
3
), (x
3
, y
2
)
∈ A
?8?A?úfi?K8???o
):9~K8?X??fl?8(x
2
,
y
2
),?1,$$,?,?¤KS?z?K$$X ??8(x
2
,
y
3
), (x
2
, y
1
),
(x
1
, y
2
),
(x
3
,
y
2
),K‰fi“?%88”o??, ?K d?(?88S?s;U?M;
K8,?8??‰8?
??8?A?fi?%88o
h?S?‰?Ifi1?m?8,??
p?Ifi1?m?8???M,??8?A
0
,?3,A
0
?fi?%88,
?|A
0
| = 2m + 2n ? 4o
????9A ? S, |A| ≥ 2m + 2n ? 3,KA‰fi?%88o
fi1 ≤ i ≤ n, A?p?Ifii?8$$k
i
K,9k
i
≥ 3,K?A? 1? ?, ?R8Sfi?8???o
??,s $$
(k
1
? 2) + (k
2
? 2) + · ·
· + (k
n
? 2) = |A| ? 2n
K8?fi??;
fi2 ≤ j ≤ m ? 1, A?‰?Ifij?8$$l
j
K,9l
j
≥ 3,K?A? ?? 1, $$R8Sfi?8?7
?;??A?‰?Ifi1?m?8???7?,??,s $$
l
1
+ (l
2
? 2) + · · · + (l
m?1
?
2) + l
m
= |A| ? 2m + 4
K8?fi7?o
??,
(|A| ? 2n) + (|A| ? 2m + 4) = (|A|
? 2n ? 2m + 3) + |A| + 1 > |A|
‰$$~8???)??,h??)7?,R?K8?‰?I?fi1?m(‰?Ifi1 ?m
?8??U?fi?
?Po
????Sfi,?8?1,$$,?,?¤KS?‰$$A???K8,?fi?%88o
?1,8?A?úfiK8? ??fi2m + 2n ? 4o
175