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2014全国初中数学联赛(初三组)初赛试卷及答案

发布时间:2014-05-24 08:16:02  

2014年全国初中数学联赛(初三组)初赛试题

3月7日下午 4:00-6:00

考生注意:

1、 本试卷共五大题,全卷满分140分

2、 用圆珠笔或钢笔作答,解题书写不要超出装订线 3、不能使用计算器。

一、选择题(本题满分42分,每小题7分)

本题共6个小题,每题均给出了代号为A、B、C、D的四个答案,其中有且只有一个选项是正确的,请将正确答案的代号填在题后的括号里,每小题选对得7分;不选或错选或选出的代号字母超过一个(不论是否写在括号内),一律得0分。 1、某件商品的标价为13200元,若以8折降价出售,仍可获利10%(相对于进货价),则该商品的进货价是为 ( ) (A)、 9504元 ( B) 、9600元 (C)、9900元 (D)、10000元

2如图,在凸四边形ABCD中,AB=BD=BC,∠ABC=80,则∠ADC等于 ( ) (A) 、800 ( B)、1000 (C) 、1400 (D)、1600 3、如果方程(x?

2)(x2

?4x?m)?0的三根可以作为一个三角形的三边

之长,那么,实数m的取值范围是

( ) (A)0?m?4 (B)m?3 (C) m?4 ( D) 3?m?4

4、如图,梯形ABCD中,AB∥CD, ∠BAD=600,∠ABC=300,AB=6且AD=CD,那么BD的长度是 ( ) (A)

(第2题) (第4题)

1 2014年全国初中数学联赛( 初三组)初赛试卷(第 1 -页共 4页)

( B) 4 (C) (D)

5、如果-2014<a<0,那么x?a?x?2014?x?a?的最小值是

( )

(A)、2014 ( B)、a+2014 (C)、4028 (D)、a+4028

6、方程x2?xy?y2?3(x?y)的整数解有

( )

(A)、3组 ( B)、4组 (C)、5 组 (D) 、6组

二、填空题(本大题满分28分,每小题7分)

1、.如图,扇形AOB的圆心角∠AOB=900,半径为5,正方形CDEF

内接于该扇形,则正方形的边长为

2、.已知四个自然数两两的和依次从小到大的次序是:23、28、33、39、x、y、则x+y=

3、.已

知的值x?y???9,

则为 。

4、有质地均匀的正方体形的红白骰子各一粒,每个骰子的六个面分别写有1、2、3、4、5、6的自然数,随机掷红白骰子各一次,红色骰子掷出向上面的点数比白色骰

子掷出向上面的点数小的概率是 。

三、(本大题满分20分)

已知2a2

的值。

2 2014年全国初中数学联赛( 初三组)初赛试卷(第 2 -页共 4页) ?a?4?0,a?b?2,求12 ?a?1b

四、(本大题满分25分)

Rt△ABC中,∠ACB=900,AE垂直于AB边的中线CD,交BC于点E

(1)求证:AC2=BCCE

(2)若CD=3,AE=4,求边AC与BC的长。

?B

3 2014年全国初中数学联赛( 初三组)初赛试卷(第 3 -页共 4页)

五、(本大题满分25分)

已知二次函数y=x2+ax+b的图像经过点A(x1,0)、B(x2,0)、C(2,m),且0<x1<x2<2

(1) 求证:m>0;

(2) 若b≥1,求证:m<1。

4 2014年全国初中数学联赛( 初三组)初赛试卷(第 4 -页共 4页)

2014年全国初中数学联赛(初三组)初赛

评 分 细 则

一、选择题(本题满分42分,每小题7分)

1、B. 2、C. 3、D. 4、C. 5、A. 6、D.

二、填空题(本大题满分28分,每小题7分)

1

2、. 3、. 54、. 12

三、(本大题满分20分)

解:由已知得b?a?2, 所以

分)

a显然a?0,由2a2?a?4?0得a2?2??. ·············································· (152

分) 12123a. ···················································· (10????2a?1ba?1a?2a?a?2所以3a3a???2, 2aa?a?2??a2

所以

分) 12····················································································· (20???2. ·a?1b

四、(本大题满分25分)

解:(1)因为CD是AB边上的中线,

所以CD=DB,

∠ABC=∠DCB=∠CAE,

∠ACB=∠ECA=90?,

B所以△ACB∽△ECA, ···················································································· (5分)

5 2014年全国初中数学联赛( 初三组)初赛试卷(第 5 -页共 4页)

所以ACCB, ?ECCA

所以AC2?BC?CE. ····················································································· (10分)

(2)因为CD是Rt△ABC的中线,

所以CD=AD=BD。

所以AB=6。

所以AC2?BC2?AB2?36. ········································································· (15分)

取BC中点F,连结DF,则DF//AC,∠DFC=∠ECA=90?,

所以△DFC∽△ECA, 所以

所以

分)

故可解得AC

分)

法2:因为CD是Rt△ABC的中线,

所以CD=AD=BD。

所以AB=6。

所以AC2?BC2?AB2?36。 ········································································· (15分)

由(1)知△ACB∽△ECA, 所以

分)

故可解得AC

分)

五、(本大题满分25分)

证:(1)由已知可得方程x2?ax?b?0的两根为x1、x2,

a所以x1?x2??a,x1?x2,所以2?x2??, ············································ (52

6 2014年全国初中数学联赛( 初三组)初赛试卷(第 6 -页共 4页) DCFC. ?EACABC2CD3···················································································· (20??, ·CAAE2························································· (25BC BCAB63··············································································· (20???。 ·CAEA42························································· (25BC

分) a由已知可得,当x??时二次函数y?x2?ax?b的值随x的增大而增大. 2

所以二次函数在x?2的函数值大于在x?x2的函数值. 即m?0. ········································································································ (10分)

法2:又由已知可得x2?ax?b?(x?x1)(x?x2), 所以m?(2?x1)(2?x2)。 ··············································································· (5分)

又因为0?x1?x2?2,

所以2?x1?0,2?x1?0, 所以m?0. ···································································································· (10分)

(2)由已知得x2?ax?b?(x?x1)(x?x2) 令x?0得b?x1x2,

令x?2得m?(2?x1)(2?x2), 所以bm?x1x2(2?x1)(2?x2)?[1?(x1?1)2][1?(x2?1)2]。····························· (15分)

因为0?x1?x2?2,所以0?1?(x1?1)2?1,0?1?(x2?1)2?1, 并且1?(x1?1)2?1和1?(x2?1)2?1不能同时成立, 所以0?bm?1. ····························································································· (20分)

又b?1,所以m?1. ····················································································· (25分)

7 2014年全国初中数学联赛( 初三组)初赛试卷(第 7 -页共 4页)

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