广东省华南师范大学数学科学学院(510631) 李湖南

(A)0 (B)1 (C)2 (D)3 (E)4

解原式=20+19=2,故(C)正确.

2.What is the hundreds digit of(20!-15!)?

(A)0 (B)1 (C)2 (D)4 (E)5

译文:(20!-15!)的百位数是多少?

解20!的标准分解式中含有个5,15!的标准分解式中含有=3个5,即它们的最后三位数均为0,从而(20!-15!)的百位数是0,故(A)正确.

3.Ana and Bonita were born on the same date in different years,nyears apart.Last year Ana was 5 times as old as Bonita.This year Ana's age is the square of Bonita's age.What isn?

(A)3 (B)5 (C)9 (D)12 (E)15

译文:安娜和博尼塔出生在相隔了n年的同一天.去年安娜的年龄是博尼塔的5倍,今年安娜的年龄是博尼塔年龄的平方.问n是多少?

解设去年博尼塔x岁,则安娜5x岁,依题意有5x+1=(x+1)2,解得x=3,于是有n=5x-x=12,故(D)正确.

4.A box contains 28 red balls,20 green balls,19 yellow balls,13 blue balls,11 white balls,and 9 black balls.What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn?

(A)75 (B)76 (C)79 (D)84 (E)91

译文:一个盒子里有28个红球,20个绿球,19个黄球,13个蓝球,11个白球和9个黑球.在不放回的情况下,必须至少从盒子中取出多少个球才能确保会取出15个同种颜色的球?

解依题意,红球、绿球、黄球得取出14个,蓝球、白球、黑球全部取出,再加任意1个就符合条件,即至少需要取出14×3+13+11+9+1=76个球,故(B)正确.

5.What is the greatest number of consecutive integers whose sum is 45?

(A)9 (B)25 (C)45 (D)90 (E)120

译文:最多有多少个连续整数的和等于45?

解设有x个连续整数,从n开始,即n+(n+1)+(n+2)+···+(n+x-1)=45,可得x(2n+x-1)=90,最大值解为x=90,此时n=-44,故(D)正确.

6.For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?

—a square

—a rectangle that is not a square

—a rhombus that is not a square

—a parallelogram that is not a rectangle or a rhombus

—an isosceles trapezoid that is not a parallelogram

(A)1 (B)2 (C)3 (D)4 (E)5

译文:下列四边形中,四边形所在平面内存在一点与四边形的四个顶点距离都相等的有多少种?

①正方形;②不是正方形的矩形;③不是正方形的菱形;④不是矩形或菱形的平行四边形;⑤不是平行四边形的等腰梯形.

解存在一点与四边形的四个顶点距离都相等,即四边形存在外接圆.矩形、正方形和等腰梯形都存在外接圆,但菱形和平行四边形就未必存在,故共有3种,(C)正确.

图1

解先写出这两条直线方程:y-和y-2=2(x-2),然后分别与方程x+y=10联立,求得交点为B(6,4)和C(4,6),设点(2,2)为A,则同理可得.如图示,作AD⊥BC于D,则D为BC中点,从而AD=于是故(C)正确.

8.The figure below shows linelwith a regular,infinite,recurring pattern of squares and line segments.How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn,other than the identity transformation,will transform this figure into itself?

—some rotation around a point of linel

—some translation in the direction parallel to linel

—the reflection across linel

—some reflection across a line perpendicular to linel

图……