2010AIME试题及参考答案
2010/03/16  1：30~4：30
1.Mayas lists all the positive divisors of.She then randomly selects two distinct divisors from this list. Let p be the probability that exactly one of the selected divisors is a perfect square. The probability p can be expressed in the form,when m and n are relatively prime positive integers. Find m+n.
解：因为，所以它的所有的因数有个，其中是完全平方数的有，所以，所以m+n=107.
2.Find the remainder when
is divided by 1000.
解：
所以余数是109.
3.Suppose the and.The quantity x+y can be expressed as a rational number,where r and s are relatively prime positive integers. Find r+s.
解：由知x>0,y>0,且，以带入得：
于是，所以，所以，所以x+y=，所以r+s=529.
4.Jackie and Phil have two fair coins and a third coin that comes up heads with probability.Jackie flips the three coins,and then Phill flips the three coins.Let be the probablity that Jackie gets the same number of heads as Phill,where m and n are relatively prime positive integers. Find m+n.
解：投掷三枚硬币的概率分布列如下：
头像朝上的次数
0
1
2
3
P
所以两人投掷硬币出现头像朝上的次数相同的概率为：
，所以m+n=515.
5.Positive integers a,b,c, and d satisfy a>b>c>d，a+b+c+d=2010,and
. Find the number of possible values of a.
解：易知，于是
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，又因为a>b>c>d，且a,b,c,d均是正整数，所以a-b=1,c-d=1,即b=a-1,d=c-1，带入a+b+c+d=2010中得，a+c=1006,但a>c，所以a可以取值504，505，…1004，所以a可能的取值为501个.
6.Let P(x) be a quadratic polynomial with real coefficients satisfying
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for all real numbers x, and suppose P(11)=181.Find P(16).
解：可化为，所以可令，由此三菱翼神改装，所以,而,所以，所以b=1，但P(11)=181,所以100a+1=181，所以a=1.8,所以P(16)=406.
解法二：直接令，易得a=1.8.
7.Define an ordered triple (A,B,C) of sets to be minimally intersecting if,. For example, ({1,2},{2,3},{1,3,4}) is a minimally intersecting triple.Let N be the number of minimally intersecting ordered triples of sets for which each set is a subset of {1,2,3,4,5,6,7}.Find the remainder when N is dividedi by 1000.
解：易知若，共有种办法，且，不妨设a=1,b=2,c=3,把剩下的4个数分成4份(有一份是不安插进来的)，安插到A,B,C中，共有种办法，故N=53760.
8.For a real number a,let [a] denote the greatest integer less than or equal to a.Let R denote the region in the coordinate plane consisting of points (x,y) such that
.
The region R is completely contained in a disk of radius r (a disk is the union of a circle and its interior). The minimum value of r can be written as,where m and n are integers and m is not divisible by the square of any prime. Find m+n.
解：易知或，或，或，于是，或，或，或
或，或
在平面直角坐标系中画出区域来如下图：容易得到此图的中心是，
边界点到其距离的最大值为，所以m+n=132.
9.Let (a,b,c) be a real solution of the system of equations
The greatest possible value of can be written in the form,where m and n are relatively prime positive integers.Find m+n.
解：易知,所以，于是,或，取尼桑颐达，得的最大值为158.
10.Let N be the number of ways to write 2010 in the form
,where the are integers, and.An example of such a represention is.Find N.
解：易知只能取0，1，2三个值，因此根据讨论如下：
（1）当=0时，asx，当时，有8种取法，当时，上海人保车险有2种取法，其余各有10种取法，共100种取法
（2）当=1时，，当时，有8种取法，当时，有2种取法，其余各有10种取法，共100种取法
（3）当=2时，，当时，有2种取法.
故N=202.
11.Let R be the region consisting of the set of points in the coordinate plane that satify both and,when R is revolved around the line whose equation is,the volume of the resulting solid is,where m,n,and p are positive integers,m and n are relatively prime,and p is not divisible by the square of any prime.Find m+n+p.
解：画出区域如右图：
易得
且C到AB的距离为
所以旋转体的体积为
，所以m+n+p=365.
12.Let be an integer an let.Find the smallest value of m such that for every partition of S into two subsets,at least one of the subsets contains integers a,b,and c(not necessarily distinct) such that ab=c.
Note:a partition of S is a pair of sets A,B such that.