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美国数学竞赛题1.Find the area of the region that lies under the gra

问题描述:

美国数学竞赛题
1.Find the area of the region that lies under the graph of f (x) =|||| 6 − x | −x | −x | −x | and
above the x axis,between x = 0 and x = 12.
2.Alex is playing a card game and tabulating the results.He calculates that he has played
1800 games and won exactly 1542 of them.Rounded to the nearest percent,this is 86%.
What is the smallest number of consecutive games he would have to win in order for his
winning percentage (rounded to the nearest percent) to be equal to 87%?
3.Find the smallest positive integer which is evenly divisible by 225 and whose digits are all zeros
or ones.
1个回答 分类:英语 2014-10-07

问题解答:

我来补答
第一题
思路是划分函数区间,单独讨论
Case1
0≤x≤6:f(x)=|||(6-x)-x|-x|-x|=|||6-2x|-x|-x|
Subcase1
0≤x≤3:f(x)=||6-2x-x|-x|=||6-3x|-x|
Subcase 1a
0≤x≤2:f(x)=|6-3x-x|=|6-4x|
Subcase 1a,a
0≤x≤1.5:f(x)=6-4x
Subcase 1a,b
1.5<x≤2:f(x)=4x-6
Subcase 1b
2<x≤3:f(x)=|3x-6-x|=|2x-6|=6-2x
Subcase b
3<x≤6:f(x)=||2x-6-x|-x|=||x-6|-x|=|6-x-x|=2x-6
Case 2
6<x≤12:y=|||x-6-x|-x|-x|=||6-x|-x|=|x-6-x|=6
Therefore:
f(x)=6-4x,0≤x≤1.5
f(x)=4x-6,1.5<x≤2
f(x)=6-2x,2<x≤3
f(x)=2x-6,3<x≤6
f(x)=6 ,6<x≤12
面积按5个矩形拼起来算就行了
A=4.5+1.5+1+9+36=51
第二题
87%是从十分位 四舍五入取到的近似值,所以我们只需要把胜率提高到86.5%
Let x be the smallest number of consecutive games he have to win.
We obtain:
(1542+x)/(1800+x)=86.5%
x=111.11
但是这里的四舍五入必须进一位,因为如果只赢111场,胜率是86.499%,无法取到87%
相反,进一位到112后,胜率将会达到86.51%≈87%
Therefore,x=112
第三题
分解225,得到225=25*9
想让25达到0和1的组合,乘上4是最小的.
想让9达到0和1的组合,各位数加起来必须是9的倍数.9最小的倍数是9,所以前面加9个1
最后的数就是11111111100
 
 
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