<>题目来源:(<a href="http://post.baidu.com/f?kz=10433388" target="_blank" ><FONT face="Times New Roman">http://post.baidu.com/f?kz=10433388</FONT></A>)
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<>(<FONT face="Times New Roman">1</FONT>)求<FONT face="Times New Roman">1287XY6</FONT>能被<FONT face="Times New Roman">72</FONT>整除的所有七位数。
(<FONT face="Times New Roman">2</FONT>)已知三角形<FONT face="Times New Roman">ABC</FONT>,角<FONT face="Times New Roman">A=80</FONT>度且<FONT face="Times New Roman">AB=AC</FONT>,<FONT face="Times New Roman">O</FONT>为三角形<FONT face="Times New Roman">ABC</FONT>内任意一点,
<FONT face="Times New Roman"> </FONT>又知角<FONT face="Times New Roman">OBC=10</FONT>度,角<FONT face="Times New Roman">OCB=20</FONT>度。求角<FONT face="Times New Roman">OAC=</FONT>?
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<>我给出的做法(仅供参考,如有不对,敬请指教)
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<P>解:(<FONT face="Times New Roman">1</FONT>)由于<FONT face="Times New Roman">72=8*9</FONT>,这样根据能被<FONT face="Times New Roman">8</FONT>整除和能被<FONT face="Times New Roman">9</FONT>整除数的特征,就可以知道有如下的结论:①<FONT face="Times New Roman">100*x+10*y+6</FONT>是<FONT face="Times New Roman">8</FONT>的正整数倍,不妨假设<FONT face="Times New Roman">100*x+10*y+6=8*m</FONT>(<FONT face="Times New Roman">m</FONT>为正整数)。
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<P><FONT face="Times New Roman"> </FONT>②<FONT face="Times New Roman">1+2+8+7+x+y+6</FONT>是<FONT face="Times New Roman">9</FONT>的正整数倍,不妨记为<FONT face="Times New Roman">1+2+8+7+x+y+6=9*n</FONT>(<FONT face="Times New Roman">n</FONT>为正整数)。
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<P>借助计算机很快就可以得到结果!(用<FONT face="Times New Roman">matlab</FONT>解决)
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<P>程序如下:<FONT face="Times New Roman">X=[];Y=[];x=1;y=1;
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<P><FONT face="Times New Roman">for i=1:9
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<P><FONT face="Times New Roman">for j=1:9
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<P><FONT face="Times New Roman"> if (rem(100*i+10*j+6,8)==0)&(rem(24+i+j,9)==0)
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<P><FONT face="Times New Roman"> X(x)=i;Y(y)=j;
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<P><FONT face="Times New Roman"> x=x+1;y=y+1;
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<P><FONT face="Times New Roman"> end
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<P><FONT face="Times New Roman"> end
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<P><FONT face="Times New Roman">end
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<P><FONT face="Times New Roman">X,Y
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<P><FONT face="Times New Roman">X =2 5 9
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<P><FONT face="Times New Roman">Y =1 7 3
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<P>也就是<FONT face="Times New Roman">1287216</FONT>,<FONT face="Times New Roman">1287576</FONT>,<FONT face="Times New Roman">1287936</FONT>。
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<P>当然用手计算也可以,也是很容易检验出来的。
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<P>(<FONT face="Times New Roman">2</FONT>)根据正弦定理可以得到:
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<P>假设角<FONT face="Times New Roman">OAC=x
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<P>则有:<FONT face="Times New Roman">OC/sin</FONT>(<FONT face="Times New Roman">x</FONT>)<FONT face="Times New Roman">=OA/sin</FONT>(<FONT face="Times New Roman">30</FONT>度),
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<P><FONT face="Times New Roman"> OB/sin</FONT>(<FONT face="Times New Roman">80</FONT>度<FONT face="Times New Roman">-x</FONT>)<FONT face="Times New Roman">=OA/sin</FONT>(<FONT face="Times New Roman">40</FONT>度)。
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<P><FONT face="Times New Roman"> OC/sin</FONT>(<FONT face="Times New Roman">10</FONT>度)<FONT face="Times New Roman">=OB/sin</FONT>(<FONT face="Times New Roman">20</FONT>度)。
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<P>这样可以得到方程,可以求出来。
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<P>结果是:角<FONT face="Times New Roman">OAC=20</FONT>度。
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<P>当然,这个方程是用计算机解出来的。
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<P>程序如下:
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<P><FONT face="Times New Roman">solve('sin(10*pi/180)*sin(80*pi/180-x)*sin(pi/6)=sin(x)*sin(40*pi/180)*sin(20*pi/180)','x')
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<P><FONT face="Times New Roman">ans =atan(sin(1/18*pi)*cos(1/18*pi)/(sin(1/18*pi)^2+2*sin(2/9*pi)*sin(1/9*pi)))
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<P>同时我也可以用几何添加辅助线的方法解决这个问题!(主要是作出这样的辅助线,过<FONT face="Times New Roman">A</FONT>做<FONT face="Times New Roman">BC</FONT>边的垂线,延长<FONT face="Times New Roman">CO</FONT>交垂线于点<FONT face="Times New Roman">O’</FONT>,则角<FONT face="Times New Roman">O’BC=20</FONT>度,只要证明<FONT face="Times New Roman">AO</FONT>平分角<FONT face="Times New Roman">O’AC</FONT>就可以了,用到了面积法和角的平分线定理)。
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<P><FONT face="Times New Roman">Levy
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<P><FONT face="Times New Roman">QQ:271021449
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<P><FONT face="Times New Roman">E-mail:leewei0329@126.com
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<P><FONT face="Times New Roman">2005 2 23
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<>为什么会有12两个推论??请解答一下!!!</P><>①<FONT face="Times New Roman">100*x+10*y+6</FONT>是<FONT face="Times New Roman">8</FONT>的正整数倍,不妨假设<FONT face="Times New Roman">100*x+10*y+6=8*m</FONT>(<FONT face="Times New Roman">m</FONT>为正整数)。
</P><><P><P>②<FONT face="Times New Roman">1+2+8+7+x+y+6</FONT>是<FONT face="Times New Roman">9</FONT>的正整数倍,不妨记为<FONT face="Times New Roman">1+2+8+7+x+y+6=9*n</FONT>(<FONT face="Times New Roman">n</FONT>为正整数)。
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