planet/doc/2019年全国高中数学联合竞赛一试试题A卷.tmu

<TMU|<tuple|1.0.3|1.2.9.1-rc2>>

<style|<tuple|generic|chinese|number-europe|number-long-article|old-lengths>>

<\body>
  \;

  <doc-data|<doc-title|2019年全国高中数学联合竞赛一试试题（A卷)>>

  <subsection*|一、填空题：本大题共8小题，每小题8分，满分64分。>

  <\enumerate>
    <item>已知正实数<math|a>满足<math|a<rsup|a>=<around*|(|9*a|)><rsup|8*a>>，则<math|log<rsub|a><around*|(|3*a|)>>的值为<underline|<space|3em>>.

    <item>若实数集合<math|<around*|{|1,2,3,x|}>>的最大元素与最小元素之差等于该集合的所有元素之和，则<math|x>的值为<underline|<space|3em>>.

    <item>平面直角坐标系中，<math|<wide|e|\<vect\>>>是单位向量，向量<math|<wide|a|\<vect\>>>满足<math|<wide|a|\<vect\>>\<cdot\><wide|e|\<vect\>>=2>，且<math|<around*|\||<wide|a|\<vect\>>|\|>\<leqslant\>5<around*|\||<wide|a|\<vect\>>+t<wide|e|\<vect\>>|\|>>对任意实数<math|t>成立，则<math|<around*|\||<wide|a|\<vect\>>|\|>>的取值范围为<underline|<space|3em>>.

    <item>设<math|A,B>为椭圆<math|\<Gamma\>>的两个焦点，<math|<around*|\||AB|\|>=4>，<math|<around*|\||AF|\|>=2+<sqrt|3>>，<math|P>为<math|\<Gamma\>>上一点，满足<math|<around*|\||PE|\|>\<cdot\><around*|\||PF|\|>=2>，则<math|\<Delta\>PEF>的面积为<underline|<space|3em>>.

    <item>在<math|1,2,3,\<ldots\>,10>中随机选出一个数<math|a>，在<math|-1,-2,-3,\<ldots\>,-10>中随机选出一个数<math|b>，则<math|a<rsup|2>+b>被3整除的概率为<underline|<space|3em>>.

    <item>对任意闭区间<math|I>，用<math|M<rsub|I>>表示函数<math|y=sin x>在<math|I>上的最大值.若正数<math|a>满足<math|M<rsub|<around*|{|0,a|}>>=2M<rsub|<around*|{|a,2a|}>>>，则<math|a>的值为<underline|<space|3em>>.

    <item>如图，正方体<math|ABCD-EFGH>的一个截面经过顶点A, C及棱EF上一点K，且将正方形分成体积比为3:1的两部分，则<math|<with|math-display|true|<frac|EK|KF>>>的值为<underline|<space|3em>>.

    <\script-output|asymptote|default>
      % -width 270

      import three;

      \;

      size (100);

      \;

      triple

      \ D= (0,0,0),

      \ A= (1,0,0),

      \ B= (1,1,0),

      \ C= (0,1,0),

      \ H= (0,0,1),

      \ E= (1,0,1),

      \ F= (1,1,1),

      \ G= (0,1,1),

      \ K= (1, 2/3, 1);

      \ 

      dot("$A$", A, NE);

      dot("$B$", B, NE);

      dot("$C$", C, NE);

      dot("$D$", D, NE);

      dot("$E$", E, NE);

      dot("$F$", F, NE);

      dot("$G$", G, NE);

      dot("$H$", H, NE);

      dot("$K$", K);

      \;

      \;

      draw (A--B--C);

      draw (D--A, dotted);

      draw (D--C, dotted);

      draw (D--H, dotted);

      draw (A--E);

      draw (B--F);

      draw (C--G);

      draw (E--F--G--H--cycle);
    </script-output|<image|<tuple|<#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

    <item>将6个数<math|2,0,1,9,20,19>按任意次序排成一行，拼成一个8位数（首位不为0），则产生的不同的8位数的个数为<underline|<space|3em>>.
  </enumerate>

  <subsection*|二、解答题：本大题共3小题，满分56分。解答应写出文字说明、证明过程或演算步骤。>

  <\enumerate>
    <item>（本题满分16分）在<math|\<Delta\>ABC>中，<math|BC=a>，<math|CA=b>，<math|AB=c>.若b是a与c的等比中项，且<math|sin A>是<math|sin<around*|(|B-A|)>>与<math|sin C>的等差中项，求<math|cos B>的值.

    <item>（本题满分20分）在平面直角坐标系<math|xOy>中，圆<math|\<Omega\>>与抛物线<math|\<Gamma\>:y<rsup|2>=4*x>恰有一个公共点，且圆<math|\<Omega\>>与<math|x>轴相切于<math|\<Gamma\>>的焦点<math|F>，求圆<math|\<Omega\>>的半径.

    <item>（本题满分20分）称一个复数数列<math|<around*|{|z<rsub|n>|}>>为“有趣的”，若<math|<around*|\||z<rsub|1>|\|>=1>，且对任意正整数n，均有<math|4z<rsub|n+1><rsup|2>+2z<rsub|n>z<rsub|n+1>+z<rsub|n><rsup|2>=0>，求最大的常数<math|C>，使得对一切有趣的数列<math|<around*|{|z<rsub|n>|}>>及任意正整数<math|m>，均有<math|<around*|\||z<rsub|1>+z<rsub|2>+\<cdots\>+z<rsub|m>|\|>\<geqslant\>C>.
  </enumerate>
</body>

<\initial>
  <\collection>
    <associate|enumerate-levels|4>
    <associate|page-even-footer|<htab|5mm>使用墨干理工套件<Xmacs-version>制作<htab|5mm>>
    <associate|page-even-header|>
    <associate|page-medium|paper>
    <associate|page-odd-footer|<htab|5mm>使用墨干理工套件<Xmacs-version>制作<htab|5mm>>
    <associate|page-odd-header|>
  </collection>
</initial>

<\references>
  <\collection>
    <associate|auto-1|<tuple|?|1>>
    <associate|auto-2|<tuple|8|1>>
  </collection>
</references>

<\auxiliary>
  <\collection>
    <\associate|toc>
      <with|par-left|<quote|1tab>|一、填空题：本大题共8小题，每小题8分，满分64分。 <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>> <no-break><pageref|auto-1>>

      <with|par-left|<quote|1tab>|二、解答题：本大题共3小题，满分56分。解答应写出文字说明、证明过程或演算步骤。 <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>> <no-break><pageref|auto-2>>
    </associate>
  </collection>
</auxiliary>