数学 每日一题（第二十六天）
$1.已知a＝\dfrac{1}{2}\sqrt{\sqrt{2}+\dfrac{1}{8}}-\dfrac{\sqrt{2}}{8}，$
$求代数式a^2+\sqrt{a^4+a+1}的值.$
$2.已知非零实数a，b，c满足\dfrac{1}{a}+\dfrac{2}{b}+\dfrac{3}{c}＝0，$
$求代数式\dfrac{3b+2c}{6a}+\dfrac{2c+6a}{3b}+\dfrac{6a+3b}{2c}的值.$
$3.已知实数a，b，c满足（a+b+c）（\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}）＝1，$
$求证：\dfrac{1}{（a+b+c）^{2027}}＝\dfrac{1}{a^{2027}}+\dfrac{1}{b^{2027}}+\dfrac{1}{c^{2027}}.$
$4.实数a，b，x，y满足：$
$ax+by＝3$
$ax^2+by^2＝7$
$ax^3+by^3＝16$
$ax^4+by^4＝42$
$求ax^5+by^5的值.$
$5.已知互异非零实数a，b，c满足｜a｜≥｜b+c｜，｜b｜≥｜c+a｜，｜c｜≥｜a+b｜，求：$
$（1）a+b+c；$
$（2）\dfrac{1}{a^2+b^2-c^2}+\dfrac{1}{b^2+c^2-a^2}+\dfrac{1}{c^2+a^2-b^2}；$
$（3）\dfrac{a^2}{2a^2+bc}+\dfrac{b^2}{2b^2+ca}+\dfrac{c^2}{2c^2+ab}；$
$（4）（\dfrac{a-b}{c}+\dfrac{b-c}{a}+\dfrac{c-a}{b}）（\dfrac{c}{a-b}+\dfrac{a}{b-c}+\dfrac{b}{c-a}）；$
$（5）\dfrac{a^5+b^5+c^5}{abc（ab+bc+ca）}.$
