几个小问题

zhangyuong

$x_{1,2}=\frac{-b+-\sqrt{b^2-4ac}}{2a}$

zhangyuong

Let A,B, and C denote complex constants, and z a complex variable. Show that
$A+ \bar A + B \bar z + bar \B z + (C+ \bar C )z\bar z =0$
is the equation of an arbitrary circle.

zhangyuong

We consider the polynomial

$P(z)=a_0 z^n + a_1 z^{n-1} + a_2 z^{n-2} + \cdots + a_n$

and assume that

$a_0 \ne 0, a_n \ne 0$.

Show that
(a) the equation

$|a_0| x^n + |a_1| x^{n-1} + \cdots + |a_{n-1}| x - |a_n|=0$

has just one positive root $r$, and
(b)$P(z)$ does not vanish in the circle $|z|<r$.

zhangyuong

Show that $z_1$, $z_2$, $z_3$ form an equilateral triangle if, and only if,

$z_1 ^2 + z_2 ^2 + z_3 ^2 = z_2 z_3 + z_3 z_1 + z_1 z_2$

zhangyuong

Show that

$|ac - \bar b d|^2 + |\bar a d + bc|^2 = (|a|^2 + |b|^2)(|c|^2 + |d|^2)$

where $a$, $b$, $c$, $d$ are all complex numbers.

zhangyuong

1986年全国高中数学联赛二试题1为：
试题A    已知实数列$a_0$, $a_1$, $a_2$, $\cdots$ 满足

$a_{i-1} + a_{i+1} = 2a_i$  $(i=1,2,3,\cdots)$.

求证：对于任何自然数$n$

$P(x)=a_0 C_n^0 (1-x)^n + a_1 C_n^1 x(1-x)^{n-1} + a_2 C_n^2 x^2 (1-x)^{n-2} + \cdots + a_{n-1} C_n^{n-1} x^{n-1}(1-x) + a_n C_n^n x^n$

是$x$的一次多项式或常数