8,9,10,11 |  |

Let $a, b$, and $c$ be three distinct numbers. Solve the system

$$
\left\{\begin{aligned}
z + a y + a^{2} x + a^{3} & =0 \\
z + b y + b^{2} x + b^{3} & =0 \\
z + c y + c^{2} x + c^{3} & =0
\end{aligned}\right.
$$
$a, b, c-$ are the roots of the polynomial $t^{3}+x t^{2}+y t+z$. Its coefficients $x, y, z$ are expressed using Vieta's formulas.

## Answer

$x=-(a+b+c), \quad y=a b+a c+b c, \quad z=-a b c$.

Using the expansion of $(1+i)^{n}$ by Newton's binomial formula, find:

a) $C_{100}^{0}-C_{100}^{2}+C_{100}^{4}-\ldots+C_{100}^{100}$;

b) $C_{99}^{1}-C_{99}^{3}+C_{99}^{5}-\ldots-C_{99}^{99}$.

## Solution

a) The given sum is equal to $\operatorname{Re}(1+i)^{100}$, and $(1+i)^{100}=(2 i)^{50}=-2^{50}$.

b) The sum is equal to $\operatorname{Im}(1+i)^{99}$, and $(1+i)^{99}=(1+i)(2 i)^{49}=(1+i) \cdot 2^{49} i^{49}=(1-i) \cdot 2^{49}$.

## Answer

a) $-2^{50} ;$ b) $2^{49}$.