2. Determine for which $a$ the system $x+y-144=0, xy-5184-0.1a^2=0$ has one solution. Answer: 0.

Solution. $\left\{\begin{array}{c}x+y=144 \\ xy=5184+0.1a^2\end{array}\right.$

This means that $x$ and $y$ are equal to the roots of the quadratic equation
$p^2 - 144p + (5184 + 0.1a^2) = 0$. Therefore, $D = 72^2 - (5184 + 0.1a^2) = -0.1a^2$.

Thus, the equation can have a solution only when $a = 0$. $p = 72$, so $x = y = 72$. Answer: 0.

## №9: Divisibility.

Find the smallest number among the natural numbers exceeding 2022 that divides the number $2021!! + 2022!!$. (The symbol $n!!$ denotes the product of all natural numbers not exceeding $n$ and having the same parity: $n!! = n \cdot (n-2) \cdot (n-4) \ldots$)
Answer. 2023.

## Solution.

We will prove that $2021!! + 2022!!$ is divisible by the number 2023. Indeed,

$$
2022!! = (2023-1)(2023-3)(2023-5) \ldots (2023-2021)
$$

the remainder of this number when divided by 2023 is the same as the remainder of the product

$$
(-1)(-3)(-5) \ldots (-2021) = (-1)^{\frac{2022}{2}} \cdot 2021!! = -2021!!
$$

when divided by 2023. Therefore, $2021!! + 2022!!$ is divisible by the number 2023. Since 2023 is the smallest number among natural numbers greater than 2022, we get the answer.

Answer. 2023.