$\underline{\text { Senerev B.A. }}$.

1999 numbers stand in a row. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of its two neighbors.

Find the last number.
Let's denote our numbers as $a_{1}, a_{2}, \ldots, a_{1999}$. By adding the equations $a_{n+1}=a_{n+2}+a_{n}$ and $a_{n+2}=a_{n+3}+a_{n+1}$, we get $a_{n+3}+a_{n}=0$, or $a_{n+3}=-a_{n}$. Therefore, $a_{n+6}=-a_{n+3}=a_{n}$, which means the sequence has a period of 6. Hence, $a_{1999}=a_{6 \cdot 333+1}=a_{1}=1$.

Answer

## 1

Author: Sergeev I.n.

When water is drained from a pool, the water level $h$ in it changes over time $t$ according to the law

$$
h(t)=a t^{2}+b t+c
$$

and at the moment $t_{0}$ when the draining is complete, the equalities $h\left(t_{0}\right)=h^{\prime}\left(t_{0}\right)=0$ are satisfied. How many hours does it take to completely drain the water from the pool if the water level decreases by half in the first hour?

## Solution

Since $h^{\prime}\left(t_{0}\right)=0$ and $h\left(t_{0}\right)=0$, we have $a \neq 0$ (otherwise, the function $h$ would be zero, and the water level in the pool, contrary to the problem's condition, could not have decreased), and the abscissa and ordinate of the vertex of the parabola $y=h(t)$ are $t_{0}$ and 0, respectively, so

$$
h(t)=a\left(t-t_{0}\right)^{2}, t \leqslant t_{0}
$$

Let $T$ be the time it takes to completely drain the pool. From the problem's condition, we have

$h\left(t_{0}-T\right)=2 h\left(t_{0}-T+1\right) \Rightarrow a T^{2}=2 a(T-1)^{2} \Rightarrow$

$$
\Rightarrow T^{2}-4 T+2=0 \Rightarrow T=2 \pm \sqrt{2}
$$

and, considering that $T>1$, we finally get $T=2+\sqrt{2}$.

Comment. From the laws of physics, it is known that the function $y=h(t)$ satisfies the differential equation

$$
y^{\prime}=-k \sqrt{y}, k>0
$$

which describes the height of the liquid level in a container with an opening at its bottom. The solution to this equation is precisely the quadratic function given in the problem (up to the moment of complete outflow of the liquid).

## Answer

$2+\sqrt{2} \cdot$