Is it only 2?

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whose roots
$$x_1, x_2 = 2 \pm \sqrt{4 + 4m^2 - 4m - 3} = 2 \pm (2m - 1),$$
i.e. $x_1 = 1 + 2m$ and $x_2 = 3 - 2m$.

The resulting roots will be valid for all real $m$. For the found values ​​of the roots, it is necessary to require the implementation of ODZ, i.e.
$$\begin{cases} 1 + 2m \geq -1, \\ 3 - 2m \geq -1; \end{cases} \quad \begin{cases} m \geq -1, \\ m \leq 2; \end{cases}$$
from here
$$-1 \leq m \leq 2.$$ (1)

In addition, according to the conditions of the problem, the roots must have different meanings. This will be the case if $x_1 \cdot x_2 < 0$ or $(1 + 2m)(3 - 2m) < 0$. After this, the inequality holds for the following values ​​of $m$:
$$m < -\frac{1}{2} \text{ и } m > \frac{3}{2}.$$ (2)

Comparing the results obtained (1) and (2) on the numerical axis (Fig. 3), we find that all the stated conditions of the problem are met for the following values ​​of $m$:
$$-1 \leq m < -\frac{1}{2} \text{ и } \frac{3}{2} < m \leq 2.$$

It remains to note that since $x_1$ and $x_2$ are solutions of a given equation, then for $x = x_1$ and $x = x_2$ its radical expression will necessarily be a perfect square, and therefore non-negative. This can also be verified by direct substitution.

A typical mistake that 70% of applicants made in this task was not taking into account the range of acceptable values ​​$x \geq -1$.

V. A. Nazaev
P. N. Torzhkov

SACK OF POTATOES

The bag contains potatoes weighing 100 kg at a humidity of 99%. The potatoes dried out, the humidity decreased to 98%.

What was her new weight?

IS IT ONLY 2?

Everyone is well aware of the following rules:

$$2 \times 2 = 4, \quad 2 + 2 = 4; \quad 0 + 0 = 0.$$

Consider these questions:

1. Are there other integers $a$ and $b$ that
   $$a + b = ab.$$

2. Are there other rational numbers?

3. Let it also
   $$a + b = ab = p,$$
   where $p$ is an integer, $p \neq 0$, $p \neq 4$. Is it possible:
   a) for real $a, b$;
   b) for rational $a, b$.

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1. A natural number is not divisible by 2, 3, or 5. Find the remainder when the 4th power of this number is divided by 480.

2. Two people bought the same notebooks: the first 5 pieces, the second 31 pieces. The change of the first with five rubles contained as many rubles and kopecks as the corresponding amount of kopecks and rubles contained the change of the second with 25 rubles. How much does a notepad cost?