Two scenes from life

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ANSWERS, INSTRUCTIONS, SOLUTIONS

For the task “Only 2?”
(see «Kvant» No. 6)

From the equality $a + b = ab$ we find $a$:

$$a = \frac{b}{b - 1}.$$

Note that $b \neq 1$.

1. Let $a - \text{целое число}.$ Since the numbers $b$ and $b - 1$ are coprime (their difference is divided by their gcd, but it is equal to 1), then $b - 1 = 1$ or $b - 1 = -1$.

Hence $b_1 = 2$, $a_1 = 2$; $b_2 = 0$, $a_2 = 0$.

2. If $b$ is an arbitrary rational number not equal to 1, then using formula (*) we determine $a$.

3. If $ab = p$, then $p = \frac{b^2}{b - 1}$, whence

$$b^2 - pb + p = 0, \quad b = \frac{p}{2} \pm \sqrt{\frac{p^2}{4} - p},$$

$$b = \frac{p}{2} \pm \sqrt{\frac{p(p - 4)}{2}}.$$

a) using formula () for any $p \geq 4$ or $p \leq 0$ one finds the corresponding $b$ and using formula () $a$;

6. we need to find whether there is a number $p$ such that

$$p(p - 4) = \frac{q^2}{s^2},$$

(***)

where $\frac{q}{s}$ is an irreducible fraction. We solve equation (*) in integers $(p, q, s$ and $m$ - integers):

$$q^2 = s^2p(p - 4),$$ but from the irreducibility of the fraction it follows that $s = 1$.

$$p^2 - 4p - q^2 = 0, \quad p = 2 \pm \sqrt{4 + q^2},$$

$$4 + q^2 = m^2,$$

$$(m - q)(m + q) = 4 \cdot 1 = 2 \cdot 2.$$

The only solution in integers $m = \pm 2$, $q = 0$ leads to solution (1).

Two scenes from life

“Knowledgeable” tour guide

Guide: …Newton’s law applies to the planets, so they orbit around Coln. Kepler’s law also applies to them, but I won’t tell you about it, since it is complicated and you won’t understand anything… To fly away from the Earth, you need to have a second escape velocity. As you can see on the poster, it is equal to 11.2 km/sec, last year it was 11.4 km/sec… It is difficult for the astronauts. When they fly through the atmosphere, it puts a lot of pressure on the ship and the cosmonauts experience overload, but behind the atmosphere it’s even worse - the space of weightlessness begins there…

In one research institute

Graduate student: I believe that the roller sags in the bitumen due to the fact that its pressure increases the speed of the molecules, and they run away from under the roller.

Scientific supervisor: What exactly are you working on now?

Graduate student: Calculate the dependence of the speed of bitumen molecules on the pressure of the roller on them.

Scientific supervisor: Okay. What do you need to finish the job?

Graduate student: I would like a hydraulic press!

Scientific supervisor: Write an application. I think we’ll get it.