# Task 1.

The Triassic Discoglossus tadpoles have five legs, while the saber-toothed frog tadpoles grow several tails (all have the same number of tails). An employee of the Jurassic Park scooped up several tadpoles with water. It turned out that the captured tadpoles had a total of 100 legs and 64 tails. How many tails does each saber-toothed frog tadpole have, if all five-legged tadpoles have one tail, and all multi-tailed tadpoles have four legs?
# Solution.

Let $x$ be the number of tails of the saber-toothed frog's pollywog. Suppose $n$ five-legged and $k$ many-tailed pollywogs were caught. Counting the total number of legs and tails gives the equations

$$
\left\{\begin{aligned}
5 n+4 k & =100 \\
n+x k & =64
\end{aligned}\right.
$$

From the first equation, we have $n \leq 100 / 5=20, \quad k \leq 100 / 4=25$.

Moreover, $4 k=5(20-n)$, so $k$ must be a multiple of 5.

Now, multiply the second equation of the system by 5 and subtract the first from the result. We get

$$
(5 x-4) k=64 \cdot 5-100=220=2 \cdot 2 \cdot 5 \cdot 11 .
$$

Since the left side is the product of two natural numbers and considering the obtained limits, we find that $k$ can only take the values

$$
k=5 \quad \text { or } \quad k=10 \quad \text { or } \quad k=20 .
$$

Substitute them one by one into the system of equations. For $k=5$, we have

$$
\left\{\begin{aligned}
5 n+20 & =100 \\
n+5 x & =64
\end{aligned}\right.
$$

from which $n=16$. But $64-16=48$ is not a multiple of 5. There are no solutions.

For $k=10$, we get

$$
\left\{\begin{array}{r}
5 n+40=100 \\
n+10 x=64
\end{array}\right.
$$

from which $n=12$. But $64-12=52$ is not a multiple of 10. There are no solutions again.

Olympiad for Schoolchildren "Hope of Energy". Final Stage. Solutions

Finally, for $k=20$, we find

$$
\left\{\begin{array}{l}
5 n+80=100 \\
n+20 x=64
\end{array}\right.
$$

from which $n=4$ and $n=3$.

Answer: 3 tails.