[Percentage and ratio problems]

In the basket, there were no more than 70 mushrooms, among which $52\%$ were white. If you throw out the three smallest mushrooms, then the white ones will make up half.

How many mushrooms are in the basket

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Let there be $n$ mushrooms in the basket, of which $m$ are white. Since $m = \frac{13}{25} n$, $n$ must be divisible by 25. Additionally, $n < 70$ and $n$ is odd (since $n-3$ is even). The only number that satisfies these conditions is $n=25$.

In this case, there are 13 white mushrooms, and they will become half of the total if two of the three removed mushrooms are white.

## Answer

25 mushrooms.

Solve the equation $\frac{x^{5}}{(x-2)^{2}} + \frac{(x-2)^{2}}{x} = 2x^{2}$.

## Solution

Given that $x \neq 0$, divide both sides of the equation by $x^{2}$: $\frac{x^{3}}{(x-2)^{2}} + \frac{(x-2)^{2}}{x^{3}} = 2$. After substituting $y = \frac{x^{3}}{(x-2)^{2}}$, we get the equation $y + \frac{1}{y} = 2$, whose only solution is $y = 1$. Therefore, $x^{3} = (x-2)^{2}$. One root, $x = 1$, is obvious. Dividing $x^{3} - (x-2)^{2}$ by $x-1$, we get $x^{2} + 4$, and the equation $x^{2} + 4 = 0$ has no real roots.

## Answer

$x = 1$.