On a plate, there are a total of 58 slices of beigli, a mixture of walnut and poppy seed. Three walnut ones can be chosen from them in the same number of ways as two poppy seed ones and one walnut one. How many poppy seed beigli are on the plate?
Solution. Let $x$ be the number of walnut-filled doughnuts, then there are $58-x$ plum-filled doughnuts.

According to the condition, $\binom{x}{3}=\binom{58-x}{2} \cdot x$, which means the number of ways to choose 3 walnut-filled doughnuts is the same as the number of ways to choose 2 plum-filled and 1 walnut-filled doughnut. That is,

$$
\frac{x(x-1)(x-2)}{1 \cdot 2 \cdot 3}=\frac{(58-x)(57-x)}{1 \cdot 2} \cdot x
$$

Simplify by $x \neq 0$, and perform the indicated operations. We arrive at the following quadratic equation: $x^{2}-$ $171 x+4958=0$, from which $x=37$, or 134, but we didn't have that many doughnuts in total.

Therefore, we placed 37 walnut-filled and 21 plum-filled doughnuts on the plate.

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Tamás Karches (Mosonmagyaróvár, Kossuth L. Gymnasium, 12th grade)