Sure, here is the translation:

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a) What is the value of the following fraction?

$$
\frac{423134 \cdot 846267 - 423133}{423133 \cdot 846267 + 423134}
$$

b) What number should be subtracted from the numerator and added to the denominator of the fraction $\frac{52367}{47633}$ so that after simplification, the value of the fraction is $\frac{17}{83}$?

c) What is the sum of the following fractions?

$$
\frac{7}{2 \cdot 9} + \frac{7}{9 \cdot 16} + \frac{7}{16 \cdot 23} + \cdots + \frac{7}{65 \cdot 72}
$$
a) The numerator's diminutive is 846267 greater than the first term of the denominator, but if we subtract the designated 423133 from it, then, since $423133+423134=846267$, we get exactly the same number in the numerator as what stands in the denominator, thus the value of the fraction is 1.

Reichlin Viktor (Bp. V., Piarist g. II. o. t.)

b) The sum of the numerator and the denominator of the fraction $\frac{52367}{47633}$ is exactly 100,000. This is obviously the same if we subtract the same number from the numerator as we add to the denominator. The value of such fractions is $\frac{17}{83}$, where the numerator is 17000 and the denominator is 83000, because here we essentially need to divide the sum of the numerator and the denominator of the previous fraction into two parts in the ratio of $17:83$. To get the fraction $\frac{17000}{83000}$, we need to subtract $52367-17000=35367$ from the original fraction's numerator and add the same amount to the denominator.

c) The sum of the given fractions can be calculated quickly using the following transformation:

$$
\begin{gathered}
\frac{7}{2 \cdot 9}+\frac{7}{9 \cdot 16}+\frac{7}{16 \cdot 23}+\ldots+\frac{7}{65 \cdot 72}=\frac{9-2}{2 \cdot 9}+\frac{16-9}{9 \cdot 16}+\frac{23-16}{16 \cdot 23}+\ldots+\frac{72-65}{65 \cdot 72}= \\
=\frac{9}{2 \cdot 9}-\frac{2}{2 \cdot 9}+\frac{16}{9 \cdot 16}-\frac{9}{9 \cdot 16}+\frac{23}{16 \cdot 23}-\frac{16}{16 \cdot 23}+\ldots+\frac{72}{65 \cdot 72}-\frac{65}{65 \cdot 72}= \\
=\frac{1}{2}-\frac{1}{9}+\frac{1}{9}-\frac{1}{16}+\frac{1}{16}-\frac{1}{23}+\ldots+\frac{1}{65}-\frac{1}{72}
\end{gathered}
$$

In the final summation, all but the two extreme terms cancel each other out in pairs, so only

$$
\frac{1}{2}-\frac{1}{72}=\frac{36}{72}-\frac{1}{72}=\frac{35}{72}
$$

remains.

Balatoni Ferenc (Bp. VII., Rákóczi közg. középisk. II. o. t.)