Task 1. Which terms should be omitted in the expression

$$
\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{12}
$$

so that the value of the new sum is 1?
Solution. Since

$$
\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{12}=\frac{60}{120}+\frac{30}{120}+\frac{20}{120}+\frac{15}{120}+\frac{12}{120}+\frac{10}{120}=\frac{147}{120} \text { and } \frac{120}{120}=1
$$

we need to remove addends that sum to $\frac{147}{120}-\frac{120}{120}=\frac{27}{120}$. Therefore, we need to remove the addends $\frac{15}{120}$ and $\frac{12}{120}$, which are the addends $\frac{1}{8}$ and $\frac{1}{10}$.