## Task 1 - 040911

Martina presents her friend with the following problem in a year that is not a leap year:

"If you add one-third of the number of remaining days of the year to half the number of days that have passed so far this year, you get the number of days that have passed. I have included today in the count of the days that have passed."

Give the date (day and month) on which this occurs!
Let $a$ be the days that have passed, and $b$ the days that remain, where $a+b=365$, assuming it is not a leap year. According to the problem statement,

$$
\frac{1}{2} a+\frac{1}{3} b=a \quad \Rightarrow \quad b=1.5 a
$$

Substituting into the first equation gives $2.5 a=365$, so $a=146$. The 146th day of the year is May 26.