## Task 4 - 150914

On December 30, 1973, when Mr. T. celebrated his birthday, he said to his wife: "Now I am exactly 8 times as old as our son, if I only count the full (completed) years of age."

To which his wife replied: "In the year 1974, the case will occur that you are 5 times as old as our son, if I also only consider the full years of age."

Investigate whether there is exactly one date for which - as the son's date of birth - all these statements are true! If this is the case, provide the exact date of birth of the son (day, month, year)!
I. Assuming all the given information about the son's date of birth is correct.

Then it follows:

If $x$ is the age of the son, expressed in full (completed) years, on December 30, 1973, then in 1974, it can only take the values $x, x+1, x+2$, and the last value only if the son has a birthday on December 31.

Furthermore, $8x$ is the age of the father, expressed in full years, on December 30, 1973, and this can only take the values $8x, 8x+1$ in 1974.

Therefore, for one of the numbers $a=0,1,2$ and one of the numbers $b=0,1$, the equation $5 \cdot (x+a) = 8x + b$ holds, which simplifies to $5a - b = 3x$.

The following table of values for $5a - b$,

| $b$ | $a=0$ | 1 | 2 |
| :---: | :---: | :---: | :---: |
| 0 | 0 | 5 | 10 |
| 1 | -1 | 4 | 9 |

shows that $5a - b$ is only divisible by 3 for $a=2, b=1$, which leads to $x=3$. Therefore, the given information can only be correct if the son has a birthday on December 31, was still 3 years old on December 30, 1973, turned 4 years old on December 31, 1973, and was thus born on December 31, 1969.

II. This birthdate of the son, together with the resulting age of the father, 24 years old on December 30, 1973, indeed satisfies all the given information; because with these data, the father was 8 times as old as his son on December 30, 1973; but on December 31, 1974, when the son turned 5 years old, the father was 25 years old, thus 5 times as old as his son.

Therefore, the given information is exactly correct for the birthdate of the son on December 31, 1969.

Solutions of the 1st Stage 1975 adopted from [5]

### 6.17.2 2nd Stage 1975, Class 9