\section*{Problem 4 - 071014}

On May 30, 1967, Mr. \(X\) notices that he uses each digit from 0 to 9 exactly once when he writes his date of birth in the format just used for date specifications and adds his age in years. Additionally, he observes that the number of his life years is a prime number.

When was Mr. \(X\) born, and how old is he?
}

Let ab.cd.efgh denote the birth date and \(i k\) the age of Mr. \(X\). Thus, \(a\) and \(b\) are the digits for the day, \(c\) and \(d\) the digits for the month, \(e, f, g\) and \(h\) the digits for the year, and \(i\) and \(k\) the digits for the age.

Then it must hold:
\(e=1\), and \(f\) can only be 9 or 8, since the number of years of Mr. \(X\)'s life must be two-digit and he was thus born after 1799.

\(c=0\), since there are only 12 months and the 1 is already assigned.

\(a=2\), since a month can have at most 31 days, the 0 and 1 are already assigned, and thus 30 and 31 are also not possible.

Case 1: Let \(f=9\)

Then it must hold
A) \(10 g+h+10 i+k=66\), if Mr. \(X\) has had his birthday after May 30th,
B) \(10 g+h+10 i+k=67\) otherwise.

From A) it follows that either
a) \(g+i=6\) and \(h+k=6\) or
b) \(g+i=5\) and \(h+k=16\) holds.

From B) it follows that either
c) \(g+i=6\) and \(h+k=7\) or
d) \(g+i=5\) and \(h+k=17\) holds.

With the remaining digits, none of the four pairs of equations is realizable. The assumption \(f=9\) is false.

Case 2: Let \(f=8\).

Then, similarly as above,
C) \(10 g+h+10 i+k=166\) or
D) \(10 g+h+10 i+k=167\).

From C) it follows that either
e) \(g+i=16\) and \(h+k=6\) or
f) \(g+i=15\) and \(h+k=16\) holds.

From D) it follows that either
g) \(g+i=16\) and \(h+k=7\) or
h) \(g+i=15\) and \(h+k=17\) holds.

With the remaining digits, only g) is realizable. The following four distributions are possible: \(g=9\) and \(i=7, h=4\) and \(k=3, g=7\) and \(i=9\) and \(h=3\) and \(k=4\).

Thus, Mr. \(X\)'s age can be 73, 74, 93, or 94 years. Of these numbers, only 73 is a prime number. This results in the following distribution:

\[
f=8, \quad i=7, \quad k=3, \quad g=9 \quad \text { and } \quad h=4
\]

For \(b\) and \(d\), only the digits 5 and 6 are available. From g) it follows, since it arose from D), that Mr. \(X\) had his birthday before May 30th. Thus, \(d=5\) and \(b=6\).

Mr. \(X\) was born on 26.05.1894 and is 73 years old.

Solutions of the I. Round 1967 taken from [5]

\subsection*{7.9.2 II. Round 1967, Class 10}