a) Is the equality $A=B$ true, if

$$
A=\frac{1+\frac{1+\frac{1+\frac{1}{2}}{4}}{2}}{2}, \quad B=\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{4}}}}}
$$

b) Which is larger, 23/31 or 35/47? Which four-digit decimal fraction deviates approximately equally from these two numbers?

a) part: Bolyai Society Fejér County Section competition for elementary school students, 1962.

b) part: I. Olympiad problem, Berlin (GDR) 1961.
I. solution. a) The expressions can be simplified by performing the additions from "inside out". Thus, the division after the addition can always be carried out immediately.

$$
\begin{aligned}
& A=\frac{1+\frac{1+\frac{3}{8}}{3}}{2}=\frac{1+\frac{11}{24}}{2}=\frac{35}{48} \\
& B=\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{4}{9}}}}=\frac{1}{1+\frac{1}{2+\frac{9}{13}}}=\frac{1}{1+\frac{13}{35}}=\frac{35}{48}
\end{aligned}
$$

According to this, the required equality holds.

b) By bringing the two numbers to a common denominator, the denominator is $31 \cdot 47=1457$, and the numerators are $23 \cdot 47=1081$, and $35 \cdot 31=1085$. The latter is larger, so the fraction $35 / 47$ is larger.

The number that deviates by the same amount from two given (different) numbers is their arithmetic mean. If the numbers are $A$ and $B$ and, for example, $A<B$, then

$$
\frac{A+B}{2}-A=\frac{B-A}{2}, \quad B-\frac{A+B}{2}=\frac{B-A}{2}
$$

and these two differences are equal.

Accordingly, in our case, the deviation of the fraction $1083/1457$ from the given fractions is equal. Its decimal form is not finite: $0,74330 \ldots$. The fifth decimal digit is 0, so we get the four-digit decimal fraction closest to it by rounding down: 0,7433. This deviates by approximately the same amount from the given two numbers.

Emese Bánkfalvi (Szeged, Ságvári E. Elementary School I. Class)

II. solution. For part b). We can reach the result with smaller numbers if we consider the complements of the given numbers (both positive real fractions) to 1:

$$
1-\frac{23}{31}=\frac{8}{31}, \quad 1-\frac{35}{47}=\frac{12}{47}
$$

and bring these to a common numerator:

$$
\frac{24}{93}, \quad \frac{24}{94}
$$

The second has a larger denominator, so the second fraction is smaller. According to this, the number $35/47$ is complemented to 1 by a smaller number than $23 / 31$, so $35 / 47$ is larger than the other number.

The decimal form of the given numbers to five digits is $0,74193 \ldots$ and $0,74468 \ldots$. From these, their arithmetic mean to five decimal places is $0,74330 \ldots$, so its four-digit rounded form is 0,7433.

Remarks. 1. We obtained the common numerator form of the complements to 1 by extending them by 3 and 2, respectively. By looking at the extended form of the original numbers, 69/93 and 70/94, the second number can be obtained from the first by increasing both the numerator and the denominator by the same amount, 1. The reader can easily verify that from a positive numerator and denominator real fraction, we always get a larger fraction this way.

Róbert Freud (Budapest, Bolyai J. Secondary School II. Class)

2. The expression $B$ is conventionally called the continued fraction expansion of the number $35 / 48$. In it, it is considered essential that only 1 is accepted as the numerator. The conversion of a given number into a continued fraction can proceed in reverse order, for example:

$$
\frac{275}{48}=5+\frac{35}{48}=5+\frac{1}{\frac{48}{35}}=5+\frac{1}{1+\frac{13}{35}}=\ldots
$$

The first, second, third, etc., approximating fraction of a continued fraction is the fraction that results if we omit the number to be added to the integer under the first, second, third, etc., denominator. In our case, the first four approximating fractions of 35/48 are

$$
\frac{1}{1}=1, \quad \frac{1}{1+\frac{1}{2}}=\frac{2}{3}, \quad \frac{1}{1+\frac{1}{2+\frac{1}{1}}}=\frac{3}{4}, \quad \frac{8}{11}
$$

the fifth approximating fraction is already the number itself.

Despite the complex form of the continued fraction expansion, it is of great importance that its approximating fractions, when converted to ordinary fractions, approximate the number represented by the continued fraction with relatively high accuracy relative to the denominator.