176. Among 150 schoolchildren, only boys collect stamps. 67 people collect stamps of the USSR, 48 people - of Africa, and 32 people - of America, 11 people only collect stamps of the USSR, 7 people - only of Africa, 4 people - only of America, and only Ivanov collected stamps of the USSR, Africa, and America. Find the maximum number of girls.
Solution. Let's represent the condition of the problem using Euler circles (Fig. 17).

We have a system of three equations:

\[
\begin{cases}
x + y + 1 = 56 & (56 = 67 - 11), \\
x + z + 1 = 41 & (41 = 48 - 7), \\
y + z + 1 = 28 & (28 = 32 - 4)
\end{cases}
\]

![](https://cdn.mathpix.com/cropped/2024_05_21_98932225530737b82071g-102.jpg?height=629&width=583&top_left_y=254&top_left_x=1139)

From this, \(2(x + y + z) = 122\), i.e., \(x + y + z = 61\).

Therefore, 84 boys collect stamps \((61 + 11 + 7 + 4 + 1)\), and the maximum number of girls is 66 \((150 - 84 = 66)\).

Answer: 66 girls.

Remark. From the system of equations obtained during the solution, it is easy to find \(x, y\), and \(z (x = 34, y = 21, z = 6)\), but for solving the problem, it is sufficient to find the sum \(x + y + z\). To reinforce the methods of solving systems of equations, it is advisable to suggest that students solve the obtained system, as well as problem 1279 from \([4]\).