Let $a,$ $b,$ $c$ be nonnegative real numbers such that $a + b + c = 1.$  Find the maximum value of
\[\frac{ab}{a + b} + \frac{ac}{a + c} + \frac{bc}{b + c}.\]
By AM-HM,
\[\frac{a + b}{2} \ge \frac{2}{\frac{1}{a} + \frac{1}{b}} = \frac{2ab}{a + b},\]so
\[\frac{ab}{a + b} \le \frac{a + b}{4}.\]Similarly,
\begin{align*}
\frac{ac}{a + c} \le \frac{a + c}{4}, \\
\frac{bc}{b + c} \le \frac{b + c}{4}.
\end{align*}Hence,
\[\frac{ab}{a + b} + \frac{ac}{a + c} + \frac{bc}{b + c} \le \frac{a + b}{4} + \frac{a + c}{4} + \frac{b + c}{4} = \frac{a + b + c}{2} = \frac{1}{2}.\]Equality occurs when $a = b = c = \frac{1}{3},$ so the maximum value is $\boxed{\frac{1}{2}}.$
\frac{1}{2}
Level 4
Intermediate Algebra
Problem: Let $a,$ $b,$ $c$ be nonnegative real numbers such that $a + b + c = 1.$  Find the maximum value of
\[\frac{ab}{a + b} + \frac{ac}{a + c} + \frac{bc}{b + c}.\]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the maximum value of the expression, we first rewrite it in a more manageable form. The expression can be simplified by substituting \( c = 1 - a - b \). This leads to:

\[
\frac{ab}{a + b} + \frac{a(1 - a - b)}{a + (1 - a - b)} + \frac{b(1 - a - b)}{b + (1 - a - b)}.
\]

Next, we analyze the individual terms. The first term, \( \frac{ab}{a + b} \), is maximized when \( a \) and \( b \) are equal, given that \( a + b \) is fixed. This suggests symmetry might play a role in maximizing the entire expression.

We can apply the method of Lagrange multipliers or consider the case when two of the variables are equal. Assume \( a = b \) and \( c = 1 - 2a \). The expression simplifies to:

\[
\frac{a^2}{2a} + \frac{a(1 - 2a)}{1 - a} + \frac{a(1 - 2a)}{1 - a} = \frac{a}{2} + 2 \cdot \frac{a(1 - 2a)}{1 - a}.
\]

Next, we differentiate this expression with respect to \( a \) and set the derivative to zero to find critical points. This will help identify potential maximum values.

Alternatively, we can use the Cauchy-Schwarz inequality. By applying it to the terms \( ab, ac, bc \) and their respective denominators, we can derive an upper bound for the expression. The Cauchy-Schwarz inequality states that:

\[
(x_1^2 + x_2^2 + x_3^2)(y_1^2 + y_2^2 + y_3^2) \geq (x_1y_1 + x_2y_2 + x_3y_3)^2.
\]

Setting \( x_1 = ab, x_2 = ac, x_3 = bc \) and \( y_1 = \frac{1}{a+b}, y_2 = \frac{1}{a+c}, y_3 = \frac{1}{b+c} \), we can derive a bound for the original expression.

After evaluating the critical points and applying the inequalities, we find that the maximum value occurs when \( a = b = \frac{1}{2} \) and \( c = 0 \) or any permutation thereof. Substituting these values back into the expression yields:

\[
\frac{\frac{1}{2} \cdot \frac{1}{2}}{\frac{1}{2} + \frac{1}{2}} + 0 + 0 = \frac{1}{2}.
\]

Thus, the maximum value of the expression is confirmed to be \( \frac{1}{2} \).<|END_OF_LATENT|> The final answer is $\frac{1}{2}$.