SOLUTION. The fractions in the given expression make sense because their denominators are positive numbers according to the problem statement. Thanks to the condition $a+b+c=1$, for the first fraction we have

$$
\frac{a+b c}{a+b}=\frac{(a+b)+(b c-b)}{a+b}=1-b \cdot \frac{1-c}{a+b}=1-b \cdot \frac{a+b}{a+b}=1-b.
$$

Similarly, the second and third fractions take the values $1-c$ and $1-a$, respectively. Therefore,

$$
\frac{a+b c}{a+b}+\frac{b+c a}{b+c}+\frac{c+a b}{c+a}=(1-b)+(1-c)+(1-a)=3-(a+b+c)=2,
$$

which is the only possible value of the given expression under the conditions of the problem.

## GUIDING AND SUPPLEMENTARY PROBLEMS:

N1. Let $a, b, c$ be non-zero real numbers whose sum is 0. Determine the values that the expression

$$
\frac{(a-b)^{2}+(b-c)^{2}+(c-a)^{2}}{3(a b+b c+c a)}
$$

can take.

[Use the identity $a^{2}+b^{2}+c^{2}=(a+b+c)^{2}-2(a b+b c+c a)$. The value of the given expression is -2.]

N2. Let $a, b, c$ be non-zero real numbers whose sum is 0. Prove that

$$
\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=\frac{a^{2}+b^{2}+c^{2}}{2 a b c}.
$$

[Replace the denominators of the fractions on the left side with $-c, -a, -b$ and after adding the thus modified fractions, apply the same identity as in the solution of N1.]

N3. Let $x, y, z$ be positive real numbers whose product is 1. Prove that the equality

$$
\frac{1}{1+x+x y}+\frac{1}{1+y+y z}+\frac{1}{1+z+z x}=1
$$

holds.

[Expand the first fraction by $z$, the second by $x z$, and use the condition $x y z=1$ three times.]

D1. If real numbers $a, b, c$ satisfy the equation

$$
a^{3}+b^{3}+c^{3}-3 a b c=0,
$$

then $a+b+c=0$ or $a=b=c$. Prove it. $[18-\mathrm{B}-\mathrm{I}-1]$

D2. Determine the smallest possible value of the expression $\left(1+a_{1}\right)\left(1+a_{2}\right)\left(1+a_{3}\right)$, where $a_{1}, a_{2}, a_{3}$ are positive real numbers whose product is 1. [Prove and then multiply the inequalities $1+a_{i} \geqq 2 \sqrt{a_{i}}$ for $\left.i=1,2,3.\right]$

D3. For non-negative real numbers $a, b, c$ such that $a+b+c=1$, find the greatest and smallest possible values of the expression

$$
(a+b)^{2}+(b+c)^{2}+(c+a)^{2}.
$$

$[69-\mathrm{C}-\mathrm{II}-4]$
4. Determine the values that the expression

$$
\frac{a+b c}{a+b}+\frac{b+c a}{b+c}+\frac{c+a b}{c+a}
$$

can take if \( a \), \( b \), and \( c \) are positive real numbers with a sum of 1. (Michal Rolínek, Pavel Calábek)