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)
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

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

$[69-\mathrm{C}-\mathrm{II}-4]$