6. Solve the equation in natural numbers

$$
a b c + a b + b c + a c + a + b + c = 164
$$

In your answer, specify the product $a b c$.
Answer: 80.

Solution: $(a+1) \times(b+1) \times(c+1)=a b c+a b+b c+a c+a+b+c+1=$ $165=3 \times 5 \times 11$, therefore, $a=2, b=4$ and $c=10$. Note that the solution is unique up to the permutation of $a, b$ and $c$, since $3,5,11$ are prime numbers.

## 2013/2014 Academic Year CRITERIA FOR DETERMINING WINNERS AND PRIZE WINNERS ${ }^{1}$

## of the school students' competition "CONQUER THE SPARROW MOUNTAINS!" IN MATHEMATICS

ELIMINATION STAGE

WINNER:

From 95 points inclusive and above.

PRIZE WINNER:

From

91

points to

94

points inclusive.

FINAL STAGE

WINNER (Diploma I degree):

From 90 points inclusive and above.

PRIZE WINNER (Diploma II degree):

From 75 points to 89 points inclusive.

PRIZE WINNER (Diploma III degree):

from 60 points to 74 points inclusive.[^0]


[^0]:    ${ }^{1}$ Approved at the meeting of the jury of the school students' competition "Conquer the Sparrow Mountains!" in mathematics